From inexact optimization to learning via gradient concentration

TL;DR

This paper explores how gradient concentration and inexact optimization techniques can be combined to provide sharp guarantees on test error in machine learning, emphasizing the implicit regularization effects of optimization.

Contribution

It introduces a framework that integrates probabilistic gradient concentration with inexact optimization to analyze learning performance and regularization.

Findings

Gradient concentration leads to sharp test error bounds.

Inexact optimization methods implicitly regularize learning.

The approach applies to unconstrained objectives, enhancing understanding of optimization effects.

Abstract

Optimization in machine learning typically deals with the minimization of empirical objectives defined by training data. However, the ultimate goal of learning is to minimize the error on future data (test error), for which the training data provides only partial information. In this view, the optimization problems that are practically feasible are based on inexact quantities that are stochastic in nature. In this paper, we show how probabilistic results, specifically gradient concentration, can be combined with results from inexact optimization to derive sharp test error guarantees. By considering unconstrained objectives we highlight the implicit regularization properties of optimization for learning.