Daily Papers

This paper presents, for the first time, an image enhancement methodology designed to enhance the clarity of small intestinal villi in Wireless Capsule Endoscopy (WCE) images. This method first separates the low-frequency and high-frequency components of small intestinal villi images using guided filtering. Subsequently, an adaptive light gain factor is generated based on the low-frequency component, and an adaptive gradient gain factor is derived from the convolution results of the Laplacian operator in different regions of small intestinal villi images. The obtained light gain factor and gradient gain factor are then combined to enhance the high-frequency components. Finally, the enhanced high-frequency component is fused with the original image to achieve adaptive sharpening of the edges of WCE small intestinal villi images. The experiments affirm that, compared to established WCE image enhancement methods, our approach not only accentuates the edge details of WCE small intestine villi images but also skillfully suppresses noise amplification, thereby preventing the occurrence of edge overshooting.

Adaptive optimization methods such as AdaGrad, RMSprop and Adam have been proposed to achieve a rapid training process with an element-wise scaling term on learning rates. Though prevailing, they are observed to generalize poorly compared with SGD or even fail to converge due to unstable and extreme learning rates. Recent work has put forward some algorithms such as AMSGrad to tackle this issue but they failed to achieve considerable improvement over existing methods. In our paper, we demonstrate that extreme learning rates can lead to poor performance. We provide new variants of Adam and AMSGrad, called AdaBound and AMSBound respectively, which employ dynamic bounds on learning rates to achieve a gradual and smooth transition from adaptive methods to SGD and give a theoretical proof of convergence. We further conduct experiments on various popular tasks and models, which is often insufficient in previous work. Experimental results show that new variants can eliminate the generalization gap between adaptive methods and SGD and maintain higher learning speed early in training at the same time. Moreover, they can bring significant improvement over their prototypes, especially on complex deep networks. The implementation of the algorithm can be found at https://github.com/Luolc/AdaBound .

We present a novel adaptive optimization algorithm for large-scale machine learning problems. Equipped with a low-cost estimate of local curvature and Lipschitz smoothness, our method dynamically adapts the search direction and step-size. The search direction contains gradient information preconditioned by a well-scaled diagonal preconditioning matrix that captures the local curvature information. Our methodology does not require the tedious task of learning rate tuning, as the learning rate is updated automatically without adding an extra hyperparameter. We provide convergence guarantees on a comprehensive collection of optimization problems, including convex, strongly convex, and nonconvex problems, in both deterministic and stochastic regimes. We also conduct an extensive empirical evaluation on standard machine learning problems, justifying our algorithm's versatility and demonstrating its strong performance compared to other start-of-the-art first-order and second-order methods.

Adaptive gradient-based optimizers, particularly Adam, have left their mark in training large-scale deep learning models. The strength of such optimizers is that they exhibit fast convergence while being more robust to hyperparameter choice. However, they often generalize worse than non-adaptive methods. Recent studies have tied this performance gap to flat minima selection: adaptive methods tend to find solutions in sharper basins of the loss landscape, which in turn hurts generalization. To overcome this issue, we propose a new memory-augmented version of Adam that promotes exploration towards flatter minima by using a buffer of critical momentum terms during training. Intuitively, the use of the buffer makes the optimizer overshoot outside the basin of attraction if it is not wide enough. We empirically show that our method improves the performance of several variants of Adam on standard supervised language modelling and image classification tasks.

Adaptive optimization methods are widely recognized as among the most popular approaches for training Deep Neural Networks (DNNs). Techniques such as Adam, AdaGrad, and AdaHessian utilize a preconditioner that modifies the search direction by incorporating information about the curvature of the objective function. However, despite their adaptive characteristics, these methods still require manual fine-tuning of the step-size. This, in turn, impacts the time required to solve a particular problem. This paper presents an optimization framework named SANIA to tackle these challenges. Beyond eliminating the need for manual step-size hyperparameter settings, SANIA incorporates techniques to address poorly scaled or ill-conditioned problems. We also explore several preconditioning methods, including Hutchinson's method, which approximates the Hessian diagonal of the loss function. We conclude with an extensive empirical examination of the proposed techniques across classification tasks, covering both convex and non-convex contexts.

Most popular optimizers for deep learning can be broadly categorized as adaptive methods (e.g. Adam) and accelerated schemes (e.g. stochastic gradient descent (SGD) with momentum). For many models such as convolutional neural networks (CNNs), adaptive methods typically converge faster but generalize worse compared to SGD; for complex settings such as generative adversarial networks (GANs), adaptive methods are typically the default because of their stability.We propose AdaBelief to simultaneously achieve three goals: fast convergence as in adaptive methods, good generalization as in SGD, and training stability. The intuition for AdaBelief is to adapt the stepsize according to the "belief" in the current gradient direction. Viewing the exponential moving average (EMA) of the noisy gradient as the prediction of the gradient at the next time step, if the observed gradient greatly deviates from the prediction, we distrust the current observation and take a small step; if the observed gradient is close to the prediction, we trust it and take a large step. We validate AdaBelief in extensive experiments, showing that it outperforms other methods with fast convergence and high accuracy on image classification and language modeling. Specifically, on ImageNet, AdaBelief achieves comparable accuracy to SGD. Furthermore, in the training of a GAN on Cifar10, AdaBelief demonstrates high stability and improves the quality of generated samples compared to a well-tuned Adam optimizer. Code is available at https://github.com/juntang-zhuang/Adabelief-Optimizer

Training deep neural networks--and more recently, large models--demands efficient and scalable optimizers. Adaptive gradient algorithms like Adam, AdamW, and their variants have been central to this task. Despite the development of numerous variance reduction algorithms in the past decade aimed at accelerating stochastic optimization in both convex and nonconvex settings, variance reduction has not found widespread success in training deep neural networks or large language models. Consequently, it has remained a less favored approach in modern AI. In this paper, to unleash the power of variance reduction for efficient training of large models, we propose a unified optimization framework, MARS (Make vAriance Reduction Shine), which reconciles preconditioned gradient methods with variance reduction via a scaled stochastic recursive momentum technique. Within our framework, we introduce three instances of MARS that leverage preconditioned gradient updates based on AdamW, Lion, and Shampoo, respectively. We also draw a connection between our algorithms and existing optimizers. Experimental results on training GPT-2 models indicate that MARS consistently outperforms AdamW by a large margin.

The choice of batch sizes in stochastic gradient optimizers is critical for model training. However, the practice of varying batch sizes throughout the training process is less explored compared to other hyperparameters. We investigate adaptive batch size strategies derived from adaptive sampling methods, traditionally applied only in stochastic gradient descent. Given the significant interplay between learning rates and batch sizes, and considering the prevalence of adaptive gradient methods in deep learning, we emphasize the need for adaptive batch size strategies in these contexts. We introduce AdAdaGrad and its scalar variant AdAdaGradNorm, which incrementally increase batch sizes during training, while model updates are performed using AdaGrad and AdaGradNorm. We prove that AdaGradNorm converges with high probability at a rate of O(1/K) for finding a first-order stationary point of smooth nonconvex functions within K iterations. AdaGrad also demonstrates similar convergence properties when integrated with a novel coordinate-wise variant of our adaptive batch size strategies. Our theoretical claims are supported by numerical experiments on various image classification tasks, highlighting the enhanced adaptability of progressive batching protocols in deep learning and the potential of such adaptive batch size strategies with adaptive gradient optimizers in large-scale model training.

SGD does not produce robust results on datasets with label noise. Because the gradients calculated according to the losses of the noisy samples cause the optimization process to go in the wrong direction. In this paper, as an alternative to SGD, we recommend using samples with loss less than a threshold value determined during the optimization process, instead of using all samples in the mini-batch. Our proposed method, Adaptive-k, aims to exclude label noise samples from the optimization process and make the process robust. On noisy datasets, we found that using a threshold-based approach, such as Adaptive-k, produces better results than using all samples or a fixed number of low-loss samples in the mini-batch. Based on our theoretical analysis and experimental results, we show that the Adaptive-k method is closest to the performance of the oracle, in which noisy samples are entirely removed from the dataset. Adaptive-k is a simple but effective method. It does not require prior knowledge of the noise ratio of the dataset, does not require additional model training, and does not increase training time significantly. The code for Adaptive-k is available at https://github.com/enesdedeoglu-TR/Adaptive-k

We study the effect of mini-batching on the loss landscape of deep neural networks using spiked, field-dependent random matrix theory. We demonstrate that the magnitude of the extremal values of the batch Hessian are larger than those of the empirical Hessian. We also derive similar results for the Generalised Gauss-Newton matrix approximation of the Hessian. As a consequence of our theorems we derive an analytical expressions for the maximal learning rates as a function of batch size, informing practical training regimens for both stochastic gradient descent (linear scaling) and adaptive algorithms, such as Adam (square root scaling), for smooth, non-convex deep neural networks. Whilst the linear scaling for stochastic gradient descent has been derived under more restrictive conditions, which we generalise, the square root scaling rule for adaptive optimisers is, to our knowledge, completely novel. %For stochastic second-order methods and adaptive methods, we derive that the minimal damping coefficient is proportional to the ratio of the learning rate to batch size. We validate our claims on the VGG/WideResNet architectures on the CIFAR-100 and ImageNet datasets. Based on our investigations of the sub-sampled Hessian we develop a stochastic Lanczos quadrature based on the fly learning rate and momentum learner, which avoids the need for expensive multiple evaluations for these key hyper-parameters and shows good preliminary results on the Pre-Residual Architecure for CIFAR-100.

Classifier-free guidance (CFG) is crucial for improving both generation quality and alignment between the input condition and final output in diffusion models. While a high guidance scale is generally required to enhance these aspects, it also causes oversaturation and unrealistic artifacts. In this paper, we revisit the CFG update rule and introduce modifications to address this issue. We first decompose the update term in CFG into parallel and orthogonal components with respect to the conditional model prediction and observe that the parallel component primarily causes oversaturation, while the orthogonal component enhances image quality. Accordingly, we propose down-weighting the parallel component to achieve high-quality generations without oversaturation. Additionally, we draw a connection between CFG and gradient ascent and introduce a new rescaling and momentum method for the CFG update rule based on this insight. Our approach, termed adaptive projected guidance (APG), retains the quality-boosting advantages of CFG while enabling the use of higher guidance scales without oversaturation. APG is easy to implement and introduces practically no additional computational overhead to the sampling process. Through extensive experiments, we demonstrate that APG is compatible with various conditional diffusion models and samplers, leading to improved FID, recall, and saturation scores while maintaining precision comparable to CFG, making our method a superior plug-and-play alternative to standard classifier-free guidance.

Despite superior training outcomes, adaptive optimization methods such as Adam, Adagrad or RMSprop have been found to generalize poorly compared to Stochastic gradient descent (SGD). These methods tend to perform well in the initial portion of training but are outperformed by SGD at later stages of training. We investigate a hybrid strategy that begins training with an adaptive method and switches to SGD when appropriate. Concretely, we propose SWATS, a simple strategy which switches from Adam to SGD when a triggering condition is satisfied. The condition we propose relates to the projection of Adam steps on the gradient subspace. By design, the monitoring process for this condition adds very little overhead and does not increase the number of hyperparameters in the optimizer. We report experiments on several standard benchmarks such as: ResNet, SENet, DenseNet and PyramidNet for the CIFAR-10 and CIFAR-100 data sets, ResNet on the tiny-ImageNet data set and language modeling with recurrent networks on the PTB and WT2 data sets. The results show that our strategy is capable of closing the generalization gap between SGD and Adam on a majority of the tasks.

Adaptive gradient methods, e.g. Adam, have achieved tremendous success in machine learning. Scaling the learning rate element-wisely by a certain form of second moment estimate of gradients, such methods are able to attain rapid training of modern deep neural networks. Nevertheless, they are observed to suffer from compromised generalization ability compared with stochastic gradient descent (SGD) and tend to be trapped in local minima at an early stage during training. Intriguingly, we discover that substituting the gradient in the second raw moment estimate term with its momentumized version in Adam can resolve the issue. The intuition is that gradient with momentum contains more accurate directional information and therefore its second moment estimation is a more favorable option for learning rate scaling than that of the raw gradient. Thereby we propose AdaMomentum as a new optimizer reaching the goal of training fast while generalizing much better. We further develop a theory to back up the improvement in generalization and provide convergence guarantees under both convex and nonconvex settings. Extensive experiments on a wide range of tasks and models demonstrate that AdaMomentum exhibits state-of-the-art performance and superior training stability consistently.

Stochastic Gradient Decent (SGD) is one of the core techniques behind the success of deep neural networks. The gradient provides information on the direction in which a function has the steepest rate of change. The main problem with basic SGD is to change by equal sized steps for all parameters, irrespective of gradient behavior. Hence, an efficient way of deep network optimization is to make adaptive step sizes for each parameter. Recently, several attempts have been made to improve gradient descent methods such as AdaGrad, AdaDelta, RMSProp and Adam. These methods rely on the square roots of exponential moving averages of squared past gradients. Thus, these methods do not take advantage of local change in gradients. In this paper, a novel optimizer is proposed based on the difference between the present and the immediate past gradient (i.e., diffGrad). In the proposed diffGrad optimization technique, the step size is adjusted for each parameter in such a way that it should have a larger step size for faster gradient changing parameters and a lower step size for lower gradient changing parameters. The convergence analysis is done using the regret bound approach of online learning framework. Rigorous analysis is made in this paper over three synthetic complex non-convex functions. The image categorization experiments are also conducted over the CIFAR10 and CIFAR100 datasets to observe the performance of diffGrad with respect to the state-of-the-art optimizers such as SGDM, AdaGrad, AdaDelta, RMSProp, AMSGrad, and Adam. The residual unit (ResNet) based Convolutional Neural Networks (CNN) architecture is used in the experiments. The experiments show that diffGrad outperforms other optimizers. Also, we show that diffGrad performs uniformly well for training CNN using different activation functions. The source code is made publicly available at https://github.com/shivram1987/diffGrad.

Conventional Low-Rank Adaptation (LoRA) methods employ a fixed rank, imposing uniform adaptation across transformer layers and attention heads despite their heterogeneous learning dynamics. This paper introduces Adaptive Rank Dynamic LoRA (ARD-LoRA), a novel framework that automates rank allocation through learnable scaling factors. These factors are optimized via a meta-objective balancing task performance and parameter efficiency, incorporating ell_1 sparsity for minimal rank and Total Variation regularization for stable rank transitions. ARD-LoRA enables continuous, differentiable, per-head rank adaptation. Experiments on LLAMA-3.1-70B and PaliGemma-2 demonstrate ARD-LoRA's efficacy, achieving up to 99.3% of full fine-tuning performance with only 0.32% trainable parameters, outperforming strong baselines like DoRA and AdaLoRA. Furthermore, it reduces multimodal adaptation memory by 41%. These results establish dynamic, fine-grained rank allocation as a critical paradigm for efficient foundation model adaptation.

Self-tuning algorithms that adapt the learning process online encourage more effective and robust learning. Among all the methods available, meta-gradients have emerged as a promising approach. They leverage the differentiability of the learning rule with respect to some hyper-parameters to adapt them in an online fashion. Although meta-gradients can be accumulated over multiple learning steps to avoid myopic updates, this is rarely used in practice. In this work, we demonstrate that whilst multi-step meta-gradients do provide a better learning signal in expectation, this comes at the cost of a significant increase in variance, hindering performance. In the light of this analysis, we introduce a novel method mixing multiple inner steps that enjoys a more accurate and robust meta-gradient signal, essentially trading off bias and variance in meta-gradient estimation. When applied to the Snake game, the mixing meta-gradient algorithm can cut the variance by a factor of 3 while achieving similar or higher performance.

We introduce two complementary techniques for efficient adaptive optimization that reduce memory requirements while accelerating training of large-scale neural networks. The first technique, Subset-Norm adaptive step size, generalizes AdaGrad-Norm and AdaGrad(-Coordinate) by reducing the second moment term's memory footprint from O(d) to O(d) through step-size sharing, where d is the model size. For non-convex smooth objectives under coordinate-wise sub-gaussian gradient noise, we prove a noise-adapted high-probability convergence guarantee showing improved dimensional dependence over existing methods. Our second technique, Subspace-Momentum, reduces the momentum state's memory footprint by operating in a low-dimensional subspace while applying standard SGD in the orthogonal complement. We establish high-probability convergence rates under similar relaxed assumptions. Empirical evaluation on LLaMA models from 60M to 1B parameters demonstrates the effectiveness of our methods, where combining subset-norm with subspace-momentum achieves Adam's validation perplexity in approximately half the training tokens (6.8B vs 13.1B) while using only 20% of the Adam's optimizer-states memory footprint and requiring minimal additional hyperparameter tuning.

We study Stochastic Gradient Descent with AdaGrad stepsizes: a popular adaptive (self-tuning) method for first-order stochastic optimization. Despite being well studied, existing analyses of this method suffer from various shortcomings: they either assume some knowledge of the problem parameters, impose strong global Lipschitz conditions, or fail to give bounds that hold with high probability. We provide a comprehensive analysis of this basic method without any of these limitations, in both the convex and non-convex (smooth) cases, that additionally supports a general ``affine variance'' noise model and provides sharp rates of convergence in both the low-noise and high-noise~regimes.

We propose a new variant of AMSGrad, a popular adaptive gradient based optimization algorithm widely used for training deep neural networks. Our algorithm adds prior knowledge about the sequence of consecutive mini-batch gradients and leverages its underlying structure making the gradients sequentially predictable. By exploiting the predictability and ideas from optimistic online learning, the proposed algorithm can accelerate the convergence and increase sample efficiency. After establishing a tighter upper bound under some convexity conditions on the regret, we offer a complimentary view of our algorithm which generalizes the offline and stochastic version of nonconvex optimization. In the nonconvex case, we establish a non-asymptotic convergence bound independently of the initialization. We illustrate the practical speedup on several deep learning models via numerical experiments.

This work proposes a procedure for designing algorithms for specific adaptive data collection tasks like active learning and pure-exploration multi-armed bandits. Unlike the design of traditional adaptive algorithms that rely on concentration of measure and careful analysis to justify the correctness and sample complexity of the procedure, our adaptive algorithm is learned via adversarial training over equivalence classes of problems derived from information theoretic lower bounds. In particular, a single adaptive learning algorithm is learned that competes with the best adaptive algorithm learned for each equivalence class. Our procedure takes as input just the available queries, set of hypotheses, loss function, and total query budget. This is in contrast to existing meta-learning work that learns an adaptive algorithm relative to an explicit, user-defined subset or prior distribution over problems which can be challenging to define and be mismatched to the instance encountered at test time. This work is particularly focused on the regime when the total query budget is very small, such as a few dozen, which is much smaller than those budgets typically considered by theoretically derived algorithms. We perform synthetic experiments to justify the stability and effectiveness of the training procedure, and then evaluate the method on tasks derived from real data including a noisy 20 Questions game and a joke recommendation task.

Label efficiency has become an increasingly important objective in deep learning applications. Active learning aims to reduce the number of labeled examples needed to train deep networks, but the empirical performance of active learning algorithms can vary dramatically across datasets and applications. It is difficult to know in advance which active learning strategy will perform well or best in a given application. To address this, we propose the first adaptive algorithm selection strategy for deep active learning. For any unlabeled dataset, our (meta) algorithm TAILOR (Thompson ActIve Learning algORithm selection) iteratively and adaptively chooses among a set of candidate active learning algorithms. TAILOR uses novel reward functions aimed at gathering class-balanced examples. Extensive experiments in multi-class and multi-label applications demonstrate TAILOR's effectiveness in achieving accuracy comparable or better than that of the best of the candidate algorithms. Our implementation of TAILOR is open-sourced at https://github.com/jifanz/TAILOR.

Stochastic Gradient Descent (SGD) and its momentum variants form the backbone of deep learning optimization, yet the underlying dynamics of their gradient behavior remain insufficiently understood. In this work, we reinterpret gradient updates through the lens of signal processing and reveal that fixed momentum coefficients inherently distort the balance between bias and variance, leading to skewed or suboptimal parameter updates. To address this, we propose SGDF (SGD with Filter), an optimizer inspired by the principles of Optimal Linear Filtering. SGDF computes an online, time-varying gain to dynamically refine gradient estimation by minimizing the mean-squared error, thereby achieving an optimal trade-off between noise suppression and signal preservation. Furthermore, our approach could extend to other optimizers, showcasing its broad applicability to optimization frameworks. Extensive experiments across diverse architectures and benchmarks demonstrate SGDF surpasses conventional momentum methods and achieves performance on par with or surpassing state-of-the-art optimizers.

Stochastic gradient descent method and its variants constitute the core optimization algorithms that achieve good convergence rates for solving machine learning problems. These rates are obtained especially when these algorithms are fine-tuned for the application at hand. Although this tuning process can require large computational costs, recent work has shown that these costs can be reduced by line search methods that iteratively adjust the stepsize. We propose an alternative approach to stochastic line search by using a new algorithm based on forward step model building. This model building step incorporates second-order information that allows adjusting not only the stepsize but also the search direction. Noting that deep learning model parameters come in groups (layers of tensors), our method builds its model and calculates a new step for each parameter group. This novel diagonalization approach makes the selected step lengths adaptive. We provide convergence rate analysis, and experimentally show that the proposed algorithm achieves faster convergence and better generalization in well-known test problems. More precisely, SMB requires less tuning, and shows comparable performance to other adaptive methods.

Training deep neural networks requires intricate initialization and careful selection of learning rates. The emergence of stochastic gradient optimization methods that use adaptive learning rates based on squared past gradients, e.g., AdaGrad, AdaDelta, and Adam, eases the job slightly. However, such methods have also been proven problematic in recent studies with their own pitfalls including non-convergence issues and so on. Alternative variants have been proposed for enhancement, such as AMSGrad, AdaShift and AdaBound. In this work, we identify a new problem of adaptive learning rate methods that exhibits at the beginning of learning where Adam produces extremely large learning rates that inhibit the start of learning. We propose the Adaptive and Momental Bound (AdaMod) method to restrict the adaptive learning rates with adaptive and momental upper bounds. The dynamic learning rate bounds are based on the exponential moving averages of the adaptive learning rates themselves, which smooth out unexpected large learning rates and stabilize the training of deep neural networks. Our experiments verify that AdaMod eliminates the extremely large learning rates throughout the training and brings significant improvements especially on complex networks such as DenseNet and Transformer, compared to Adam. Our implementation is available at: https://github.com/lancopku/AdaMod

We consider the problem of estimating the learning rate in adaptive methods, such as AdaGrad and Adam. We propose Prodigy, an algorithm that provably estimates the distance to the solution D, which is needed to set the learning rate optimally. At its core, Prodigy is a modification of the D-Adaptation method for learning-rate-free learning. It improves upon the convergence rate of D-Adaptation by a factor of O(log(D/d_0)), where d_0 is the initial estimate of D. We test Prodigy on 12 common logistic-regression benchmark datasets, VGG11 and ResNet-50 training on CIFAR10, ViT training on Imagenet, LSTM training on IWSLT14, DLRM training on Criteo dataset, VarNet on Knee MRI dataset, as well as RoBERTa and GPT transformer training on BookWiki. Our experimental results show that our approach consistently outperforms D-Adaptation and reaches test accuracy values close to that of hand-tuned Adam.

Adaptive optimization is critical in federated learning, where enabling adaptivity on both the server and client sides has proven essential for achieving optimal performance. However, the scalability of such jointly adaptive systems is often hindered by resource limitations in communication and memory. In this paper, we introduce a class of efficient adaptive algorithms, named FedAda^2 and its enhanced version FedAda^2++, designed specifically for large-scale, cross-device federated environments. FedAda^2 optimizes communication efficiency by avoiding the transfer of preconditioners between the server and clients. Additionally, FedAda^2++ extends this approach by incorporating memory-efficient adaptive optimizers on the client side, further reducing on-device memory usage. Theoretically, we demonstrate that FedAda^2 and FedAda^2++ achieve the same convergence rates for general, non-convex objectives as its more resource-intensive counterparts that directly integrate joint adaptivity. Extensive empirical evaluations on image and text datasets demonstrate both the advantages of joint adaptivity and the effectiveness of FedAda^2/FedAda^2++.

Supervised fine-tuning (SFT) on chain-of-thought (CoT) trajectories demonstrations is a common approach for enabling reasoning in large language models. Standard practices typically only retain trajectories with correct final answers (positives) while ignoring the rest (negatives). We argue that this paradigm discards substantial supervision and exacerbates overfitting, limiting out-of-domain (OOD) generalization. Specifically, we surprisingly find that incorporating negative trajectories into SFT yields substantial OOD generalization gains over positive-only training, as these trajectories often retain valid intermediate reasoning despite incorrect final answers. To understand this effect in depth, we systematically analyze data, training dynamics, and inference behavior, identifying 22 recurring patterns in negative chains that serve a dual role: they moderate loss descent to mitigate overfitting during training and boost policy entropy by 35.67% during inference to facilitate exploration. Motivated by these observations, we further propose Gain-based LOss Weighting (GLOW), an adaptive, sample-aware scheme that exploits such distinctive training dynamics by rescaling per-sample loss based on inter-epoch progress. Empirically, GLOW efficiently leverages unfiltered trajectories, yielding a 5.51% OOD gain over positive-only SFT on Qwen2.5-7B and boosting MMLU from 72.82% to 76.47% as an RL initialization.

Neural network training is commonly accelerated by using multiple synchronized workers to compute gradient updates in parallel. Asynchronous methods remove synchronization overheads and improve hardware utilization at the cost of introducing gradient delay, which impedes optimization and can lead to lower final model performance. We introduce Adaptive Braking (AB), a modification for momentum-based optimizers that mitigates the effects of gradient delay. AB dynamically scales the gradient based on the alignment of the gradient and the velocity. This can dampen oscillations along high curvature directions of the loss surface, stabilizing and accelerating asynchronous training. We show that applying AB on top of SGD with momentum enables training ResNets on CIFAR-10 and ImageNet-1k with delays D geq 32 update steps with minimal drop in final test accuracy.

A range of recent optimizers have emerged that approximate the same "matrix-whitening" transformation in various ways. In this work, we systematically deconstruct such optimizers, aiming to disentangle the key components that explain performance. Across tuned hyperparameters across the board, all flavors of matrix-whitening methods reliably outperform elementwise counterparts, such as Adam. Matrix-whitening is often related to spectral descent -- however, experiments reveal that performance gains are *not explained solely by accurate spectral normalization* -- particularly, SOAP displays the largest per-step gain, even though Muon more accurately descends along the steepest spectral descent direction. Instead, we argue that matrix-whitening serves two purposes, and the variance adaptation component of matrix-whitening is the overlooked ingredient explaining this performance gap. Experiments show that variance-adapted versions of optimizers consistently outperform their sign-descent counterparts, including an adaptive version of Muon. We further ablate variance adaptation strategies, finding that while lookahead style approximations are not as effective, low-rank variance estimators can effectively reduce memory costs without a performance loss.

Federated learning is a distributed machine learning paradigm in which a large number of clients coordinate with a central server to learn a model without sharing their own training data. Standard federated optimization methods such as Federated Averaging (FedAvg) are often difficult to tune and exhibit unfavorable convergence behavior. In non-federated settings, adaptive optimization methods have had notable success in combating such issues. In this work, we propose federated versions of adaptive optimizers, including Adagrad, Adam, and Yogi, and analyze their convergence in the presence of heterogeneous data for general non-convex settings. Our results highlight the interplay between client heterogeneity and communication efficiency. We also perform extensive experiments on these methods and show that the use of adaptive optimizers can significantly improve the performance of federated learning.

This paper presents a novel technique based on gradient boosting to train the final layers of a neural network (NN). Gradient boosting is an additive expansion algorithm in which a series of models are trained sequentially to approximate a given function. A neural network can also be seen as an additive expansion where the scalar product of the responses of the last hidden layer and its weights provide the final output of the network. Instead of training the network as a whole, the proposed algorithm trains the network sequentially in T steps. First, the bias term of the network is initialized with a constant approximation that minimizes the average loss of the data. Then, at each step, a portion of the network, composed of J neurons, is trained to approximate the pseudo-residuals on the training data computed from the previous iterations. Finally, the T partial models and bias are integrated as a single NN with T times J neurons in the hidden layer. Extensive experiments in classification and regression tasks, as well as in combination with deep neural networks, are carried out showing a competitive generalization performance with respect to neural networks trained with different standard solvers, such as Adam, L-BFGS, SGD and deep models. Furthermore, we show that the proposed method design permits to switch off a number of hidden units during test (the units that were last trained) without a significant reduction of its generalization ability. This permits the adaptation of the model to different classification speed requirements on the fly.

Forward Gradients - the idea of using directional derivatives in forward differentiation mode - have recently been shown to be utilizable for neural network training while avoiding problems generally associated with backpropagation gradient computation, such as locking and memorization requirements. The cost is the requirement to guess the step direction, which is hard in high dimensions. While current solutions rely on weighted averages over isotropic guess vector distributions, we propose to strongly bias our gradient guesses in directions that are much more promising, such as feedback obtained from small, local auxiliary networks. For a standard computer vision neural network, we conduct a rigorous study systematically covering a variety of combinations of gradient targets and gradient guesses, including those previously presented in the literature. We find that using gradients obtained from a local loss as a candidate direction drastically improves on random noise in Forward Gradient methods.

The AdamW optimizer, while standard for LLM pretraining, is a critical memory bottleneck, consuming optimizer states equivalent to twice the model's size. Although light-state optimizers like SinkGD attempt to address this issue, we identify the embedding layer dilemma: these methods fail to handle the sparse, high-variance gradients inherent to embeddings, forcing a hybrid design that reverts to AdamW and partially negates the memory gains. We propose SAGE (Sign Adaptive GradiEnt), a novel optimizer that resolves this dilemma by replacing AdamW in this hybrid structure. SAGE combines a Lion-style update direction with a new, memory-efficient O(d) adaptive scale. This scale acts as a "safe damper," provably bounded by 1.0, which tames high-variance dimensions more effectively than existing methods. This superior stability allows SAGE to achieve better convergence. On Llama models up to 1.3B parameters, our SAGE-based hybrid achieves new state-of-the-art perplexity, outperforming all baselines, including SinkGD hybrid, while significantly reducing optimizer state memory.

Statistical whitening transformations play a fundamental role in many computational systems, and may also play an important role in biological sensory systems. Existing neural circuit models of adaptive whitening operate by modifying synaptic interactions; however, such modifications would seem both too slow and insufficiently reversible. Motivated by the extensive neuroscience literature on gain modulation, we propose an alternative model that adaptively whitens its responses by modulating the gains of individual neurons. Starting from a novel whitening objective, we derive an online algorithm that whitens its outputs by adjusting the marginal variances of an overcomplete set of projections. We map the algorithm onto a recurrent neural network with fixed synaptic weights and gain-modulating interneurons. We demonstrate numerically that sign-constraining the gains improves robustness of the network to ill-conditioned inputs, and a generalization of the circuit achieves a form of local whitening in convolutional populations, such as those found throughout the visual or auditory systems.

The vast majority of modern deep learning models are trained with momentum-based first-order optimizers. The momentum term governs the optimizer's memory by determining how much each past gradient contributes to the current convergence direction. Fundamental momentum methods, such as Nesterov Accelerated Gradient and the Heavy Ball method, as well as more recent optimizers such as AdamW and Lion, all rely on the momentum coefficient that is customarily set to β= 0.9 and kept constant during model training, a strategy widely used by practitioners, yet suboptimal. In this paper, we introduce an adaptive memory mechanism that replaces constant momentum with a dynamic momentum coefficient that is adjusted online during optimization. We derive our method by approximating the objective function using two planes: one derived from the gradient at the current iterate and the other obtained from the accumulated memory of the past gradients. To the best of our knowledge, such a proximal framework was never used for momentum-based optimization. Our proposed approach is novel, extremely simple to use, and does not rely on extra assumptions or hyperparameter tuning. We implement adaptive memory variants of both SGD and AdamW across a wide range of learning tasks, from simple convex problems to large-scale deep learning scenarios, demonstrating that our approach can outperform standard SGD and Adam with hand-tuned momentum coefficients. Finally, our work opens doors for new ways of inducing adaptivity in optimization.

Existing reinforcement learning (RL)-based post-training methods for large language models have advanced rapidly, yet their design has largely been guided by heuristics rather than systematic theoretical principles. This gap limits our understanding of the properties of the gradient estimators and the associated optimization algorithms, thereby constraining opportunities to improve training stability and overall performance. In this work, we provide a unified theoretical framework that characterizes the statistical properties of commonly used policy-gradient estimators under mild assumptions. Our analysis establishes unbiasedness, derives exact variance expressions, and yields an optimization-loss upper bound that enables principled reasoning about learning dynamics. Building on these results, we prove convergence guarantees and derive an adaptive learning-rate schedule governed by the signal-to-noise ratio (SNR) of gradients. We further show that the variance-optimal baseline is a gradient-weighted estimator, offering a new principle for variance reduction and naturally enhancing stability beyond existing methods. These insights motivate Optimal Baseline and Learning-Rate Policy Optimization (OBLR-PO), an algorithm that jointly adapts learning rates and baselines in a theoretically grounded manner. Experiments on Qwen3-4B-Base and Qwen3-8B-Base demonstrate consistent gains over existing policy optimization methods, validating that our theoretical contributions translate into practical improvements in large-scale post-training.

In several recently proposed stochastic optimization methods (e.g. RMSProp, Adam, Adadelta), parameter updates are scaled by the inverse square roots of exponential moving averages of squared past gradients. Maintaining these per-parameter second-moment estimators requires memory equal to the number of parameters. For the case of neural network weight matrices, we propose maintaining only the per-row and per-column sums of these moving averages, and estimating the per-parameter second moments based on these sums. We demonstrate empirically that this method produces similar results to the baseline. Secondly, we show that adaptive methods can produce larger-than-desired updates when the decay rate of the second moment accumulator is too slow. We propose update clipping and a gradually increasing decay rate scheme as remedies. Combining these methods and dropping momentum, we achieve comparable results to the published Adam regime in training the Transformer model on the WMT 2014 English-German machine translation task, while using very little auxiliary storage in the optimizer. Finally, we propose scaling the parameter updates based on the scale of the parameters themselves.

This paper presents the Adaptive Personalized Control System (APECS) architecture, a novel framework for human-in-the-loop control. An architecture is developed which defines appropriate constraints for the system objectives. A method for enacting Lipschitz and sector bounds on the resulting controller is derived to ensure desirable control properties. An analysis of worst-case loss functions and the optimal loss function weighting is made to implement an effective training scheme. Finally, simulations are carried out to demonstrate the effectiveness of the proposed architecture. This architecture resulted in a 4.5% performance increase compared to the human operator and 9% to an unconstrained feedforward neural network trained in the same way.

Gradient regularization (GR) is a method that penalizes the gradient norm of the training loss during training. While some studies have reported that GR can improve generalization performance, little attention has been paid to it from the algorithmic perspective, that is, the algorithms of GR that efficiently improve the performance. In this study, we first reveal that a specific finite-difference computation, composed of both gradient ascent and descent steps, reduces the computational cost of GR. Next, we show that the finite-difference computation also works better in the sense of generalization performance. We theoretically analyze a solvable model, a diagonal linear network, and clarify that GR has a desirable implicit bias to so-called rich regime and finite-difference computation strengthens this bias. Furthermore, finite-difference GR is closely related to some other algorithms based on iterative ascent and descent steps for exploring flat minima. In particular, we reveal that the flooding method can perform finite-difference GR in an implicit way. Thus, this work broadens our understanding of GR for both practice and theory.

In machine learning applications, it is well known that carefully designed learning rate (step size) schedules can significantly improve the convergence of commonly used first-order optimization algorithms. Therefore how to set step size adaptively becomes an important research question. A popular and effective method is the Polyak step size, which sets step size adaptively for gradient descent or stochastic gradient descent without the need to estimate the smoothness parameter of the objective function. However, there has not been a principled way to generalize the Polyak step size for algorithms with momentum accelerations. This paper presents a general framework to set the learning rate adaptively for first-order optimization methods with momentum, motivated by the derivation of Polyak step size. It is shown that the resulting methods are much less sensitive to the choice of momentum parameter and may avoid the oscillation of the heavy-ball method on ill-conditioned problems. These adaptive step sizes are further extended to the stochastic settings, which are attractive choices for stochastic gradient descent with momentum. Our methods are demonstrated to be more effective for stochastic gradient methods than prior adaptive step size algorithms in large-scale machine learning tasks.

This paper develops a policy learning method for tuning a pre-trained policy to adapt to additional tasks without altering the original task. A method named Adaptive Policy Gradient (APG) is proposed in this paper, which combines Bellman's principle of optimality with the policy gradient approach to improve the convergence rate. This paper provides theoretical analysis which guarantees the convergence rate and sample complexity of O(1/T) and O(1/epsilon), respectively, where T denotes the number of iterations and epsilon denotes the accuracy of the resulting stationary policy. Furthermore, several challenging numerical simulations, including cartpole, lunar lander, and robot arm, are provided to show that APG obtains similar performance compared to existing deterministic policy gradient methods while utilizing much less data and converging at a faster rate.

Training large language models (LLMs) poses challenges due to their massive scale and heterogeneous architectures. While adaptive optimizers like AdamW help address gradient variations, they still struggle with efficient and effective parameter-wise learning rate estimation, resulting in training instability, slow convergence, and poor compatibility with parameter-efficient fine-tuning (PEFT) techniques. This work introduces Scaling with Gradient Grouping (SGG), an optimizer wrapper that improves adaptive learning rate estimation by dynamic grouping and group-specific scaling. SGG first groups gradient statistics in each layer into clusters and then applies cluster-specific scaling to calibrate learning rates for each parameter, thus imposing collective group-wise constraints while maintaining precise per-parameter adaptation. Experiments on diverse (M)LLM benchmarks show that SGG integrates seamlessly with existing optimizers, and offers consistent gains and faster convergence over baselines, with various model sizes. Its stability across varying batch sizes and learning rates establishes SGG as a robust choice for LLM optimization.

The learning rate warmup heuristic achieves remarkable success in stabilizing training, accelerating convergence and improving generalization for adaptive stochastic optimization algorithms like RMSprop and Adam. Here, we study its mechanism in details. Pursuing the theory behind warmup, we identify a problem of the adaptive learning rate (i.e., it has problematically large variance in the early stage), suggest warmup works as a variance reduction technique, and provide both empirical and theoretical evidence to verify our hypothesis. We further propose RAdam, a new variant of Adam, by introducing a term to rectify the variance of the adaptive learning rate. Extensive experimental results on image classification, language modeling, and neural machine translation verify our intuition and demonstrate the effectiveness and robustness of our proposed method. All implementations are available at: https://github.com/LiyuanLucasLiu/RAdam.

This paper presents a comprehensive study on the role of Classifier-Free Guidance (CFG) in text-conditioned diffusion models from the perspective of inference efficiency. In particular, we relax the default choice of applying CFG in all diffusion steps and instead search for efficient guidance policies. We formulate the discovery of such policies in the differentiable Neural Architecture Search framework. Our findings suggest that the denoising steps proposed by CFG become increasingly aligned with simple conditional steps, which renders the extra neural network evaluation of CFG redundant, especially in the second half of the denoising process. Building upon this insight, we propose "Adaptive Guidance" (AG), an efficient variant of CFG, that adaptively omits network evaluations when the denoising process displays convergence. Our experiments demonstrate that AG preserves CFG's image quality while reducing computation by 25%. Thus, AG constitutes a plug-and-play alternative to Guidance Distillation, achieving 50% of the speed-ups of the latter while being training-free and retaining the capacity to handle negative prompts. Finally, we uncover further redundancies of CFG in the first half of the diffusion process, showing that entire neural function evaluations can be replaced by simple affine transformations of past score estimates. This method, termed LinearAG, offers even cheaper inference at the cost of deviating from the baseline model. Our findings provide insights into the efficiency of the conditional denoising process that contribute to more practical and swift deployment of text-conditioned diffusion models.

This paper explores Large Batch Training techniques using layer-wise adaptive scaling ratio (LARS) across diverse settings, uncovering insights. LARS algorithms with warm-up tend to be trapped in sharp minimizers early on due to redundant ratio scaling. Additionally, a fixed steep decline in the latter phase restricts deep neural networks from effectively navigating early-phase sharp minimizers. Building on these findings, we propose Time Varying LARS (TVLARS), a novel algorithm that replaces warm-up with a configurable sigmoid-like function for robust training in the initial phase. TVLARS promotes gradient exploration early on, surpassing sharp optimizers and gradually transitioning to LARS for robustness in later phases. Extensive experiments demonstrate that TVLARS consistently outperforms LARS and LAMB in most cases, with up to 2\% improvement in classification scenarios. Notably, in all self-supervised learning cases, TVLARS dominates LARS and LAMB with performance improvements of up to 10\%.

While backpropagation (BP) is the mainstream approach for gradient computation in neural network training, its heavy reliance on the chain rule of differentiation constrains the designing flexibility of network architecture and training pipelines. We avoid the recursive computation in BP and develop a unified likelihood ratio (ULR) method for gradient estimation with just one forward propagation. Not only can ULR be extended to train a wide variety of neural network architectures, but the computation flow in BP can also be rearranged by ULR for better device adaptation. Moreover, we propose several variance reduction techniques to further accelerate the training process. Our experiments offer numerical results across diverse aspects, including various neural network training scenarios, computation flow rearrangement, and fine-tuning of pre-trained models. All findings demonstrate that ULR effectively enhances the flexibility of neural network training by permitting localized module training without compromising the global objective and significantly boosts the network robustness.

Gradient-based meta-learning has proven to be highly effective at learning model initializations, representations, and update rules that allow fast adaptation from a few samples. The core idea behind these approaches is to use fast adaptation and generalization -- two second-order metrics -- as training signals on a meta-training dataset. However, little attention has been given to other possible second-order metrics. In this paper, we investigate a different training signal -- robustness to catastrophic interference -- and demonstrate that representations learned by directing minimizing interference are more conducive to incremental learning than those learned by just maximizing fast adaptation.

We present a method to formulate algorithm discovery as program search, and apply it to discover optimization algorithms for deep neural network training. We leverage efficient search techniques to explore an infinite and sparse program space. To bridge the large generalization gap between proxy and target tasks, we also introduce program selection and simplification strategies. Our method discovers a simple and effective optimization algorithm, Lion (Evo\textbf{Lved Sign Momentum}). It is more memory-efficient than Adam as it only keeps track of the momentum. Different from adaptive optimizers, its update has the same magnitude for each parameter calculated through the sign operation. We compare Lion with widely used optimizers, such as Adam and Adafactor, for training a variety of models on different tasks. On image classification, Lion boosts the accuracy of ViT by up to 2% on ImageNet and saves up to 5x the pre-training compute on JFT. On vision-language contrastive learning, we achieve 88.3% zero-shot and 91.1% fine-tuning accuracy on ImageNet, surpassing the previous best results by 2% and 0.1%, respectively. On diffusion models, Lion outperforms Adam by achieving a better FID score and reducing the training compute by up to 2.3x. For autoregressive, masked language modeling, and fine-tuning, Lion exhibits a similar or better performance compared to Adam. Our analysis of Lion reveals that its performance gain grows with the training batch size. It also requires a smaller learning rate than Adam due to the larger norm of the update produced by the sign function. Additionally, we examine the limitations of Lion and identify scenarios where its improvements are small or not statistically significant. The implementation of Lion is publicly available.

Matrix-based optimizers have attracted growing interest for improving LLM training efficiency, with significant progress centered on orthogonalization/whitening based methods. While yielding substantial performance gains, a fundamental question arises: can we develop new paradigms beyond orthogonalization, pushing the efficiency frontier further? We present Adaptively Rotated Optimization (ARO, a new matrix optimization framework that treats gradient rotation as a first class design principle. ARO accelerates LLM training by performing normed steepest descent in a rotated coordinate system, where the rotation is determined by a novel norm-informed policy. This perspective yields update rules that go beyond existing orthogonalization and whitening optimizers, improving sample efficiency in practice. To make comparisons reliable, we propose a rigorously controlled benchmarking protocol that reduces confounding and bias. Under this protocol, ARO consistently outperforms AdamW (by 1.3 sim1.35times) and orthogonalization methods (by 1.1sim1.15times) in LLM pretraining at up to 8B activated parameters, and up to 8times overtrain budget, without evidence of diminishing returns. Finally, we discuss how ARO can be reformulated as a symmetry-aware optimizer grounded in rotational symmetries of residual streams, motivating advanced designs that enable computationally efficient exploitation of cross-layer/cross module couplings.

Most downstream adaptation methods tune all or part of the parameters of pre-trained models (PTMs) through gradient descent, where the tuning cost increases linearly with the growth of the model size. By contrast, gradient-free methods only require the forward computation of the PTM to tune the prompt, retaining the benefits of efficient tuning and deployment. Though, past work on gradient-free tuning often introduces gradient descent to seek a good initialization of prompt and lacks versatility across tasks and PTMs. In this paper, we present BBTv2, an improved version of Black-Box Tuning, to drive PTMs for few-shot learning. We prepend continuous prompts to every layer of the PTM and propose a divide-and-conquer gradient-free algorithm to optimize the prompts at different layers alternately. Extensive experiments across various tasks and PTMs show that BBTv2 can achieve comparable performance to full model tuning and state-of-the-art parameter-efficient methods (e.g., Adapter, LoRA, BitFit, etc.) under few-shot settings while maintaining much fewer tunable parameters.

While prior research has proposed a plethora of methods that build neural classifiers robust against adversarial robustness, practitioners are still reluctant to adopt them due to their unacceptably severe clean accuracy penalties. This paper significantly alleviates this accuracy-robustness trade-off by mixing the output probabilities of a standard classifier and a robust classifier, where the standard network is optimized for clean accuracy and is not robust in general. We show that the robust base classifier's confidence difference for correct and incorrect examples is the key to this improvement. In addition to providing intuitions and empirical evidence, we theoretically certify the robustness of the mixed classifier under realistic assumptions. Furthermore, we adapt an adversarial input detector into a mixing network that adaptively adjusts the mixture of the two base models, further reducing the accuracy penalty of achieving robustness. The proposed flexible method, termed "adaptive smoothing", can work in conjunction with existing or even future methods that improve clean accuracy, robustness, or adversary detection. Our empirical evaluation considers strong attack methods, including AutoAttack and adaptive attack. On the CIFAR-100 dataset, our method achieves an 85.21% clean accuracy while maintaining a 38.72% ell_infty-AutoAttacked (epsilon = 8/255) accuracy, becoming the second most robust method on the RobustBench CIFAR-100 benchmark as of submission, while improving the clean accuracy by ten percentage points compared with all listed models. The code that implements our method is available at https://github.com/Bai-YT/AdaptiveSmoothing.

We derive a simple and model-independent formula for the change in the generalization gap due to a gradient descent update. We then compare the change in the test error for stochastic gradient descent to the change in test error from an equivalent number of gradient descent updates and show explicitly that stochastic gradient descent acts to regularize generalization error by decorrelating nearby updates. These calculations depends on the details of the model only through the mean and covariance of the gradient distribution, which may be readily measured for particular models of interest. We discuss further improvements to these calculations and comment on possible implications for stochastic optimization.

A common way to speed up training of large convolutional networks is to add computational units. Training is then performed using data-parallel synchronous Stochastic Gradient Descent (SGD) with mini-batch divided between computational units. With an increase in the number of nodes, the batch size grows. But training with large batch size often results in the lower model accuracy. We argue that the current recipe for large batch training (linear learning rate scaling with warm-up) is not general enough and training may diverge. To overcome this optimization difficulties we propose a new training algorithm based on Layer-wise Adaptive Rate Scaling (LARS). Using LARS, we scaled Alexnet up to a batch size of 8K, and Resnet-50 to a batch size of 32K without loss in accuracy.

Large Language Models (LLMs) face increasing loss spikes during scaling, undermining training stability and final performance. While gradient clipping mitigates this issue, traditional global approaches poorly handle parameter-specific gradient variations and decaying gradient norms. We propose **AdaGC**, an adaptive gradient clipping framework that automatically adjusts local thresholds per parameter through exponential moving average of gradient norms. Theoretical analysis proves AdaGC's convergence under non-convex conditions. Extensive experiments demonstrate significant improvements: On Llama-2 7B/13B, AdaGC completely eliminates loss spikes while reducing WikiText perplexity by 3.5% (+0.14pp LAMBADA accuracy) for 7B and achieving 0.65% lower training loss with 1.47% reduced validation perplexity for 13B compared to global clipping. For CLIP ViT-Base, AdaGC converges 25% faster than StableAdamW with full spike elimination. The method shows universal effectiveness across architectures (Llama-2 7B/13B) and modalities (CLIP), with successful integration into diverse optimizers like AdamW and Lion. Source code will be released on GitHub.

We propose ScaledGD(\lambda), a preconditioned gradient descent method to tackle the low-rank matrix sensing problem when the true rank is unknown, and when the matrix is possibly ill-conditioned. Using overparametrized factor representations, ScaledGD(\lambda) starts from a small random initialization, and proceeds by gradient descent with a specific form of damped preconditioning to combat bad curvatures induced by overparameterization and ill-conditioning. At the expense of light computational overhead incurred by preconditioners, ScaledGD(\lambda) is remarkably robust to ill-conditioning compared to vanilla gradient descent (GD) even with overprameterization. Specifically, we show that, under the Gaussian design, ScaledGD(\lambda) converges to the true low-rank matrix at a constant linear rate after a small number of iterations that scales only logarithmically with respect to the condition number and the problem dimension. This significantly improves over the convergence rate of vanilla GD which suffers from a polynomial dependency on the condition number. Our work provides evidence on the power of preconditioning in accelerating the convergence without hurting generalization in overparameterized learning.

We propose an adaptive step size with an energy approach for a suitable class of preconditioned gradient descent methods. We focus on settings where the preconditioning is applied to address the constraints in optimization problems, such as the Hessian-Riemannian and natural gradient descent methods. More specifically, we incorporate these preconditioned gradient descent algorithms in the recently introduced Adaptive Energy Gradient Descent (AEGD) framework. In particular, we discuss theoretical results on the unconditional energy-stability and convergence rates across three classes of objective functions. Furthermore, our numerical results demonstrate excellent performance of the proposed method on several test bed optimization problems.

Effective regularization techniques are highly desired in deep learning for alleviating overfitting and improving generalization. This work proposes a new regularization scheme, based on the understanding that the flat local minima of the empirical risk cause the model to generalize better. This scheme is referred to as adversarial model perturbation (AMP), where instead of directly minimizing the empirical risk, an alternative "AMP loss" is minimized via SGD. Specifically, the AMP loss is obtained from the empirical risk by applying the "worst" norm-bounded perturbation on each point in the parameter space. Comparing with most existing regularization schemes, AMP has strong theoretical justifications, in that minimizing the AMP loss can be shown theoretically to favour flat local minima of the empirical risk. Extensive experiments on various modern deep architectures establish AMP as a new state of the art among regularization schemes. Our code is available at https://github.com/hiyouga/AMP-Regularizer.

We propose a method for adapting neural networks to distribution shifts at test-time. In contrast to training-time robustness mechanisms that attempt to anticipate and counter the shift, we create a closed-loop system and make use of a test-time feedback signal to adapt a network on the fly. We show that this loop can be effectively implemented using a learning-based function, which realizes an amortized optimizer for the network. This leads to an adaptation method, named Rapid Network Adaptation (RNA), that is notably more flexible and orders of magnitude faster than the baselines. Through a broad set of experiments using various adaptation signals and target tasks, we study the efficiency and flexibility of this method. We perform the evaluations using various datasets (Taskonomy, Replica, ScanNet, Hypersim, COCO, ImageNet), tasks (depth, optical flow, semantic segmentation, classification), and distribution shifts (Cross-datasets, 2D and 3D Common Corruptions) with promising results. We end with a discussion on general formulations for handling distribution shifts and our observations from comparing with similar approaches from other domains.

Adaptive gradient methods, such as Adam and LAMB, have demonstrated excellent performance in the training of large language models. Nevertheless, the need for adaptivity requires maintaining second-moment estimates of the per-parameter gradients, which entails a high cost of extra memory overheads. To solve this problem, several memory-efficient optimizers (e.g., Adafactor) have been proposed to obtain a drastic reduction in auxiliary memory usage, but with a performance penalty. In this paper, we first study a confidence-guided strategy to reduce the instability of existing memory efficient optimizers. Based on this strategy, we propose CAME to simultaneously achieve two goals: fast convergence as in traditional adaptive methods, and low memory usage as in memory-efficient methods. Extensive experiments demonstrate the training stability and superior performance of CAME across various NLP tasks such as BERT and GPT-2 training. Notably, for BERT pre-training on the large batch size of 32,768, our proposed optimizer attains faster convergence and higher accuracy compared with the Adam optimizer. The implementation of CAME is publicly available.

Low-rank adaptation, also known as LoRA, has emerged as a prominent method for parameter-efficient fine-tuning of foundation models. Despite its computational efficiency, LoRA still yields inferior performance compared to full fine-tuning. In this paper, we first uncover a fundamental connection between the optimization processes of LoRA and full fine-tuning: using LoRA for optimization is mathematically equivalent to full fine-tuning using a low-rank gradient for parameter updates. And this low-rank gradient can be expressed in terms of the gradients of the two low-rank matrices in LoRA. Leveraging this insight, we introduce LoRA-Pro, a method that enhances LoRA's performance by strategically adjusting the gradients of these low-rank matrices. This adjustment allows the low-rank gradient to more accurately approximate the full fine-tuning gradient, thereby narrowing the performance gap between LoRA and full fine-tuning. Furthermore, we theoretically derive the optimal solutions for adjusting the gradients of the low-rank matrices, applying them during fine-tuning in LoRA-Pro. We conduct extensive experiments across natural language understanding, dialogue generation, mathematical reasoning, code generation, and image classification tasks, demonstrating that LoRA-Pro substantially improves LoRA's performance, effectively narrowing the gap with full fine-tuning. Code is publicly available at https://github.com/mrflogs/LoRA-Pro.

Geometry-aware optimization algorithms, such as Muon, have achieved remarkable success in training deep neural networks (DNNs). These methods leverage the underlying geometry of DNNs by selecting appropriate norms for different layers and updating parameters via norm-constrained linear minimization oracles (LMOs). However, even within a group of layers associated with the same norm, the local curvature can be heterogeneous across layers and vary dynamically over the course of training. For example, recent work shows that sharpness varies substantially across transformer layers and throughout training, yet standard geometry-aware optimizers impose fixed learning rates to layers within the same group, which may be inefficient for DNN training. In this paper, we introduce a noise-adaptive layerwise learning rate scheme on top of geometry-aware optimization algorithms and substantially accelerate DNN training compared to methods that use fixed learning rates within each group. Our method estimates gradient variance in the dual norm induced by the chosen LMO on the fly, and uses it to assign time-varying noise-adaptive layerwise learning rates within each group. We provide a theoretical analysis showing that our algorithm achieves a sharp convergence rate. Empirical results on transformer architectures such as LLaMA and GPT demonstrate that our approach achieves faster convergence than state-of-the-art optimizers.

We introduce Gradient Descent with Adaptive Momentum Scaling (Grams), a novel optimization algorithm that decouples the direction and magnitude of parameter updates in deep learning. Unlike traditional optimizers that directly integrate momentum into updates, Grams separates the update direction, derived from current gradients, from momentum, which is used solely for adaptive magnitude scaling. This approach enables Grams to achieve improved loss descent compared to state-of-the-art cautious and momentum-based optimizers. We establish a global convergence guarantee for Grams and validate its effectiveness through extensive empirical evaluations. The results demonstrate Grams' superior performance, including faster convergence and better generalization, compared to widely-used optimizers such as Adam, Lion, and their cautious variants. Our results highlight Grams' potential as a transformative approach for efficient optimization in large-scale machine learning.

Gradient inversion attacks can leak data privacy when clients share weight updates with the server in federated learning (FL). Existing studies mainly use L2 or cosine distance as the loss function for gradient matching in the attack. Our empirical investigation shows that the vulnerability ranking varies with the loss function used. Gradient norm, which is commonly used as a vulnerability proxy for gradient inversion attack, cannot explain this as it remains constant regardless of the loss function for gradient matching. In this paper, we propose a loss-aware vulnerability proxy (LAVP) for the first time. LAVP refers to either the maximum or minimum eigenvalue of the Hessian with respect to gradient matching loss at ground truth. This suggestion is based on our theoretical findings regarding the local optimization of the gradient inversion in proximity to the ground truth, which corresponds to the worst case attack scenario. We demonstrate the effectiveness of LAVP on various architectures and datasets, showing its consistent superiority over the gradient norm in capturing sample vulnerabilities. The performance of each proxy is measured in terms of Spearman's rank correlation with respect to several similarity scores. This work will contribute to enhancing FL security against any potential loss functions beyond L2 or cosine distance in the future.

The rapid development of machine learning and deep learning has introduced increasingly complex optimization challenges that must be addressed. Indeed, training modern, advanced models has become difficult to implement without leveraging multiple computing nodes in a distributed environment. Distributed optimization is also fundamental to emerging fields such as federated learning. Specifically, there is a need to organize the training process to minimize the time lost due to communication. A widely used and extensively researched technique to mitigate the communication bottleneck involves performing local training before communication. This approach is the focus of our paper. Concurrently, adaptive methods that incorporate scaling, notably led by Adam, have gained significant popularity in recent years. Therefore, this paper aims to merge the local training technique with the adaptive approach to develop efficient distributed learning methods. We consider the classical Local SGD method and enhance it with a scaling feature. A crucial aspect is that the scaling is described generically, allowing us to analyze various approaches, including Adam, RMSProp, and OASIS, in a unified manner. In addition to theoretical analysis, we validate the performance of our methods in practice by training a neural network.

We present EGN, a stochastic second-order optimization algorithm that combines the generalized Gauss-Newton (GN) Hessian approximation with low-rank linear algebra to compute the descent direction. Leveraging the Duncan-Guttman matrix identity, the parameter update is obtained by factorizing a matrix which has the size of the mini-batch. This is particularly advantageous for large-scale machine learning problems where the dimension of the neural network parameter vector is several orders of magnitude larger than the batch size. Additionally, we show how improvements such as line search, adaptive regularization, and momentum can be seamlessly added to EGN to further accelerate the algorithm. Moreover, under mild assumptions, we prove that our algorithm converges to an epsilon-stationary point at a linear rate. Finally, our numerical experiments demonstrate that EGN consistently exceeds, or at most matches the generalization performance of well-tuned SGD, Adam, and SGN optimizers across various supervised and reinforcement learning tasks.

We propose NovoGrad, an adaptive stochastic gradient descent method with layer-wise gradient normalization and decoupled weight decay. In our experiments on neural networks for image classification, speech recognition, machine translation, and language modeling, it performs on par or better than well tuned SGD with momentum and Adam or AdamW. Additionally, NovoGrad (1) is robust to the choice of learning rate and weight initialization, (2) works well in a large batch setting, and (3) has two times smaller memory footprint than Adam.

Foundation models (FMs) are pre-trained on large-scale datasets and then fine-tuned on a downstream task for a specific application. The most successful and most commonly used fine-tuning method is to update the pre-trained weights via a low-rank adaptation (LoRA). LoRA introduces new weight matrices that are usually initialized at random with a uniform rank distribution across model weights. Recent works focus on weight-driven initialization or learning of adaptive ranks during training. Both approaches have only been investigated in isolation, resulting in slow convergence or a uniform rank distribution, in turn leading to sub-optimal performance. We propose to enhance LoRA by initializing the new weights in a data-driven manner by computing singular value decomposition on minibatches of activation vectors. Then, we initialize the LoRA matrices with the obtained right-singular vectors and re-distribute ranks among all weight matrices to explain the maximal amount of variance and continue the standard LoRA fine-tuning procedure. This results in our new method Explained Variance Adaptation (EVA). We apply EVA to a variety of fine-tuning tasks ranging from language generation and understanding to image classification and reinforcement learning. EVA exhibits faster convergence than competitors and attains the highest average score across a multitude of tasks per domain.

Learning to Rank (LTR) algorithms are usually evaluated using Information Retrieval metrics like Normalised Discounted Cumulative Gain (NDCG) or Mean Average Precision. As these metrics rely on sorting predicted items' scores (and thus, on items' ranks), their derivatives are either undefined or zero everywhere. This makes them unsuitable for gradient-based optimisation, which is the usual method of learning appropriate scoring functions. Commonly used LTR loss functions are only loosely related to the evaluation metrics, causing a mismatch between the optimisation objective and the evaluation criterion. In this paper, we address this mismatch by proposing NeuralNDCG, a novel differentiable approximation to NDCG. Since NDCG relies on the non-differentiable sorting operator, we obtain NeuralNDCG by relaxing that operator using NeuralSort, a differentiable approximation of sorting. As a result, we obtain a new ranking loss function which is an arbitrarily accurate approximation to the evaluation metric, thus closing the gap between the training and the evaluation of LTR models. We introduce two variants of the proposed loss function. Finally, the empirical evaluation shows that our proposed method outperforms previous work aimed at direct optimisation of NDCG and is competitive with the state-of-the-art methods.

In this paper, we propose a learning rule based on a back-propagation (BP) algorithm that can be applied to a hardware-based deep neural network (HW-DNN) using electronic devices that exhibit discrete and limited conductance characteristics. This adaptive learning rule, which enables forward, backward propagation, as well as weight updates in hardware, is helpful during the implementation of power-efficient and high-speed deep neural networks. In simulations using a three-layer perceptron network, we evaluate the learning performance according to various conductance responses of electronic synapse devices and weight-updating methods. It is shown that the learning accuracy is comparable to that obtained when using a software-based BP algorithm when the electronic synapse device has a linear conductance response with a high dynamic range. Furthermore, the proposed unidirectional weight-updating method is suitable for electronic synapse devices which have nonlinear and finite conductance responses. Because this weight-updating method can compensate the demerit of asymmetric weight updates, we can obtain better accuracy compared to other methods. This adaptive learning rule, which can be applied to full hardware implementation, can also compensate the degradation of learning accuracy due to the probable device-to-device variation in an actual electronic synapse device.

In this empirical article, we introduce INNAprop, an optimization algorithm that combines the INNA method with the RMSprop adaptive gradient scaling. It leverages second-order information and rescaling while keeping the memory requirements of standard DL methods as AdamW or SGD with momentum.After having recalled our geometrical motivations, we provide quite extensive experiments. On image classification (CIFAR-10, ImageNet) and language modeling (GPT-2), INNAprop consistently matches or outperforms AdamW both in training speed and accuracy, with minimal hyperparameter tuning in large-scale settings. Our code is publicly available at https://github.com/innaprop/innaprop.

Empirical scaling laws prescribe how to allocate parameters, data, and compute, while maximal-update parameterization (muP) enables learning-rate transfer across widths by equalizing early-time update magnitudes. However, in modern scale-invariant architectures, training quickly enters an optimizer-governed steady state where normalization layers create backward scale sensitivity and the effective learning rate becomes width dependent, degrading muP transfer. We address this by introducing a weight-decay scaling rule for AdamW that preserves sublayer gain across widths. Empirically, the singular-value spectrum of each matrix parameter scales in norm as eta/lambda with an approximately invariant shape; under width scaling d, we observe that the top singular value scales approximately as eta/lambdacdot d^{0.75}. Combining this observation with the muP learning-rate rule eta_2propto d^{-1} for matrix-like parameters implies an empirical weight-decay scaling rule lambda_2propto d that approximately keeps sublayer gains width invariant. Together with vector-like parameters trained at eta_1=Theta_d(1) and lambda_1=0, this yields zero-shot transfer of both learning rate and weight decay from proxy to target widths, removing per-width sweeps. We validate the rule on LLaMA-style Transformers and in a minimal synthetic setting, and we provide a simple diagnostic, matching top singular values, to check sublayer-gain invariance. Our results extend muP beyond the near-init regime by explicitly controlling steady-state scales set by the optimizer, offering a practical recipe for width-robust hyperparameter transfer under AdamW.

Exemplar-free class-incremental learning (EFCIL) presents a significant challenge as the old class samples are absent for new task learning. Due to the severe imbalance between old and new class samples, the learned classifiers can be easily biased toward the new ones. Moreover, continually updating the feature extractor under EFCIL can compromise the discriminative power of old class features, e.g., leading to less compact and more overlapping distributions across classes. Existing methods mainly focus on handling biased classifier learning. In this work, both cases are considered using the proposed method. Specifically, we first introduce a Distribution-Based Global Classifier (DBGC) to avoid bias factors in existing methods, such as data imbalance and sampling. More importantly, the compromised distributions of old classes are simulated via a simple operation, variance enlarging (VE). Incorporating VE based on DBGC results in a novel classification loss for EFCIL. This loss is proven equivalent to an Adaptive Margin Softmax Cross Entropy (AMarX). The proposed method is thus called Adaptive Margin Global Classifier (AMGC). AMGC is simple yet effective. Extensive experiments show that AMGC achieves superior image classification results on its own under a challenging EFCIL setting. Detailed analysis is also provided for further demonstration.

We investigate the convergence rates and data sample sizes required for training a machine learning model using a stochastic gradient descent (SGD) algorithm, where data points are sampled based on either their loss value or uncertainty value. These training methods are particularly relevant for active learning and data subset selection problems. For SGD with a constant step size update, we present convergence results for linear classifiers and linearly separable datasets using squared hinge loss and similar training loss functions. Additionally, we extend our analysis to more general classifiers and datasets, considering a wide range of loss-based sampling strategies and smooth convex training loss functions. We propose a novel algorithm called Adaptive-Weight Sampling (AWS) that utilizes SGD with an adaptive step size that achieves stochastic Polyak's step size in expectation. We establish convergence rate results for AWS for smooth convex training loss functions. Our numerical experiments demonstrate the efficiency of AWS on various datasets by using either exact or estimated loss values.

Estimating the test performance of a model, possibly under distribution shift, without having access to the ground-truth labels is a challenging, yet very important problem for the safe deployment of machine learning algorithms in the wild. Existing works mostly rely on information from either the outputs or the extracted features of neural networks to estimate a score that correlates with the ground-truth test accuracy. In this paper, we investigate -- both empirically and theoretically -- how the information provided by the gradients can be predictive of the ground-truth test accuracy even under distribution shifts. More specifically, we use the norm of classification-layer gradients, backpropagated from the cross-entropy loss after only one gradient step over test data. Our intuition is that these gradients should be of higher magnitude when the model generalizes poorly. We provide the theoretical insights behind our approach and the key ingredients that ensure its empirical success. Extensive experiments conducted with various architectures on diverse distribution shifts demonstrate that our method significantly outperforms current state-of-the-art approaches. The code is available at https://github.com/Renchunzi-Xie/GdScore

We consider solving equality-constrained nonlinear, nonconvex optimization problems. This class of problems appears widely in a variety of applications in machine learning and engineering, ranging from constrained deep neural networks, to optimal control, to PDE-constrained optimization. We develop an adaptive inexact Newton method for this problem class. In each iteration, we solve the Lagrangian Newton system inexactly via a randomized iterative sketching solver, and select a suitable stepsize by performing line search on an exact augmented Lagrangian merit function. The randomized solvers have advantages over deterministic linear system solvers by significantly reducing per-iteration flops complexity and storage cost, when equipped with suitable sketching matrices. Our method adaptively controls the accuracy of the randomized solver and the penalty parameters of the exact augmented Lagrangian, to ensure that the inexact Newton direction is a descent direction of the exact augmented Lagrangian. This allows us to establish a global almost sure convergence. We also show that a unit stepsize is admissible locally, so that our method exhibits a local linear convergence. Furthermore, we prove that the linear convergence can be strengthened to superlinear convergence if we gradually sharpen the adaptive accuracy condition on the randomized solver. We demonstrate the superior performance of our method on benchmark nonlinear problems in CUTEst test set, constrained logistic regression with data from LIBSVM, and a PDE-constrained problem.

Deep learning-based speech enhancement methods have significantly improved speech quality and intelligibility. Convolutional neural networks (CNNs) have been proven to be essential components of many high-performance models. In this paper, we introduce adaptive convolution, an efficient and versatile convolutional module that enhances the model's capability to adaptively represent speech signals. Adaptive convolution performs frame-wise causal dynamic convolution, generating time-varying kernels for each frame by assembling multiple parallel candidate kernels. A lightweight attention mechanism is proposed for adaptive convolution, leveraging both current and historical information to assign adaptive weights to each candidate kernel. This enables the convolution operation to adapt to frame-level speech spectral features, leading to more efficient extraction and reconstruction. We integrate adaptive convolution into various CNN-based models, highlighting its generalizability. Experimental results demonstrate that adaptive convolution significantly improves the performance with negligible increases in computational complexity, especially for lightweight models. Moreover, we present an intuitive analysis revealing a strong correlation between kernel selection and signal characteristics. Furthermore, we propose the adaptive convolutional recurrent network (AdaptCRN), an ultra-lightweight model that incorporates adaptive convolution and an efficient encoder-decoder design, achieving superior performance compared to models with similar or even higher computational costs.

Low-Rank Adaptation (LoRA) is the bread and butter of Large Language Model (LLM) finetuning. LoRA learns an additive low-rank perturbation, AB, of a pretrained matrix parameter W to align the model to a new task or dataset with W+AB. We identify three core limitations to LoRA for finetuning--a setting that employs limited amount of data and training steps. First, LoRA employs Dropout to prevent overfitting. We prove that Dropout is only suitable for long training episodes but fails to converge to a reliable regularizer for short training episodes. Second, LoRA's initialization of B at 0 creates a slow training dynamic between A and B. That dynamic is also exacerbated by Dropout that further slows the escape from 0 for B which is particularly harmful for short training episodes. Third, the scaling factor multiplying each LoRA additive perturbation creates ``short-sighted'' interactions between the LoRA modules of different layers. Motivated by principled analysis of those limitations, we find an elegant solution: a Dropout-free, scaling-free, LoRA with Adaptive Learning rate--coined ALLoRA. By scaling the per sample and per parameter gradients with a coefficient inversely proportional to parameters' ell_2 norm, ALLoRA alleviates those three limitations. As a by-product, ALLoRA removes two hyper-parameters from LoRA: the scaling factor and the dropout rate. Empirical results show that ALLoRA admits better accuracy than LoRA on various settings, including against recent LoRA variants such as Weight-Decomposed Low-Rank Adaptation (DoRA). Ablation studies show our solution is the optimal in a family of weight-dependent / output-dependent approaches on various LLMs including the latest Llama3.

Applying weight decay (WD) to matrix layers is standard practice in large-language-model pretraining. Prior work suggests that stochastic gradient noise induces a Brownian-like expansion of the weight matrices W, whose growth is counteracted by WD, leading to a WD-noise equilibrium with a certain weight norm ||W||. In this work, we view the equilibrium norm as a harmful artifact of the training procedure, and address it by introducing learnable multipliers to learn the optimal scale. First, we attach a learnable scalar multiplier to W and confirm that the WD-noise equilibrium norm is suboptimal: the learned scale adapts to data and improves performance. We then argue that individual row and column norms are similarly constrained, and free their scale by introducing learnable per-row and per-column multipliers. Our method can be viewed as a learnable, more expressive generalization of muP multipliers. It outperforms a well-tuned muP baseline, reduces the computational overhead of multiplier tuning, and surfaces practical questions such as forward-pass symmetries and the width-scaling of the learned multipliers. Finally, we validate learnable multipliers with both Adam and Muon optimizers, where it shows improvement in downstream evaluations matching the improvement of the switching from Adam to Muon.

Normalization techniques are a boon for modern deep learning. They let weights converge more quickly with often better generalization performances. It has been argued that the normalization-induced scale invariance among the weights provides an advantageous ground for gradient descent (GD) optimizers: the effective step sizes are automatically reduced over time, stabilizing the overall training procedure. It is often overlooked, however, that the additional introduction of momentum in GD optimizers results in a far more rapid reduction in effective step sizes for scale-invariant weights, a phenomenon that has not yet been studied and may have caused unwanted side effects in the current practice. This is a crucial issue because arguably the vast majority of modern deep neural networks consist of (1) momentum-based GD (e.g. SGD or Adam) and (2) scale-invariant parameters. In this paper, we verify that the widely-adopted combination of the two ingredients lead to the premature decay of effective step sizes and sub-optimal model performances. We propose a simple and effective remedy, SGDP and AdamP: get rid of the radial component, or the norm-increasing direction, at each optimizer step. Because of the scale invariance, this modification only alters the effective step sizes without changing the effective update directions, thus enjoying the original convergence properties of GD optimizers. Given the ubiquity of momentum GD and scale invariance in machine learning, we have evaluated our methods against the baselines on 13 benchmarks. They range from vision tasks like classification (e.g. ImageNet), retrieval (e.g. CUB and SOP), and detection (e.g. COCO) to language modelling (e.g. WikiText) and audio classification (e.g. DCASE) tasks. We verify that our solution brings about uniform gains in those benchmarks. Source code is available at https://github.com/clovaai/AdamP.

While stochastic gradient descent (SGD) is still the most popular optimization algorithm in deep learning, adaptive algorithms such as Adam have established empirical advantages over SGD in some deep learning applications such as training transformers. However, it remains a question that why Adam converges significantly faster than SGD in these scenarios. In this paper, we propose one explanation of why Adam converges faster than SGD using a new concept directional sharpness. We argue that the performance of optimization algorithms is closely related to the directional sharpness of the update steps, and show SGD has much worse directional sharpness compared to adaptive algorithms. We further observe that only a small fraction of the coordinates causes the bad sharpness and slow convergence of SGD, and propose to use coordinate-wise clipping as a solution to SGD and other optimization algorithms. We demonstrate the effect of coordinate-wise clipping on sharpness reduction and speeding up the convergence of optimization algorithms under various settings. We show that coordinate-wise clipping improves the local loss reduction when only a small fraction of the coordinates has bad sharpness. We conclude that the sharpness reduction effect of adaptive coordinate-wise scaling is the reason for Adam's success in practice and suggest the use of coordinate-wise clipping as a universal technique to speed up deep learning optimization.

In this article, we introduce a novel normalization technique for neural network weight matrices, which we term weight conditioning. This approach aims to narrow the gap between the smallest and largest singular values of the weight matrices, resulting in better-conditioned matrices. The inspiration for this technique partially derives from numerical linear algebra, where well-conditioned matrices are known to facilitate stronger convergence results for iterative solvers. We provide a theoretical foundation demonstrating that our normalization technique smoothens the loss landscape, thereby enhancing convergence of stochastic gradient descent algorithms. Empirically, we validate our normalization across various neural network architectures, including Convolutional Neural Networks (CNNs), Vision Transformers (ViT), Neural Radiance Fields (NeRF), and 3D shape modeling. Our findings indicate that our normalization method is not only competitive but also outperforms existing weight normalization techniques from the literature.

Learning rate schedules used in practice bear little resemblance to those recommended by theory. We close much of this theory/practice gap, and as a consequence are able to derive new problem-adaptive learning rate schedules. Our key technical contribution is a refined analysis of learning rate schedules for a wide class of optimization algorithms (including SGD). In contrast to most prior works that study the convergence of the average iterate, we study the last iterate, which is what most people use in practice. When considering only worst-case analysis, our theory predicts that the best choice is the linear decay schedule: a popular choice in practice that sets the stepsize proportionally to 1 - t/T, where t is the current iteration and T is the total number of steps. To go beyond this worst-case analysis, we use the observed gradient norms to derive schedules refined for any particular task. These refined schedules exhibit learning rate warm-up and rapid learning rate annealing near the end of training. Ours is the first systematic approach to automatically yield both of these properties. We perform the most comprehensive evaluation of learning rate schedules to date, evaluating across 10 diverse deep learning problems, a series of LLMs, and a suite of logistic regression problems. We validate that overall, the linear-decay schedule matches or outperforms all commonly used default schedules including cosine annealing, and that our schedule refinement method gives further improvements.

We introduce Gradient Agreement Filtering (GAF) to improve on gradient averaging in distributed deep learning optimization. Traditional distributed data-parallel stochastic gradient descent involves averaging gradients of microbatches to calculate a macrobatch gradient that is then used to update model parameters. We find that gradients across microbatches are often orthogonal or negatively correlated, especially in late stages of training, which leads to memorization of the training set, reducing generalization. In this paper, we introduce a simple, computationally effective way to reduce gradient variance by computing the cosine distance between micro-gradients during training and filtering out conflicting updates prior to averaging. We improve validation accuracy with significantly smaller microbatch sizes. We also show this reduces memorizing noisy labels. We demonstrate the effectiveness of this technique on standard image classification benchmarks including CIFAR-100 and CIFAR-100N-Fine. We show this technique consistently outperforms validation accuracy, in some cases by up to 18.2\% compared to traditional training approaches while reducing the computation required nearly an order of magnitude because we can now rely on smaller microbatch sizes without destabilizing training.

D-Adaptation is an approach to automatically setting the learning rate which asymptotically achieves the optimal rate of convergence for minimizing convex Lipschitz functions, with no back-tracking or line searches, and no additional function value or gradient evaluations per step. Our approach is the first hyper-parameter free method for this class without additional multiplicative log factors in the convergence rate. We present extensive experiments for SGD and Adam variants of our method, where the method automatically matches hand-tuned learning rates across more than a dozen diverse machine learning problems, including large-scale vision and language problems. An open-source implementation is available.

We present Gabliteration, a novel neural weight modification technique that advances beyond traditional abliteration methods by implementing adaptive multi-directional projections with regularized layer selection. Our approach addresses the fundamental limitation of existing methods that compromise model quality while attempting to modify specific behavioral patterns. Through dynamic layer optimization, regularized projection matrices, and adaptive scaling mechanisms, we achieve theoretically superior weight modification while minimizing quality degradation in unrelated domains. We validate our method through the gabliterated-v1 model series (0.6B to 4B parameters) available on Hugging Face, demonstrating practical applicability across multiple model scales.

Training large deep neural networks on massive datasets is computationally very challenging. There has been recent surge in interest in using large batch stochastic optimization methods to tackle this issue. The most prominent algorithm in this line of research is LARS, which by employing layerwise adaptive learning rates trains ResNet on ImageNet in a few minutes. However, LARS performs poorly for attention models like BERT, indicating that its performance gains are not consistent across tasks. In this paper, we first study a principled layerwise adaptation strategy to accelerate training of deep neural networks using large mini-batches. Using this strategy, we develop a new layerwise adaptive large batch optimization technique called LAMB; we then provide convergence analysis of LAMB as well as LARS, showing convergence to a stationary point in general nonconvex settings. Our empirical results demonstrate the superior performance of LAMB across various tasks such as BERT and ResNet-50 training with very little hyperparameter tuning. In particular, for BERT training, our optimizer enables use of very large batch sizes of 32868 without any degradation of performance. By increasing the batch size to the memory limit of a TPUv3 Pod, BERT training time can be reduced from 3 days to just 76 minutes (Table 1). The LAMB implementation is available at https://github.com/tensorflow/addons/blob/master/tensorflow_addons/optimizers/lamb.py

In class incremental learning, neural networks typically suffer from catastrophic forgetting. We show that an MLP featuring a sparse activation function and an adaptive learning rate optimizer can compete with established regularization techniques in the Split-MNIST task. We highlight the effectiveness of the Adaptive SwisH (ASH) activation function in this context and introduce a novel variant, Hard Adaptive SwisH (Hard ASH) to further enhance the learning retention.

Sharpness of minima is a promising quantity that can correlate with generalization in deep networks and, when optimized during training, can improve generalization. However, standard sharpness is not invariant under reparametrizations of neural networks, and, to fix this, reparametrization-invariant sharpness definitions have been proposed, most prominently adaptive sharpness (Kwon et al., 2021). But does it really capture generalization in modern practical settings? We comprehensively explore this question in a detailed study of various definitions of adaptive sharpness in settings ranging from training from scratch on ImageNet and CIFAR-10 to fine-tuning CLIP on ImageNet and BERT on MNLI. We focus mostly on transformers for which little is known in terms of sharpness despite their widespread usage. Overall, we observe that sharpness does not correlate well with generalization but rather with some training parameters like the learning rate that can be positively or negatively correlated with generalization depending on the setup. Interestingly, in multiple cases, we observe a consistent negative correlation of sharpness with out-of-distribution error implying that sharper minima can generalize better. Finally, we illustrate on a simple model that the right sharpness measure is highly data-dependent, and that we do not understand well this aspect for realistic data distributions. The code of our experiments is available at https://github.com/tml-epfl/sharpness-vs-generalization.

A major challenge in training large-scale machine learning models is configuring the training process to maximize model performance, i.e., finding the best training setup from a vast design space. In this work, we unlock a gradient-based approach to this problem. We first introduce an algorithm for efficiently calculating metagradients -- gradients through model training -- at scale. We then introduce a "smooth model training" framework that enables effective optimization using metagradients. With metagradient descent (MGD), we greatly improve on existing dataset selection methods, outperform accuracy-degrading data poisoning attacks by an order of magnitude, and automatically find competitive learning rate schedules.

When an agent encounters a continual stream of new tasks in the lifelong learning setting, it leverages the knowledge it gained from the earlier tasks to help learn the new tasks better. In such a scenario, identifying an efficient knowledge representation becomes a challenging problem. Most research works propose to either store a subset of examples from the past tasks in a replay buffer, dedicate a separate set of parameters to each task or penalize excessive updates over parameters by introducing a regularization term. While existing methods employ the general task-agnostic stochastic gradient descent update rule, we propose a task-aware optimizer that adapts the learning rate based on the relatedness among tasks. We utilize the directions taken by the parameters during the updates by accumulating the gradients specific to each task. These task-based accumulated gradients act as a knowledge base that is maintained and updated throughout the stream. We empirically show that our proposed adaptive learning rate not only accounts for catastrophic forgetting but also allows positive backward transfer. We also show that our method performs better than several state-of-the-art methods in lifelong learning on complex datasets with a large number of tasks.

Low-Rank Adaptation (LoRA) offers a parameter-efficient paradigm for tuning large models. While recent spectral initialization methods improve convergence and performance over the naive "Noise & Zeros" scheme, their extra computational and storage overhead undermines efficiency. In this paper, we establish update magnitude as the fundamental driver of LoRA performance and propose LoRAM, a magnitude-driven "Basis & Basis" initialization scheme that matches spectral methods without their inefficiencies. Our key contributions are threefold: (i) Magnitude of weight updates determines convergence. We prove low-rank structures intrinsically bound update magnitudes, unifying hyperparameter tuning in learning rate, scaling factor, and initialization as mechanisms to optimize magnitude regulation. (ii) Spectral initialization succeeds via magnitude amplification. We demystify that the presumed knowledge-driven benefit of the spectral component essentially arises from the boost in the weight update magnitude. (iii) A novel and compact initialization strategy, LoRAM, scales deterministic orthogonal bases using pretrained weight magnitudes to simulate spectral gains. Extensive experiments show that LoRAM serves as a strong baseline, retaining the full efficiency of LoRA while matching or outperforming spectral initialization across benchmarks.

Adam is one of the most popular optimization algorithms in deep learning. However, it is known that Adam does not converge in theory unless choosing a hyperparameter, i.e., beta_2, in a problem-dependent manner. There have been many attempts to fix the non-convergence (e.g., AMSGrad), but they require an impractical assumption that the gradient noise is uniformly bounded. In this paper, we propose a new adaptive gradient method named ADOPT, which achieves the optimal convergence rate of O ( 1 / T ) with any choice of beta_2 without depending on the bounded noise assumption. ADOPT addresses the non-convergence issue of Adam by removing the current gradient from the second moment estimate and changing the order of the momentum update and the normalization by the second moment estimate. We also conduct intensive numerical experiments, and verify that our ADOPT achieves superior results compared to Adam and its variants across a wide range of tasks, including image classification, generative modeling, natural language processing, and deep reinforcement learning. The implementation is available at https://github.com/iShohei220/adopt.

How do we transfer the relevant knowledge from ever larger foundation models into small, task-specific downstream models that can run at much lower costs? Standard transfer learning using pre-trained weights as the initialization transfers limited information and commits us to often massive pre-trained architectures. This procedure also precludes combining multiple pre-trained models that learn complementary information. To address these shortcomings, we introduce Adaptive Feature Transfer (AFT). Instead of transferring weights, AFT operates purely on features, thereby decoupling the choice of the pre-trained model from the smaller downstream model. Rather than indiscriminately compressing all pre-trained features, AFT adaptively transfers pre-trained features that are most useful for performing the downstream task, using a simple regularization that adds minimal overhead. Across multiple vision, language, and multi-modal datasets, AFT achieves significantly better downstream performance compared to alternatives with a similar computational cost. Furthermore, AFT reliably translates improvement in pre-trained models into improvement in downstream performance, even if the downstream model is over 50times smaller, and can effectively transfer complementary information learned by multiple pre-trained models.

Sensory systems across all modalities and species exhibit adaptation to continuously changing input statistics. Individual neurons have been shown to modulate their response gains so as to maximize information transmission in different stimulus contexts. Experimental measurements have revealed additional, nuanced sensory adaptation effects including changes in response maxima and minima, tuning curve repulsion from the adapter stimulus, and stimulus-driven response decorrelation. Existing explanations of these phenomena rely on changes in inter-neuronal synaptic efficacy, which, while more flexible, are unlikely to operate as rapidly or reversibly as single neuron gain modulations. Using published V1 population adaptation data, we show that propagation of single neuron gain changes in a recurrent network is sufficient to capture the entire set of observed adaptation effects. We propose a novel adaptive efficient coding objective with which single neuron gains are modulated, maximizing the fidelity of the stimulus representation while minimizing overall activity in the network. From this objective, we analytically derive a set of gains that optimize the trade-off between preserving information about the stimulus and conserving metabolic resources. Our model generalizes well-established concepts of single neuron adaptive gain control to recurrent populations, and parsimoniously explains experimental adaptation data.

Continual Instruction Tuning (CIT) is adopted to continually instruct Large Models to follow human intent data by data. It is observed that existing gradient update would heavily destroy the performance on previous datasets during CIT process. Instead, Exponential Moving Average (EMA), owns the ability to trace previous parameters, which can aid in decreasing forgetting. Nonetheless, its stable balance weight fails to deal with the ever-changing datasets, leading to the out-of-balance between plasticity and stability. In this paper, we propose a general continual instruction tuning framework to address the challenge. Starting from the trade-off prerequisite and EMA update, we propose the plasticity and stability ideal condition. Based on Taylor expansion in the loss function, we find the optimal balance weight can be automatically determined by the gradients and learned parameters. Therefore, we propose a stable-plasticity balanced coefficient to avoid knowledge interference. Based on the semantic similarity of the instructions, we can determine whether to retrain or expand the training parameters and allocate the most suitable parameters for the testing instances. Extensive experiments across multiple continual instruction tuning benchmarks demonstrate that our approach not only enhances anti-forgetting capabilities but also significantly improves overall continual tuning performance. Our code is available at https://github.com/JingyangQiao/CoIN.

Over-parameterized deep networks trained using gradient-based optimizers are a popular choice for solving classification and ranking problems. Without appropriately tuned ell_2 regularization or weight decay, such networks have the tendency to make output scores (logits) and network weights large, causing training loss to become too small and the network to lose its adaptivity (ability to move around) in the parameter space. Although regularization is typically understood from an overfitting perspective, we highlight its role in making the network more adaptive and enabling it to escape more easily from weights that generalize poorly. To provide such a capability, we propose a method called Logit Attenuating Weight Normalization (LAWN), that can be stacked onto any gradient-based optimizer. LAWN controls the logits by constraining the weight norms of layers in the final homogeneous sub-network. Empirically, we show that the resulting LAWN variant of the optimizer makes a deep network more adaptive to finding minimas with superior generalization performance on large-scale image classification and recommender systems. While LAWN is particularly impressive in improving Adam, it greatly improves all optimizers when used with large batch sizes

SoftMax is a ubiquitous ingredient of modern machine learning algorithms. It maps an input vector onto a probability simplex and reweights the input by concentrating the probability mass at large entries. Yet, as a smooth approximation to the Argmax function, a significant amount of probability mass is distributed to other, residual entries, leading to poor interpretability and noise. Although sparsity can be achieved by a family of SoftMax variants, they often require an alternative loss function and do not preserve multi-modality. We show that this trade-off between multi-modality and sparsity limits the expressivity of SoftMax as well as its variants. We provide a solution to this tension between objectives by proposing a piece-wise differentiable function, termed MultiMax, which adaptively modulates the output distribution according to input entry range. Through comprehensive analysis and evaluation, we show that MultiMax successfully produces a distribution that supresses irrelevant entries while preserving multimodality, with benefits in image classification, language modeling and machine translation. The code is available at https://github.com/ZhouYuxuanYX/MultiMax.

Real-image super-resolution (Real-ISR) seeks to recover HR images from LR inputs with mixed, unknown degradations. While diffusion models surpass GANs in perceptual quality, they under-reconstruct high-frequency (HF) details due to a low-frequency (LF) bias and a depth-wise "low-first, high-later" hierarchy. We introduce FRAMER, a plug-and-play training scheme that exploits diffusion priors without changing the backbone or inference. At each denoising step, the final-layer feature map teaches all intermediate layers. Teacher and student feature maps are decomposed into LF/HF bands via FFT masks to align supervision with the model's internal frequency hierarchy. For LF, an Intra Contrastive Loss (IntraCL) stabilizes globally shared structure. For HF, an Inter Contrastive Loss (InterCL) sharpens instance-specific details using random-layer and in-batch negatives. Two adaptive modulators, Frequency-based Adaptive Weight (FAW) and Frequency-based Alignment Modulation (FAM), reweight per-layer LF/HF signals and gate distillation by current similarity. Across U-Net and DiT backbones (e.g., Stable Diffusion 2, 3), FRAMER consistently improves PSNR/SSIM and perceptual metrics (LPIPS, NIQE, MANIQA, MUSIQ). Ablations validate the final-layer teacher and random-layer negatives.

Continual learning aims to avoid catastrophic forgetting and effectively leverage learned experiences to master new knowledge. Existing gradient projection approaches impose hard constraints on the optimization space for new tasks to minimize interference, which simultaneously hinders forward knowledge transfer. To address this issue, recent methods reuse frozen parameters with a growing network, resulting in high computational costs. Thus, it remains a challenge whether we can improve forward knowledge transfer for gradient projection approaches using a fixed network architecture. In this work, we propose the Restricted Orthogonal Gradient prOjection (ROGO) framework. The basic idea is to adopt a restricted orthogonal constraint allowing parameters optimized in the direction oblique to the whole frozen space to facilitate forward knowledge transfer while consolidating previous knowledge. Our framework requires neither data buffers nor extra parameters. Extensive experiments have demonstrated the superiority of our framework over several strong baselines. We also provide theoretical guarantees for our relaxing strategy.

Optimal configuration of the learning rate (LR) is a fundamental yet formidable challenge in large-scale pre-training. Given the stringent trade-off between training costs and model performance, the pivotal question is whether the optimal LR can be accurately extrapolated from low-cost experiments. In this paper, we formalize this investigation into two distinct research paradigms: Fitting and Transfer. Within the Fitting Paradigm, we innovatively introduce a Scaling Law for search factor, effectively reducing the search complexity from O(n^3) to O(n*C_D*C_η) via predictive modeling. Within the Transfer Paradigm, we extend the principles of μTransfer to the Mixture of Experts (MoE) architecture, broadening its applicability to encompass model depth, weight decay, and token horizons. By pushing the boundaries of existing hyperparameter research in terms of scale, we conduct a comprehensive comparison between these two paradigms. Our empirical results challenge the scalability of the widely adopted μ Transfer in large-scale pre-training scenarios. Furthermore, we provide a rigorous analysis through the dual lenses of training stability and feature learning to elucidate the underlying reasons why module-wise parameter tuning underperforms in large-scale settings. This work offers systematic practical guidelines and a fresh theoretical perspective for optimizing industrial-level pre-training.

The Sharpness Aware Minimization (SAM) optimization algorithm has been shown to control large eigenvalues of the loss Hessian and provide generalization benefits in a variety of settings. The original motivation for SAM was a modified loss function which penalized sharp minima; subsequent analyses have also focused on the behavior near minima. However, our work reveals that SAM provides a strong regularization of the eigenvalues throughout the learning trajectory. We show that in a simplified setting, SAM dynamically induces a stabilization related to the edge of stability (EOS) phenomenon observed in large learning rate gradient descent. Our theory predicts the largest eigenvalue as a function of the learning rate and SAM radius parameters. Finally, we show that practical models can also exhibit this EOS stabilization, and that understanding SAM must account for these dynamics far away from any minima.

This paper provides the first tight convergence analyses for RMSProp and Adam in non-convex optimization under the most relaxed assumptions of coordinate-wise generalized smoothness and affine noise variance. We first analyze RMSProp, which is a special case of Adam with adaptive learning rates but without first-order momentum. Specifically, to solve the challenges due to dependence among adaptive update, unbounded gradient estimate and Lipschitz constant, we demonstrate that the first-order term in the descent lemma converges and its denominator is upper bounded by a function of gradient norm. Based on this result, we show that RMSProp with proper hyperparameters converges to an epsilon-stationary point with an iteration complexity of mathcal O(epsilon^{-4}). We then generalize our analysis to Adam, where the additional challenge is due to a mismatch between the gradient and first-order momentum. We develop a new upper bound on the first-order term in the descent lemma, which is also a function of the gradient norm. We show that Adam with proper hyperparameters converges to an epsilon-stationary point with an iteration complexity of mathcal O(epsilon^{-4}). Our complexity results for both RMSProp and Adam match with the complexity lower bound established in arjevani2023lower.

The ability to learn new concepts continually is necessary in this ever-changing world. However, deep neural networks suffer from catastrophic forgetting when learning new categories. Many works have been proposed to alleviate this phenomenon, whereas most of them either fall into the stability-plasticity dilemma or take too much computation or storage overhead. Inspired by the gradient boosting algorithm to gradually fit the residuals between the target model and the previous ensemble model, we propose a novel two-stage learning paradigm FOSTER, empowering the model to learn new categories adaptively. Specifically, we first dynamically expand new modules to fit the residuals between the target and the output of the original model. Next, we remove redundant parameters and feature dimensions through an effective distillation strategy to maintain the single backbone model. We validate our method FOSTER on CIFAR-100 and ImageNet-100/1000 under different settings. Experimental results show that our method achieves state-of-the-art performance. Code is available at: https://github.com/G-U-N/ECCV22-FOSTER.

We study the problem of optimizing a function under a budgeted number of evaluations. We only assume that the function is locally smooth around one of its global optima. The difficulty of optimization is measured in terms of 1) the amount of noise b of the function evaluation and 2) the local smoothness, d, of the function. A smaller d results in smaller optimization error. We come with a new, simple, and parameter-free approach. First, for all values of b and d, this approach recovers at least the state-of-the-art regret guarantees. Second, our approach additionally obtains these results while being agnostic to the values of both b and d. This leads to the first algorithm that naturally adapts to an unknown range of noise b and leads to significant improvements in a moderate and low-noise regime. Third, our approach also obtains a remarkable improvement over the state-of-the-art SOO algorithm when the noise is very low which includes the case of optimization under deterministic feedback (b=0). There, under our minimal local smoothness assumption, this improvement is of exponential magnitude and holds for a class of functions that covers the vast majority of functions that practitioners optimize (d=0). We show that our algorithmic improvement is borne out in experiments as we empirically show faster convergence on common benchmarks.

Real-world data streams exhibit inherent non-stationarity characterized by concept drift, posing significant challenges for adaptive learning systems. While existing methods address isolated distribution shifts, they overlook the critical co-evolution of label spaces and distributions under limited supervision and persistent uncertainty. To address this, we formalize Generalized Incremental Learning under Concept Drift (GILCD), characterizing the joint evolution of distributions and label spaces in open-environment streaming contexts, and propose a novel framework called Calibrated Source-Free Adaptation (CSFA). First, CSFA introduces a training-free prototype calibration mechanism that dynamically fuses emerging prototypes with base representations, enabling stable new-class identification without optimization overhead. Second, we design a novel source-free adaptation algorithm, i.e., Reliable Surrogate Gap Sharpness-aware (RSGS) minimization. It integrates sharpness-aware perturbation loss optimization with surrogate gap minimization, while employing entropy-based uncertainty filtering to discard unreliable samples. This mechanism ensures robust distribution alignment and mitigates generalization degradation caused by uncertainties. Therefore, CSFA establishes a unified framework for stable adaptation to evolving semantics and distributions in open-world streaming scenarios. Extensive experiments validate the superior performance and effectiveness of CSFA compared to state-of-the-art approaches.

In this work, we question the necessity of adaptive gradient methods for training deep neural networks. SGD-SaI is a simple yet effective enhancement to stochastic gradient descent with momentum (SGDM). SGD-SaI performs learning rate Scaling at Initialization (SaI) to distinct parameter groups, guided by their respective gradient signal-to-noise ratios (g-SNR). By adjusting learning rates without relying on adaptive second-order momentum, SGD-SaI helps prevent training imbalances from the very first iteration and cuts the optimizer's memory usage by half compared to AdamW. Despite its simplicity and efficiency, SGD-SaI consistently matches or outperforms AdamW in training a variety of Transformer-based tasks, effectively overcoming a long-standing challenge of using SGD for training Transformers. SGD-SaI excels in ImageNet-1K classification with Vision Transformers(ViT) and GPT-2 pretraining for large language models (LLMs, transformer decoder-only), demonstrating robustness to hyperparameter variations and practicality for diverse applications. We further tested its robustness on tasks like LoRA fine-tuning for LLMs and diffusion models, where it consistently outperforms state-of-the-art optimizers. From a memory efficiency perspective, SGD-SaI achieves substantial memory savings for optimizer states, reducing memory usage by 5.93 GB for GPT-2 (1.5B parameters) and 25.15 GB for Llama2-7B compared to AdamW in full-precision training settings.

Fine-tuning over large pretrained language models (PLMs) has established many state-of-the-art results. Despite its superior performance, such fine-tuning can be unstable, resulting in significant variance in performance and potential risks for practical applications. Previous works have attributed such instability to the catastrophic forgetting problem in the top layers of PLMs, which indicates iteratively that fine-tuning layers in a top-down manner is a promising solution. In this paper, we first point out that this method does not always work out due to the different convergence speeds of different layers/modules. Inspired by this observation, we propose a simple component-wise gradient norm clipping method to adjust the convergence speed for different components. Experiment results demonstrate that our method achieves consistent improvements in terms of generalization performance, convergence speed, and training stability. The codebase can be found at https://github.com/yangalan123/FineTuningStability.

Post-training processes are essential phases in grounding pre-trained language models to real-world tasks, with learning from demonstrations or preference signals playing a crucial role in this adaptation. We present a unified theoretical framework bridging Supervised Fine-Tuning (SFT) and preference learning in Large Language Model (LLM) post-training. Through rigorous mathematical derivation, we demonstrate that both SFT and preference learning methods like Direct Preference Optimization (DPO) operate within the same optimal policy-reward subspace, with SFT representing a special case of implicit reward learning. Our analysis reveals a critical limitation in conventional SFT: the KL divergence term in distribution matching becomes constant with respect to the policy during optimization, failing to constrain model updates. To address this, we propose a simple yet effective learning rate reduction approach that yields significant performance improvements (up to 25\% relative gain and 6\% absolute win rate increase in instruction following tasks. Additionally, we derive alternative SFT objectives from various f-divergence functions that preserve the KL term during optimization, further enhancing post-DPO model performance. Finally, we extend the theoretical relationship between LLM logits and Q-functions from preference learning to the SFT context, providing mathematical derivations and experimental validation.

We propose a gradient flow procedure for generative modeling by transporting particles from an initial source distribution to a target distribution, where the gradient field on the particles is given by a noise-adaptive Wasserstein Gradient of the Maximum Mean Discrepancy (MMD). The noise-adaptive MMD is trained on data distributions corrupted by increasing levels of noise, obtained via a forward diffusion process, as commonly used in denoising diffusion probabilistic models. The result is a generalization of MMD Gradient Flow, which we call Diffusion-MMD-Gradient Flow or DMMD. The divergence training procedure is related to discriminator training in Generative Adversarial Networks (GAN), but does not require adversarial training. We obtain competitive empirical performance in unconditional image generation on CIFAR10, MNIST, CELEB-A (64 x64) and LSUN Church (64 x 64). Furthermore, we demonstrate the validity of the approach when MMD is replaced by a lower bound on the KL divergence.

Is the standard weight decay in AdamW truly optimal? Although AdamW decouples weight decay from adaptive gradient scaling, a fundamental conflict remains: the Radial Tug-of-War. In deep learning, gradients tend to increase parameter norms to expand effective capacity while steering directions to learn features, whereas weight decay indiscriminately suppresses norm growth. This push--pull interaction induces radial oscillations, injecting noise into Adam's second-moment estimates and potentially degrading delicate tangential feature learning. We argue that magnitude and direction play distinct roles and should be decoupled in optimizer dynamics. We propose Orthogonal Dynamics Decoupling and instantiate it as AdamO: an SGD-style update handles the one-dimensional norm control, while Adam's adaptive preconditioning is confined to the tangential subspace. AdamO further incorporates curvature-adaptive radial step sizing and architecture-aware rules and projections for scale-invariant layers and low-dimensional parameters. Experiments on vision and language tasks show that AdamO improves generalization and stability over AdamW without introducing additional complex constraints.

In deep learning, different kinds of deep networks typically need different optimizers, which have to be chosen after multiple trials, making the training process inefficient. To relieve this issue and consistently improve the model training speed across deep networks, we propose the ADAptive Nesterov momentum algorithm, Adan for short. Adan first reformulates the vanilla Nesterov acceleration to develop a new Nesterov momentum estimation (NME) method, which avoids the extra overhead of computing gradient at the extrapolation point. Then Adan adopts NME to estimate the gradient's first- and second-order moments in adaptive gradient algorithms for convergence acceleration. Besides, we prove that Adan finds an epsilon-approximate first-order stationary point within O(epsilon^{-3.5}) stochastic gradient complexity on the non-convex stochastic problems (e.g., deep learning problems), matching the best-known lower bound. Extensive experimental results show that Adan consistently surpasses the corresponding SoTA optimizers on vision, language, and RL tasks and sets new SoTAs for many popular networks and frameworks, e.g., ResNet, ConvNext, ViT, Swin, MAE, DETR, GPT-2, Transformer-XL, and BERT. More surprisingly, Adan can use half of the training cost (epochs) of SoTA optimizers to achieve higher or comparable performance on ViT, GPT-2, MAE, e.t.c., and also shows great tolerance to a large range of minibatch size, e.g., from 1k to 32k. Code is released at https://github.com/sail-sg/Adan, and has been used in multiple popular deep learning frameworks or projects.

Federated learning (FL) is a distributed machine learning framework where the global model of a central server is trained via multiple collaborative steps by participating clients without sharing their data. While being a flexible framework, where the distribution of local data, participation rate, and computing power of each client can greatly vary, such flexibility gives rise to many new challenges, especially in the hyperparameter tuning on the client side. We propose Delta-SGD, a simple step size rule for SGD that enables each client to use its own step size by adapting to the local smoothness of the function each client is optimizing. We provide theoretical and empirical results where the benefit of the client adaptivity is shown in various FL scenarios.

We present a genetic algorithm framework for automatically discovering deep learning optimization algorithms. Our approach encodes optimizers as genomes that specify combinations of primitive update terms (gradient, momentum, RMS normalization, Adam-style adaptive terms, and sign-based updates) along with hyperparameters and scheduling options. Through evolutionary search over 50 generations with a population of 50 individuals, evaluated across multiple vision tasks, we discover an evolved optimizer that outperforms Adam by 2.6% in aggregate fitness and achieves a 7.7% relative improvement on CIFAR-10. The evolved optimizer combines sign-based gradient terms with adaptive moment estimation, uses lower momentum coefficients than Adam (β_1=0.86, β_2=0.94), and notably disables bias correction while enabling learning rate warmup and cosine decay. Our results demonstrate that evolutionary search can discover competitive optimization algorithms and reveal design principles that differ from hand-crafted optimizers. Code is available at https://github.com/mmarfinetz/evo-optimizer.

Low-Rank Adaptation (LoRA) has become a widely adopted technique for fine-tuning large-scale pre-trained models with minimal parameter updates. However, existing methods rely on fixed ranks or focus solely on either rank pruning or expansion, failing to adapt ranks dynamically to match the importance of different layers during training. In this work, we propose ElaLoRA, an adaptive low-rank adaptation framework that dynamically prunes and expands ranks based on gradient-derived importance scores. To the best of our knowledge, ElaLoRA is the first method that enables both rank pruning and expansion during fine-tuning. Experiments across multiple benchmarks demonstrate that ElaLoRA consistently outperforms existing PEFT methods across different parameter budgets. Furthermore, our studies validate that layers receiving higher rank allocations contribute more significantly to model performance, providing theoretical justification for our adaptive strategy. By introducing a principled and adaptive rank allocation mechanism, ElaLoRA offers a scalable and efficient fine-tuning solution, particularly suited for resource-constrained environments.

Stochastic Gradient Descent (SGD) stands as a cornerstone optimization algorithm with proven real-world empirical successes but relatively limited theoretical understanding. Recent research has illuminated a key factor contributing to its practical efficacy: the implicit regularization it instigates. Several studies have investigated the linear stability property of SGD in the vicinity of a stationary point as a predictive proxy for sharpness and generalization error in overparameterized neural networks (Wu et al., 2022; Jastrzebski et al., 2019; Cohen et al., 2021). In this paper, we delve deeper into the relationship between linear stability and sharpness. More specifically, we meticulously delineate the necessary and sufficient conditions for linear stability, contingent on hyperparameters of SGD and the sharpness at the optimum. Towards this end, we introduce a novel coherence measure of the loss Hessian that encapsulates pertinent geometric properties of the loss function that are relevant to the linear stability of SGD. It enables us to provide a simplified sufficient condition for identifying linear instability at an optimum. Notably, compared to previous works, our analysis relies on significantly milder assumptions and is applicable for a broader class of loss functions than known before, encompassing not only mean-squared error but also cross-entropy loss.

It is believed that Gradient Descent (GD) induces an implicit bias towards good generalization in training machine learning models. This paper provides a fine-grained analysis of the dynamics of GD for the matrix sensing problem, whose goal is to recover a low-rank ground-truth matrix from near-isotropic linear measurements. It is shown that GD with small initialization behaves similarly to the greedy low-rank learning heuristics (Li et al., 2020) and follows an incremental learning procedure (Gissin et al., 2019): GD sequentially learns solutions with increasing ranks until it recovers the ground truth matrix. Compared to existing works which only analyze the first learning phase for rank-1 solutions, our result provides characterizations for the whole learning process. Moreover, besides the over-parameterized regime that many prior works focused on, our analysis of the incremental learning procedure also applies to the under-parameterized regime. Finally, we conduct numerical experiments to confirm our theoretical findings.

We propose an algorithm for inexpensive gradient-based hyperparameter optimization that combines the implicit function theorem (IFT) with efficient inverse Hessian approximations. We present results about the relationship between the IFT and differentiating through optimization, motivating our algorithm. We use the proposed approach to train modern network architectures with millions of weights and millions of hyper-parameters. For example, we learn a data-augmentation network - where every weight is a hyperparameter tuned for validation performance - outputting augmented training examples. Jointly tuning weights and hyperparameters with our approach is only a few times more costly in memory and compute than standard training.

Low-Rank Adaptation (LoRA) has gained popularity for fine-tuning large foundation models, leveraging low-rank matrices A and B to represent weight changes (i.e., Delta W = B A). This method reduces trainable parameters and mitigates heavy memory consumption associated with full delta matrices by sequentially multiplying A and B with the activation. Despite its success, the intrinsic low-rank characteristic may limit its performance. Although several variants have been proposed to address this issue, they often overlook the crucial computational and memory efficiency brought by LoRA. In this paper, we propose Circular Convolution Adaptation (C^3A), which not only achieves high-rank adaptation with enhanced performance but also excels in both computational power and memory utilization. Extensive experiments demonstrate that C^3A consistently outperforms LoRA and its variants across various fine-tuning tasks.

Deep ensemble methods often improve predictive performance, yet they suffer from three practical limitations: redundancy among base models that inflates computational cost and degrades conditioning, unstable weighting under multicollinearity, and overfitting in meta-learning pipelines. We propose a regularized meta-learning framework that addresses these challenges through a four-stage pipeline combining redundancy-aware projection, statistical meta-feature augmentation, and cross-validated regularized meta-models (Ridge, Lasso, and ElasticNet). Our multi-metric de-duplication strategy removes near-collinear predictors using correlation and MSE thresholds (τ_{corr}=0.95), reducing the effective condition number of the meta-design matrix while preserving predictive diversity. Engineered ensemble statistics and interaction terms recover higher-order structure unavailable to raw prediction columns. A final inverse-RMSE blending stage mitigates regularizer-selection variance. On the Playground Series S6E1 benchmark (100K samples, 72 base models), the proposed framework achieves an out-of-fold RMSE of 8.582, improving over simple averaging (8.894) and conventional Ridge stacking (8.627), while matching greedy hill climbing (8.603) with substantially lower runtime (4 times faster). Conditioning analysis shows a 53.7\% reduction in effective matrix condition number after redundancy projection. Comprehensive ablations demonstrate consistent contributions from de-duplication, statistical meta-features, and meta-ensemble blending. These results position regularized meta-learning as a stable and deployment-efficient stacking strategy for high-dimensional ensemble systems.

The theoretical landscape of federated learning (FL) undergoes rapid evolution, but its practical application encounters a series of intricate challenges, and hyperparameter optimization is one of these critical challenges. Amongst the diverse adjustments in hyperparameters, the adaptation of the learning rate emerges as a crucial component, holding the promise of significantly enhancing the efficacy of FL systems. In response to this critical need, this paper presents FedHyper, a novel hypergradient-based learning rate adaptation algorithm specifically designed for FL. FedHyper serves as a universal learning rate scheduler that can adapt both global and local rates as the training progresses. In addition, FedHyper not only showcases unparalleled robustness to a spectrum of initial learning rate configurations but also significantly alleviates the necessity for laborious empirical learning rate adjustments. We provide a comprehensive theoretical analysis of FedHyper's convergence rate and conduct extensive experiments on vision and language benchmark datasets. The results demonstrate that FEDHYPER consistently converges 1.1-3x faster than FedAvg and the competing baselines while achieving superior final accuracy. Moreover, FedHyper catalyzes a remarkable surge in accuracy, augmenting it by up to 15% compared to FedAvg under suboptimal initial learning rate settings.