# Copyright 2018-2020 The GPflow Contributors. 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.
import abc
import functools
from typing import Any, Callable, Dict, Optional, Sequence, Tuple, Union
import numpy as np
import tensorflow as tf
from ..base import Parameter, _to_constrained
Scalar = Union[float, tf.Tensor, np.ndarray]
LossClosure = Callable[[], tf.Tensor]
NatGradParameters = Union[Tuple[Parameter, Parameter], Tuple[Parameter, Parameter, "XiTransform"]]
__all__ = [
"NaturalGradient",
"XiTransform",
"XiNat",
"XiSqrtMeanVar",
]
#
# Xi transformations necessary for natural gradient optimizer.
# Abstract class and two implementations: XiNat and XiSqrtMeanVar.
#
[docs]class XiNat(XiTransform):
"""
This is the default transform. Using the natural directly saves the forward mode
gradient, and also gives the analytic optimal solution for gamma=1 in the case
of Gaussian likelihood.
"""
[docs] @staticmethod
def meanvarsqrt_to_xi(mean: tf.Tensor, varsqrt: tf.Tensor) -> Tuple[tf.Tensor, tf.Tensor]:
return meanvarsqrt_to_natural(mean, varsqrt)
[docs] @staticmethod
def xi_to_meanvarsqrt(xi1: tf.Tensor, xi2: tf.Tensor) -> Tuple[tf.Tensor, tf.Tensor]:
return natural_to_meanvarsqrt(xi1, xi2)
[docs] @staticmethod
def naturals_to_xi(nat1: tf.Tensor, nat2: tf.Tensor) -> Tuple[tf.Tensor, tf.Tensor]:
return nat1, nat2
[docs]class XiSqrtMeanVar(XiTransform):
"""
This transformation will perform natural gradient descent on the model parameters,
so saves the conversion to and from Xi.
"""
[docs] @staticmethod
def meanvarsqrt_to_xi(mean: tf.Tensor, varsqrt: tf.Tensor) -> Tuple[tf.Tensor, tf.Tensor]:
return mean, varsqrt
[docs] @staticmethod
def xi_to_meanvarsqrt(xi1: tf.Tensor, xi2: tf.Tensor) -> Tuple[tf.Tensor, tf.Tensor]:
return xi1, xi2
[docs] @staticmethod
def naturals_to_xi(nat1: tf.Tensor, nat2: tf.Tensor) -> Tuple[tf.Tensor, tf.Tensor]:
return natural_to_meanvarsqrt(nat1, nat2)
[docs]class NaturalGradient(tf.optimizers.Optimizer):
"""
Implements a natural gradient descent optimizer for variational models
that are based on a distribution q(u) = N(q_mu, q_sqrt q_sqrtᵀ) that is
parameterized by mean q_mu and lower-triangular Cholesky factor q_sqrt
of the covariance.
Note that this optimizer does not implement the standard API of
tf.optimizers.Optimizer. Its only public method is minimize(), which has
a custom signature (var_list needs to be a list of (q_mu, q_sqrt) tuples,
where q_mu and q_sqrt are gpflow.Parameter instances, not tf.Variable).
Note furthermore that the natural gradients are implemented only for the
full covariance case (i.e., q_diag=True is NOT supported).
When using in your work, please cite
@inproceedings{salimbeni18,
title={Natural Gradients in Practice: Non-Conjugate Variational Inference in Gaussian Process Models},
author={Salimbeni, Hugh and Eleftheriadis, Stefanos and Hensman, James},
booktitle={AISTATS},
year={2018}
"""
def __init__(
self, gamma: Scalar, xi_transform: XiTransform = XiNat(), name: Optional[str] = None
) -> None:
"""
:param gamma: natgrad step length
:param xi_transform: default ξ transform (can be overridden in the call to minimize())
The XiNat default choice works well in general.
"""
name = self.__class__.__name__ if name is None else name
super().__init__(name)
self.gamma = gamma
self.xi_transform = xi_transform
[docs] def minimize(
self,
loss_fn: LossClosure,
var_list: Sequence[NatGradParameters],
) -> None:
"""
Minimizes objective function of the model.
Natural Gradient optimizer works with variational parameters only.
:param loss_fn: Loss function.
:param var_list: List of pair tuples of variational parameters or
triplet tuple with variational parameters and ξ transformation.
If ξ is not specified, will use self.xi_transform.
For example, `var_list` could be
```
var_list = [
(q_mu1, q_sqrt1),
(q_mu2, q_sqrt2, XiSqrtMeanVar())
]
```
GPflow implements the `XiNat` (default) and `XiSqrtMeanVar` transformations
for parameters. Custom transformations that implement the `XiTransform`
interface are also possible.
"""
parameters = [(v[0], v[1], (v[2] if len(v) > 2 else None)) for v in var_list] # type: ignore
self._natgrad_steps(loss_fn, parameters)
def _natgrad_steps(
self,
loss_fn: LossClosure,
parameters: Sequence[Tuple[Parameter, Parameter, Optional[XiTransform]]],
) -> None:
"""
Computes gradients of loss_fn() w.r.t. q_mu and q_sqrt, and updates
these parameters using the natgrad backwards step, for all sets of
variational parameters passed in.
:param loss_fn: Loss function.
:param parameters: List of tuples (q_mu, q_sqrt, xi_transform)
"""
q_mus, q_sqrts, xis = zip(*parameters)
q_mu_vars = [p.unconstrained_variable for p in q_mus]
q_sqrt_vars = [p.unconstrained_variable for p in q_sqrts]
with tf.GradientTape(watch_accessed_variables=False) as tape:
tape.watch(q_mu_vars + q_sqrt_vars)
loss = loss_fn()
q_mu_grads, q_sqrt_grads = tape.gradient(loss, [q_mu_vars, q_sqrt_vars])
# NOTE that these are the gradients in *unconstrained* space
with tf.name_scope(f"{self._name}/natural_gradient_steps"):
for q_mu_grad, q_sqrt_grad, q_mu, q_sqrt, xi_transform in zip(
q_mu_grads, q_sqrt_grads, q_mus, q_sqrts, xis
):
self._natgrad_apply_gradients(q_mu_grad, q_sqrt_grad, q_mu, q_sqrt, xi_transform)
def _assert_shapes(self, q_mu: tf.Tensor, q_sqrt: tf.Tensor) -> None:
tf.debugging.assert_shapes(
[
(q_mu, ["M", "L"]),
(q_sqrt, ["L", "M", "M"]),
]
)
def _natgrad_apply_gradients(
self,
q_mu_grad: tf.Tensor,
q_sqrt_grad: tf.Tensor,
q_mu: Parameter,
q_sqrt: Parameter,
xi_transform: Optional[XiTransform] = None,
) -> None:
"""
This function does the backward step on the q_mu and q_sqrt parameters,
given the gradients of the loss function with respect to their unconstrained
variables. I.e., it expects the arguments to come from
with tf.GradientTape() as tape:
loss = loss_function()
q_mu_grad, q_mu_sqrt = tape.gradient(loss, [q_mu, q_sqrt])
(Note that tape.gradient() returns the gradients in *unconstrained* space!)
Implements equation [10] from
@inproceedings{salimbeni18,
title={Natural Gradients in Practice: Non-Conjugate Variational Inference in Gaussian Process Models},
author={Salimbeni, Hugh and Eleftheriadis, Stefanos and Hensman, James},
booktitle={AISTATS},
year={2018}
In addition, for convenience with the rest of GPflow, this code computes ∂L/∂η using
the chain rule (the following assumes a numerator layout where the gradient is a row
vector; note that TensorFlow actually returns a column vector), where L is the loss:
∂L/∂η = (∂L / ∂[q_mu, q_sqrt])(∂[q_mu, q_sqrt] / ∂η)
In total there are three derivative calculations:
natgrad of L w.r.t ξ = (∂ξ / ∂θ) [(∂L / ∂[q_mu, q_sqrt]) (∂[q_mu, q_sqrt] / ∂η)]ᵀ
Note that if ξ = θ (i.e. [q_mu, q_sqrt]) some of these calculations are the identity.
In the code η = eta, ξ = xi, θ = nat.
:param q_mu_grad: gradient of loss w.r.t. q_mu (in unconstrained space)
:param q_sqrt_grad: gradient of loss w.r.t. q_sqrt (in unconstrained space)
:param q_mu: parameter for the mean of q(u) with shape [M, L]
:param q_sqrt: parameter for the square root of the covariance of q(u)
with shape [L, M, M] (the diagonal parametrization, q_diag=True, is NOT supported)
:param xi_transform: the ξ transform to use (self.xi_transform if not specified)
"""
self._assert_shapes(q_mu, q_sqrt)
if xi_transform is None:
xi_transform = self.xi_transform
# 1) the ordinary gpflow gradient
dL_dmean = _to_constrained(q_mu_grad, q_mu.transform)
dL_dvarsqrt = _to_constrained(q_sqrt_grad, q_sqrt.transform)
with tf.GradientTape(persistent=True, watch_accessed_variables=False) as tape:
tape.watch([q_mu.unconstrained_variable, q_sqrt.unconstrained_variable])
# the three parameterizations as functions of [q_mu, q_sqrt]
eta1, eta2 = meanvarsqrt_to_expectation(q_mu, q_sqrt)
# we need these to calculate the relevant gradients
meanvarsqrt = expectation_to_meanvarsqrt(eta1, eta2)
if not isinstance(xi_transform, XiNat):
nat1, nat2 = meanvarsqrt_to_natural(q_mu, q_sqrt)
xi1_nat, xi2_nat = xi_transform.naturals_to_xi(nat1, nat2)
dummy_tensors = tf.ones_like(xi1_nat), tf.ones_like(xi2_nat)
with tf.GradientTape(watch_accessed_variables=False) as forward_tape:
forward_tape.watch(dummy_tensors)
dummy_gradients = tape.gradient(
[xi1_nat, xi2_nat], [nat1, nat2], output_gradients=dummy_tensors
)
# 2) the chain rule to get ∂L/∂η, where η (eta) are the expectation parameters
dL_deta1, dL_deta2 = tape.gradient(
meanvarsqrt, [eta1, eta2], output_gradients=[dL_dmean, dL_dvarsqrt]
)
if not isinstance(xi_transform, XiNat):
nat_dL_xi1, nat_dL_xi2 = forward_tape.gradient(
dummy_gradients, dummy_tensors, output_gradients=[dL_deta1, dL_deta2]
)
else:
nat_dL_xi1, nat_dL_xi2 = dL_deta1, dL_deta2
del tape # Remove "persistent" tape
xi1, xi2 = xi_transform.meanvarsqrt_to_xi(q_mu, q_sqrt)
xi1_new = xi1 - self.gamma * nat_dL_xi1
xi2_new = xi2 - self.gamma * nat_dL_xi2
# Transform back to the model parameters [q_mu, q_sqrt]
mean_new, varsqrt_new = xi_transform.xi_to_meanvarsqrt(xi1_new, xi2_new)
q_mu.assign(mean_new)
q_sqrt.assign(varsqrt_new)
[docs] def get_config(self) -> Dict[str, Any]:
config = super().get_config()
config.update({"gamma": self._serialize_hyperparameter("gamma")})
return config
#
# Auxiliary gaussian parameter conversion functions.
#
# The following functions expect their first and second inputs to have shape
# [D, N, 1] and [D, N, N], respectively. Return values are also of shapes [D, N, 1] and [D, N, N].
[docs]def swap_dimensions(
method: Callable[[tf.Tensor, tf.Tensor], Tuple[tf.Tensor, tf.Tensor]]
) -> Callable[..., Tuple[tf.Tensor, tf.Tensor]]:
"""
Converts between GPflow indexing and tensorflow indexing
`method` is a function that broadcasts over the first dimension (i.e. like all tensorflow matrix ops):
`method` inputs [D, N, 1], [D, N, N]
`method` outputs [D, N, 1], [D, N, N]
:return: Function that broadcasts over the final dimension (i.e. compatible with GPflow):
inputs: [N, D], [D, N, N]
outputs: [N, D], [D, N, N]
"""
@functools.wraps(method)
def wrapper(
a_nd: tf.Tensor, b_dnn: tf.Tensor, swap: bool = True
) -> Tuple[tf.Tensor, tf.Tensor]:
if swap:
if a_nd.shape.ndims != 2: # pragma: no cover
raise ValueError("The mean parametrization must have 2 dimensions.")
if b_dnn.shape.ndims != 3: # pragma: no cover
raise ValueError("The covariance parametrization must have 3 dimensions.")
a_dn1 = tf.linalg.adjoint(a_nd)[:, :, None]
A_dn1, B_dnn = method(a_dn1, b_dnn)
A_nd = tf.linalg.adjoint(A_dn1[:, :, 0])
return A_nd, B_dnn
else:
return method(a_nd, b_dnn)
return wrapper
[docs]@swap_dimensions
def natural_to_meanvarsqrt(nat1: tf.Tensor, nat2: tf.Tensor) -> Tuple[tf.Tensor, tf.Tensor]:
var_sqrt_inv = tf.linalg.cholesky(-2 * nat2)
var_sqrt = _inverse_lower_triangular(var_sqrt_inv)
S = tf.linalg.matmul(var_sqrt, var_sqrt, transpose_a=True)
mu = tf.linalg.matmul(S, nat1)
# We need the decomposition of S as L L^T, not as L^T L,
# hence we need another cholesky.
return mu, tf.linalg.cholesky(S)
[docs]@swap_dimensions
def meanvarsqrt_to_natural(mu: tf.Tensor, s_sqrt: tf.Tensor) -> Tuple[tf.Tensor, tf.Tensor]:
s_sqrt_inv = _inverse_lower_triangular(s_sqrt)
s_inv = tf.linalg.matmul(s_sqrt_inv, s_sqrt_inv, transpose_a=True)
return tf.linalg.matmul(s_inv, mu), -0.5 * s_inv
[docs]@swap_dimensions
def natural_to_expectation(nat1: tf.Tensor, nat2: tf.Tensor) -> Tuple[tf.Tensor, tf.Tensor]:
args = natural_to_meanvarsqrt(nat1, nat2, swap=False)
return meanvarsqrt_to_expectation(*args, swap=False)
[docs]@swap_dimensions
def expectation_to_natural(eta1: tf.Tensor, eta2: tf.Tensor) -> Tuple[tf.Tensor, tf.Tensor]:
args = expectation_to_meanvarsqrt(eta1, eta2, swap=False)
return meanvarsqrt_to_natural(*args, swap=False)
[docs]@swap_dimensions
def expectation_to_meanvarsqrt(eta1: tf.Tensor, eta2: tf.Tensor) -> Tuple[tf.Tensor, tf.Tensor]:
var = eta2 - tf.linalg.matmul(eta1, eta1, transpose_b=True)
return eta1, tf.linalg.cholesky(var)
[docs]@swap_dimensions
def meanvarsqrt_to_expectation(m: tf.Tensor, v_sqrt: tf.Tensor) -> Tuple[tf.Tensor, tf.Tensor]:
v = tf.linalg.matmul(v_sqrt, v_sqrt, transpose_b=True)
return m, v + tf.linalg.matmul(m, m, transpose_b=True)
def _inverse_lower_triangular(M: tf.Tensor) -> tf.Tensor:
"""
Take inverse of lower triangular (e.g. Cholesky) matrix. This function
broadcasts over the first index.
:param M: Tensor with lower triangular structure of shape [D, N, N]
:return: The inverse of the Cholesky decomposition. Same shape as input.
"""
if M.shape.ndims != 3: # pragma: no cover
raise ValueError("Number of dimensions for input is required to be 3.")
D, N = tf.shape(M)[0], tf.shape(M)[1]
I_dnn = tf.eye(N, dtype=M.dtype)[None, :, :] * tf.ones((D, 1, 1), dtype=M.dtype)
return tf.linalg.triangular_solve(M, I_dnn)