High Dimensional Inference with Random Maximum A-Posteriori Perturbations

TL;DR

This paper introduces a perturb-max framework for high-dimensional inference that uses low-dimensional random perturbations to efficiently generate unbiased samples from Gibbs distributions, with theoretical bounds on entropy and concentration.

Contribution

It demonstrates that low-dimensional perturbations enable efficient sampling and provides bounds on entropy and deviation, advancing high-dimensional statistical inference methods.

Findings

Low-dimensional perturbations allow efficient sampling similar to MAP optimization.

Expected perturb-max value bounds the entropy of the model.

Sample averages of perturb-max values concentrate exponentially around the expectation.

Abstract

This paper presents a new approach, called perturb-max, for high-dimensional statistical inference that is based on applying random perturbations followed by optimization. This framework injects randomness to maximum a-posteriori (MAP) predictors by randomly perturbing the potential function for the input. A classic result from extreme value statistics asserts that perturb-max operations generate unbiased samples from the Gibbs distribution using high-dimensional perturbations. Unfortunately, the computational cost of generating so many high-dimensional random variables can be prohibitive. However, when the perturbations are of low dimension, sampling the perturb-max prediction is as efficient as MAP optimization. This paper shows that the expected value of perturb-max inference with low dimensional perturbations can be used sequentially to generate unbiased samples from the Gibbs distribution. Furthermore the expected value of the maximal perturbations is a natural bound on the entropy of such perturb-max models. A measure concentration result for perturb-max values shows that the deviation of their sampled average from its expectation decays exponentially in the number of samples, allowing effective approximation of the expectation.