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Source code for torch.distributions.kl

import math
import warnings
from functools import total_ordering
from typing import Type, Dict, Callable, Tuple

import torch
from torch._six import inf

from .bernoulli import Bernoulli
from .beta import Beta
from .binomial import Binomial
from .categorical import Categorical
from .cauchy import Cauchy
from .continuous_bernoulli import ContinuousBernoulli
from .dirichlet import Dirichlet
from .distribution import Distribution
from .exponential import Exponential
from .exp_family import ExponentialFamily
from .gamma import Gamma
from .geometric import Geometric
from .gumbel import Gumbel
from .half_normal import HalfNormal
from .independent import Independent
from .laplace import Laplace
from .lowrank_multivariate_normal import (LowRankMultivariateNormal, _batch_lowrank_logdet,
                                          _batch_lowrank_mahalanobis)
from .multivariate_normal import (MultivariateNormal, _batch_mahalanobis)
from .normal import Normal
from .one_hot_categorical import OneHotCategorical
from .pareto import Pareto
from .poisson import Poisson
from .transformed_distribution import TransformedDistribution
from .uniform import Uniform
from .utils import _sum_rightmost

_KL_REGISTRY = {}  # Source of truth mapping a few general (type, type) pairs to functions.
_KL_MEMOIZE: Dict[Tuple[Type, Type], Callable] = {}  # Memoized version mapping many specific (type, type) pairs to functions.


[docs]def register_kl(type_p, type_q): """ Decorator to register a pairwise function with :meth:`kl_divergence`. Usage:: @register_kl(Normal, Normal) def kl_normal_normal(p, q): # insert implementation here Lookup returns the most specific (type,type) match ordered by subclass. If the match is ambiguous, a `RuntimeWarning` is raised. For example to resolve the ambiguous situation:: @register_kl(BaseP, DerivedQ) def kl_version1(p, q): ... @register_kl(DerivedP, BaseQ) def kl_version2(p, q): ... you should register a third most-specific implementation, e.g.:: register_kl(DerivedP, DerivedQ)(kl_version1) # Break the tie. Args: type_p (type): A subclass of :class:`~torch.distributions.Distribution`. type_q (type): A subclass of :class:`~torch.distributions.Distribution`. """ if not isinstance(type_p, type) and issubclass(type_p, Distribution): raise TypeError('Expected type_p to be a Distribution subclass but got {}'.format(type_p)) if not isinstance(type_q, type) and issubclass(type_q, Distribution): raise TypeError('Expected type_q to be a Distribution subclass but got {}'.format(type_q)) def decorator(fun): _KL_REGISTRY[type_p, type_q] = fun _KL_MEMOIZE.clear() # reset since lookup order may have changed return fun return decorator
@total_ordering class _Match(object): __slots__ = ['types'] def __init__(self, *types): self.types = types def __eq__(self, other): return self.types == other.types def __le__(self, other): for x, y in zip(self.types, other.types): if not issubclass(x, y): return False if x is not y: break return True def _dispatch_kl(type_p, type_q): """ Find the most specific approximate match, assuming single inheritance. """ matches = [(super_p, super_q) for super_p, super_q in _KL_REGISTRY if issubclass(type_p, super_p) and issubclass(type_q, super_q)] if not matches: return NotImplemented # Check that the left- and right- lexicographic orders agree. # mypy isn't smart enough to know that _Match implements __lt__ # see: https://github.com/python/typing/issues/760#issuecomment-710670503 left_p, left_q = min(_Match(*m) for m in matches).types # type: ignore right_q, right_p = min(_Match(*reversed(m)) for m in matches).types # type: ignore left_fun = _KL_REGISTRY[left_p, left_q] right_fun = _KL_REGISTRY[right_p, right_q] if left_fun is not right_fun: warnings.warn('Ambiguous kl_divergence({}, {}). Please register_kl({}, {})'.format( type_p.__name__, type_q.__name__, left_p.__name__, right_q.__name__), RuntimeWarning) return left_fun def _infinite_like(tensor): """ Helper function for obtaining infinite KL Divergence throughout """ return torch.full_like(tensor, inf) def _x_log_x(tensor): """ Utility function for calculating x log x """ return tensor * tensor.log() def _batch_trace_XXT(bmat): """ Utility function for calculating the trace of XX^{T} with X having arbitrary trailing batch dimensions """ n = bmat.size(-1) m = bmat.size(-2) flat_trace = bmat.reshape(-1, m * n).pow(2).sum(-1) return flat_trace.reshape(bmat.shape[:-2])
[docs]def kl_divergence(p, q): r""" Compute Kullback-Leibler divergence :math:`KL(p \| q)` between two distributions. .. math:: KL(p \| q) = \int p(x) \log\frac {p(x)} {q(x)} \,dx Args: p (Distribution): A :class:`~torch.distributions.Distribution` object. q (Distribution): A :class:`~torch.distributions.Distribution` object. Returns: Tensor: A batch of KL divergences of shape `batch_shape`. Raises: NotImplementedError: If the distribution types have not been registered via :meth:`register_kl`. """ try: fun = _KL_MEMOIZE[type(p), type(q)] except KeyError: fun = _dispatch_kl(type(p), type(q)) _KL_MEMOIZE[type(p), type(q)] = fun if fun is NotImplemented: raise NotImplementedError return fun(p, q)
################################################################################ # KL Divergence Implementations ################################################################################ _euler_gamma = 0.57721566490153286060 # Same distributions @register_kl(Bernoulli, Bernoulli) def _kl_bernoulli_bernoulli(p, q): t1 = p.probs * (p.probs / q.probs).log() t1[q.probs == 0] = inf t1[p.probs == 0] = 0 t2 = (1 - p.probs) * ((1 - p.probs) / (1 - q.probs)).log() t2[q.probs == 1] = inf t2[p.probs == 1] = 0 return t1 + t2 @register_kl(Beta, Beta) def _kl_beta_beta(p, q): sum_params_p = p.concentration1 + p.concentration0 sum_params_q = q.concentration1 + q.concentration0 t1 = q.concentration1.lgamma() + q.concentration0.lgamma() + (sum_params_p).lgamma() t2 = p.concentration1.lgamma() + p.concentration0.lgamma() + (sum_params_q).lgamma() t3 = (p.concentration1 - q.concentration1) * torch.digamma(p.concentration1) t4 = (p.concentration0 - q.concentration0) * torch.digamma(p.concentration0) t5 = (sum_params_q - sum_params_p) * torch.digamma(sum_params_p) return t1 - t2 + t3 + t4 + t5 @register_kl(Binomial, Binomial) def _kl_binomial_binomial(p, q): # from https://math.stackexchange.com/questions/2214993/ # kullback-leibler-divergence-for-binomial-distributions-p-and-q if (p.total_count < q.total_count).any(): raise NotImplementedError('KL between Binomials where q.total_count > p.total_count is not implemented') kl = p.total_count * (p.probs * (p.logits - q.logits) + (-p.probs).log1p() - (-q.probs).log1p()) inf_idxs = p.total_count > q.total_count kl[inf_idxs] = _infinite_like(kl[inf_idxs]) return kl @register_kl(Categorical, Categorical) def _kl_categorical_categorical(p, q): t = p.probs * (p.logits - q.logits) t[(q.probs == 0).expand_as(t)] = inf t[(p.probs == 0).expand_as(t)] = 0 return t.sum(-1) @register_kl(ContinuousBernoulli, ContinuousBernoulli) def _kl_continuous_bernoulli_continuous_bernoulli(p, q): t1 = p.mean * (p.logits - q.logits) t2 = p._cont_bern_log_norm() + torch.log1p(-p.probs) t3 = - q._cont_bern_log_norm() - torch.log1p(-q.probs) return t1 + t2 + t3 @register_kl(Dirichlet, Dirichlet) def _kl_dirichlet_dirichlet(p, q): # From http://bariskurt.com/kullback-leibler-divergence-between-two-dirichlet-and-beta-distributions/ sum_p_concentration = p.concentration.sum(-1) sum_q_concentration = q.concentration.sum(-1) t1 = sum_p_concentration.lgamma() - sum_q_concentration.lgamma() t2 = (p.concentration.lgamma() - q.concentration.lgamma()).sum(-1) t3 = p.concentration - q.concentration t4 = p.concentration.digamma() - sum_p_concentration.digamma().unsqueeze(-1) return t1 - t2 + (t3 * t4).sum(-1) @register_kl(Exponential, Exponential) def _kl_exponential_exponential(p, q): rate_ratio = q.rate / p.rate t1 = -rate_ratio.log() return t1 + rate_ratio - 1 @register_kl(ExponentialFamily, ExponentialFamily) def _kl_expfamily_expfamily(p, q): if not type(p) == type(q): raise NotImplementedError("The cross KL-divergence between different exponential families cannot \ be computed using Bregman divergences") p_nparams = [np.detach().requires_grad_() for np in p._natural_params] q_nparams = q._natural_params lg_normal = p._log_normalizer(*p_nparams) gradients = torch.autograd.grad(lg_normal.sum(), p_nparams, create_graph=True) result = q._log_normalizer(*q_nparams) - lg_normal for pnp, qnp, g in zip(p_nparams, q_nparams, gradients): term = (qnp - pnp) * g result -= _sum_rightmost(term, len(q.event_shape)) return result @register_kl(Gamma, Gamma) def _kl_gamma_gamma(p, q): t1 = q.concentration * (p.rate / q.rate).log() t2 = torch.lgamma(q.concentration) - torch.lgamma(p.concentration) t3 = (p.concentration - q.concentration) * torch.digamma(p.concentration) t4 = (q.rate - p.rate) * (p.concentration / p.rate) return t1 + t2 + t3 + t4 @register_kl(Gumbel, Gumbel) def _kl_gumbel_gumbel(p, q): ct1 = p.scale / q.scale ct2 = q.loc / q.scale ct3 = p.loc / q.scale t1 = -ct1.log() - ct2 + ct3 t2 = ct1 * _euler_gamma t3 = torch.exp(ct2 + (1 + ct1).lgamma() - ct3) return t1 + t2 + t3 - (1 + _euler_gamma) @register_kl(Geometric, Geometric) def _kl_geometric_geometric(p, q): return -p.entropy() - torch.log1p(-q.probs) / p.probs - q.logits @register_kl(HalfNormal, HalfNormal) def _kl_halfnormal_halfnormal(p, q): return _kl_normal_normal(p.base_dist, q.base_dist) @register_kl(Laplace, Laplace) def _kl_laplace_laplace(p, q): # From http://www.mast.queensu.ca/~communications/Papers/gil-msc11.pdf scale_ratio = p.scale / q.scale loc_abs_diff = (p.loc - q.loc).abs() t1 = -scale_ratio.log() t2 = loc_abs_diff / q.scale t3 = scale_ratio * torch.exp(-loc_abs_diff / p.scale) return t1 + t2 + t3 - 1 @register_kl(LowRankMultivariateNormal, LowRankMultivariateNormal) def _kl_lowrankmultivariatenormal_lowrankmultivariatenormal(p, q): if p.event_shape != q.event_shape: raise ValueError("KL-divergence between two Low Rank Multivariate Normals with\ different event shapes cannot be computed") term1 = (_batch_lowrank_logdet(q._unbroadcasted_cov_factor, q._unbroadcasted_cov_diag, q._capacitance_tril) - _batch_lowrank_logdet(p._unbroadcasted_cov_factor, p._unbroadcasted_cov_diag, p._capacitance_tril)) term3 = _batch_lowrank_mahalanobis(q._unbroadcasted_cov_factor, q._unbroadcasted_cov_diag, q.loc - p.loc, q._capacitance_tril) # Expands term2 according to # inv(qcov) @ pcov = [inv(qD) - inv(qD) @ qW @ inv(qC) @ qW.T @ inv(qD)] @ (pW @ pW.T + pD) # = [inv(qD) - A.T @ A] @ (pD + pW @ pW.T) qWt_qDinv = (q._unbroadcasted_cov_factor.transpose(-1, -2) / q._unbroadcasted_cov_diag.unsqueeze(-2)) A = torch.triangular_solve(qWt_qDinv, q._capacitance_tril, upper=False)[0] term21 = (p._unbroadcasted_cov_diag / q._unbroadcasted_cov_diag).sum(-1) term22 = _batch_trace_XXT(p._unbroadcasted_cov_factor * q._unbroadcasted_cov_diag.rsqrt().unsqueeze(-1)) term23 = _batch_trace_XXT(A * p._unbroadcasted_cov_diag.sqrt().unsqueeze(-2)) term24 = _batch_trace_XXT(A.matmul(p._unbroadcasted_cov_factor)) term2 = term21 + term22 - term23 - term24 return 0.5 * (term1 + term2 + term3 - p.event_shape[0]) @register_kl(MultivariateNormal, LowRankMultivariateNormal) def _kl_multivariatenormal_lowrankmultivariatenormal(p, q): if p.event_shape != q.event_shape: raise ValueError("KL-divergence between two (Low Rank) Multivariate Normals with\ different event shapes cannot be computed") term1 = (_batch_lowrank_logdet(q._unbroadcasted_cov_factor, q._unbroadcasted_cov_diag, q._capacitance_tril) - 2 * p._unbroadcasted_scale_tril.diagonal(dim1=-2, dim2=-1).log().sum(-1)) term3 = _batch_lowrank_mahalanobis(q._unbroadcasted_cov_factor, q._unbroadcasted_cov_diag, q.loc - p.loc, q._capacitance_tril) # Expands term2 according to # inv(qcov) @ pcov = [inv(qD) - inv(qD) @ qW @ inv(qC) @ qW.T @ inv(qD)] @ p_tril @ p_tril.T # = [inv(qD) - A.T @ A] @ p_tril @ p_tril.T qWt_qDinv = (q._unbroadcasted_cov_factor.transpose(-1, -2) / q._unbroadcasted_cov_diag.unsqueeze(-2)) A = torch.triangular_solve(qWt_qDinv, q._capacitance_tril, upper=False)[0] term21 = _batch_trace_XXT(p._unbroadcasted_scale_tril * q._unbroadcasted_cov_diag.rsqrt().unsqueeze(-1)) term22 = _batch_trace_XXT(A.matmul(p._unbroadcasted_scale_tril)) term2 = term21 - term22 return 0.5 * (term1 + term2 + term3 - p.event_shape[0]) @register_kl(LowRankMultivariateNormal, MultivariateNormal) def _kl_lowrankmultivariatenormal_multivariatenormal(p, q): if p.event_shape != q.event_shape: raise ValueError("KL-divergence between two (Low Rank) Multivariate Normals with\ different event shapes cannot be computed") term1 = (2 * q._unbroadcasted_scale_tril.diagonal(dim1=-2, dim2=-1).log().sum(-1) - _batch_lowrank_logdet(p._unbroadcasted_cov_factor, p._unbroadcasted_cov_diag, p._capacitance_tril)) term3 = _batch_mahalanobis(q._unbroadcasted_scale_tril, (q.loc - p.loc)) # Expands term2 according to # inv(qcov) @ pcov = inv(q_tril @ q_tril.T) @ (pW @ pW.T + pD) combined_batch_shape = torch._C._infer_size(q._unbroadcasted_scale_tril.shape[:-2], p._unbroadcasted_cov_factor.shape[:-2]) n = p.event_shape[0] q_scale_tril = q._unbroadcasted_scale_tril.expand(combined_batch_shape + (n, n)) p_cov_factor = p._unbroadcasted_cov_factor.expand(combined_batch_shape + (n, p.cov_factor.size(-1))) p_cov_diag = (torch.diag_embed(p._unbroadcasted_cov_diag.sqrt()) .expand(combined_batch_shape + (n, n))) term21 = _batch_trace_XXT(torch.triangular_solve(p_cov_factor, q_scale_tril, upper=False)[0]) term22 = _batch_trace_XXT(torch.triangular_solve(p_cov_diag, q_scale_tril, upper=False)[0]) term2 = term21 + term22 return 0.5 * (term1 + term2 + term3 - p.event_shape[0]) @register_kl(MultivariateNormal, MultivariateNormal) def _kl_multivariatenormal_multivariatenormal(p, q): # From https://en.wikipedia.org/wiki/Multivariate_normal_distribution#Kullback%E2%80%93Leibler_divergence if p.event_shape != q.event_shape: raise ValueError("KL-divergence between two Multivariate Normals with\ different event shapes cannot be computed") half_term1 = (q._unbroadcasted_scale_tril.diagonal(dim1=-2, dim2=-1).log().sum(-1) - p._unbroadcasted_scale_tril.diagonal(dim1=-2, dim2=-1).log().sum(-1)) combined_batch_shape = torch._C._infer_size(q._unbroadcasted_scale_tril.shape[:-2], p._unbroadcasted_scale_tril.shape[:-2]) n = p.event_shape[0] q_scale_tril = q._unbroadcasted_scale_tril.expand(combined_batch_shape + (n, n)) p_scale_tril = p._unbroadcasted_scale_tril.expand(combined_batch_shape + (n, n)) term2 = _batch_trace_XXT(torch.triangular_solve(p_scale_tril, q_scale_tril, upper=False)[0]) term3 = _batch_mahalanobis(q._unbroadcasted_scale_tril, (q.loc - p.loc)) return half_term1 + 0.5 * (term2 + term3 - n) @register_kl(Normal, Normal) def _kl_normal_normal(p, q): var_ratio = (p.scale / q.scale).pow(2) t1 = ((p.loc - q.loc) / q.scale).pow(2) return 0.5 * (var_ratio + t1 - 1 - var_ratio.log()) @register_kl(OneHotCategorical, OneHotCategorical) def _kl_onehotcategorical_onehotcategorical(p, q): return _kl_categorical_categorical(p._categorical, q._categorical) @register_kl(Pareto, Pareto) def _kl_pareto_pareto(p, q): # From http://www.mast.queensu.ca/~communications/Papers/gil-msc11.pdf scale_ratio = p.scale / q.scale alpha_ratio = q.alpha / p.alpha t1 = q.alpha * scale_ratio.log() t2 = -alpha_ratio.log() result = t1 + t2 + alpha_ratio - 1 result[p.support.lower_bound < q.support.lower_bound] = inf return result @register_kl(Poisson, Poisson) def _kl_poisson_poisson(p, q): return p.rate * (p.rate.log() - q.rate.log()) - (p.rate - q.rate) @register_kl(TransformedDistribution, TransformedDistribution) def _kl_transformed_transformed(p, q): if p.transforms != q.transforms: raise NotImplementedError if p.event_shape != q.event_shape: raise NotImplementedError # extra_event_dim = len(p.event_shape) - len(p.base_dist.event_shape) extra_event_dim = len(p.event_shape) base_kl_divergence = kl_divergence(p.base_dist, q.base_dist) return _sum_rightmost(base_kl_divergence, extra_event_dim) @register_kl(Uniform, Uniform) def _kl_uniform_uniform(p, q): result = ((q.high - q.low) / (p.high - p.low)).log() result[(q.low > p.low) | (q.high < p.high)] = inf return result # Different distributions @register_kl(Bernoulli, Poisson) def _kl_bernoulli_poisson(p, q): return -p.entropy() - (p.probs * q.rate.log() - q.rate) @register_kl(Beta, ContinuousBernoulli) def _kl_beta_continuous_bernoulli(p, q): return -p.entropy() - p.mean * q.logits - torch.log1p(-q.probs) - q._cont_bern_log_norm() @register_kl(Beta, Pareto) def _kl_beta_infinity(p, q): return _infinite_like(p.concentration1) @register_kl(Beta, Exponential) def _kl_beta_exponential(p, q): return -p.entropy() - q.rate.log() + q.rate * (p.concentration1 / (p.concentration1 + p.concentration0)) @register_kl(Beta, Gamma) def _kl_beta_gamma(p, q): t1 = -p.entropy() t2 = q.concentration.lgamma() - q.concentration * q.rate.log() t3 = (q.concentration - 1) * (p.concentration1.digamma() - (p.concentration1 + p.concentration0).digamma()) t4 = q.rate * p.concentration1 / (p.concentration1 + p.concentration0) return t1 + t2 - t3 + t4 # TODO: Add Beta-Laplace KL Divergence @register_kl(Beta, Normal) def _kl_beta_normal(p, q): E_beta = p.concentration1 / (p.concentration1 + p.concentration0) var_normal = q.scale.pow(2) t1 = -p.entropy() t2 = 0.5 * (var_normal * 2 * math.pi).log() t3 = (E_beta * (1 - E_beta) / (p.concentration1 + p.concentration0 + 1) + E_beta.pow(2)) * 0.5 t4 = q.loc * E_beta t5 = q.loc.pow(2) * 0.5 return t1 + t2 + (t3 - t4 + t5) / var_normal @register_kl(Beta, Uniform) def _kl_beta_uniform(p, q): result = -p.entropy() + (q.high - q.low).log() result[(q.low > p.support.lower_bound) | (q.high < p.support.upper_bound)] = inf return result # Note that the KL between a ContinuousBernoulli and Beta has no closed form @register_kl(ContinuousBernoulli, Pareto) def _kl_continuous_bernoulli_infinity(p, q): return _infinite_like(p.probs) @register_kl(ContinuousBernoulli, Exponential) def _kl_continuous_bernoulli_exponential(p, q): return -p.entropy() - torch.log(q.rate) + q.rate * p.mean # Note that the KL between a ContinuousBernoulli and Gamma has no closed form # TODO: Add ContinuousBernoulli-Laplace KL Divergence @register_kl(ContinuousBernoulli, Normal) def _kl_continuous_bernoulli_normal(p, q): t1 = -p.entropy() t2 = 0.5 * (math.log(2. * math.pi) + torch.square(q.loc / q.scale)) + torch.log(q.scale) t3 = (p.variance + torch.square(p.mean) - 2. * q.loc * p.mean) / (2.0 * torch.square(q.scale)) return t1 + t2 + t3 @register_kl(ContinuousBernoulli, Uniform) def _kl_continuous_bernoulli_uniform(p, q): result = -p.entropy() + (q.high - q.low).log() return torch.where(torch.max(torch.ge(q.low, p.support.lower_bound), torch.le(q.high, p.support.upper_bound)), torch.ones_like(result) * inf, result) @register_kl(Exponential, Beta) @register_kl(Exponential, ContinuousBernoulli) @register_kl(Exponential, Pareto) @register_kl(Exponential, Uniform) def _kl_exponential_infinity(p, q): return _infinite_like(p.rate) @register_kl(Exponential, Gamma) def _kl_exponential_gamma(p, q): ratio = q.rate / p.rate t1 = -q.concentration * torch.log(ratio) return t1 + ratio + q.concentration.lgamma() + q.concentration * _euler_gamma - (1 + _euler_gamma) @register_kl(Exponential, Gumbel) def _kl_exponential_gumbel(p, q): scale_rate_prod = p.rate * q.scale loc_scale_ratio = q.loc / q.scale t1 = scale_rate_prod.log() - 1 t2 = torch.exp(loc_scale_ratio) * scale_rate_prod / (scale_rate_prod + 1) t3 = scale_rate_prod.reciprocal() return t1 - loc_scale_ratio + t2 + t3 # TODO: Add Exponential-Laplace KL Divergence @register_kl(Exponential, Normal) def _kl_exponential_normal(p, q): var_normal = q.scale.pow(2) rate_sqr = p.rate.pow(2) t1 = 0.5 * torch.log(rate_sqr * var_normal * 2 * math.pi) t2 = rate_sqr.reciprocal() t3 = q.loc / p.rate t4 = q.loc.pow(2) * 0.5 return t1 - 1 + (t2 - t3 + t4) / var_normal @register_kl(Gamma, Beta) @register_kl(Gamma, ContinuousBernoulli) @register_kl(Gamma, Pareto) @register_kl(Gamma, Uniform) def _kl_gamma_infinity(p, q): return _infinite_like(p.concentration) @register_kl(Gamma, Exponential) def _kl_gamma_exponential(p, q): return -p.entropy() - q.rate.log() + q.rate * p.concentration / p.rate @register_kl(Gamma, Gumbel) def _kl_gamma_gumbel(p, q): beta_scale_prod = p.rate * q.scale loc_scale_ratio = q.loc / q.scale t1 = (p.concentration - 1) * p.concentration.digamma() - p.concentration.lgamma() - p.concentration t2 = beta_scale_prod.log() + p.concentration / beta_scale_prod t3 = torch.exp(loc_scale_ratio) * (1 + beta_scale_prod.reciprocal()).pow(-p.concentration) - loc_scale_ratio return t1 + t2 + t3 # TODO: Add Gamma-Laplace KL Divergence @register_kl(Gamma, Normal) def _kl_gamma_normal(p, q): var_normal = q.scale.pow(2) beta_sqr = p.rate.pow(2) t1 = 0.5 * torch.log(beta_sqr * var_normal * 2 * math.pi) - p.concentration - p.concentration.lgamma() t2 = 0.5 * (p.concentration.pow(2) + p.concentration) / beta_sqr t3 = q.loc * p.concentration / p.rate t4 = 0.5 * q.loc.pow(2) return t1 + (p.concentration - 1) * p.concentration.digamma() + (t2 - t3 + t4) / var_normal @register_kl(Gumbel, Beta) @register_kl(Gumbel, ContinuousBernoulli) @register_kl(Gumbel, Exponential) @register_kl(Gumbel, Gamma) @register_kl(Gumbel, Pareto) @register_kl(Gumbel, Uniform) def _kl_gumbel_infinity(p, q): return _infinite_like(p.loc) # TODO: Add Gumbel-Laplace KL Divergence @register_kl(Gumbel, Normal) def _kl_gumbel_normal(p, q): param_ratio = p.scale / q.scale t1 = (param_ratio / math.sqrt(2 * math.pi)).log() t2 = (math.pi * param_ratio * 0.5).pow(2) / 3 t3 = ((p.loc + p.scale * _euler_gamma - q.loc) / q.scale).pow(2) * 0.5 return -t1 + t2 + t3 - (_euler_gamma + 1) @register_kl(Laplace, Beta) @register_kl(Laplace, ContinuousBernoulli) @register_kl(Laplace, Exponential) @register_kl(Laplace, Gamma) @register_kl(Laplace, Pareto) @register_kl(Laplace, Uniform) def _kl_laplace_infinity(p, q): return _infinite_like(p.loc) @register_kl(Laplace, Normal) def _kl_laplace_normal(p, q): var_normal = q.scale.pow(2) scale_sqr_var_ratio = p.scale.pow(2) / var_normal t1 = 0.5 * torch.log(2 * scale_sqr_var_ratio / math.pi) t2 = 0.5 * p.loc.pow(2) t3 = p.loc * q.loc t4 = 0.5 * q.loc.pow(2) return -t1 + scale_sqr_var_ratio + (t2 - t3 + t4) / var_normal - 1 @register_kl(Normal, Beta) @register_kl(Normal, ContinuousBernoulli) @register_kl(Normal, Exponential) @register_kl(Normal, Gamma) @register_kl(Normal, Pareto) @register_kl(Normal, Uniform) def _kl_normal_infinity(p, q): return _infinite_like(p.loc) @register_kl(Normal, Gumbel) def _kl_normal_gumbel(p, q): mean_scale_ratio = p.loc / q.scale var_scale_sqr_ratio = (p.scale / q.scale).pow(2) loc_scale_ratio = q.loc / q.scale t1 = var_scale_sqr_ratio.log() * 0.5 t2 = mean_scale_ratio - loc_scale_ratio t3 = torch.exp(-mean_scale_ratio + 0.5 * var_scale_sqr_ratio + loc_scale_ratio) return -t1 + t2 + t3 - (0.5 * (1 + math.log(2 * math.pi))) # TODO: Add Normal-Laplace KL Divergence @register_kl(Pareto, Beta) @register_kl(Pareto, ContinuousBernoulli) @register_kl(Pareto, Uniform) def _kl_pareto_infinity(p, q): return _infinite_like(p.scale) @register_kl(Pareto, Exponential) def _kl_pareto_exponential(p, q): scale_rate_prod = p.scale * q.rate t1 = (p.alpha / scale_rate_prod).log() t2 = p.alpha.reciprocal() t3 = p.alpha * scale_rate_prod / (p.alpha - 1) result = t1 - t2 + t3 - 1 result[p.alpha <= 1] = inf return result @register_kl(Pareto, Gamma) def _kl_pareto_gamma(p, q): common_term = p.scale.log() + p.alpha.reciprocal() t1 = p.alpha.log() - common_term t2 = q.concentration.lgamma() - q.concentration * q.rate.log() t3 = (1 - q.concentration) * common_term t4 = q.rate * p.alpha * p.scale / (p.alpha - 1) result = t1 + t2 + t3 + t4 - 1 result[p.alpha <= 1] = inf return result # TODO: Add Pareto-Laplace KL Divergence @register_kl(Pareto, Normal) def _kl_pareto_normal(p, q): var_normal = 2 * q.scale.pow(2) common_term = p.scale / (p.alpha - 1) t1 = (math.sqrt(2 * math.pi) * q.scale * p.alpha / p.scale).log() t2 = p.alpha.reciprocal() t3 = p.alpha * common_term.pow(2) / (p.alpha - 2) t4 = (p.alpha * common_term - q.loc).pow(2) result = t1 - t2 + (t3 + t4) / var_normal - 1 result[p.alpha <= 2] = inf return result @register_kl(Poisson, Bernoulli) @register_kl(Poisson, Binomial) def _kl_poisson_infinity(p, q): return _infinite_like(p.rate) @register_kl(Uniform, Beta) def _kl_uniform_beta(p, q): common_term = p.high - p.low t1 = torch.log(common_term) t2 = (q.concentration1 - 1) * (_x_log_x(p.high) - _x_log_x(p.low) - common_term) / common_term t3 = (q.concentration0 - 1) * (_x_log_x((1 - p.high)) - _x_log_x((1 - p.low)) + common_term) / common_term t4 = q.concentration1.lgamma() + q.concentration0.lgamma() - (q.concentration1 + q.concentration0).lgamma() result = t3 + t4 - t1 - t2 result[(p.high > q.support.upper_bound) | (p.low < q.support.lower_bound)] = inf return result @register_kl(Uniform, ContinuousBernoulli) def _kl_uniform_continuous_bernoulli(p, q): result = -p.entropy() - p.mean * q.logits - torch.log1p(-q.probs) - q._cont_bern_log_norm() return torch.where(torch.max(torch.ge(p.high, q.support.upper_bound), torch.le(p.low, q.support.lower_bound)), torch.ones_like(result) * inf, result) @register_kl(Uniform, Exponential) def _kl_uniform_exponetial(p, q): result = q.rate * (p.high + p.low) / 2 - ((p.high - p.low) * q.rate).log() result[p.low < q.support.lower_bound] = inf return result @register_kl(Uniform, Gamma) def _kl_uniform_gamma(p, q): common_term = p.high - p.low t1 = common_term.log() t2 = q.concentration.lgamma() - q.concentration * q.rate.log() t3 = (1 - q.concentration) * (_x_log_x(p.high) - _x_log_x(p.low) - common_term) / common_term t4 = q.rate * (p.high + p.low) / 2 result = -t1 + t2 + t3 + t4 result[p.low < q.support.lower_bound] = inf return result @register_kl(Uniform, Gumbel) def _kl_uniform_gumbel(p, q): common_term = q.scale / (p.high - p.low) high_loc_diff = (p.high - q.loc) / q.scale low_loc_diff = (p.low - q.loc) / q.scale t1 = common_term.log() + 0.5 * (high_loc_diff + low_loc_diff) t2 = common_term * (torch.exp(-high_loc_diff) - torch.exp(-low_loc_diff)) return t1 - t2 # TODO: Uniform-Laplace KL Divergence @register_kl(Uniform, Normal) def _kl_uniform_normal(p, q): common_term = p.high - p.low t1 = (math.sqrt(math.pi * 2) * q.scale / common_term).log() t2 = (common_term).pow(2) / 12 t3 = ((p.high + p.low - 2 * q.loc) / 2).pow(2) return t1 + 0.5 * (t2 + t3) / q.scale.pow(2) @register_kl(Uniform, Pareto) def _kl_uniform_pareto(p, q): support_uniform = p.high - p.low t1 = (q.alpha * q.scale.pow(q.alpha) * (support_uniform)).log() t2 = (_x_log_x(p.high) - _x_log_x(p.low) - support_uniform) / support_uniform result = t2 * (q.alpha + 1) - t1 result[p.low < q.support.lower_bound] = inf return result @register_kl(Independent, Independent) def _kl_independent_independent(p, q): if p.reinterpreted_batch_ndims != q.reinterpreted_batch_ndims: raise NotImplementedError result = kl_divergence(p.base_dist, q.base_dist) return _sum_rightmost(result, p.reinterpreted_batch_ndims) @register_kl(Cauchy, Cauchy) def _kl_cauchy_cauchy(p, q): # From https://arxiv.org/abs/1905.10965 t1 = ((p.scale + q.scale).pow(2) + (p.loc - q.loc).pow(2)).log() t2 = (4 * p.scale * q.scale).log() return t1 - t2

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