Daily Papers

In modern large language models (LLMs), increasing sequence lengths is a crucial challenge for enhancing their comprehension and coherence in handling complex tasks such as multi-modal question answering. However, handling long context sequences with LLMs is prohibitively costly due to the conventional attention mechanism's quadratic time and space complexity, and the context window size is limited by the GPU memory. Although recent works have proposed linear and sparse attention mechanisms to address this issue, their real-world applicability is often limited by the need to re-train pre-trained models. In response, we propose a novel approach, Hierarchically Pruned Attention (HiP), which simultaneously reduces the training and inference time complexity from O(T^2) to O(T log T) and the space complexity from O(T^2) to O(T). To this end, we devise a dynamic sparse attention mechanism that generates an attention mask through a novel tree-search-like algorithm for a given query on the fly. HiP is training-free as it only utilizes the pre-trained attention scores to spot the positions of the top-k most significant elements for each query. Moreover, it ensures that no token is overlooked, unlike the sliding window-based sub-quadratic attention methods, such as StreamingLLM. Extensive experiments on diverse real-world benchmarks demonstrate that HiP significantly reduces prompt (i.e., prefill) and decoding latency and memory usage while maintaining high generation performance with little or no degradation. As HiP allows pretrained LLMs to scale to millions of tokens on commodity GPUs with no additional engineering due to its easy plug-and-play deployment, we believe that our work will have a large practical impact, opening up the possibility to many long-context LLM applications previously infeasible.

Recent advancements in Large Language Models (LLMs) have yielded remarkable success across diverse fields. However, handling long contexts remains a significant challenge for LLMs due to the quadratic time and space complexity of attention mechanisms and the growing memory consumption of the key-value cache during generation. This work introduces MemLong: Memory-Augmented Retrieval for Long Text Generation, a method designed to enhance the capabilities of long-context language modeling by utilizing an external retriever for historical information retrieval. MemLong combines a non-differentiable ``ret-mem'' module with a partially trainable decoder-only language model and introduces a fine-grained, controllable retrieval attention mechanism that leverages semantic-level relevant chunks. Comprehensive evaluations on multiple long-context language modeling benchmarks demonstrate that MemLong consistently outperforms other state-of-the-art LLMs. More importantly, MemLong can extend the context length on a single 3090 GPU from 4k up to 80k. Our code is available at https://github.com/Bui1dMySea/MemLong

The transformer model is known to be computationally demanding, and prohibitively costly for long sequences, as the self-attention module uses a quadratic time and space complexity with respect to sequence length. Many researchers have focused on designing new forms of self-attention or introducing new parameters to overcome this limitation, however a large portion of them prohibits the model to inherit weights from large pretrained models. In this work, the transformer's inefficiency has been taken care of from another perspective. We propose Fourier Transformer, a simple yet effective approach by progressively removing redundancies in hidden sequence using the ready-made Fast Fourier Transform (FFT) operator to perform Discrete Cosine Transformation (DCT). Fourier Transformer is able to significantly reduce computational costs while retain the ability to inherit from various large pretrained models. Experiments show that our model achieves state-of-the-art performances among all transformer-based models on the long-range modeling benchmark LRA with significant improvement in both speed and space. For generative seq-to-seq tasks including CNN/DailyMail and ELI5, by inheriting the BART weights our model outperforms the standard BART and other efficient models. Our code is publicly available at \url{https://github.com/LUMIA-Group/FourierTransformer}

One limitation of existing Transformer-based models is that they cannot handle very long sequences as input since their self-attention operations exhibit quadratic time and space complexity. This problem becomes especially acute when Transformers are deployed on hardware platforms equipped only with CPUs. To address this issue, we propose a novel method for accelerating self-attention at inference time that works with pretrained Transformer models out-of-the-box without requiring retraining. We experiment using our method to accelerate various long-sequence Transformers, including a leading LLaMA 2-based LLM, on various benchmarks and demonstrate a greater speedup of 2.73x - 7.63x while retaining 98.6% - 99.6% of the accuracy of the original pretrained models. The code is available on our project website at https://yuzhenmao.github.io/IceFormer/.

We present Neighborhood Attention (NA), the first efficient and scalable sliding-window attention mechanism for vision. NA is a pixel-wise operation, localizing self attention (SA) to the nearest neighboring pixels, and therefore enjoys a linear time and space complexity compared to the quadratic complexity of SA. The sliding-window pattern allows NA's receptive field to grow without needing extra pixel shifts, and preserves translational equivariance, unlike Swin Transformer's Window Self Attention (WSA). We develop NATTEN (Neighborhood Attention Extension), a Python package with efficient C++ and CUDA kernels, which allows NA to run up to 40% faster than Swin's WSA while using up to 25% less memory. We further present Neighborhood Attention Transformer (NAT), a new hierarchical transformer design based on NA that boosts image classification and downstream vision performance. Experimental results on NAT are competitive; NAT-Tiny reaches 83.2% top-1 accuracy on ImageNet, 51.4% mAP on MS-COCO and 48.4% mIoU on ADE20K, which is 1.9% ImageNet accuracy, 1.0% COCO mAP, and 2.6% ADE20K mIoU improvement over a Swin model with similar size. To support more research based on sliding-window attention, we open source our project and release our checkpoints at: https://github.com/SHI-Labs/Neighborhood-Attention-Transformer .

In this work we introduce KERNELIZED TRANSFORMER, a generic, scalable, data driven framework for learning the kernel function in Transformers. Our framework approximates the Transformer kernel as a dot product between spectral feature maps and learns the kernel by learning the spectral distribution. This not only helps in learning a generic kernel end-to-end, but also reduces the time and space complexity of Transformers from quadratic to linear. We show that KERNELIZED TRANSFORMERS achieve performance comparable to existing efficient Transformer architectures, both in terms of accuracy as well as computational efficiency. Our study also demonstrates that the choice of the kernel has a substantial impact on performance, and kernel learning variants are competitive alternatives to fixed kernel Transformers, both in long as well as short sequence tasks.

Large transformer models have shown extraordinary success in achieving state-of-the-art results in many natural language processing applications. However, training and deploying these models can be prohibitively costly for long sequences, as the standard self-attention mechanism of the Transformer uses O(n^2) time and space with respect to sequence length. In this paper, we demonstrate that the self-attention mechanism can be approximated by a low-rank matrix. We further exploit this finding to propose a new self-attention mechanism, which reduces the overall self-attention complexity from O(n^2) to O(n) in both time and space. The resulting linear transformer, the Linformer, performs on par with standard Transformer models, while being much more memory- and time-efficient.

Transformer has shown great successes in natural language processing, computer vision, and audio processing. As one of its core components, the softmax attention helps to capture long-range dependencies yet prohibits its scale-up due to the quadratic space and time complexity to the sequence length. Kernel methods are often adopted to reduce the complexity by approximating the softmax operator. Nevertheless, due to the approximation errors, their performances vary in different tasks/corpus and suffer crucial performance drops when compared with the vanilla softmax attention. In this paper, we propose a linear transformer called cosFormer that can achieve comparable or better accuracy to the vanilla transformer in both casual and cross attentions. cosFormer is based on two key properties of softmax attention: i). non-negativeness of the attention matrix; ii). a non-linear re-weighting scheme that can concentrate the distribution of the attention matrix. As its linear substitute, cosFormer fulfills these properties with a linear operator and a cosine-based distance re-weighting mechanism. Extensive experiments on language modeling and text understanding tasks demonstrate the effectiveness of our method. We further examine our method on long sequences and achieve state-of-the-art performance on the Long-Range Arena benchmark. The source code is available at https://github.com/OpenNLPLab/cosFormer.

We introduce Performers, Transformer architectures which can estimate regular (softmax) full-rank-attention Transformers with provable accuracy, but using only linear (as opposed to quadratic) space and time complexity, without relying on any priors such as sparsity or low-rankness. To approximate softmax attention-kernels, Performers use a novel Fast Attention Via positive Orthogonal Random features approach (FAVOR+), which may be of independent interest for scalable kernel methods. FAVOR+ can be also used to efficiently model kernelizable attention mechanisms beyond softmax. This representational power is crucial to accurately compare softmax with other kernels for the first time on large-scale tasks, beyond the reach of regular Transformers, and investigate optimal attention-kernels. Performers are linear architectures fully compatible with regular Transformers and with strong theoretical guarantees: unbiased or nearly-unbiased estimation of the attention matrix, uniform convergence and low estimation variance. We tested Performers on a rich set of tasks stretching from pixel-prediction through text models to protein sequence modeling. We demonstrate competitive results with other examined efficient sparse and dense attention methods, showcasing effectiveness of the novel attention-learning paradigm leveraged by Performers.

Efficient long-context modeling remains a critical challenge for natural language processing (NLP), as the time complexity of the predominant Transformer architecture scales quadratically with the sequence length. While state-space models (SSMs) offer alternative sub-quadratic solutions, they struggle to capture long-range dependencies effectively. In this work, we focus on analyzing and improving the long-context modeling capabilities of SSMs. We show that the widely used synthetic task, associative recall, which requires a model to recall a value associated with a single key without context, insufficiently represents the complexities of real-world long-context modeling. To address this limitation, we extend the associative recall to a novel synthetic task, joint recall, which requires a model to recall the value associated with a key given in a specified context. Theoretically, we prove that SSMs do not have the expressiveness to solve multi-query joint recall in sub-quadratic time complexity. To resolve this issue, we propose a solution based on integrating SSMs with Context-Dependent Sparse Attention (CDSA), which has the expressiveness to solve multi-query joint recall with sub-quadratic computation. To bridge the gap between theoretical analysis and real-world applications, we propose locality-sensitive Hashing Attention with sparse Key Selection (HAX), which instantiates the theoretical solution and is further tailored to natural language domains. Extensive experiments on both synthetic and real-world long-context benchmarks show that HAX consistently outperforms SSM baselines and SSMs integrated with context-independent sparse attention (CISA).

Transformers have revolutionized deep learning across various tasks, including audio representation learning, due to their powerful modeling capabilities. However, they often suffer from quadratic complexity in both GPU memory usage and computational inference time, affecting their efficiency. Recently, state space models (SSMs) like Mamba have emerged as a promising alternative, offering a more efficient approach by avoiding these complexities. Given these advantages, we explore the potential of SSM-based models in audio tasks. In this paper, we introduce Self-Supervised Audio Mamba (SSAMBA), the first self-supervised, attention-free, and SSM-based model for audio representation learning. SSAMBA leverages the bidirectional Mamba to capture complex audio patterns effectively. We incorporate a self-supervised pretraining framework that optimizes both discriminative and generative objectives, enabling the model to learn robust audio representations from large-scale, unlabeled datasets. We evaluated SSAMBA on various tasks such as audio classification, keyword spotting, and speaker identification. Our results demonstrate that SSAMBA outperforms the Self-Supervised Audio Spectrogram Transformer (SSAST) in most tasks. Notably, SSAMBA is approximately 92.7% faster in batch inference speed and 95.4% more memory-efficient than SSAST for the tiny model size with an input token size of 22k. These efficiency gains, combined with superior performance, underscore the effectiveness of SSAMBA's architectural innovation, making it a compelling choice for a wide range of audio processing applications.

Selective state-space models (SSMs) like Mamba overcome some of the shortcomings of Transformers, such as quadratic computational complexity with sequence length and large inference-time memory requirements from the key-value cache. Moreover, recent studies have shown that SSMs can match or exceed the language modeling capabilities of Transformers, making them an attractive alternative. In a controlled setting (e.g., same data), however, studies so far have only presented small scale experiments comparing SSMs to Transformers. To understand the strengths and weaknesses of these architectures at larger scales, we present a direct comparison between 8B-parameter Mamba, Mamba-2, and Transformer models trained on the same datasets of up to 3.5T tokens. We also compare these models to a hybrid architecture consisting of 43% Mamba-2, 7% attention, and 50% MLP layers (Mamba-2-Hybrid). Using a diverse set of tasks, we answer the question of whether Mamba models can match Transformers at larger training budgets. Our results show that while pure SSMs match or exceed Transformers on many tasks, they lag behind Transformers on tasks which require strong copying or in-context learning abilities (e.g., 5-shot MMLU, Phonebook) or long-context reasoning. In contrast, we find that the 8B Mamba-2-Hybrid exceeds the 8B Transformer on all 12 standard tasks we evaluated (+2.65 points on average) and is predicted to be up to 8x faster when generating tokens at inference time. To validate long-context capabilities, we provide additional experiments evaluating variants of the Mamba-2-Hybrid and Transformer extended to support 16K, 32K, and 128K sequences. On an additional 23 long-context tasks, the hybrid model continues to closely match or exceed the Transformer on average. To enable further study, we release the checkpoints as well as the code used to train our models as part of NVIDIA's Megatron-LM project.

Timeseries analytics is of great importance in many real-world applications. Recently, the Transformer model, popular in natural language processing, has been leveraged to learn high quality feature embeddings from timeseries, core to the performance of various timeseries analytics tasks. However, the quadratic time and space complexities limit Transformers' scalability, especially for long timeseries. To address these issues, we develop a timeseries analytics tool, RITA, which uses a novel attention mechanism, named group attention, to address this scalability issue. Group attention dynamically clusters the objects based on their similarity into a small number of groups and approximately computes the attention at the coarse group granularity. It thus significantly reduces the time and space complexity, yet provides a theoretical guarantee on the quality of the computed attention. The dynamic scheduler of RITA continuously adapts the number of groups and the batch size in the training process, ensuring group attention always uses the fewest groups needed to meet the approximation quality requirement. Extensive experiments on various timeseries datasets and analytics tasks demonstrate that RITA outperforms the state-of-the-art in accuracy and is significantly faster -- with speedups of up to 63X.

We propose Scalable Mechanistic Neural Network (S-MNN), an enhanced neural network framework designed for scientific machine learning applications involving long temporal sequences. By reformulating the original Mechanistic Neural Network (MNN) (Pervez et al., 2024), we reduce the computational time and space complexities from cubic and quadratic with respect to the sequence length, respectively, to linear. This significant improvement enables efficient modeling of long-term dynamics without sacrificing accuracy or interpretability. Extensive experiments demonstrate that S-MNN matches the original MNN in precision while substantially reducing computational resources. Consequently, S-MNN can drop-in replace the original MNN in applications, providing a practical and efficient tool for integrating mechanistic bottlenecks into neural network models of complex dynamical systems.

Ever since their conception, Transformers have taken over traditional sequence models in many tasks, such as NLP, image classification, and video/audio processing, for their fast training and superior performance. Much of the merit is attributable to positional encoding and multi-head attention. However, Transformers fall short in learning long-range dependencies mainly due to the quadratic complexity scaled with context length, in terms of both time and space. Consequently, over the past five years, a myriad of methods has been proposed to make Transformers more efficient. In this work, we first take a step back, study and compare existing solutions to long-sequence modeling in terms of their pure mathematical formulation. Specifically, we summarize them using a unified template, given their shared nature of token mixing. Through benchmarks, we then demonstrate that long context length does yield better performance, albeit application-dependent, and traditional Transformer models fall short in taking advantage of long-range dependencies. Next, inspired by emerging sparse models of huge capacity, we propose a machine learning system for handling million-scale dependencies. As a proof of concept, we evaluate the performance of one essential component of this system, namely, the distributed multi-head attention. We show that our algorithm can scale up attention computation by almost 40times using four GeForce RTX 4090 GPUs, compared to vanilla multi-head attention mechanism. We believe this study is an instrumental step towards modeling million-scale dependencies.

Recently, Visual Autoregressive (VAR) Models introduced a groundbreaking advancement in the field of image generation, offering a scalable approach through a coarse-to-fine "next-scale prediction" paradigm. However, the state-of-the-art algorithm of VAR models in [Tian, Jiang, Yuan, Peng and Wang, NeurIPS 2024] takes O(n^4) time, which is computationally inefficient. In this work, we analyze the computational limits and efficiency criteria of VAR Models through a fine-grained complexity lens. Our key contribution is identifying the conditions under which VAR computations can achieve sub-quadratic time complexity. Specifically, we establish a critical threshold for the norm of input matrices used in VAR attention mechanisms. Above this threshold, assuming the Strong Exponential Time Hypothesis (SETH) from fine-grained complexity theory, a sub-quartic time algorithm for VAR models is impossible. To substantiate our theoretical findings, we present efficient constructions leveraging low-rank approximations that align with the derived criteria. This work initiates the study of the computational efficiency of the VAR model from a theoretical perspective. Our technique will shed light on advancing scalable and efficient image generation in VAR frameworks.

Sublinear time complexity is required by the massively parallel computation (MPC) model. Breaking dynamic programs into a set of sparse dynamic programs that can be divided, solved, and merged in sublinear time. The rectangle escape problem (REP) is defined as follows: For n axis-aligned rectangles inside an axis-aligned bounding box B, extend each rectangle in only one of the four directions: up, down, left, or right until it reaches B and the density k is minimized, where k is the maximum number of extensions of rectangles to the boundary that pass through a point inside bounding box B. REP is NP-hard for k>1. If the rectangles are points of a grid (or unit squares of a grid), the problem is called the square escape problem (SEP) and it is still NP-hard. We give a 2-approximation algorithm for SEP with kgeq2 with time complexity O(n^{3/2}k^2). This improves the time complexity of existing algorithms which are at least quadratic. Also, the approximation ratio of our algorithm for kgeq 3 is 3/2 which is tight. We also give a 8-approximation algorithm for REP with time complexity O(nlog n+nk) and give a MPC version of this algorithm for k=O(1) which is the first parallel algorithm for this problem.

We show that for all functions t(n) geq n, every multitape Turing machine running in time t can be simulated in space only O(t log t). This is a substantial improvement over Hopcroft, Paul, and Valiant's simulation of time t in O(t/log t) space from 50 years ago [FOCS 1975, JACM 1977]. Among other results, our simulation implies that bounded fan-in circuits of size s can be evaluated on any input in only s cdot poly(log s) space, and that there are explicit problems solvable in O(n) space which require n^{2-varepsilon} time on a multitape Turing machine for all varepsilon > 0, thereby making a little progress on the P versus PSPACE problem. Our simulation reduces the problem of simulating time-bounded multitape Turing machines to a series of implicitly-defined Tree Evaluation instances with nice parameters, leveraging the remarkable space-efficient algorithm for Tree Evaluation recently found by Cook and Mertz [STOC 2024].

k-Clustering in R^d (e.g., k-median and k-means) is a fundamental machine learning problem. While near-linear time approximation algorithms were known in the classical setting for a dataset with cardinality n, it remains open to find sublinear-time quantum algorithms. We give quantum algorithms that find coresets for k-clustering in R^d with O(nkd^{3/2}) query complexity. Our coreset reduces the input size from n to poly(kepsilon^{-1}d), so that existing alpha-approximation algorithms for clustering can run on top of it and yield (1 + epsilon)alpha-approximation. This eventually yields a quadratic speedup for various k-clustering approximation algorithms. We complement our algorithm with a nearly matching lower bound, that any quantum algorithm must make Omega(nk) queries in order to achieve even O(1)-approximation for k-clustering.

We study the problem of learning hierarchical polynomials over the standard Gaussian distribution with three-layer neural networks. We specifically consider target functions of the form h = g circ p where p : R^d rightarrow R is a degree k polynomial and g: R rightarrow R is a degree q polynomial. This function class generalizes the single-index model, which corresponds to k=1, and is a natural class of functions possessing an underlying hierarchical structure. Our main result shows that for a large subclass of degree k polynomials p, a three-layer neural network trained via layerwise gradient descent on the square loss learns the target h up to vanishing test error in mathcal{O}(d^k) samples and polynomial time. This is a strict improvement over kernel methods, which require widetilde Theta(d^{kq}) samples, as well as existing guarantees for two-layer networks, which require the target function to be low-rank. Our result also generalizes prior works on three-layer neural networks, which were restricted to the case of p being a quadratic. When p is indeed a quadratic, we achieve the information-theoretically optimal sample complexity mathcal{O}(d^2), which is an improvement over prior work~nichani2023provable requiring a sample size of widetildeTheta(d^4). Our proof proceeds by showing that during the initial stage of training the network performs feature learning to recover the feature p with mathcal{O}(d^k) samples. This work demonstrates the ability of three-layer neural networks to learn complex features and as a result, learn a broad class of hierarchical functions.

Motivated by the problem of fast processing of attention matrices, we study fast algorithms for computing matrix-vector products for asymmetric Gaussian Kernel matrices Kin R^{ntimes n}. K's columns are indexed by a set of n keys k_1,k_2ldots, k_nin R^d, rows by a set of n queries q_1,q_2,ldots,q_nin R^d , and its i,j entry is K_{ij} = e^{-|q_i-k_j|_2^2/2sigma^2} for some bandwidth parameter sigma>0. Given a vector xin R^n and error parameter epsilon>0, our task is to output a yin R^n such that |Kx-y|_2leq epsilon |x|_2 in time subquadratic in n and linear in d. Our algorithms rely on the following modelling assumption about the matrices K: the sum of the entries of K scales linearly in n, as opposed to worst case quadratic growth. We validate this assumption experimentally, for Gaussian kernel matrices encountered in various settings such as fast attention computation in LLMs. We obtain the first subquadratic-time algorithm that works under this assumption, for unrestricted vectors.

Alternating Direction Method of Multipliers (ADMM) has been used successfully in many conventional machine learning applications and is considered to be a useful alternative to Stochastic Gradient Descent (SGD) as a deep learning optimizer. However, as an emerging domain, several challenges remain, including 1) The lack of global convergence guarantees, 2) Slow convergence towards solutions, and 3) Cubic time complexity with regard to feature dimensions. In this paper, we propose a novel optimization framework for deep learning via ADMM (dlADMM) to address these challenges simultaneously. The parameters in each layer are updated backward and then forward so that the parameter information in each layer is exchanged efficiently. The time complexity is reduced from cubic to quadratic in (latent) feature dimensions via a dedicated algorithm design for subproblems that enhances them utilizing iterative quadratic approximations and backtracking. Finally, we provide the first proof of global convergence for an ADMM-based method (dlADMM) in a deep neural network problem under mild conditions. Experiments on benchmark datasets demonstrated that our proposed dlADMM algorithm outperforms most of the comparison methods.

This article concerns the computational complexity of a fundamental problem in number theory: counting points on curves and surfaces over finite fields. There is no subexponential-time algorithm known and it is unclear if it can be NP-hard. Given a curve, we present the first efficient Arthur-Merlin protocol to certify its point-count, its Jacobian group structure, and its Hasse-Weil zeta function. We extend this result to a smooth projective surface to certify the factor P_{1}(T), corresponding to the first Betti number, of the zeta function; by using the counting oracle. We give the first algorithm to compute P_{1}(T) that is poly(log q)-time if the degree D of the input surface is fixed; and in quantum poly(Dlog q)-time in general. Our technique in the curve case, is to sample hash functions using the Weil and Riemann-Roch bounds, to certify the group order of its Jacobian. For higher dimension varieties, we first reduce to the case of a surface, which is fibred as a Lefschetz pencil of hyperplane sections over P^{1}. The formalism of vanishing cycles, and the inherent big monodromy, enable us to prove an effective version of Deligne's `theoreme du pgcd' using the hard-Lefschetz theorem and an equidistribution result due to Katz. These reduce our investigations to that of computing the zeta function of a curve, defined over a finite field extension F_{Q}/F_{q} of poly-bounded degree. This explicitization of the theory yields the first nontrivial upper bounds on the computational complexity.

When rows of an n times d matrix A are given in a stream, we study algorithms for approximating the top eigenvector of the matrix {A}^TA (equivalently, the top right singular vector of A). We consider worst case inputs A but assume that the rows are presented to the streaming algorithm in a uniformly random order. We show that when the gap parameter R = σ_1(A)^2/σ_2(A)^2 = Ω(1), then there is a randomized algorithm that uses O(h cdot d cdot polylog(d)) bits of space and outputs a unit vector v that has a correlation 1 - O(1/R) with the top eigenvector v_1. Here h denotes the number of heavy rows in the matrix, defined as the rows with Euclidean norm at least |{A}|_F/d cdot operatorname{polylog(d)}. We also provide a lower bound showing that any algorithm using O(hd/R) bits of space can obtain at most 1 - Ω(1/R^2) correlation with the top eigenvector. Thus, parameterizing the space complexity in terms of the number of heavy rows is necessary for high accuracy solutions. Our results improve upon the R = Ω(log n cdot log d) requirement in a recent work of Price and Xun (FOCS 2024). We note that the algorithm of Price and Xun works for arbitrary order streams whereas our algorithm requires a stronger assumption that the rows are presented in a uniformly random order. We additionally show that the gap requirements in their analysis can be brought down to R = Ω(log^2 d) for arbitrary order streams and R = Ω(log d) for random order streams. The requirement of R = Ω(log d) for random order streams is nearly tight for their analysis as we obtain a simple instance with R = Ω(log d/loglog d) for which their algorithm, with any fixed learning rate, cannot output a vector approximating the top eigenvector v_1.

We study the representation complexity of model-based and model-free reinforcement learning (RL) in the context of circuit complexity. We prove theoretically that there exists a broad class of MDPs such that their underlying transition and reward functions can be represented by constant depth circuits with polynomial size, while the optimal Q-function suffers an exponential circuit complexity in constant-depth circuits. By drawing attention to the approximation errors and building connections to complexity theory, our theory provides unique insights into why model-based algorithms usually enjoy better sample complexity than model-free algorithms from a novel representation complexity perspective: in some cases, the ground-truth rule (model) of the environment is simple to represent, while other quantities, such as Q-function, appear complex. We empirically corroborate our theory by comparing the approximation error of the transition kernel, reward function, and optimal Q-function in various Mujoco environments, which demonstrates that the approximation errors of the transition kernel and reward function are consistently lower than those of the optimal Q-function. To the best of our knowledge, this work is the first to study the circuit complexity of RL, which also provides a rigorous framework for future research.

Code based Language Models (LMs) have shown very promising results in the field of software engineering with applications such as code refinement, code completion and generation. However, the task of time and space complexity classification from code has not been extensively explored due to a lack of datasets, with prior endeavors being limited to Java. In this project, we aim to address these gaps by creating a labelled dataset of code snippets spanning multiple languages (Python and C++ datasets currently, with C, C#, and JavaScript datasets being released shortly). We find that existing time complexity calculation libraries and tools only apply to a limited number of use-cases. The lack of a well-defined rule based system motivates the application of several recently proposed code-based LMs. We demonstrate the effectiveness of dead code elimination and increasing the maximum sequence length of LMs. In addition to time complexity, we propose to use LMs to find space complexities from code, and to the best of our knowledge, this is the first attempt to do so. Furthermore, we introduce a novel code comprehension task, called cross-language transfer, where we fine-tune the LM on one language and run inference on another. Finally, we visualize the activation of the attention fed classification head of our LMs using Non-negative Matrix Factorization (NMF) to interpret our results.

Sphere packing, Hilbert's eighteenth problem, asks for the densest arrangement of congruent spheres in n-dimensional Euclidean space. Although relevant to areas such as cryptography, crystallography, and medical imaging, the problem remains unresolved: beyond a few special dimensions, neither optimal packings nor tight upper bounds are known. Even a major breakthrough in dimension n=8, later recognised with a Fields Medal, underscores its difficulty. A leading technique for upper bounds, the three-point method, reduces the problem to solving large, high-precision semidefinite programs (SDPs). Because each candidate SDP may take days to evaluate, standard data-intensive AI approaches are infeasible. We address this challenge by formulating SDP construction as a sequential decision process, the SDP game, in which a policy assembles SDP formulations from a set of admissible components. Using a sample-efficient model-based framework that combines Bayesian optimisation with Monte Carlo Tree Search, we obtain new state-of-the-art upper bounds in dimensions 4-16, showing that model-based search can advance computational progress in longstanding geometric problems. Together, these results demonstrate that sample-efficient, model-based search can make tangible progress on mathematically rigid, evaluation limited problems, pointing towards a complementary direction for AI-assisted discovery beyond large-scale LLM-driven exploration.

We study the conditions under which the convex relaxation of a mixed-integer linear programming formulation for ordered optimization problems, where sorting is part of the decision process, yields integral optimal solutions. Thereby solving the problem exactly in polynomial time. Our analysis identifies structural properties of the input data that influence the integrality of the relaxation. We show that incorporating ordered components introduces additional layers of combinatorial complexity that invalidate the exactness observed in classical (non-ordered) settings. In particular, for certain ordered problems such as the min--max case, the linear relaxation never recovers the integral solution. These results clarify the intrinsic hardness introduced by sorting and reveal that the strength of the relaxation depends critically on the ``proximity'' of the ordered problem to its classical counterpart: problems closer to the non-ordered case tend to admit tighter relaxations, while those further away exhibit substantially weaker behavior. Computational experiments on benchmark instances confirm the predictive value of the integrality conditions and demonstrate the practical implications of exact relaxations for ordered location problems.

Modern machine learning algorithms, especially deep learning based techniques, typically involve careful hyperparameter tuning to achieve the best performance. Despite the surge of intense interest in practical techniques like Bayesian optimization and random search based approaches to automating this laborious and compute intensive task, the fundamental learning theoretic complexity of tuning hyperparameters for deep neural networks is poorly understood. Inspired by this glaring gap, we initiate the formal study of hyperparameter tuning complexity in deep learning through a recently introduced data driven setting. We assume that we have a series of deep learning tasks, and we have to tune hyperparameters to do well on average over the distribution of tasks. A major difficulty is that the utility function as a function of the hyperparameter is very volatile and furthermore, it is given implicitly by an optimization problem over the model parameters. To tackle this challenge, we introduce a new technique to characterize the discontinuities and oscillations of the utility function on any fixed problem instance as we vary the hyperparameter; our analysis relies on subtle concepts including tools from differential/algebraic geometry and constrained optimization. This can be used to show that the learning theoretic complexity of the corresponding family of utility functions is bounded. We instantiate our results and provide sample complexity bounds for concrete applications tuning a hyperparameter that interpolates neural activation functions and setting the kernel parameter in graph neural networks.

The complexity of learning a concept class under Gaussian marginals in the difficult agnostic model is closely related to its L_1-approximability by low-degree polynomials. For any concept class with Gaussian surface area at most Γ, Klivans et al. (2008) show that degree d = O(Γ^2 / varepsilon^4) suffices to achieve an varepsilon-approximation. This leads to the best-known bounds on the complexity of learning a variety of concept classes. In this note, we improve their analysis by showing that degree d = tilde O (Γ^2 / varepsilon^2) is enough. In light of lower bounds due to Diakonikolas et al. (2021), this yields (near) optimal bounds on the complexity of agnostically learning polynomial threshold functions in the statistical query model. Our proof relies on a direct analogue of a construction of Feldman et al. (2020), who considered L_1-approximation on the Boolean hypercube.

Large language model (LLM) agents exhibit strong mathematical problem-solving abilities and can even solve International Mathematical Olympiad (IMO) level problems with the assistance of formal proof systems. However, due to weak heuristics for auxiliary constructions, AI for geometry problem solving remains dominated by expert models such as AlphaGeometry 2, which rely heavily on large-scale data synthesis and search for both training and evaluation. In this work, we make the first attempt to build a medalist-level LLM agent for geometry and present InternGeometry. InternGeometry overcomes the heuristic limitations in geometry by iteratively proposing propositions and auxiliary constructions, verifying them with a symbolic engine, and reflecting on the engine's feedback to guide subsequent proposals. A dynamic memory mechanism enables InternGeometry to conduct more than two hundred interactions with the symbolic engine per problem. To further accelerate learning, we introduce Complexity-Boosting Reinforcement Learning (CBRL), which gradually increases the complexity of synthesized problems across training stages. Built on InternThinker-32B, InternGeometry solves 44 of 50 IMO geometry problems (2000-2024), exceeding the average gold medalist score (40.9), using only 13K training examples, just 0.004% of the data used by AlphaGeometry 2, demonstrating the potential of LLM agents on expert-level geometry tasks. InternGeometry can also propose novel auxiliary constructions for IMO problems that do not appear in human solutions. We will release the model, data, and symbolic engine to support future research.

We prove that any Turing machine running on inputs of arbitrary length can be simulated by a constant bit-size transformer, as long as the context window is sufficiently long. This improves previous works, which require scaling up either the model's precision or the number of parameters on longer inputs. Furthermore, we prove that the complexity class SPACE[s(n)] exactly characterizes the expressive power of a constant bit-size transformer with a context window of length s(n). Our approach relies on simulating Post machines, a Turing-complete computational model. Post machines can be modeled as automata equipped with a queue, exhibiting computational behaviors naturally aligned with those of transformers. The behavioral similarity between transformers and Post machines may offer new insights into the mechanisms underlying the reasoning abilities of transformers.

We present an approximate attention mechanism named HyperAttention to address the computational challenges posed by the growing complexity of long contexts used in Large Language Models (LLMs). Recent work suggests that in the worst-case scenario, quadratic time is necessary unless the entries of the attention matrix are bounded or the matrix has low stable rank. We introduce two parameters which measure: (1) the max column norm in the normalized attention matrix, and (2) the ratio of row norms in the unnormalized attention matrix after detecting and removing large entries. We use these fine-grained parameters to capture the hardness of the problem. Despite previous lower bounds, we are able to achieve a linear time sampling algorithm even when the matrix has unbounded entries or a large stable rank, provided the above parameters are small. HyperAttention features a modular design that easily accommodates integration of other fast low-level implementations, particularly FlashAttention. Empirically, employing Locality Sensitive Hashing (LSH) to identify large entries, HyperAttention outperforms existing methods, giving significant speed improvements compared to state-of-the-art solutions like FlashAttention. We validate the empirical performance of HyperAttention on a variety of different long-context length datasets. For example, HyperAttention makes the inference time of ChatGLM2 50\% faster on 32k context length while perplexity increases from 5.6 to 6.3. On larger context length, e.g., 131k, with causal masking, HyperAttention offers 5-fold speedup on a single attention layer.

An established measure of the expressive power of a given ReLU neural network is the number of linear regions into which it partitions the input space. There exist many different, non-equivalent definitions of what a linear region actually is. We systematically assess which papers use which definitions and discuss how they relate to each other. We then analyze the computational complexity of counting the number of such regions for the various definitions. Generally, this turns out to be an intractable problem. We prove NP- and #P-hardness results already for networks with one hidden layer and strong hardness of approximation results for two or more hidden layers. Finally, on the algorithmic side, we demonstrate that counting linear regions can at least be achieved in polynomial space for some common definitions.

Maximizing a monotone submodular function under cardinality constraint k is a core problem in machine learning and database with many basic applications, including video and data summarization, recommendation systems, feature extraction, exemplar clustering, and coverage problems. We study this classic problem in the fully dynamic model where a stream of insertions and deletions of elements of an underlying ground set is given and the goal is to maintain an approximate solution using a fast update time. A recent paper at NeurIPS'20 by Lattanzi, Mitrovic, Norouzi{-}Fard, Tarnawski, Zadimoghaddam claims to obtain a dynamic algorithm for this problem with a 1{2} -epsilon approximation ratio and a query complexity bounded by poly(log(n),log(k),epsilon^{-1}). However, as we explain in this paper, the analysis has some important gaps. Having a dynamic algorithm for the problem with polylogarithmic update time is even more important in light of a recent result by Chen and Peng at STOC'22 who show a matching lower bound for the problem -- any randomized algorithm with a 1{2}+epsilon approximation ratio must have an amortized query complexity that is polynomial in n. In this paper, we develop a simpler algorithm for the problem that maintains a (1{2}-epsilon)-approximate solution for submodular maximization under cardinality constraint k using a polylogarithmic amortized update time.

Time series play a fundamental role in many domains, capturing a plethora of information about the underlying data-generating processes. When a process generates multiple synchronized signals we are faced with multidimensional time series. In this context a fundamental problem is that of motif mining, where we seek patterns repeating twice with minor variations, spanning some of the dimensions. State of the art exact solutions for this problem run in time quadratic in the length of the input time series. We provide a scalable method to find the top-k motifs in multidimensional time series with probabilistic guarantees on the quality of the results. Our algorithm runs in time subquadratic in the length of the input, and returns the exact solution with probability at least 1-δ, where δ is a user-defined parameter. The algorithm is designed to be adaptive to the input distribution, self-tuning its parameters while respecting user-defined limits on the memory to use. Our theoretical analysis is complemented by an extensive experimental evaluation, showing that our algorithm is orders of magnitude faster than the state of the art.

In the context of distributed synchronous computing, processors perform in rounds, and the time-complexity of a distributed algorithm is classically defined as the number of rounds before all computing nodes have output. Hence, this complexity measure captures the running time of the slowest node(s). In this paper, we are interested in the running time of the ordinary nodes, to be compared with the running time of the slowest nodes. The node-averaged time-complexity of a distributed algorithm on a given instance is defined as the average, taken over every node of the instance, of the number of rounds before that node output. We compare the node-averaged time-complexity with the classical one in the standard LOCAL model for distributed network computing. We show that there can be an exponential gap between the node-averaged time-complexity and the classical time-complexity, as witnessed by, e.g., leader election. Our first main result is a positive one, stating that, in fact, the two time-complexities behave the same for a large class of problems on very sparse graphs. In particular, we show that, for LCL problems on cycles, the node-averaged time complexity is of the same order of magnitude as the slowest node time-complexity. In addition, in the LOCAL model, the time-complexity is computed as a worst case over all possible identity assignments to the nodes of the network. In this paper, we also investigate the ID-averaged time-complexity, when the number of rounds is averaged over all possible identity assignments. Our second main result is that the ID-averaged time-complexity is essentially the same as the expected time-complexity of randomized algorithms (where the expectation is taken over all possible random bits used by the nodes, and the number of rounds is measured for the worst-case identity assignment). Finally, we study the node-averaged ID-averaged time-complexity.

In this paper, we analyze the computational limitations of Mamba and State-space Models (SSMs) by using the circuit complexity framework. Despite Mamba's stateful design and recent attention as a strong candidate to outperform Transformers, we have demonstrated that both Mamba and SSMs with poly(n)-precision and constant-depth layers reside within the DLOGTIME-uniform TC^0 complexity class. This result indicates Mamba has the same computational capabilities as Transformer theoretically, and it cannot solve problems like arithmetic formula problems, boolean formula value problems, and permutation composition problems if TC^0 neq NC^1. Therefore, it challenges the assumption Mamba is more computationally expressive than Transformers. Our contributions include rigorous proofs showing that Selective SSM and Mamba architectures can be simulated by DLOGTIME-uniform TC^0 circuits, and they cannot solve problems outside TC^0.

Scaling inference-time compute has emerged as an important driver of LLM performance, making inference efficiency a central focus of model design alongside model quality. While the current Transformer-based models deliver strong model quality, their quadratic compute and linear memory make inference expensive. This has spurred the development of sub-quadratic models with reduced linear compute and constant memory requirements. However, many recent linear models trade off model quality and capability for algorithmic efficiency, failing on tasks such as state tracking. Moreover, their theoretically linear inference remains hardware-inefficient in practice. Guided by an inference-first perspective, we introduce three core methodological improvements inspired by the state space model (SSM) viewpoint of linear models. We combine: (1) a more expressive recurrence derived from SSM discretization, (2) a complex-valued state update rule that enables richer state tracking, and (3) a multi-input, multi-output (MIMO) formulation for better model performance without increasing decode latency. Together with architectural refinements, our Mamba-3 model achieves significant gains across retrieval, state-tracking, and downstream language modeling tasks. At the 1.5B scale, Mamba-3 improves average downstream accuracy by 0.6 percentage points compared to the next best model (Gated DeltaNet), with Mamba-3's MIMO variant further improving accuracy by another 1.2 points for a total 1.8 point gain. Across state-size experiments, Mamba-3 achieves comparable perplexity to Mamba-2 despite using half of its predecessor's state size. Our evaluations demonstrate Mamba-3's ability to advance the performance-efficiency Pareto frontier.

The non-convexity of the artificial neural network (ANN) training landscape brings inherent optimization difficulties. While the traditional back-propagation stochastic gradient descent (SGD) algorithm and its variants are effective in certain cases, they can become stuck at spurious local minima and are sensitive to initializations and hyperparameters. Recent work has shown that the training of an ANN with ReLU activations can be reformulated as a convex program, bringing hope to globally optimizing interpretable ANNs. However, naively solving the convex training formulation has an exponential complexity, and even an approximation heuristic requires cubic time. In this work, we characterize the quality of this approximation and develop two efficient algorithms that train ANNs with global convergence guarantees. The first algorithm is based on the alternating direction method of multiplier (ADMM). It solves both the exact convex formulation and the approximate counterpart. Linear global convergence is achieved, and the initial several iterations often yield a solution with high prediction accuracy. When solving the approximate formulation, the per-iteration time complexity is quadratic. The second algorithm, based on the "sampled convex programs" theory, is simpler to implement. It solves unconstrained convex formulations and converges to an approximately globally optimal classifier. The non-convexity of the ANN training landscape exacerbates when adversarial training is considered. We apply the robust convex optimization theory to convex training and develop convex formulations that train ANNs robust to adversarial inputs. Our analysis explicitly focuses on one-hidden-layer fully connected ANNs, but can extend to more sophisticated architectures.

We study the computational complexity of approximating general constrained Markov decision processes. Our primary contribution is the design of a polynomial time (0,ε)-additive bicriteria approximation algorithm for finding optimal constrained policies across a broad class of recursively computable constraints, including almost-sure, chance, expectation, and their anytime variants. Matching lower bounds imply our approximation guarantees are optimal so long as P neq NP. The generality of our approach results in answers to several long-standing open complexity questions in the constrained reinforcement learning literature. Specifically, we are the first to prove polynomial-time approximability for the following settings: policies under chance constraints, deterministic policies under multiple expectation constraints, policies under non-homogeneous constraints (i.e., constraints of different types), and policies under constraints for continuous-state processes.

The class of tree-adjoining languages can be characterized by various two-level formalisms, consisting of a context-free grammar (CFG) or pushdown automaton (PDA) controlling another CFG or PDA. These four formalisms are equivalent to tree-adjoining grammars (TAG), linear indexed grammars (LIG), pushdown-adjoining automata (PAA), and embedded pushdown automata (EPDA). We define semiring-weighted versions of the above two-level formalisms, and we design new algorithms for computing their stringsums (the weight of all derivations of a string) and allsums (the weight of all derivations). From these, we also immediately obtain stringsum and allsum algorithms for TAG, LIG, PAA, and EPDA. For LIG, our algorithm is more time-efficient by a factor of O(n|N|) (where n is the string length and |N| is the size of the nonterminal set) and more space-efficient by a factor of O(|Gamma|) (where |Gamma| is the size of the stack alphabet) than the algorithm of Vijay-Shanker and Weir (1989). For EPDA, our algorithm is both more space-efficient and time-efficient than the algorithm of Alonso et al. (2001) by factors of O(|Gamma|^2) and O(|Gamma|^3), respectively. Finally, we give the first PAA stringsum and allsum algorithms.

Inspired by the diversity of biological neurons, quadratic artificial neurons can play an important role in deep learning models. The type of quadratic neurons of our interest replaces the inner-product operation in the conventional neuron with a quadratic function. Despite promising results so far achieved by networks of quadratic neurons, there are important issues not well addressed. Theoretically, the superior expressivity of a quadratic network over either a conventional network or a conventional network via quadratic activation is not fully elucidated, which makes the use of quadratic networks not well grounded. Practically, although a quadratic network can be trained via generic backpropagation, it can be subject to a higher risk of collapse than the conventional counterpart. To address these issues, we first apply the spline theory and a measure from algebraic geometry to give two theorems that demonstrate better model expressivity of a quadratic network than the conventional counterpart with or without quadratic activation. Then, we propose an effective training strategy referred to as ReLinear to stabilize the training process of a quadratic network, thereby unleashing the full potential in its associated machine learning tasks. Comprehensive experiments on popular datasets are performed to support our findings and confirm the performance of quadratic deep learning. We have shared our code in https://github.com/FengleiFan/ReLinear.

Recent works have demonstrated reasonable success of representation learning in hypercomplex space. Specifically, "fully-connected layers with Quaternions" (4D hypercomplex numbers), which replace real-valued matrix multiplications in fully-connected layers with Hamilton products of Quaternions, both enjoy parameter savings with only 1/4 learnable parameters and achieve comparable performance in various applications. However, one key caveat is that hypercomplex space only exists at very few predefined dimensions (4D, 8D, and 16D). This restricts the flexibility of models that leverage hypercomplex multiplications. To this end, we propose parameterizing hypercomplex multiplications, allowing models to learn multiplication rules from data regardless of whether such rules are predefined. As a result, our method not only subsumes the Hamilton product, but also learns to operate on any arbitrary nD hypercomplex space, providing more architectural flexibility using arbitrarily 1/n learnable parameters compared with the fully-connected layer counterpart. Experiments of applications to the LSTM and Transformer models on natural language inference, machine translation, text style transfer, and subject verb agreement demonstrate architectural flexibility and effectiveness of the proposed approach.

We study the classic Text-to-Pattern Hamming Distances problem: given a pattern P of length m and a text T of length n, both over a polynomial-size alphabet, compute the Hamming distance between P and T[i, ., . , i+m-1] for every shift i, under the standard Word-RAM model with Theta(log n)-bit words. - We provide an O(nm) time Las Vegas randomized algorithm for this problem, beating the decades-old O(n m log m) running time [Abrahamson, SICOMP 1987]. We also obtain a deterministic algorithm, with a slightly higher O(nm(log mloglog m)^{1/4}) running time. Our randomized algorithm extends to the k-bounded setting, with running time Obig(n+nk{m}big), removing all the extra logarithmic factors from earlier algorithms [Gawrychowski and Uzna\'{n}ski, ICALP 2018; Chan, Golan, Kociumaka, Kopelowitz and Porat, STOC 2020]. - For the (1+epsilon)-approximate version of Text-to-Pattern Hamming Distances, we give an O(epsilon^{-0.93}n) time Monte Carlo randomized algorithm, beating the previous O(epsilon^{-1}n) running time [Kopelowitz and Porat, FOCS 2015; Kopelowitz and Porat, SOSA 2018]. Our approximation algorithm exploits a connection with 3SUM, and uses a combination of Fredman's trick, equality matrix product, and random sampling; in particular, we obtain new results on approximate counting versions of 3SUM and Exact Triangle, which may be of independent interest. Our exact algorithms use a novel combination of hashing, bit-packed FFT, and recursion; in particular, we obtain a faster algorithm for computing the sumset of two integer sets, in the regime when the universe size is close to quadratic in the number of elements. We also prove a fine-grained equivalence between the exact Text-to-Pattern Hamming Distances problem and a range-restricted, counting version of 3SUM.

In distributed deep learning with data parallelism, synchronizing gradients at each training step can cause a huge communication overhead, especially when many nodes work together to train large models. Local gradient methods, such as Local SGD, address this issue by allowing workers to compute locally for H steps without synchronizing with others, hence reducing communication frequency. While H has been viewed as a hyperparameter to trade optimization efficiency for communication cost, recent research indicates that setting a proper H value can lead to generalization improvement. Yet, selecting a proper H is elusive. This work proposes a theory-grounded method for determining H, named the Quadratic Synchronization Rule (QSR), which recommends dynamically setting H in proportion to 1{eta^2} as the learning rate eta decays over time. Extensive ImageNet experiments on ResNet and ViT show that local gradient methods with QSR consistently improve the test accuracy over other synchronization strategies. Compared with the standard data parallel training, QSR enables Local AdamW on ViT-B to cut the training time on 16 or 64 GPUs down from 26.7 to 20.2 hours or from 8.6 to 5.5 hours and, at the same time, achieves 1.16% or 0.84% higher top-1 validation accuracy.

Using a long-standing conjecture from combinatorial group theory, we explore, from multiple perspectives, the challenges of finding rare instances carrying disproportionately high rewards. Based on lessons learned in the context defined by the Andrews-Curtis conjecture, we propose algorithmic enhancements and a topological hardness measure with implications for a broad class of search problems. As part of our study, we also address several open mathematical questions. Notably, we demonstrate the length reducibility of all but two presentations in the Akbulut-Kirby series (1981), and resolve various potential counterexamples in the Miller-Schupp series (1991), including three infinite subfamilies.

In this work, we consider the optimization process of minibatch stochastic gradient descent (SGD) on a 2-layer neural network with data separated by a quadratic ground truth function. We prove that with data drawn from the d-dimensional Boolean hypercube labeled by the quadratic ``XOR'' function y = -x_ix_j, it is possible to train to a population error o(1) with d :polylog(d) samples. Our result considers simultaneously training both layers of the two-layer-neural network with ReLU activations via standard minibatch SGD on the logistic loss. To our knowledge, this work is the first to give a sample complexity of O(d) for efficiently learning the XOR function on isotropic data on a standard neural network with standard training. Our main technique is showing that the network evolves in two phases: a signal-finding phase where the network is small and many of the neurons evolve independently to find features, and a signal-heavy phase, where SGD maintains and balances the features. We leverage the simultaneous training of the layers to show that it is sufficient for only a small fraction of the neurons to learn features, since those neurons will be amplified by the simultaneous growth of their second layer weights.

We study the task of agnostically learning halfspaces under the Gaussian distribution. Specifically, given labeled examples (x,y) from an unknown distribution on R^n times { pm 1}, whose marginal distribution on x is the standard Gaussian and the labels y can be arbitrary, the goal is to output a hypothesis with 0-1 loss OPT+epsilon, where OPT is the 0-1 loss of the best-fitting halfspace. We prove a near-optimal computational hardness result for this task, under the widely believed sub-exponential time hardness of the Learning with Errors (LWE) problem. Prior hardness results are either qualitatively suboptimal or apply to restricted families of algorithms. Our techniques extend to yield near-optimal lower bounds for related problems, including ReLU regression.

We consider a family of convex quadratic programs in which the coefficients of the linear objective term and the righthand side of the constraints are affine functions of a parameter. It is well known that the solution of such a parametrized quadratic program is a piecewise affine function of the parameter. The number of (polyhedral) regions in the solution map can grow exponentially in problem size, but when the number of regions is moderate, a so-called explicit solver is practical. Such a solver computes the coefficients of the affine functions and the linear inequalities defining the polyhedral regions offline; to solve a problem instance online it simply evaluates this explicit solution map. Potential advantages of an explicit solver over a more general purpose iterative solver can include transparency, interpretability, reliability, and speed. In this paper we describe how code generation can be used to automatically generate an explicit solver from a high level description of a parametrized quadratic program. Our method has been implemented in the open-source software CVXPYgen, which is part of CVXPY, a domain specific language for general convex optimization.

We study optimization problems on Hadamard manifolds, motivated by recent advances in geometric approaches to optimization on curved spaces, particularly those involving the structure of Busemann functions. We introduce a projection based variant of the proximal point algorithm, termed the Busemann hybrid projection proximal point algorithm, which replaces Euclidean hyperplanes with horospheres defined via convex Busemann functions. The algorithm performs projections in closed form using the gradients of these functions, resulting in a geometrically intrinsic scheme that requires no tangent space linear solves. We allow for inexact subgradient evaluations and prove global convergence under controlled inexactness, with a relative error level strictly below one. We establish a Fejér type descent and sublinear complexity with a rate proportional to the inverse square root of the iteration count, and show that the exact variant coincides with the classical Riemannian proximal point algorithm. The framework clarifies the role of Busemann based subdifferentials in optimization on spaces of nonpositive curvature.

We compare the (1,lambda)-EA and the (1 + lambda)-EA on the recently introduced benchmark DisOM, which is the OneMax function with randomly planted local optima. Previous work showed that if all local optima have the same relative height, then the plus strategy never loses more than a factor O(nlog n) compared to the comma strategy. Here we show that even small random fluctuations in the heights of the local optima have a devastating effect for the plus strategy and lead to super-polynomial runtimes. On the other hand, due to their ability to escape local optima, comma strategies are unaffected by the height of the local optima and remain efficient. Our results hold for a broad class of possible distortions and show that the plus strategy, but not the comma strategy, is generally deceived by sparse unstructured fluctuations of a smooth landscape.

This paper introduces bucket calculus, a novel mathematical framework that fundamentally transforms the computational complexity landscape of parallel machine scheduling optimization. We address the strongly NP-hard problem P2|r_j|C_{max} through an innovative adaptive temporal discretization methodology that achieves exponential complexity reduction from O(T^n) to O(B^n) where B ll T, while maintaining near-optimal solution quality. Our bucket-indexed mixed-integer linear programming (MILP) formulation exploits dimensional complexity heterogeneity through precision-aware discretization, reducing decision variables by 94.4\% and achieving a theoretical speedup factor 2.75 times 10^{37} for 20-job instances. Theoretical contributions include partial discretization theory, fractional bucket calculus operators, and quantum-inspired mechanisms for temporal constraint modeling. Empirical validation on instances with 20--400 jobs demonstrates 97.6\% resource utilization, near-perfect load balancing (σ/μ= 0.006), and sustained performance across problem scales with optimality gaps below 5.1\%. This work represents a paradigm shift from fine-grained temporal discretization to multi-resolution precision allocation, bridging the fundamental gap between exact optimization and computational tractability for industrial-scale NP-hard scheduling problems.

In the classical transformer attention scheme, we are given three n times d size matrices Q, K, V (the query, key, and value tokens), and the goal is to compute a new n times d size matrix D^{-1} exp(QK^top) V where D = diag( exp(QK^top) {bf 1}_n ). In this work, we study a generalization of attention which captures triple-wise correlations. This generalization is able to solve problems about detecting triple-wise connections that were shown to be impossible for transformers. The potential downside of this generalization is that it appears as though computations are even more difficult, since the straightforward algorithm requires cubic time in n. However, we show that in the bounded-entry setting (which arises in practice, and which is well-studied in both theory and practice), there is actually a near-linear time algorithm. More precisely, we show that bounded entries are both necessary and sufficient for quickly performing generalized computations: bullet On the positive side, if all entries of the input matrices are bounded above by o(sqrt[3]{log n}) then we show how to approximate the ``tensor-type'' attention matrix in n^{1+o(1)} time. bullet On the negative side, we show that if the entries of the input matrices may be as large as Omega(sqrt[3]{log n}), then there is no algorithm that runs faster than n^{3-o(1)} (assuming the Strong Exponential Time Hypothesis from fine-grained complexity theory). We also show that our construction, algorithms, and lower bounds naturally generalize to higher-order tensors and correlations. Interestingly, the higher the order of the tensors, the lower the bound on the entries needs to be for an efficient algorithm. Our results thus yield a natural tradeoff between the boundedness of the entries, and order of the tensor one may use for more expressive, efficient attention computation.

Quantum optimization holds promise for addressing classically intractable combinatorial problems, yet a standardized framework for benchmarking its performance, particularly in terms of solution quality, computational speed, and scalability is still lacking. In this work, we introduce a comprehensive benchmarking framework designed to systematically evaluate a range of quantum optimization techniques against well-established NP-hard combinatorial problems. Our framework focuses on key problem classes, including the Multi-Dimensional Knapsack Problem (MDKP), Maximum Independent Set (MIS), Quadratic Assignment Problem (QAP), and Market Share Problem (MSP). Our study evaluates gate-based quantum approaches, including the Variational Quantum Eigensolver (VQE) and its CVaR-enhanced variant, alongside advanced quantum algorithms such as the Quantum Approximate Optimization Algorithm (QAOA) and its extensions. To address resource constraints, we incorporate qubit compression techniques like Pauli Correlation Encoding (PCE) and Quantum Random Access Optimization (QRAO). Experimental results, obtained from simulated quantum environments and classical solvers, provide key insights into feasibility, optimality gaps, and scalability. Our findings highlight both the promise and current limitations of quantum optimization, offering a structured pathway for future research and practical applications in quantum-enhanced decision-making.

While recent works (e.g. o1, DeepSeek R1) have demonstrated great promise of using long Chain-of-Thought (CoT) to improve reasoning capabilities of language models, scaling it up during test-time is challenging due to inefficient memory usage -- intermediate computations accumulate indefinitely in context even no longer needed for future thoughts. We propose PENCIL, which incorporates a reduction mechanism into the autoregressive generation process, allowing the model to recursively clean up intermediate thoughts based on patterns learned from training. With this reduction mechanism, PENCIL significantly reduces the maximal context length required during generation, and thus can generate longer thoughts with limited memory, solving larger-scale problems given more thinking time. For example, we demonstrate PENCIL achieves 97\% accuracy on the challenging Einstein's puzzle -- a task even large models like GPT-4 struggle with -- using only a small 25M-parameter transformer with 2048 context length. Theoretically, we prove PENCIL can perform universal space-efficient computation by simulating Turing machines with optimal time and space complexity, and thus can solve arbitrary computational tasks that would otherwise be intractable given context window constraints.

Sequential learning with feedback graphs is a natural extension of the multi-armed bandit problem where the problem is equipped with an underlying graph structure that provides additional information - playing an action reveals the losses of all the neighbors of the action. This problem was introduced by mannor2011 and received considerable attention in recent years. It is generally stated in the literature that the minimax regret rate for this problem is of order alpha T, where alpha is the independence number of the graph, and T is the time horizon. However, this is proven only when the number of rounds T is larger than alpha^3, which poses a significant restriction for the usability of this result in large graphs. In this paper, we define a new quantity R^*, called the problem complexity, and prove that the minimax regret is proportional to R^* for any graph and time horizon T. Introducing an intricate exploration strategy, we define the \mainAlgorithm algorithm that achieves the minimax optimal regret bound and becomes the first provably optimal algorithm for this setting, even if T is smaller than alpha^3.

An open-source C++ framework for discovering fast matrix multiplication schemes using the flip graph approach is presented. The framework supports multiple coefficient rings -- binary (Z_2), modular ternary (Z_3) and integer ternary (Z_T = {-1,0,1}) -- and implements both fixed-dimension and meta-dimensional search operators. Using efficient bit-level encoding of coefficient vectors and OpenMP parallelism, the tools enable large-scale exploration on commodity hardware. The study covers 680 schemes ranging from (2 times 2 times 2) to (16 times 16 times 16), with 276 schemes now in Z_T coefficients and 117 in integer coefficients. With this framework, the multiplicative complexity (rank) is improved for 79 matrix multiplication schemes. Notably, a new 4 times 4 times 10 scheme requiring only 115 multiplications is discovered, achieving ωapprox 2.80478 and beating Strassen's exponent for this specific size. Additionally, 93 schemes are rediscovered in ternary coefficients that were previously known only over rationals or integers, and 68 schemes in integer coefficients that previously required fractions. All tools and discovered schemes are made publicly available to enable reproducible research.

As first-order optimization methods become the method of choice for solving large-scale optimization problems, optimization solvers based on first-order algorithms are being built. Such general-purpose solvers must robustly detect infeasible or misspecified problem instances, but the computational complexity of first-order methods for doing so has yet to be formally studied. In this work, we characterize the optimal accelerated rate of infeasibility detection. We show that the standard fixed-point iteration achieves a O(1/k^2) and O(1/k) rates, respectively, on the normalized iterates and the fixed-point residual converging to the infimal displacement vector, while the accelerated fixed-point iteration achieves O(1/k^2) and mathcal{O}(1/k^2) rates. We then provide a matching complexity lower bound to establish that Theta(1/k^2) is indeed the optimal accelerated rate.

Hyperbolic geometry is gaining traction in machine learning for its effectiveness at capturing hierarchical structures in real-world data. Hyperbolic spaces, where neighborhoods grow exponentially, offer substantial advantages and consistently deliver state-of-the-art results across diverse applications. However, hyperbolic classifiers often grapple with computational challenges. Methods reliant on Riemannian optimization frequently exhibit sluggishness, stemming from the increased computational demands of operations on Riemannian manifolds. In response to these challenges, we present hyperDT, a novel extension of decision tree algorithms into hyperbolic space. Crucially, hyperDT eliminates the need for computationally intensive Riemannian optimization, numerically unstable exponential and logarithmic maps, or pairwise comparisons between points by leveraging inner products to adapt Euclidean decision tree algorithms to hyperbolic space. Our approach is conceptually straightforward and maintains constant-time decision complexity while mitigating the scalability issues inherent in high-dimensional Euclidean spaces. Building upon hyperDT we introduce hyperRF, a hyperbolic random forest model. Extensive benchmarking across diverse datasets underscores the superior performance of these models, providing a swift, precise, accurate, and user-friendly toolkit for hyperbolic data analysis.

The adversarial vulnerability of Deep Neural Networks (DNNs) has been well-known and widely concerned, often under the context of learning top-1 attacks (e.g., fooling a DNN to classify a cat image as dog). This paper shows that the concern is much more serious by learning significantly more aggressive ordered top-K clear-box~ This is often referred to as white/black-box attacks in the literature. We choose to adopt neutral terminology, clear/opaque-box attacks in this paper, and omit the prefix clear-box for simplicity. targeted attacks proposed in Adversarial Distillation. We propose a novel and rigorous quadratic programming (QP) method of learning ordered top-K attacks with low computing cost, dubbed as QuadAttacK. Our QuadAttacK directly solves the QP to satisfy the attack constraint in the feature embedding space (i.e., the input space to the final linear classifier), which thus exploits the semantics of the feature embedding space (i.e., the principle of class coherence). With the optimized feature embedding vector perturbation, it then computes the adversarial perturbation in the data space via the vanilla one-step back-propagation. In experiments, the proposed QuadAttacK is tested in the ImageNet-1k classification using ResNet-50, DenseNet-121, and Vision Transformers (ViT-B and DEiT-S). It successfully pushes the boundary of successful ordered top-K attacks from K=10 up to K=20 at a cheap budget (1times 60) and further improves attack success rates for K=5 for all tested models, while retaining the performance for K=1.

We study the complexity of sampling from a distribution over all index subsets of the set {1,...,n} with the probability of a subset S proportional to the determinant of the submatrix L_S of some ntimes n p.s.d. matrix L, where L_S corresponds to the entries of L indexed by S. Known as a determinantal point process, this distribution is used in machine learning to induce diversity in subset selection. In practice, we often wish to sample multiple subsets S with small expected size k = E[|S|] ll n from a very large matrix L, so it is important to minimize the preprocessing cost of the procedure (performed once) as well as the sampling cost (performed repeatedly). For this purpose, we propose a new algorithm which, given access to L, samples exactly from a determinantal point process while satisfying the following two properties: (1) its preprocessing cost is n cdot poly(k), i.e., sublinear in the size of L, and (2) its sampling cost is poly(k), i.e., independent of the size of L. Prior to our results, state-of-the-art exact samplers required O(n^3) preprocessing time and sampling time linear in n or dependent on the spectral properties of L. We also give a reduction which allows using our algorithm for exact sampling from cardinality constrained determinantal point processes with ncdotpoly(k) time preprocessing.

This work proposes a Momentum-Enabled Kronecker-Factor-Based Optimizer Using Rank-1 updates, called MKOR, that improves the training time and convergence properties of deep neural networks (DNNs). Second-order techniques, while enjoying higher convergence rates vs first-order counterparts, have cubic complexity with respect to either the model size and/or the training batch size. Hence they exhibit poor scalability and performance in transformer models, e.g. large language models (LLMs), because the batch sizes in these models scale by the attention mechanism sequence length, leading to large model size and batch sizes. MKOR's complexity is quadratic with respect to the model size, alleviating the computation bottlenecks in second-order methods. Because of their high computation complexity, state-of-the-art implementations of second-order methods can only afford to update the second order information infrequently, and thus do not fully exploit the promise of better convergence from these updates. By reducing the communication complexity of the second-order updates as well as achieving a linear communication complexity, MKOR increases the frequency of second order updates. We also propose a hybrid version of MKOR (called MKOR-H) that mid-training falls backs to a first order optimizer if the second order updates no longer accelerate convergence. Our experiments show that MKOR outperforms state -of-the-art first order methods, e.g. the LAMB optimizer, and best implementations of second-order methods, i.e. KAISA/KFAC, up to 2.57x and 1.85x respectively on BERT-Large-Uncased on 64 GPUs.

We introduce the Mixed-Integer Quadratically Constrained Quadratic Programming framework for the quantum compilation problem and apply it in the context of topological quantum computing. In this setting, quantum gates are realized by sequences of elementary braids of quasiparticles with exotic fractional statistics in certain two-dimensional topological condensed matter systems, described by effective topological quantum field theories. We specifically focus on a non-semisimple version of topological field theory, which provides a foundation for an extended theory of Ising anyons and which has recently been shown by Iulianelli et al., Nature Communications {\bf 16}, 6408 (2025), to permit universal quantum computation. While the proofs of this pioneering result are existential in nature, the mixed integer programming provides an approach to explicitly construct quantum gates in topological systems. We demonstrate this by focusing specifically on the entangling controlled-NOT operation, and its local equivalence class, using braiding operations in the non-semisimple Ising system. This illustrates the utility of the Mixed-Integer Quadratically Constrained Quadratic Programming for topological quantum compilation.

Given the exponential growth of the volume of the ball w.r.t. its radius, the hyperbolic space is capable of embedding trees with arbitrarily small distortion and hence has received wide attention for representing hierarchical datasets. However, this exponential growth property comes at a price of numerical instability such that training hyperbolic learning models will sometimes lead to catastrophic NaN problems, encountering unrepresentable values in floating point arithmetic. In this work, we carefully analyze the limitation of two popular models for the hyperbolic space, namely, the Poincar\'e ball and the Lorentz model. We first show that, under the 64 bit arithmetic system, the Poincar\'e ball has a relatively larger capacity than the Lorentz model for correctly representing points. Then, we theoretically validate the superiority of the Lorentz model over the Poincar\'e ball from the perspective of optimization. Given the numerical limitations of both models, we identify one Euclidean parametrization of the hyperbolic space which can alleviate these limitations. We further extend this Euclidean parametrization to hyperbolic hyperplanes and exhibits its ability in improving the performance of hyperbolic SVM.

In this article, we study the parameterized complexity of the Set Cover problem restricted to semi-ladder-free hypergraphs, a class defined by Fabianski et al. [Proceedings of STACS 2019]. We observe that two algorithms introduced by Langerman and Morin [Discrete & Computational Geometry 2005] in the context of geometric covering problems can be adapted to this setting, yielding simple FPT and kernelization algorithms for Set Cover in semi-ladder-free hypergraphs. We complement our algorithmic results with a compression lower bound for the problem, which proves the tightness of our kernelization under standard complexity-theoretic assumptions.

In this paper, we find a sample complexity bound for learning a simplex from noisy samples. Assume a dataset of size n is given which includes i.i.d. samples drawn from a uniform distribution over an unknown simplex in R^K, where samples are assumed to be corrupted by a multi-variate additive Gaussian noise of an arbitrary magnitude. We prove the existence of an algorithm that with high probability outputs a simplex having a ell_2 distance of at most varepsilon from the true simplex (for any varepsilon>0). Also, we theoretically show that in order to achieve this bound, it is sufficient to have ngeleft(K^2/varepsilon^2right)e^{Omegaleft(K/SNR^2right)} samples, where SNR stands for the signal-to-noise ratio. This result solves an important open problem and shows as long as SNRgeOmegaleft(K^{1/2}right), the sample complexity of the noisy regime has the same order to that of the noiseless case. Our proofs are a combination of the so-called sample compression technique in ashtiani2018nearly, mathematical tools from high-dimensional geometry, and Fourier analysis. In particular, we have proposed a general Fourier-based technique for recovery of a more general class of distribution families from additive Gaussian noise, which can be further used in a variety of other related problems.

Linear time-invariant state space models (SSM) are a classical model from engineering and statistics, that have recently been shown to be very promising in machine learning through the Structured State Space sequence model (S4). A core component of S4 involves initializing the SSM state matrix to a particular matrix called a HiPPO matrix, which was empirically important for S4's ability to handle long sequences. However, the specific matrix that S4 uses was actually derived in previous work for a particular time-varying dynamical system, and the use of this matrix as a time-invariant SSM had no known mathematical interpretation. Consequently, the theoretical mechanism by which S4 models long-range dependencies actually remains unexplained. We derive a more general and intuitive formulation of the HiPPO framework, which provides a simple mathematical interpretation of S4 as a decomposition onto exponentially-warped Legendre polynomials, explaining its ability to capture long dependencies. Our generalization introduces a theoretically rich class of SSMs that also lets us derive more intuitive S4 variants for other bases such as the Fourier basis, and explains other aspects of training S4, such as how to initialize the important timescale parameter. These insights improve S4's performance to 86% on the Long Range Arena benchmark, with 96% on the most difficult Path-X task.

Sparse linear regression is one of the most basic questions in machine learning and statistics. Here, we are given as input a design matrix X in R^{N times d} and measurements or labels {y} in R^N where {y} = {X} {w}^* + {xi}, and {xi} is the noise in the measurements. Importantly, we have the additional constraint that the unknown signal vector {w}^* is sparse: it has k non-zero entries where k is much smaller than the ambient dimension. Our goal is to output a prediction vector {w} that has small prediction error: 1{N}cdot |{X} {w}^* - {X} {w}|^2_2. Information-theoretically, we know what is best possible in terms of measurements: under most natural noise distributions, we can get prediction error at most epsilon with roughly N = O(k log d/epsilon) samples. Computationally, this currently needs d^{Omega(k)} run-time. Alternately, with N = O(d), we can get polynomial-time. Thus, there is an exponential gap (in the dependence on d) between the two and we do not know if it is possible to get d^{o(k)} run-time and o(d) samples. We give the first generic positive result for worst-case design matrices {X}: For any {X}, we show that if the support of {w}^* is chosen at random, we can get prediction error epsilon with N = poly(k, log d, 1/epsilon) samples and run-time poly(d,N). This run-time holds for any design matrix {X} with condition number up to 2^{poly(d)}. Previously, such results were known for worst-case {w}^*, but only for random design matrices from well-behaved families, matrices that have a very low condition number (poly(log d); e.g., as studied in compressed sensing), or those with special structural properties.

The quadratic computational complexity in the self-attention mechanism of popular transformer architectures poses significant challenges for training and inference, particularly in terms of efficiency and memory requirements. Towards addressing these challenges, this paper introduces a novel fast computation method for gradient calculation in multi-layer transformer models. Our approach enables the computation of gradients for the entire multi-layer transformer model in almost linear time n^{1+o(1)}, where n is the input sequence length. This breakthrough significantly reduces the computational bottleneck associated with the traditional quadratic time complexity. Our theory holds for any loss function and maintains a bounded approximation error across the entire model. Furthermore, our analysis can hold when the multi-layer transformer model contains many practical sub-modules, such as residual connection, casual mask, and multi-head attention. By improving the efficiency of gradient computation in large language models, we hope that our work will facilitate the more effective training and deployment of long-context language models based on our theoretical results.

Let T be a rooted and weighted tree, where the weight of any node is equal to the sum of the weights of its children. The popular Treemap algorithm visualizes such a tree as a hierarchical partition of a square into rectangles, where the area of the rectangle corresponding to any node in T is equal to the weight of that node. The aspect ratio of the rectangles in such a rectangular partition necessarily depends on the weights and can become arbitrarily high. We introduce a new hierarchical partition scheme, called a polygonal partition, which uses convex polygons rather than just rectangles. We present two methods for constructing polygonal partitions, both having guarantees on the worst-case aspect ratio of the constructed polygons; in particular, both methods guarantee a bound on the aspect ratio that is independent of the weights of the nodes. We also consider rectangular partitions with slack, where the areas of the rectangles may differ slightly from the weights of the corresponding nodes. We show that this makes it possible to obtain partitions with constant aspect ratio. This result generalizes to hyper-rectangular partitions in R^d. We use these partitions with slack for embedding ultrametrics into d-dimensional Euclidean space: we give a rm polylog(Delta)-approximation algorithm for embedding n-point ultrametrics into R^d with minimum distortion, where Delta denotes the spread of the metric, i.e., the ratio between the largest and the smallest distance between two points. The previously best-known approximation ratio for this problem was polynomial in n. This is the first algorithm for embedding a non-trivial family of weighted-graph metrics into a space of constant dimension that achieves polylogarithmic approximation ratio.

We analyze Newton's method with lazy Hessian updates for solving general possibly non-convex optimization problems. We propose to reuse a previously seen Hessian for several iterations while computing new gradients at each step of the method. This significantly reduces the overall arithmetical complexity of second-order optimization schemes. By using the cubic regularization technique, we establish fast global convergence of our method to a second-order stationary point, while the Hessian does not need to be updated each iteration. For convex problems, we justify global and local superlinear rates for lazy Newton steps with quadratic regularization, which is easier to compute. The optimal frequency for updating the Hessian is once every d iterations, where d is the dimension of the problem. This provably improves the total arithmetical complexity of second-order algorithms by a factor d.

This paper uses the concept of algorithmic efficiency to present a unified theory of intelligence. Intelligence is defined informally, formally, and computationally. We introduce the concept of Dimensional complexity in algorithmic efficiency and deduce that an optimally efficient algorithm has zero Time complexity, zero Space complexity, and an infinite Dimensional complexity. This algorithm is used to generate the number line.

Reinforcement Learning algorithms that learn from human feedback (RLHF) need to be efficient in terms of statistical complexity, computational complexity, and query complexity. In this work, we consider the RLHF setting where the feedback is given in the format of preferences over pairs of trajectories. In the linear MDP model, using randomization in algorithm design, we present an algorithm that is sample efficient (i.e., has near-optimal worst-case regret bounds) and has polynomial running time (i.e., computational complexity is polynomial with respect to relevant parameters). Our algorithm further minimizes the query complexity through a novel randomized active learning procedure. In particular, our algorithm demonstrates a near-optimal tradeoff between the regret bound and the query complexity. To extend the results to more general nonlinear function approximation, we design a model-based randomized algorithm inspired by the idea of Thompson sampling. Our algorithm minimizes Bayesian regret bound and query complexity, again achieving a near-optimal tradeoff between these two quantities. Computation-wise, similar to the prior Thompson sampling algorithms under the regular RL setting, the main computation primitives of our algorithm are Bayesian supervised learning oracles which have been heavily investigated on the empirical side when applying Thompson sampling algorithms to RL benchmark problems.

Time series modeling is a well-established problem, which often requires that methods (1) expressively represent complicated dependencies, (2) forecast long horizons, and (3) efficiently train over long sequences. State-space models (SSMs) are classical models for time series, and prior works combine SSMs with deep learning layers for efficient sequence modeling. However, we find fundamental limitations with these prior approaches, proving their SSM representations cannot express autoregressive time series processes. We thus introduce SpaceTime, a new state-space time series architecture that improves all three criteria. For expressivity, we propose a new SSM parameterization based on the companion matrix -- a canonical representation for discrete-time processes -- which enables SpaceTime's SSM layers to learn desirable autoregressive processes. For long horizon forecasting, we introduce a "closed-loop" variation of the companion SSM, which enables SpaceTime to predict many future time-steps by generating its own layer-wise inputs. For efficient training and inference, we introduce an algorithm that reduces the memory and compute of a forward pass with the companion matrix. With sequence length ell and state-space size d, we go from O(d ell) na\"ively to O(d + ell). In experiments, our contributions lead to state-of-the-art results on extensive and diverse benchmarks, with best or second-best AUROC on 6 / 7 ECG and speech time series classification, and best MSE on 14 / 16 Informer forecasting tasks. Furthermore, we find SpaceTime (1) fits AR(p) processes that prior deep SSMs fail on, (2) forecasts notably more accurately on longer horizons than prior state-of-the-art, and (3) speeds up training on real-world ETTh1 data by 73% and 80% relative wall-clock time over Transformers and LSTMs.

We study multi-agent reinforcement learning in the setting of episodic Markov decision processes, where multiple agents cooperate via communication through a central server. We propose a provably efficient algorithm based on value iteration that enable asynchronous communication while ensuring the advantage of cooperation with low communication overhead. With linear function approximation, we prove that our algorithm enjoys an mathcal{O}(d^{3/2}H^2K) regret with mathcal{O}(dHM^2) communication complexity, where d is the feature dimension, H is the horizon length, M is the total number of agents, and K is the total number of episodes. We also provide a lower bound showing that a minimal Omega(dM) communication complexity is required to improve the performance through collaboration.

Solving the inverse problem is the key step in evaluating the capacity of a physical model to describe real phenomena. In medical image computing, it aligns with the classical theme of image-based model personalization. Traditionally, a solution to the problem is obtained by performing either sampling or variational inference based methods. Both approaches aim to identify a set of free physical model parameters that results in a simulation best matching an empirical observation. When applied to brain tumor modeling, one of the instances of image-based model personalization in medical image computing, the overarching drawback of the methods is the time complexity for finding such a set. In a clinical setting with limited time between imaging and diagnosis or even intervention, this time complexity may prove critical. As the history of quantitative science is the history of compression, we align in this paper with the historical tendency and propose a method compressing complex traditional strategies for solving an inverse problem into a simple database query task. We evaluated different ways of performing the database query task assessing the trade-off between accuracy and execution time. On the exemplary task of brain tumor growth modeling, we prove that the proposed method achieves one order speed-up compared to existing approaches for solving the inverse problem. The resulting compute time offers critical means for relying on more complex and, hence, realistic models, for integrating image preprocessing and inverse modeling even deeper, or for implementing the current model into a clinical workflow.

Machine learning approaches relying on such criteria as adversarial robustness or multi-agent settings have raised the need for solving game-theoretic equilibrium problems. Of particular relevance to these applications are methods targeting finite-sum structure, which generically arises in empirical variants of learning problems in these contexts. Further, methods with computable approximation errors are highly desirable, as they provide verifiable exit criteria. Motivated by these applications, we study finite-sum monotone inclusion problems, which model broad classes of equilibrium problems. Our main contributions are variants of the classical Halpern iteration that employ variance reduction to obtain improved complexity guarantees in which n component operators in the finite sum are ``on average'' either cocoercive or Lipschitz continuous and monotone, with parameter L. The resulting oracle complexity of our methods, which provide guarantees for the last iterate and for a (computable) operator norm residual, is mathcal{O}( n + nLvarepsilon^{-1}), which improves upon existing methods by a factor up to n. This constitutes the first variance reduction-type result for general finite-sum monotone inclusions and for more specific problems such as convex-concave optimization when operator norm residual is the optimality measure. We further argue that, up to poly-logarithmic factors, this complexity is unimprovable in the monotone Lipschitz setting; i.e., the provided result is near-optimal.

In deep learning, performance is strongly affected by the choice of architecture and hyperparameters. While there has been extensive work on automatic hyperparameter optimization for simple spaces, complex spaces such as the space of deep architectures remain largely unexplored. As a result, the choice of architecture is done manually by the human expert through a slow trial and error process guided mainly by intuition. In this paper we describe a framework for automatically designing and training deep models. We propose an extensible and modular language that allows the human expert to compactly represent complex search spaces over architectures and their hyperparameters. The resulting search spaces are tree-structured and therefore easy to traverse. Models can be automatically compiled to computational graphs once values for all hyperparameters have been chosen. We can leverage the structure of the search space to introduce different model search algorithms, such as random search, Monte Carlo tree search (MCTS), and sequential model-based optimization (SMBO). We present experiments comparing the different algorithms on CIFAR-10 and show that MCTS and SMBO outperform random search. In addition, these experiments show that our framework can be used effectively for model discovery, as it is possible to describe expressive search spaces and discover competitive models without much effort from the human expert. Code for our framework and experiments has been made publicly available.

While machine learning has advanced through massive parallelization, we identify a critical blind spot: some problems are fundamentally sequential. These "inherently serial" problems-from mathematical reasoning to physical simulations to sequential decision-making-require dependent computational steps that cannot be parallelized. Drawing from complexity theory, we formalize this distinction and demonstrate that current parallel-centric architectures face fundamental limitations on such tasks. We argue that recognizing the serial nature of computation holds profound implications on machine learning, model design, hardware development. As AI tackles increasingly complex reasoning, deliberately scaling serial computation-not just parallel computation-is essential for continued progress.

This monograph presents the main complexity theorems in convex optimization and their corresponding algorithms. Starting from the fundamental theory of black-box optimization, the material progresses towards recent advances in structural optimization and stochastic optimization. Our presentation of black-box optimization, strongly influenced by Nesterov's seminal book and Nemirovski's lecture notes, includes the analysis of cutting plane methods, as well as (accelerated) gradient descent schemes. We also pay special attention to non-Euclidean settings (relevant algorithms include Frank-Wolfe, mirror descent, and dual averaging) and discuss their relevance in machine learning. We provide a gentle introduction to structural optimization with FISTA (to optimize a sum of a smooth and a simple non-smooth term), saddle-point mirror prox (Nemirovski's alternative to Nesterov's smoothing), and a concise description of interior point methods. In stochastic optimization we discuss stochastic gradient descent, mini-batches, random coordinate descent, and sublinear algorithms. We also briefly touch upon convex relaxation of combinatorial problems and the use of randomness to round solutions, as well as random walks based methods.

We show that a variety of modern deep learning tasks exhibit a "double-descent" phenomenon where, as we increase model size, performance first gets worse and then gets better. Moreover, we show that double descent occurs not just as a function of model size, but also as a function of the number of training epochs. We unify the above phenomena by defining a new complexity measure we call the effective model complexity and conjecture a generalized double descent with respect to this measure. Furthermore, our notion of model complexity allows us to identify certain regimes where increasing (even quadrupling) the number of train samples actually hurts test performance.

Classification of time series data is an important task for many application domains. One of the best existing methods for this task, in terms of accuracy and computation time, is MiniROCKET. In this work, we extend this approach to provide better global temporal encodings using hyperdimensional computing (HDC) mechanisms. HDC (also known as Vector Symbolic Architectures, VSA) is a general method to explicitly represent and process information in high-dimensional vectors. It has previously been used successfully in combination with deep neural networks and other signal processing algorithms. We argue that the internal high-dimensional representation of MiniROCKET is well suited to be complemented by the algebra of HDC. This leads to a more general formulation, HDC-MiniROCKET, where the original algorithm is only a special case. We will discuss and demonstrate that HDC-MiniROCKET can systematically overcome catastrophic failures of MiniROCKET on simple synthetic datasets. These results are confirmed by experiments on the 128 datasets from the UCR time series classification benchmark. The extension with HDC can achieve considerably better results on datasets with high temporal dependence without increasing the computational effort for inference.

We study whether an aperiodic hierarchy can provide a structural advantage for lossless compression over periodic alternatives. We show that Fibonacci quasicrystal tilings avoid the finite-depth collapse that affects periodic hierarchies: usable n-gram lookup positions remain non-zero at every level, while periodic tilings collapse after O(log p) levels for period p. This yields an aperiodic hierarchy advantage: dictionary reuse remains available across all scales instead of vanishing beyond a finite depth. Our analysis gives four main consequences. First, the Golden Compensation property shows that the exponential decay in the number of positions is exactly balanced by the exponential growth in phrase length, so potential coverage remains scale-invariant with asymptotic value Wvarphi/5. Second, using the Sturmian complexity law p(n)=n+1, we show that Fibonacci/Sturmian hierarchies maximize codebook coverage efficiency among binary aperiodic tilings. Third, under long-range dependence, the resulting hierarchy achieves lower coding entropy than comparable periodic hierarchies. Fourth, redundancy decays super-exponentially with depth, whereas periodic systems remain locked at the depth where collapse occurs. We validate these results with Quasicryth, a lossless text compressor built on a ten-level Fibonacci hierarchy with phrase lengths {2,3,5,8,13,21,34,55,89,144}. In controlled A/B experiments with identical codebooks, the aperiodic advantage over a Period-5 baseline grows from 36{,}243 B at 3 MB to 11{,}089{,}469 B at 1 GB, explained by the activation of deeper hierarchy levels. On enwik9, Quasicryth achieves 225{,}918{,}349 B (22.59%), with 20{,}735{,}733 B saved by the Fibonacci tiling relative to no tiling.

Magnitude of a finite metric space and the related notion of magnitude functions on metric spaces is an active area of research in algebraic topology. Magnitude originally arose in the context of biology, where it represents the number of effective species in an environment; when applied to a one-parameter family of metric spaces tX with scale parameter t, the magnitude captures much of the underlying geometry of the space. Prior work has mostly focussed on properties of magnitude in a global sense; in this paper we restrict the sets to finite subsets of Euclidean space and investigate its individual components. We give an explicit formula for the corrected inclusion-exclusion principle, and define a quantity associated with each point, called the moment which gives an intrinsic ordering to the points. We exploit this in order to form an algorithm which approximates the convex hull.

Given a matrix Min R^{mtimes n}, the low rank matrix completion problem asks us to find a rank-k approximation of M as UV^top for Uin R^{mtimes k} and Vin R^{ntimes k} by only observing a few entries specified by a set of entries Omegasubseteq [m]times [n]. In particular, we examine an approach that is widely used in practice -- the alternating minimization framework. Jain, Netrapalli and Sanghavi~jns13 showed that if M has incoherent rows and columns, then alternating minimization provably recovers the matrix M by observing a nearly linear in n number of entries. While the sample complexity has been subsequently improved~glz17, alternating minimization steps are required to be computed exactly. This hinders the development of more efficient algorithms and fails to depict the practical implementation of alternating minimization, where the updates are usually performed approximately in favor of efficiency. In this paper, we take a major step towards a more efficient and error-robust alternating minimization framework. To this end, we develop an analytical framework for alternating minimization that can tolerate moderate amount of errors caused by approximate updates. Moreover, our algorithm runs in time widetilde O(|Omega| k), which is nearly linear in the time to verify the solution while preserving the sample complexity. This improves upon all prior known alternating minimization approaches which require widetilde O(|Omega| k^2) time.

In pursuit of faster computation, Efficient Transformers demonstrate an impressive variety of approaches -- models attaining sub-quadratic attention complexity can utilize a notion of sparsity or a low-rank approximation of inputs to reduce the number of attended keys; other ways to reduce complexity include locality-sensitive hashing, key pooling, additional memory to store information in compacted or hybridization with other architectures, such as CNN. Often based on a strong mathematical basis, kernelized approaches allow for the approximation of attention with linear complexity while retaining high accuracy. Therefore, in the present paper, we aim to expand the idea of trainable kernel methods to approximate the self-attention mechanism of the Transformer architecture.

A range of quantum algorithms, especially those leveraging variational parameterization and circuit-based optimization, are being studied as alternatives for solving classically intractable combinatorial optimization problems (COPs). However, their applicability is limited by hardware constraints, including shallow circuit depth, limited qubit counts, and noise. To mitigate these issues, we propose a hybrid classical--quantum framework based on graph shrinking to reduce the number of variables and constraints in QUBO formulations of COPs, while preserving problem structure. Our approach introduces three key ideas: (i) constraint-aware shrinking that prevents merges that will likely violate problem-specific feasibility constraints, (ii) a verification-and-repair pipeline to correct infeasible solutions post-optimization, and (iii) adaptive strategies for recalculating correlations and controlling the graph shrinking process. We apply our approach to three standard benchmark problems: Multidimensional Knapsack (MDKP), Maximum Independent Set (MIS), and the Quadratic Assignment Problem (QAP). Empirical results show that our approach improves solution feasibility, reduces repair complexity, and enhances quantum optimization quality on hardware-limited instances. These findings demonstrate a scalable pathway for applying near-term quantum algorithms to classically challenging constrained optimization problems.

In this paper, we explore the potential for quantum annealing to solve realistic routing problems. We focus on two NP-Hard problems, including the Traveling Salesman Problem with Time Windows and the Capacitated Vehicle Routing Problem with Time Windows. We utilize D-Wave's Quantum Annealer and Constrained Quadratic Model (CQM) solver within a hybrid framework to solve these problems. We demonstrate that while the CQM solver effectively minimizes route costs, it struggles to maintain time window feasibility as the problem size increases. To address this limitation, we implement a heuristic method that fixes infeasible solutions through a series of swapping operations. Testing on benchmark instances shows our method achieves promising results with an average optimality gap of 3.86%.

This paper develops a hierarchical clustering algorithm for weighted projective spaces P_{q}, utilizing a Finsler metric d_F([z], [w]) and its rational analogue d_{F,Q}([z], [w]) to define distances that preserve the non-Euclidean geometry of these quotient manifolds. Defined via geodesic integrals of a scaling invariant Finsler norm weighted by the grades q = (q_0, q_1, dots, q_n), these metrics satisfy true metric properties including the triangle inequality, overcoming the limitations of the non-metric dissimilarity measure from prior work.

We study the realizability of simplicial complexes with a given pair of integer sequences, representing the node degree distribution and the facet size distribution, respectively. While the s-uniform variant of the problem is NP-complete when s geq 3, we identify two populations of input sequences, most of which can be solved in polynomial time using a recursive algorithm that we contribute. Combining with a sampler for the simplicial configuration model [J.-G. Young et al., Phys. Rev. E 96, 032312 (2017)], we facilitate the efficient sampling of simplicial ensembles from arbitrary degree and size distributions. We find that, contrary to expectations based on dyadic networks, increasing the nodes' degrees reduces the number of loops in simplicial complexes. Our work unveils a fundamental constraint on the degree-size sequences and sheds light on further analysis of higher-order phenomena based on local structures.

The concept class of low-degree polynomial threshold functions (PTFs) plays a fundamental role in machine learning. In this paper, we study PAC learning of K-sparse degree-d PTFs on R^n, where any such concept depends only on K out of n attributes of the input. Our main contribution is a new algorithm that runs in time ({nd}/{epsilon})^{O(d)} and under the Gaussian marginal distribution, PAC learns the class up to error rate epsilon with O(K^{4d}{epsilon^{2d}} cdot log^{5d} n) samples even when an eta leq O(epsilon^d) fraction of them are corrupted by the nasty noise of Bshouty et al. (2002), possibly the strongest corruption model. Prior to this work, attribute-efficient robust algorithms are established only for the special case of sparse homogeneous halfspaces. Our key ingredients are: 1) a structural result that translates the attribute sparsity to a sparsity pattern of the Chow vector under the basis of Hermite polynomials, and 2) a novel attribute-efficient robust Chow vector estimation algorithm which uses exclusively a restricted Frobenius norm to either certify a good approximation or to validate a sparsity-induced degree-2d polynomial as a filter to detect corrupted samples.

In this paper, we propose new problem-independent lower bounds on the sample complexity and regret in episodic MDPs, with a particular focus on the non-stationary case in which the transition kernel is allowed to change in each stage of the episode. Our main contribution is a novel lower bound of Ω((H^3SA/ε^2)log(1/δ)) on the sample complexity of an (varepsilon,δ)-PAC algorithm for best policy identification in a non-stationary MDP. This lower bound relies on a construction of "hard MDPs" which is different from the ones previously used in the literature. Using this same class of MDPs, we also provide a rigorous proof of the Ω(H^3SAT) regret bound for non-stationary MDPs. Finally, we discuss connections to PAC-MDP lower bounds.

In this paper we present a novel probabilistic sampling-based motion planning algorithm called the Fast Marching Tree algorithm (FMT*). The algorithm is specifically aimed at solving complex motion planning problems in high-dimensional configuration spaces. This algorithm is proven to be asymptotically optimal and is shown to converge to an optimal solution faster than its state-of-the-art counterparts, chiefly PRM* and RRT*. The FMT* algorithm performs a "lazy" dynamic programming recursion on a predetermined number of probabilistically-drawn samples to grow a tree of paths, which moves steadily outward in cost-to-arrive space. As a departure from previous analysis approaches that are based on the notion of almost sure convergence, the FMT* algorithm is analyzed under the notion of convergence in probability: the extra mathematical flexibility of this approach allows for convergence rate bounds--the first in the field of optimal sampling-based motion planning. Specifically, for a certain selection of tuning parameters and configuration spaces, we obtain a convergence rate bound of order O(n^{-1/d+ρ}), where n is the number of sampled points, d is the dimension of the configuration space, and ρ is an arbitrarily small constant. We go on to demonstrate asymptotic optimality for a number of variations on FMT*, namely when the configuration space is sampled non-uniformly, when the cost is not arc length, and when connections are made based on the number of nearest neighbors instead of a fixed connection radius. Numerical experiments over a range of dimensions and obstacle configurations confirm our theoretical and heuristic arguments by showing that FMT*, for a given execution time, returns substantially better solutions than either PRM* or RRT*, especially in high-dimensional configuration spaces and in scenarios where collision-checking is expensive.

One might think that, once we know something is computable, how efficiently it can be computed is a practical question with little further philosophical importance. In this essay, I offer a detailed case that one would be wrong. In particular, I argue that computational complexity theory -- the field that studies the resources (such as time, space, and randomness) needed to solve computational problems -- leads to new perspectives on the nature of mathematical knowledge, the strong AI debate, computationalism, the problem of logical omniscience, Hume's problem of induction, Goodman's grue riddle, the foundations of quantum mechanics, economic rationality, closed timelike curves, and several other topics of philosophical interest. I end by discussing aspects of complexity theory itself that could benefit from philosophical analysis.

Kernel online convex optimization (KOCO) is a framework combining the expressiveness of non-parametric kernel models with the regret guarantees of online learning. First-order KOCO methods such as functional gradient descent require only O(t) time and space per iteration, and, when the only information on the losses is their convexity, achieve a minimax optimal O(T) regret. Nonetheless, many common losses in kernel problems, such as squared loss, logistic loss, and squared hinge loss posses stronger curvature that can be exploited. In this case, second-order KOCO methods achieve O(log(Det(K))) regret, which we show scales as O(d_{eff}log T), where d_{eff} is the effective dimension of the problem and is usually much smaller than O(T). The main drawback of second-order methods is their much higher O(t^2) space and time complexity. In this paper, we introduce kernel online Newton step (KONS), a new second-order KOCO method that also achieves O(d_{eff}log T) regret. To address the computational complexity of second-order methods, we introduce a new matrix sketching algorithm for the kernel matrix K_t, and show that for a chosen parameter γleq 1 our Sketched-KONS reduces the space and time complexity by a factor of γ^2 to O(t^2γ^2) space and time per iteration, while incurring only 1/γ times more regret.

Constraint handling remains a key bottleneck in quantum combinatorial optimization. While slack-variable-based encodings are straightforward, they significantly increase qubit counts and circuit depth, challenging the scalability of quantum solvers. In this work, we investigate a suite of Lagrangian-based optimization techniques including dual ascent, bundle methods, cutting plane approaches, and augmented Lagrangian formulations for solving constrained combinatorial problems on quantum simulators and hardware. Our framework is applied to three representative NP-hard problems: the Travelling Salesman Problem (TSP), the Multi-Dimensional Knapsack Problem (MDKP), and the Maximum Independent Set (MIS). We demonstrate that MDKP and TSP, with their inequality-based or degree-constrained structures, allow for slack-free reformulations, leading to significant qubit savings without compromising performance. In contrast, MIS does not inherently benefit from slack elimination but still gains in feasibility and objective quality from principled Lagrangian updates. We benchmark these methods across classically hard instances, analyzing trade-offs in qubit usage, feasibility, and optimality gaps. Our results highlight the flexibility of Lagrangian formulations as a scalable alternative to naive QUBO penalization, even when qubit savings are not always achievable. This work provides practical insights for deploying constraint-aware quantum optimization pipelines, with applications in logistics, network design, and resource allocation.

Matrix multiplication optimization remains a fundamental challenge in computational mathematics. This work introduces a novel approach that discovers matrix multiplication schemes in the ternary field (Z_T), where coefficients are restricted to {-1, 0, 1} to minimize naive additive complexity. The core of the method is a GPU-accelerated meta flip graph algorithm that maintains ternary safety through specialized arithmetic operations and sign symmetry breaking. Key results include new best ranks for the formats 4 times 5 times 12, 5 times 6 times 10, and 6 times 7 times 9, the independent discovery of 32 schemes in Z_T that match known optimal ranks (including 8 previously known only with rational coefficients), and 30 rank improvements in the binary field. The analysis of 164 known schemes shows that 92 can be implemented in Z_T, while 72 could not be found in the ternary field with current methods, defining the current boundaries of this approach. All software, results, and discovered schemes are provided as open-source.

In this work, we consider the problem of collaborative multi-user reinforcement learning. In this setting there are multiple users with the same state-action space and transition probabilities but with different rewards. Under the assumption that the reward matrix of the N users has a low-rank structure -- a standard and practically successful assumption in the offline collaborative filtering setting -- the question is can we design algorithms with significantly lower sample complexity compared to the ones that learn the MDP individually for each user. Our main contribution is an algorithm which explores rewards collaboratively with N user-specific MDPs and can learn rewards efficiently in two key settings: tabular MDPs and linear MDPs. When N is large and the rank is constant, the sample complexity per MDP depends logarithmically over the size of the state-space, which represents an exponential reduction (in the state-space size) when compared to the standard ``non-collaborative'' algorithms.

In contrast to entropy, which increases monotonically, the "complexity" or "interestingness" of closed systems seems intuitively to increase at first and then decrease as equilibrium is approached. For example, our universe lacked complex structures at the Big Bang and will also lack them after black holes evaporate and particles are dispersed. This paper makes an initial attempt to quantify this pattern. As a model system, we use a simple, two-dimensional cellular automaton that simulates the mixing of two liquids ("coffee" and "cream"). A plausible complexity measure is then the Kolmogorov complexity of a coarse-grained approximation of the automaton's state, which we dub the "apparent complexity." We study this complexity measure, and show analytically that it never becomes large when the liquid particles are non-interacting. By contrast, when the particles do interact, we give numerical evidence that the complexity reaches a maximum comparable to the "coffee cup's" horizontal dimension. We raise the problem of proving this behavior analytically.

We study the polynomial-time approximability of the optimal online stochastic bipartite matching algorithm, initiated by Papadimitriou et al. (EC'21). Here, nodes on one side of the graph are given upfront, while at each time t, an online node and its edge weights are drawn from a time-dependent distribution. The optimal algorithm is PSPACE-hard to approximate within some universal constant. We refer to this optimal algorithm, which requires time to think (compute), as a philosopher, and refer to polynomial-time online approximations of the above as philosopher inequalities. The best known philosopher inequality for online matching yields a 0.652-approximation. In contrast, the best possible prophet inequality, or approximation of the optimum offline solution, is 0.5. Our main results are a 0.678-approximate algorithm and a 0.685-approximation for a vertex-weighted special case. Notably, both bounds exceed the 0.666-approximation of the offline optimum obtained by Tang, Wu, and Wu (STOC'22) for the vertex-weighted problem. Building on our algorithms and the recent black-box reduction of Banihashem et al. (SODA'24), we provide polytime (pricing-based) truthful mechanisms which 0.678-approximate the social welfare of the optimal online allocation for bipartite matching markets. Our online allocation algorithm relies on the classic pivotal sampling algorithm (Srinivasan FOCS'01, Gandhi et al. J.ACM'06), along with careful discarding to obtain negative correlations between offline nodes. Consequently, the analysis boils down to examining the distribution of a weighted sum X of negatively correlated Bernoulli variables, specifically lower bounding its mass below a threshold, E[min(1,X)], of possible independent interest. Interestingly, our bound relies on an imaginary invocation of pivotal sampling.

Maximizing monotone submodular functions under a matroid constraint is a classic algorithmic problem with multiple applications in data mining and machine learning. We study this classic problem in the fully dynamic setting, where elements can be both inserted and deleted in real-time. Our main result is a randomized algorithm that maintains an efficient data structure with an O(k^2) amortized update time (in the number of additions and deletions) and yields a 4-approximate solution, where k is the rank of the matroid.

The transformer architecture has driven breakthroughs in recent years on tasks which require modeling pairwise relationships between sequential elements, as is the case in natural language understanding. However, long seqeuences pose a problem due to the quadratic complexity of the attention operation. Previous research has aimed to lower the complexity by sparsifying or linearly approximating the attention matrix. Yet, these approaches cannot straightforwardly distill knowledge from a teacher's attention matrix and often require complete retraining from scratch. Furthermore, previous sparse and linear approaches lose interpretability if they cannot produce full attention matrices. To address these challenges, we propose SEA: Sparse linear attention with an Estimated Attention mask. SEA estimates the attention matrix with linear complexity via kernel-based linear attention, then subsequently creates a sparse attention matrix with a top-k selection to perform a sparse attention operation. For language modeling tasks (Wikitext2), previous linear and sparse attention methods show roughly two-fold worse perplexity scores over the quadratic OPT-1.3B baseline, while SEA achieves better perplexity than OPT-1.3B, using roughly half the memory of OPT-1.3B, providing interpretable attention matrix. We believe that our work will have a large practical impact, as it opens the possibility of running large transformers on resource-limited devices with less memory.

Gaussian processes (GP) are a well studied Bayesian approach for the optimization of black-box functions. Despite their effectiveness in simple problems, GP-based algorithms hardly scale to high-dimensional functions, as their per-iteration time and space cost is at least quadratic in the number of dimensions d and iterations t. Given a set of A alternatives to choose from, the overall runtime O(t^3A) is prohibitive. In this paper we introduce BKB (budgeted kernelized bandit), a new approximate GP algorithm for optimization under bandit feedback that achieves near-optimal regret (and hence near-optimal convergence rate) with near-constant per-iteration complexity and remarkably no assumption on the input space or covariance of the GP. We combine a kernelized linear bandit algorithm (GP-UCB) with randomized matrix sketching based on leverage score sampling, and we prove that randomly sampling inducing points based on their posterior variance gives an accurate low-rank approximation of the GP, preserving variance estimates and confidence intervals. As a consequence, BKB does not suffer from variance starvation, an important problem faced by many previous sparse GP approximations. Moreover, we show that our procedure selects at most O(d_{eff}) points, where d_{eff} is the effective dimension of the explored space, which is typically much smaller than both d and t. This greatly reduces the dimensionality of the problem, thus leading to a O(TAd_{eff}^2) runtime and O(A d_{eff}) space complexity.

Finding the mode of a high dimensional probability distribution D is a fundamental algorithmic problem in statistics and data analysis. There has been particular interest in efficient methods for solving the problem when D is represented as a mixture model or kernel density estimate, although few algorithmic results with worst-case approximation and runtime guarantees are known. In this work, we significantly generalize a result of (LeeLiMusco:2021) on mode approximation for Gaussian mixture models. We develop randomized dimensionality reduction methods for mixtures involving a broader class of kernels, including the popular logistic, sigmoid, and generalized Gaussian kernels. As in Lee et al.'s work, our dimensionality reduction results yield quasi-polynomial algorithms for mode finding with multiplicative accuracy (1-epsilon) for any epsilon > 0. Moreover, when combined with gradient descent, they yield efficient practical heuristics for the problem. In addition to our positive results, we prove a hardness result for box kernels, showing that there is no polynomial time algorithm for finding the mode of a kernel density estimate, unless P = NP. Obtaining similar hardness results for kernels used in practice (like Gaussian or logistic kernels) is an interesting future direction.

The trade-off between regret and computational cost is a fundamental problem for online kernel regression, and previous algorithms worked on the trade-off can not keep optimal regret bounds at a sublinear computational complexity. In this paper, we propose two new algorithms, AOGD-ALD and NONS-ALD, which can keep nearly optimal regret bounds at a sublinear computational complexity, and give sufficient conditions under which our algorithms work. Both algorithms dynamically maintain a group of nearly orthogonal basis used to approximate the kernel mapping, and keep nearly optimal regret bounds by controlling the approximate error. The number of basis depends on the approximate error and the decay rate of eigenvalues of the kernel matrix. If the eigenvalues decay exponentially, then AOGD-ALD and NONS-ALD separately achieves a regret of O(L(f)) and O(d_{eff}(mu)T) at a computational complexity in O(ln^2{T}). If the eigenvalues decay polynomially with degree pgeq 1, then our algorithms keep the same regret bounds at a computational complexity in o(T) in the case of p>4 and pgeq 10, respectively. L(f) is the cumulative losses of f and d_{eff}(mu) is the effective dimension of the problem. The two regret bounds are nearly optimal and are not comparable.

Many real-world machine learning problems involve inherently hierarchical data, yet traditional approaches rely on Euclidean metrics that fail to capture the discrete, branching nature of hierarchical relationships. We present a theoretical foundation for machine learning in p-adic metric spaces, which naturally respect hierarchical structure. Our main result proves that an n-dimensional plane minimizing the p-adic sum of distances to points in a dataset must pass through at least n + 1 of those points -- a striking contrast to Euclidean regression that highlights how p-adic metrics better align with the discrete nature of hierarchical data. As a corollary, a polynomial of degree n constructed to minimise the p-adic sum of residuals will pass through at least n + 1 points. As a further corollary, a polynomial of degree n approximating a higher degree polynomial at a finite number of points will yield a difference polynomial that has distinct rational roots. We demonstrate the practical significance of this result through two applications in natural language processing: analyzing hierarchical taxonomies and modeling grammatical morphology. These results suggest that p-adic metrics may be fundamental to properly handling hierarchical data structures in machine learning. In hierarchical data, interpolation between points often makes less sense than selecting actual observed points as representatives.

We study risk-sensitive Reinforcement Learning (RL), where we aim to maximize the Conditional Value at Risk (CVaR) with a fixed risk tolerance tau. Prior theoretical work studying risk-sensitive RL focuses on the tabular Markov Decision Processes (MDPs) setting. To extend CVaR RL to settings where state space is large, function approximation must be deployed. We study CVaR RL in low-rank MDPs with nonlinear function approximation. Low-rank MDPs assume the underlying transition kernel admits a low-rank decomposition, but unlike prior linear models, low-rank MDPs do not assume the feature or state-action representation is known. We propose a novel Upper Confidence Bound (UCB) bonus-driven algorithm to carefully balance the interplay between exploration, exploitation, and representation learning in CVaR RL. We prove that our algorithm achieves a sample complexity of Oleft(H^7 A^2 d^4{tau^2 epsilon^2}right) to yield an epsilon-optimal CVaR, where H is the length of each episode, A is the capacity of action space, and d is the dimension of representations. Computational-wise, we design a novel discretized Least-Squares Value Iteration (LSVI) algorithm for the CVaR objective as the planning oracle and show that we can find the near-optimal policy in a polynomial running time with a Maximum Likelihood Estimation oracle. To our knowledge, this is the first provably efficient CVaR RL algorithm in low-rank MDPs.

Hyperbolic embeddings offer excellent quality with few dimensions when embedding hierarchical data structures like synonym or type hierarchies. Given a tree, we give a combinatorial construction that embeds the tree in hyperbolic space with arbitrarily low distortion without using optimization. On WordNet, our combinatorial embedding obtains a mean-average-precision of 0.989 with only two dimensions, while Nickel et al.'s recent construction obtains 0.87 using 200 dimensions. We provide upper and lower bounds that allow us to characterize the precision-dimensionality tradeoff inherent in any hyperbolic embedding. To embed general metric spaces, we propose a hyperbolic generalization of multidimensional scaling (h-MDS). We show how to perform exact recovery of hyperbolic points from distances, provide a perturbation analysis, and give a recovery result that allows us to reduce dimensionality. The h-MDS approach offers consistently low distortion even with few dimensions across several datasets. Finally, we extract lessons from the algorithms and theory above to design a PyTorch-based implementation that can handle incomplete information and is scalable.

Regularized optimal transport (OT) is now increasingly used as a loss or as a matching layer in neural networks. Entropy-regularized OT can be computed using the Sinkhorn algorithm but it leads to fully-dense transportation plans, meaning that all sources are (fractionally) matched with all targets. To address this issue, several works have investigated quadratic regularization instead. This regularization preserves sparsity and leads to unconstrained and smooth (semi) dual objectives, that can be solved with off-the-shelf gradient methods. Unfortunately, quadratic regularization does not give direct control over the cardinality (number of nonzeros) of the transportation plan. We propose in this paper a new approach for OT with explicit cardinality constraints on the transportation plan. Our work is motivated by an application to sparse mixture of experts, where OT can be used to match input tokens such as image patches with expert models such as neural networks. Cardinality constraints ensure that at most k tokens are matched with an expert, which is crucial for computational performance reasons. Despite the nonconvexity of cardinality constraints, we show that the corresponding (semi) dual problems are tractable and can be solved with first-order gradient methods. Our method can be thought as a middle ground between unregularized OT (recovered in the limit case k=1) and quadratically-regularized OT (recovered when k is large enough). The smoothness of the objectives increases as k increases, giving rise to a trade-off between convergence speed and sparsity of the optimal plan.

In this paper, we propose new randomized algorithms for estimating the two-to-infinity and one-to-two norms in a matrix-free setting, using only matrix-vector multiplications. Our methods are based on appropriate modifications of Hutchinson's diagonal estimator and its Hutch++ version. We provide oracle complexity bounds for both modifications. We further illustrate the practical utility of our algorithms for Jacobian-based regularization in deep neural network training on image classification tasks. We also demonstrate that our methodology can be applied to mitigate the effect of adversarial attacks in the domain of recommender systems.

Contrastive learning is a highly successful technique for learning representations of data from labeled tuples, specifying the distance relations within the tuple. We study the sample complexity of contrastive learning, i.e. the minimum number of labeled tuples sufficient for getting high generalization accuracy. We give tight bounds on the sample complexity in a variety of settings, focusing on arbitrary distance functions, both general ell_p-distances, and tree metrics. Our main result is an (almost) optimal bound on the sample complexity of learning ell_p-distances for integer p. For any p ge 1 we show that tilde Theta(min(nd,n^2)) labeled tuples are necessary and sufficient for learning d-dimensional representations of n-point datasets. Our results hold for an arbitrary distribution of the input samples and are based on giving the corresponding bounds on the Vapnik-Chervonenkis/Natarajan dimension of the associated problems. We further show that the theoretical bounds on sample complexity obtained via VC/Natarajan dimension can have strong predictive power for experimental results, in contrast with the folklore belief about a substantial gap between the statistical learning theory and the practice of deep learning.

We prove rich algebraic structures of the solution space for 2-layer neural networks with quadratic activation and L_2 loss, trained on reasoning tasks in Abelian group (e.g., modular addition). Such a rich structure enables analytical construction of global optimal solutions from partial solutions that only satisfy part of the loss, despite its high nonlinearity. We coin the framework as CoGO (Composing Global Optimizers). Specifically, we show that the weight space over different numbers of hidden nodes of the 2-layer network is equipped with a semi-ring algebraic structure, and the loss function to be optimized consists of monomial potentials, which are ring homomorphism, allowing partial solutions to be composed into global ones by ring addition and multiplication. Our experiments show that around 95% of the solutions obtained by gradient descent match exactly our theoretical constructions. Although the global optimizers constructed only required a small number of hidden nodes, our analysis on gradient dynamics shows that over-parameterization asymptotically decouples training dynamics and is beneficial. We further show that training dynamics favors simpler solutions under weight decay, and thus high-order global optimizers such as perfect memorization are unfavorable.

Triangle centrality is introduced for finding important vertices in a graph based on the concentration of triangles surrounding each vertex. It has the distinct feature of allowing a vertex to be central if it is in many triangles or none at all. We show experimentally that triangle centrality is broadly applicable to many different types of networks. Our empirical results demonstrate that 30% of the time triangle centrality identified central vertices that differed with those found by five well-known centrality measures, which suggests novelty without being overly specialized. It is also asymptotically faster to compute on sparse graphs than all but the most trivial of these other measures. We introduce optimal algorithms that compute triangle centrality in O(mbarδ) time and O(m+n) space, where barδle O(m) is the average degeneracy introduced by Burkhardt, Faber, and Harris (2020). In practical applications, barδ is much smaller than m so triangle centrality can be computed in nearly linear time. On a Concurrent Read Exclusive Write (CREW) Parallel Random Access Machine (PRAM), we give a near work-optimal parallel algorithm that takes O(log n) time using O(mm) CREW PRAM processors. In MapReduce, we show it takes four rounds using O(mm) communication bits and is therefore optimal. We also derive a linear algebraic formulation of triangle centrality which can be computed in O(mbarδ) time on sparse graphs.

This article shows yet another proof of NP=CoNP$. In a previous article, we proved that NP=PSPACE and from it we can conclude that NP=CoNP immediately. The former proof shows how to obtain polynomial and, polynomial in time checkable Dag-like proofs for all purely implicational Minimal logic tautologies. From the fact that Minimal implicational logic is PSPACE-complete we get the proof that NP=PSPACE. This first proof of NP=CoNP uses Hudelmaier linear upper-bound on the height of Sequent Calculus minimal implicational logic proofs. In an addendum to the proof of NP=PSPACE, we observe that we do not need to use Hudelmaier upper-bound since any proof of non-hamiltonicity for any graph is linear upper-bounded. By the CoNP-completeness of non-hamiltonicity, we obtain NP=CoNP as a corollary of the first proof. In this article we show the third proof of CoNP=NP, also providing polynomial size and polynomial verifiable certificates that are Dags. They are generated from normal Natural Deduction proofs, linear height upper-bounded too, by removing redundancy, i.e., repeated parts. The existence of repeated parts is a consequence of the redundancy theorem for a family of super-polynomial proofs in the purely implicational Minimal logic. It is mandatory to read at least two previous articles to get the details of the proof presented here. The article that proves the redundancy theorem and the article that shows how to remove the repeated parts of a normal Natural Deduction proof to have a polynomial Dag certificate for minimal implicational logic tautologies.

Given a finite poset mathcal P, the hypercube-height, denoted by h^*(mathcal P), is defined to be the largest h such that, for any natural number n, the subsets of [n] of size less than h do not contain an induced copy of mathcal P. The hypercube-width, denoted by w^*(mathcal P), is the smallest w such that the subsets of [w] of size at most h^*(mathcal P) contain an induced copy of mathcal P. In other words, h^*(mathcal P) asks how `low' can a poset be embedded, and w^*(mathcal P) asks for the first hypercube in which such an `optimal' embedding occurs. These notions were introduced by Bastide, Groenland, Ivan and Johnston in connection to upper bounds for the poset saturation numbers. While it is not hard to see that h^*(mathcal P)leq |mathcal P|-1 (and this bound can be tight), the hypercube-width has proved to be much more elusive. It was shown by the authors mentioned above that w^*(mathcal P)leq|mathcal P|^2/4, but they conjectured that in fact w^*(mathcal P)leq |mathcal P| for any finite poset mathcal P. In this paper we prove this conjecture. The proof uses Hall's theorem for bipartite graphs as a precision tool for modifing an existing copy of our poset.

The enduring legacy of Euclidean geometry underpins classical machine learning, which, for decades, has been primarily developed for data lying in Euclidean space. Yet, modern machine learning increasingly encounters richly structured data that is inherently nonEuclidean. This data can exhibit intricate geometric, topological and algebraic structure: from the geometry of the curvature of space-time, to topologically complex interactions between neurons in the brain, to the algebraic transformations describing symmetries of physical systems. Extracting knowledge from such non-Euclidean data necessitates a broader mathematical perspective. Echoing the 19th-century revolutions that gave rise to non-Euclidean geometry, an emerging line of research is redefining modern machine learning with non-Euclidean structures. Its goal: generalizing classical methods to unconventional data types with geometry, topology, and algebra. In this review, we provide an accessible gateway to this fast-growing field and propose a graphical taxonomy that integrates recent advances into an intuitive unified framework. We subsequently extract insights into current challenges and highlight exciting opportunities for future development in this field.

We study first-order methods with preconditioning for solving structured nonlinear convex optimization problems. We propose a new family of preconditioners generated by symmetric polynomials. They provide first-order optimization methods with a provable improvement of the condition number, cutting the gaps between highest eigenvalues, without explicit knowledge of the actual spectrum. We give a stochastic interpretation of this preconditioning in terms of coordinate volume sampling and compare it with other classical approaches, including the Chebyshev polynomials. We show how to incorporate a polynomial preconditioning into the Gradient and Fast Gradient Methods and establish the corresponding global complexity bounds. Finally, we propose a simple adaptive search procedure that automatically chooses the best possible polynomial preconditioning for the Gradient Method, minimizing the objective along a low-dimensional Krylov subspace. Numerical experiments confirm the efficiency of our preconditioning strategies for solving various machine learning problems.

Gradient-free/zeroth-order methods for black-box convex optimization have been extensively studied in the last decade with the main focus on oracle calls complexity. In this paper, besides the oracle complexity, we focus also on iteration complexity, and propose a generic approach that, based on optimal first-order methods, allows to obtain in a black-box fashion new zeroth-order algorithms for non-smooth convex optimization problems. Our approach not only leads to optimal oracle complexity, but also allows to obtain iteration complexity similar to first-order methods, which, in turn, allows to exploit parallel computations to accelerate the convergence of our algorithms. We also elaborate on extensions for stochastic optimization problems, saddle-point problems, and distributed optimization.

Combinatorial optimization finds an optimal solution within a discrete set of variables and constraints. The field has seen tremendous progress both in research and industry. With the success of deep learning in the past decade, a recent trend in combinatorial optimization has been to improve state-of-the-art combinatorial optimization solvers by replacing key heuristic components with machine learning (ML) models. In this paper, we investigate two essential aspects of machine learning algorithms for combinatorial optimization: temporal characteristics and attention. We argue that for the task of variable selection in the branch-and-bound (B&B) algorithm, incorporating the temporal information as well as the bipartite graph attention improves the solver's performance. We support our claims with intuitions and numerical results over several standard datasets used in the literature and competitions. Code is available at: https://developer.huaweicloud.com/develop/aigallery/notebook/detail?id=047c6cf2-8463-40d7-b92f-7b2ca998e935