|
|
|
import math |
|
import warnings |
|
from functools import total_ordering |
|
from typing import Callable |
|
|
|
import torch |
|
from torch import inf, Tensor |
|
|
|
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 .exp_family import ExponentialFamily |
|
from .exponential import Exponential |
|
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 ( |
|
_batch_lowrank_logdet, |
|
_batch_lowrank_mahalanobis, |
|
LowRankMultivariateNormal, |
|
) |
|
from .multivariate_normal import _batch_mahalanobis, MultivariateNormal |
|
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, euler_constant as _euler_gamma |
|
|
|
|
|
_KL_REGISTRY: dict[ |
|
tuple[type, type], Callable |
|
] = {} |
|
_KL_MEMOIZE: dict[ |
|
tuple[type, type], Callable |
|
] = {} |
|
|
|
__all__ = ["register_kl", "kl_divergence"] |
|
|
|
|
|
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( |
|
f"Expected type_p to be a Distribution subclass but got {type_p}" |
|
) |
|
if not isinstance(type_q, type) and issubclass(type_q, Distribution): |
|
raise TypeError( |
|
f"Expected type_q to be a Distribution subclass but got {type_q}" |
|
) |
|
|
|
def decorator(fun): |
|
_KL_REGISTRY[type_p, type_q] = fun |
|
_KL_MEMOIZE.clear() |
|
return fun |
|
|
|
return decorator |
|
|
|
|
|
@total_ordering |
|
class _Match: |
|
__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 |
|
|
|
|
|
|
|
left_p, left_q = min(_Match(*m) for m in matches).types |
|
right_q, right_p = min(_Match(*reversed(m)) for m in matches).types |
|
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( |
|
f"Ambiguous kl_divergence({type_p.__name__}, {type_q.__name__}). " |
|
f"Please register_kl({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 torch.special.xlogy(tensor, tensor) |
|
|
|
|
|
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]) |
|
|
|
|
|
def kl_divergence(p: Distribution, q: Distribution) -> Tensor: |
|
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( |
|
f"No KL(p || q) is implemented for p type {p.__class__.__name__} and q type {q.__class__.__name__}" |
|
) |
|
return fun(p, q) |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
@register_kl(Bernoulli, Bernoulli) |
|
def _kl_bernoulli_bernoulli(p, q): |
|
t1 = p.probs * ( |
|
torch.nn.functional.softplus(-q.logits) |
|
- torch.nn.functional.softplus(-p.logits) |
|
) |
|
t1[q.probs == 0] = inf |
|
t1[p.probs == 0] = 0 |
|
t2 = (1 - p.probs) * ( |
|
torch.nn.functional.softplus(q.logits) - torch.nn.functional.softplus(p.logits) |
|
) |
|
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): |
|
|
|
|
|
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): |
|
|
|
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): |
|
|
|
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, |
|
) |
|
|
|
|
|
|
|
qWt_qDinv = q._unbroadcasted_cov_factor.mT / q._unbroadcasted_cov_diag.unsqueeze(-2) |
|
A = torch.linalg.solve_triangular(q._capacitance_tril, qWt_qDinv, upper=False) |
|
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, |
|
) |
|
|
|
|
|
|
|
qWt_qDinv = q._unbroadcasted_cov_factor.mT / q._unbroadcasted_cov_diag.unsqueeze(-2) |
|
A = torch.linalg.solve_triangular(q._capacitance_tril, qWt_qDinv, upper=False) |
|
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)) |
|
|
|
|
|
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.linalg.solve_triangular(q_scale_tril, p_cov_factor, upper=False) |
|
) |
|
term22 = _batch_trace_XXT( |
|
torch.linalg.solve_triangular(q_scale_tril, p_cov_diag, upper=False) |
|
) |
|
term2 = term21 + term22 |
|
return 0.5 * (term1 + term2 + term3 - p.event_shape[0]) |
|
|
|
|
|
@register_kl(MultivariateNormal, MultivariateNormal) |
|
def _kl_multivariatenormal_multivariatenormal(p, q): |
|
|
|
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.linalg.solve_triangular(q_scale_tril, p_scale_tril, upper=False) |
|
) |
|
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): |
|
|
|
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 |
|
return kl_divergence(p.base_dist, q.base_dist) |
|
|
|
|
|
@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 |
|
|
|
|
|
|
|
@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 |
|
|
|
|
|
|
|
|
|
|
|
@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 |
|
|
|
|
|
|
|
|
|
|
|
@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 |
|
|
|
|
|
|
|
|
|
|
|
|
|
@register_kl(ContinuousBernoulli, Normal) |
|
def _kl_continuous_bernoulli_normal(p, q): |
|
t1 = -p.entropy() |
|
t2 = 0.5 * (math.log(2.0 * math.pi) + torch.square(q.loc / q.scale)) + torch.log( |
|
q.scale |
|
) |
|
t3 = (p.variance + torch.square(p.mean) - 2.0 * 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 |
|
|
|
|
|
|
|
|
|
|
|
@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 |
|
|
|
|
|
|
|
|
|
|
|
@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) |
|
|
|
|
|
|
|
|
|
|
|
@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))) |
|
|
|
|
|
@register_kl(Normal, Laplace) |
|
def _kl_normal_laplace(p, q): |
|
loc_diff = p.loc - q.loc |
|
scale_ratio = p.scale / q.scale |
|
loc_diff_scale_ratio = loc_diff / p.scale |
|
t1 = torch.log(scale_ratio) |
|
t2 = ( |
|
math.sqrt(2 / math.pi) * p.scale * torch.exp(-0.5 * loc_diff_scale_ratio.pow(2)) |
|
) |
|
t3 = loc_diff * torch.erf(math.sqrt(0.5) * loc_diff_scale_ratio) |
|
return -t1 + (t2 + t3) / q.scale - (0.5 * (1 + math.log(0.5 * math.pi))) |
|
|
|
|
|
@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 |
|
|
|
|
|
|
|
|
|
|
|
@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 |
|
|
|
|
|
|
|
|
|
|
|
@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): |
|
|
|
t1 = ((p.scale + q.scale).pow(2) + (p.loc - q.loc).pow(2)).log() |
|
t2 = (4 * p.scale * q.scale).log() |
|
return t1 - t2 |
|
|
|
|
|
def _add_kl_info(): |
|
"""Appends a list of implemented KL functions to the doc for kl_divergence.""" |
|
rows = [ |
|
"KL divergence is currently implemented for the following distribution pairs:" |
|
] |
|
for p, q in sorted( |
|
_KL_REGISTRY, key=lambda p_q: (p_q[0].__name__, p_q[1].__name__) |
|
): |
|
rows.append( |
|
f"* :class:`~torch.distributions.{p.__name__}` and :class:`~torch.distributions.{q.__name__}`" |
|
) |
|
kl_info = "\n\t".join(rows) |
|
if kl_divergence.__doc__: |
|
kl_divergence.__doc__ += kl_info |
|
|