File size: 2,529 Bytes
09d8e80 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 | # Copyright 2019 DeepMind Technologies Limited. All Rights Reserved.
#
# Licensed under the Apache License, Version 2.0 (the "License");
# you may not use this file except in compliance with the License.
# You may obtain a copy of the License at
#
# http://www.apache.org/licenses/LICENSE-2.0
#
# Unless required by applicable law or agreed to in writing, software
# distributed under the License is distributed on an "AS IS" BASIS,
# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
# See the License for the specific language governing permissions and
# limitations under the License.
# ==============================================================================
"""Functions for computing diagonals of the Hessian wrt to a set of parameters.
Computing the Hessian for neural networks is typically intractible due to the
quadratic memory requirements. Solving for the diagonal can be done cheaply,
with sub-quadratic memory requirements.
"""
from typing import Any
import jax
from jax import flatten_util
import jax.numpy as jnp
from optax.second_order import _base
def _ravel(p: Any) -> jax.Array:
return flatten_util.ravel_pytree(p)[0]
def hvp(
loss: _base.LossFn,
v: jax.Array,
params: Any,
inputs: jax.Array,
targets: jax.Array,
) -> jax.Array:
"""Performs an efficient vector-Hessian (of `loss`) product.
Args:
loss: the loss function.
v: a vector of size `ravel(params)`.
params: model parameters.
inputs: inputs at which `loss` is evaluated.
targets: targets at which `loss` is evaluated.
Returns:
An Array corresponding to the product of `v` and the Hessian of `loss`
evaluated at `(params, inputs, targets)`.
"""
_, unravel_fn = flatten_util.ravel_pytree(params)
loss_fn = lambda p: loss(p, inputs, targets)
return jax.jvp(jax.grad(loss_fn), [params], [unravel_fn(v)])[1]
def hessian_diag(
loss: _base.LossFn,
params: Any,
inputs: jax.Array,
targets: jax.Array,
) -> jax.Array:
"""Computes the diagonal hessian of `loss` at (`inputs`, `targets`).
Args:
loss: the loss function.
params: model parameters.
inputs: inputs at which `loss` is evaluated.
targets: targets at which `loss` is evaluated.
Returns:
A DeviceArray corresponding to the product to the Hessian of `loss`
evaluated at `(params, inputs, targets)`.
"""
vs = jnp.eye(_ravel(params).size)
comp = lambda v: jnp.vdot(v, _ravel(hvp(loss, v, params, inputs, targets)))
return jax.vmap(comp)(vs)
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