A Minimax Approach to Supervised Learning

TL;DR

This paper introduces a minimax framework for supervised learning based on a generalized maximum entropy principle, leading to new classifiers like the maximum entropy machine that can outperform traditional methods in worst-case scenarios.

Contribution

It develops a novel minimax approach for supervised learning using a generalized maximum entropy principle, including a new linear classifier for 0-1 loss.

Findings

Maximum entropy machine outperforms SVM in experiments

The approach recovers known models for squared-error and log loss

Provides a bound on worst-case generalization error

Abstract

Given a task of predicting $Y$ from $X$, a loss function $L$, and a set of probability distributions $\Gamma$ on $(X,Y)$, what is the optimal decision rule minimizing the worst-case expected loss over $\Gamma$? In this paper, we address this question by introducing a generalization of the principle of maximum entropy. Applying this principle to sets of distributions with marginal on $X$ constrained to be the empirical marginal from the data, we develop a general minimax approach for supervised learning problems. While for some loss functions such as squared-error and log loss, the minimax approach rederives well-knwon regression models, for the 0-1 loss it results in a new linear classifier which we call the maximum entropy machine. The maximum entropy machine minimizes the worst-case 0-1 loss over the structured set of distribution, and by our numerical experiments can outperform other well-known linear classifiers such as SVM. We also prove a bound on the generalization worst-case error in the minimax approach.