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Stochastic Backpropagation through Mixture Density Distributions
Alex Graves
TL;DR
The paper addresses how to compute expectation derivatives with respect to normalized mixture weights. It uses a multivariate quantile transform and Monte Carlo estimation; the resulting estimator does not require the transform to be known when samples from the distribution are available.
Problem
The section addresses how to obtain derivatives of an expectation with respect to normalized mixture weights.
Source: Application to Mixture Density Weights · p. 3Method
The approach uses a multivariate quantile transform and Monte Carlo estimation, with conditional sampling and tractable conditional CDFs for the intractable integral.
Source: General Result · p. 2Results
The Monte Carlo estimator does not require the quantile transform to be known, provided samples from f are available.
Source: General Result · p. 3Takeaways & Limitations
The derivation obtains expectation derivatives for normalized mixture weights, including weights normalized by softmax.
Source: Application to Mixture Density Weights · p. 3The mixture-weight derivation assumes diagonal-covariance components; extension to non-diagonal components is left for future work.
Source: Application to Mixture Density Weights · p. 3
Abstract
from arXiv · showhide
The ability to backpropagate stochastic gradients through continuous latent distributions has been crucial to the emergence of variational autoencoders and stochastic gradient variational Bayes. The key ingredient is an unbiased and low-variance way of estimating gradients with respect to distribution parameters from gradients evaluated at distribution samples. The "reparameterization trick" provides a class of transforms yielding such estimators for many continuous distributions, including the Gaussian and other members of the location-scale family. However the trick does not readily extend to mixture density models, due to the difficulty of reparameterizing the discrete distribution over mixture weights. This report describes an alternative transform, applicable to any continuous multivariate distribution with a differentiable density function from which samples can be drawn, and uses it to derive an unbiased estimator for mixture density weight derivatives. Combined with the reparameterization trick applied to the individual mixture components, this estimator makes it straightforward to train variational autoencoders with mixture-distributed latent variables, or to perform stochastic variational inference with a mixture density variational posterior.
General Result
The section develops a Monte Carlo estimator for derivatives of an expectation over a continuous distribution, using conditional quantiles to describe its sampling transform. The estimator does not require the quantile transform itself to be known when samples from the distribution are available.
General Result
- The multivariate quantile transform constructs a sample recursively from independent uniform draws using conditional marginal inverse CDFs.Source: General Result · pp. 1–2
- For a distribution depending on a parameter, the derivation invokes the Leibniz integral rule to obtain the expectation derivative.Source: General Result · p. 2
- When the integral is intractable, Monte Carlo estimation requires sampling from each conditional density and a tractable conditional CDF; rejection sampling discards draws above the sampled coordinate.Source: General Result · p. 2
- The Monte Carlo estimator for an expectation of a differentiable function does not require the quantile transform to be known, provided samples from f are available.Source: General Result · pp. 2–3
Application to Mixture Density Weights
The section derives expectation derivatives with respect to normalized mixture weights and describes a recursive computation. Its treatment assumes diagonal-covariance components, while non-diagonal components are left for future work.
Application to Mixture Density Weights
- The component posterior responsibility is conditioned on the prior mixture weight and the observed coordinate prefix.Source: Application to Mixture Density Weights · p. 3
- The derivation assumes diagonal-covariance mixture components; extending it to non-diagonal components is left for future work.The diagonal covariance assumption is used to simplify the expression in Eq. 16.Source: Application to Mixture Density Weights · p. 3
- The derivation targets derivatives of the expectation with respect to normalized mixture weights, such as weights normalized by softmax.It sets a coordinate to t, differentiates the mixture expression, and evaluates at a sample from f.Source: Application to Mixture Density Weights · pp. 3–4
- The weight derivatives are computed by a joint recursion, with the complete computation given in Algorithm 1.Source: Application to Mixture Density Weights · pp. 3–5