Stochastic Gradient Descent with Exponential Convergence Rates of Expected Classification Errors

TL;DR

This paper demonstrates that stochastic gradient descent and its averaging variant can achieve exponential convergence rates for expected classification errors in binary classification, under certain conditions, improving upon traditional sublinear rates.

Contribution

It extends exponential convergence results to a broader class of loss functions beyond squared loss, applicable to binary classification with theoretical guarantees.

Findings

Exponential convergence of expected classification error shown for a wide class of loss functions.

Averaged stochastic gradient descent achieves exponential convergence from early training phase.

Experimental verification on L2-regularized logistic regression supports theoretical results.

Abstract

We consider stochastic gradient descent and its averaging variant for binary classification problems in a reproducing kernel Hilbert space. In the traditional analysis using a consistency property of loss functions, it is known that the expected classification error converges more slowly than the expected risk even when assuming a low-noise condition on the conditional label probabilities. Consequently, the resulting rate is sublinear. Therefore, it is important to consider whether much faster convergence of the expected classification error can be achieved. In recent research, an exponential convergence rate for stochastic gradient descent was shown under a strong low-noise condition but provided theoretical analysis was limited to the squared loss function, which is somewhat inadequate for binary classification tasks. In this paper, we show an exponential convergence of the expected classification error in the final phase of the stochastic gradient descent for a wide class of differentiable convex loss functions under similar assumptions. As for the averaged stochastic gradient descent, we show that the same convergence rate holds from the early phase of training. In experiments, we verify our analyses on the $L_2$-regularized logistic regression.