# Source code for neural_tangents._src.monte_carlo

```
# Copyright 2019 Google LLC
#
# 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
#
# https://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.
"""Function to compute Monte Carlo NNGP and NTK estimates.
This module contains a function `monte_carlo_kernel_fn` that allow to compute
Monte Carlo estimates of NNGP and NTK kernels of arbitrary functions. For more
details on how individual samples are computed, refer to `utils/empirical.py`.
Note that the `monte_carlo_kernel_fn` accepts arguments like `batch_size`,
`device_count`, and `store_on_device`, and is appropriately batched /
parallelized. You don't need to apply the :obj:`~neural_tangents.batch` or
:obj:`jax.jit` decorators to it. Further, you do not need to apply
:obj:`jax.jit` to the input `apply_fn` function, as the resulting empirical
kernel function is JITted internally.
"""
from functools import partial
import operator
from typing import Generator, Iterable, Optional, Set, Tuple, Union
from .batching import batch
from .empirical import empirical_kernel_fn, NtkImplementation, DEFAULT_NTK_IMPLEMENTATION, _DEFAULT_NTK_FWD, _DEFAULT_NTK_S_RULES, _DEFAULT_NTK_J_RULES
from jax import random
import jax.numpy as np
from jax.tree_util import tree_map
from .utils import utils
from .utils.typing import ApplyFn, Axes, EmpiricalGetKernelFn, Get, InitFn, MonteCarloKernelFn, NTTree, PyTree, VMapAxes
def _sample_once_kernel_fn(
kernel_fn: EmpiricalGetKernelFn,
init_fn: InitFn,
batch_size: int = 0,
device_count: int = -1,
store_on_device: bool = True
):
@partial(batch,
batch_size=batch_size,
device_count=device_count,
store_on_device=store_on_device)
def kernel_fn_sample_once(
x1: NTTree[np.ndarray],
x2: Optional[NTTree[np.ndarray]],
key: random.KeyArray,
get: Get,
**apply_fn_kwargs):
init_key, dropout_key = random.split(key, 2)
shape = tree_map(lambda x: x.shape, x1)
_, params = init_fn(init_key, shape)
return kernel_fn(x1, x2, get, params, rng=dropout_key, **apply_fn_kwargs)
return kernel_fn_sample_once
def _sample_many_kernel_fn(
kernel_fn_sample_once,
key: random.KeyArray,
n_samples: Set[int],
get_generator: bool):
def normalize(sample: PyTree, n: int) -> PyTree:
return tree_map(lambda sample: sample / n, sample)
def get_samples(
x1: NTTree[np.ndarray],
x2: Optional[NTTree[np.ndarray]],
get: Get,
**apply_fn_kwargs):
_key = key
ker_sampled = None
for n in range(1, max(n_samples) + 1):
_key, split = random.split(_key)
one_sample = kernel_fn_sample_once(x1, x2, split, get, **apply_fn_kwargs)
if ker_sampled is None:
ker_sampled = one_sample
else:
ker_sampled = tree_map(operator.add, ker_sampled, one_sample)
yield n, ker_sampled
if get_generator:
@utils.get_namedtuple('MonteCarloKernel')
def get_sampled_kernel(
x1: np.ndarray,
x2: np.ndarray,
get: Optional[Get] = None,
**apply_fn_kwargs
) -> Generator[Union[np.ndarray, Tuple[np.ndarray, ...]], None, None]:
for n, sample in get_samples(x1, x2, get, **apply_fn_kwargs):
if n in n_samples:
yield normalize(sample, n)
else:
@utils.get_namedtuple('MonteCarloKernel')
def get_sampled_kernel(
x1: np.ndarray,
x2: np.ndarray,
get: Optional[Get] = None,
**apply_fn_kwargs
) -> Union[np.ndarray, Tuple[np.ndarray, ...]]:
for n, sample in get_samples(x1, x2, get, **apply_fn_kwargs):
pass
return normalize(sample, n)
return get_sampled_kernel
[docs]def monte_carlo_kernel_fn(
init_fn: InitFn,
apply_fn: ApplyFn,
key: random.KeyArray,
n_samples: Union[int, Iterable[int]],
batch_size: int = 0,
device_count: int = -1,
store_on_device: bool = True,
trace_axes: Axes = (-1,),
diagonal_axes: Axes = (),
vmap_axes: Optional[VMapAxes] = None,
implementation: Union[int, NtkImplementation] = DEFAULT_NTK_IMPLEMENTATION,
_j_rules: bool = _DEFAULT_NTK_J_RULES,
_s_rules: bool = _DEFAULT_NTK_S_RULES,
_fwd: Optional[bool] = _DEFAULT_NTK_FWD,
) -> MonteCarloKernelFn:
r"""Return a Monte Carlo sampler of NTK and NNGP kernels of a given function.
Note that the returned function is appropriately batched / parallelized. You
don't need to apply the `nt.batch` or `jax.jit` decorators to it. Further,
you do not need to apply `jax.jit` to the input `apply_fn` function, as the
resulting empirical kernel function is JITted internally.
Args:
init_fn:
a function initializing parameters of the neural network. From
:obj:`jax.example_libraries.stax`: "takes an rng key and an input shape
and returns an `(output_shape, params)` pair".
apply_fn:
a function computing the output of the neural network.
From :obj:`jax.example_libraries.stax`: "takes params, inputs, and an
rng key and applies the layer".
key:
RNG (`jax.random.PRNGKey`) for sampling random networks. Must have
shape `(2,)`.
n_samples:
number of Monte Carlo samples. Can be either an integer or an
iterable of integers at which the resulting generator will yield
estimates. Example: use `n_samples=[2**k for k in range(10)]` for the
generator to yield estimates using 1, 2, 4, ..., 512 Monte Carlo samples.
batch_size: an integer making the kernel computed in batches of `x1` and
`x2` of this size. `0` means computing the whole kernel. Must divide
`x1.shape[0]` and `x2.shape[0]`.
device_count:
an integer making the kernel be computed in parallel across
this number of devices (e.g. GPUs or TPU cores). `-1` means use all
available devices. `0` means compute on a single device sequentially. If
not `0`, must divide `x1.shape[0]`.
store_on_device:
a boolean, indicating whether to store the resulting
kernel on the device (e.g. GPU or TPU), or in the CPU RAM, where larger
kernels may fit.
trace_axes:
output axes to trace the output kernel over, i.e. compute only the trace
of the covariance along the respective pair of axes (one pair for each
axis in `trace_axes`). This allows to save space and compute if you are
only interested in the respective trace, but also improve approximation
accuracy if you know that covariance along these pairs of axes converges
to a `constant * identity matrix` in the limit of interest (e.g.
infinite width or infinite `n_samples`). A common use case is the channel
/ feature / logit axis, since activation slices along such axis are i.i.d.
and the respective covariance along the respective pair of axes indeed
converges to a constant-diagonal matrix in the infinite width or infinite
`n_samples` limit.
Also related to "contracting dimensions" in XLA terms.
(https://www.tensorflow.org/xla/operation_semantics#dotgeneral)
diagonal_axes:
output axes to diagonalize the output kernel over, i.e. compute only the
diagonal of the covariance along the respective pair of axes (one pair for
each axis in `diagonal_axes`). This allows to save space and compute, if
off-diagonal values along these axes are not needed, but also improve
approximation accuracy if their limiting value is known theoretically,
e.g. if they vanish in the limit of interest (e.g. infinite
width or infinite `n_samples`). If you further know that on-diagonal
values converge to the same constant in your limit of interest, you should
specify these axes in `trace_axes` instead, to save even more compute and
gain even more accuracy. A common use case is computing the variance
(instead of covariance) along certain axes.
Also related to "batch dimensions" in XLA terms.
(https://www.tensorflow.org/xla/operation_semantics#dotgeneral)
vmap_axes:
applicable only to NTK. A triple of `(in_axes, out_axes, kwargs_axes)`
passed to `vmap` to evaluate the empirical NTK in parallel ove these axes.
Precisely, providing this argument implies that `f(params, x, **kwargs)`
equals to a concatenation along `out_axes` of `f` applied to slices of
`x` and `**kwargs` along `in_axes` and `kwargs_axes`, i.e. `f` can be
evaluated as a `vmap`. This allows to evaluate Jacobians much more
efficiently. If `vmap_axes` is not a triple, it is interpreted as
`in_axes = out_axes = vmap_axes, kwargs_axes = {}`. For example a very
common usecase is `vmap_axes=0` for a neural network with leading (`0`)
batch dimension, both for inputs and outputs, and no interactions between
different elements of the batch (e.g. no BatchNorm, and, in the case of
`nt.stax`, also no Dropout). However, if there is interaction between
batch elements or no concept of a batch axis at all, `vmap_axes` must be
set to `None`, to avoid wrong (and potentially silent) results.
implementation:
Applicable only to NTK, an :class:`NtkImplementation` value (or an
:class:`int` `0`, `1`, `2`, or `3`). See the :class:`NtkImplementation`
docstring for details.
_j_rules:
Internal debugging parameter, applicable only to NTK when
`implementation` is :attr:`~NtkImplementation.STRUCTURED_DERIVATIVES`
(`3`) or :attr:`~NtkImplementation.AUTO` (`0`). Set to `True` to allow
custom Jacobian rules for intermediary primitive `dy/dw` computations for
MJJMPs (matrix-Jacobian-Jacobian-matrix products). Set to `False` to use
JVPs or VJPs, via JAX's :obj:`jax.jacfwd` or :obj:`jax.jacrev`. Custom
Jacobian rules (`True`) are expected to be not worse, and sometimes better
than automated alternatives, but in case of a suboptimal implementation
setting it to `False` could improve performance.
_s_rules:
Internal debugging parameter, applicable only to NTK when
`implementation` is :attr:`~NtkImplementation.STRUCTURED_DERIVATIVES`
(`3`) or :attr:`~NtkImplementation.AUTO` (`0`). Set to `True` to allow
efficient MJJMp rules for structured `dy/dw` primitive Jacobians. In
practice should be set to `True`, and setting it to `False` can lead to
dramatic deterioration of performance.
_fwd:
Internal debugging parameter, applicable only to NTK when
`implementation` is :attr:`~NtkImplementation.STRUCTURED_DERIVATIVES`
(`3`) or :attr:`~NtkImplementation.AUTO` (`0`). Set to `True` to allow
:obj:`jax.jvp` in intermediary primitive Jacobian `dy/dw` computations,
`False` to always use :obj:`jax.vjp`. `None` to decide automatically
based on input/output sizes. Applicable when `_j_rules=False`, or when a
primitive does not have a Jacobian rule. Should be set to `None` for best
performance.
Returns:
If `n_samples` is an integer, returns a function of signature
`kernel_fn(x1, x2, get)` that returns an MC estimation of the kernel using
`n_samples`. If `n_samples` is a collection of integers,
`kernel_fn(x1, x2, get)` returns a generator that yields estimates using
`n` samples for `n in n_samples`.
Example:
>>> from jax import random
>>> import neural_tangents as nt
>>> from neural_tangents import stax
>>> #
>>> key1, key2 = random.split(random.PRNGKey(1), 2)
>>> x_train = random.normal(key1, (20, 32, 32, 3))
>>> y_train = random.uniform(key1, (20, 10))
>>> x_test = random.normal(key2, (5, 32, 32, 3))
>>> #
>>> init_fn, apply_fn, _ = stax.serial(
>>> stax.Conv(128, (3, 3)),
>>> stax.Relu(),
>>> stax.Conv(256, (3, 3)),
>>> stax.Relu(),
>>> stax.Conv(512, (3, 3)),
>>> stax.Flatten(),
>>> stax.Dense(10)
>>> )
>>> #
>>> n_samples = 200
>>> kernel_fn = nt.monte_carlo_kernel_fn(init_fn, apply_fn, key1, n_samples)
>>> kernel = kernel_fn(x_train, x_test, get=('nngp', 'ntk'))
>>> # `kernel` is a tuple of NNGP and NTK MC estimate using `n_samples`.
>>> #
>>> n_samples = [1, 10, 100, 1000]
>>> kernel_fn_generator = nt.monte_carlo_kernel_fn(init_fn, apply_fn, key1,
>>> n_samples)
>>> kernel_samples = kernel_fn_generator(x_train, x_test,
>>> get=('nngp', 'ntk'))
>>> for n, kernel in zip(n_samples, kernel_samples):
>>> print(n, kernel)
>>> # `kernel` is a tuple of NNGP and NTK MC estimate using `n` samples.
"""
kwargs = dict(
f=apply_fn,
trace_axes=trace_axes,
diagonal_axes=diagonal_axes,
vmap_axes=vmap_axes,
implementation=implementation,
_s_rules=_s_rules,
_j_rules=_j_rules,
_fwd=_fwd
)
kernel_fn = empirical_kernel_fn(**kwargs)
kernel_fn_sample_once = _sample_once_kernel_fn(
kernel_fn=kernel_fn,
init_fn=init_fn,
batch_size=batch_size,
device_count=device_count,
store_on_device=store_on_device
)
n_samples, get_generator = _canonicalize_n_samples(n_samples)
kernel_fn = _sample_many_kernel_fn(
kernel_fn_sample_once=kernel_fn_sample_once,
key=key,
n_samples=n_samples,
get_generator=get_generator
)
return kernel_fn
def _canonicalize_n_samples(
n_samples: Union[int, Iterable[int]]) -> Tuple[Set[int], bool]:
get_generator = True
if isinstance(n_samples, int):
get_generator = False
n_samples = (n_samples,)
if hasattr(n_samples, '__iter__'):
n_samples = set(n_samples)
if not all(isinstance(n, int) for n in n_samples):
raise ValueError(f'`n_samples` must contain only integers, '
f'got {n_samples}.')
if any(n <= 0 for n in n_samples):
raise ValueError(f'`n_samples` must be positive, got {n_samples}.')
else:
raise TypeError(f'`n_samples` must be either an integer of a set of '
f'integers, got {type(n_samples)}.')
return n_samples, get_generator
```