kl.py

# Copyright (c) 2021 PaddlePaddle Authors. 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 functools
import warnings

import paddle
from paddle.distribution.bernoulli import Bernoulli
from paddle.distribution.beta import Beta
from paddle.distribution.categorical import Categorical
from paddle.distribution.cauchy import Cauchy
from paddle.distribution.dirichlet import Dirichlet
from paddle.distribution.distribution import Distribution
from paddle.distribution.exponential_family import ExponentialFamily
from paddle.distribution.geometric import Geometric
from paddle.distribution.laplace import Laplace
from paddle.distribution.lognormal import LogNormal
from paddle.distribution.normal import Normal
from paddle.distribution.uniform import Uniform
from paddle.framework import in_dynamic_mode

__all__ = ["register_kl", "kl_divergence"]

_REGISTER_TABLE = {}


def kl_divergence(p, q):
    r"""
    Kullback-Leibler divergence between distribution p and q.

    .. math::

        KL(p||q) = \int p(x)log\frac{p(x)}{q(x)} \mathrm{d}x

    Args:
        p (Distribution): ``Distribution`` object. Inherits from the Distribution Base class.
        q (Distribution): ``Distribution`` object. Inherits from the Distribution Base class.

    Returns:
        Tensor, Batchwise KL-divergence between distribution p and q.

    Examples:

        .. code-block:: python

            import paddle

            p = paddle.distribution.Beta(alpha=0.5, beta=0.5)
            q = paddle.distribution.Beta(alpha=0.3, beta=0.7)

            print(paddle.distribution.kl_divergence(p, q))
            # Tensor(shape=[], dtype=float32, place=CUDAPlace(0), stop_gradient=True,
            #        0.21193528)

    """
    return _dispatch(type(p), type(q))(p, q)


def register_kl(cls_p, cls_q):
    """Decorator for register a KL divergence implemention function.

    The ``kl_divergence(p, q)`` function will search concrete implemention
    functions registered by ``register_kl``, according to multi-dispatch pattern.
    If an implemention function is found, it will return the result, otherwise,
    it will raise ``NotImplementError`` exception. Users can register
    implemention function by the decorator.

    Args:
        cls_p (Distribution): The Distribution type of Instance p. Subclass derived from ``Distribution``.
        cls_q (Distribution): The Distribution type of Instance q. Subclass derived from ``Distribution``.

    Examples:
        .. code-block:: python

            import paddle

            @paddle.distribution.register_kl(paddle.distribution.Beta, paddle.distribution.Beta)
            def kl_beta_beta():
                pass # insert implementation here
    """
    if not issubclass(cls_p, Distribution) or not issubclass(
        cls_q, Distribution
    ):
        raise TypeError('cls_p and cls_q must be subclass of Distribution')

    def decorator(f):
        _REGISTER_TABLE[cls_p, cls_q] = f
        return f

    return decorator


def _dispatch(cls_p, cls_q):
    """Multiple dispatch into concrete implement function."""

    # find all matched super class pair of p and q
    matchs = [
        (super_p, super_q)
        for super_p, super_q in _REGISTER_TABLE
        if issubclass(cls_p, super_p) and issubclass(cls_q, super_q)
    ]
    if not matchs:
        raise NotImplementedError

    left_p, left_q = min(_Compare(*m) for m in matchs).classes
    right_p, right_q = min(_Compare(*reversed(m)) for m in matchs).classes

    if _REGISTER_TABLE[left_p, left_q] is not _REGISTER_TABLE[right_p, right_q]:
        warnings.warn(
            'Ambiguous kl_divergence({}, {}). Please register_kl({}, {})'.format(
                cls_p.__name__,
                cls_q.__name__,
                left_p.__name__,
                right_q.__name__,
            ),
            RuntimeWarning,
        )

    return _REGISTER_TABLE[left_p, left_q]


@functools.total_ordering
class _Compare:
    def __init__(self, *classes):
        self.classes = classes

    def __eq__(self, other):
        return self.classes == other.classes

    def __le__(self, other):
        for cls_x, cls_y in zip(self.classes, other.classes):
            if not issubclass(cls_x, cls_y):
                return False
            if cls_x is not cls_y:
                break
        return True


@register_kl(Bernoulli, Bernoulli)
def _kl_bernoulli_bernoulli(p, q):
    return p.kl_divergence(q)


@register_kl(Beta, Beta)
def _kl_beta_beta(p, q):
    return (
        (q.alpha.lgamma() + q.beta.lgamma() + (p.alpha + p.beta).lgamma())
        - (p.alpha.lgamma() + p.beta.lgamma() + (q.alpha + q.beta).lgamma())
        + ((p.alpha - q.alpha) * p.alpha.digamma())
        + ((p.beta - q.beta) * p.beta.digamma())
        + (
            ((q.alpha + q.beta) - (p.alpha + p.beta))
            * (p.alpha + p.beta).digamma()
        )
    )


@register_kl(Dirichlet, Dirichlet)
def _kl_dirichlet_dirichlet(p, q):
    return (
        (p.concentration.sum(-1).lgamma() - q.concentration.sum(-1).lgamma())
        - ((p.concentration.lgamma() - q.concentration.lgamma()).sum(-1))
        + (
            (
                (p.concentration - q.concentration)
                * (
                    p.concentration.digamma()
                    - p.concentration.sum(-1).digamma().unsqueeze(-1)
                )
            ).sum(-1)
        )
    )


@register_kl(Categorical, Categorical)
def _kl_categorical_categorical(p, q):
    return p.kl_divergence(q)


@register_kl(Cauchy, Cauchy)
def _kl_cauchy_cauchy(p, q):
    return p.kl_divergence(q)


@register_kl(Normal, Normal)
def _kl_normal_normal(p, q):
    return p.kl_divergence(q)


@register_kl(Uniform, Uniform)
def _kl_uniform_uniform(p, q):
    return p.kl_divergence(q)


@register_kl(Laplace, Laplace)
def _kl_laplace_laplace(p, q):
    return p.kl_divergence(q)


@register_kl(Geometric, Geometric)
def _kl_geometric_geometric(p, q):
    return p.kl_divergence(q)


@register_kl(ExponentialFamily, ExponentialFamily)
def _kl_expfamily_expfamily(p, q):
    """Compute kl-divergence using `Bregman divergences <https://www.lix.polytechnique.fr/~nielsen/EntropyEF-ICIP2010.pdf>`_"""
    if not type(p) == type(q):
        raise NotImplementedError

    p_natural_params = []
    for param in p._natural_parameters:
        param = param.detach()
        param.stop_gradient = False
        p_natural_params.append(param)

    q_natural_params = q._natural_parameters

    p_log_norm = p._log_normalizer(*p_natural_params)

    try:
        if in_dynamic_mode():
            p_grads = paddle.grad(
                p_log_norm, p_natural_params, create_graph=True
            )
        else:
            p_grads = paddle.static.gradients(p_log_norm, p_natural_params)
    except RuntimeError as e:
        raise TypeError(
            "Cann't compute kl_divergence({cls_p}, {cls_q}) use bregman divergence. Please register_kl({cls_p}, {cls_q}).".format(
                cls_p=type(p).__name__, cls_q=type(q).__name__
            )
        ) from e

    kl = q._log_normalizer(*q_natural_params) - p_log_norm
    for p_param, q_param, p_grad in zip(
        p_natural_params, q_natural_params, p_grads
    ):
        term = (q_param - p_param) * p_grad
        kl -= _sum_rightmost(term, len(q.event_shape))

    return kl


@register_kl(LogNormal, LogNormal)
def _kl_lognormal_lognormal(p, q):
    return p._base.kl_divergence(q._base)


def _sum_rightmost(value, n):
    return value.sum(list(range(-n, 0))) if n > 0 else value