Convergence of Unregularized Online Learning Algorithms

TL;DR

This paper analyzes the convergence behavior of unregularized online gradient descent algorithms in RKHSs, providing new high-probability convergence rates for both averaged and last iterates without strong convexity assumptions.

Contribution

It establishes the first high-probability convergence rate for the last iterate of online gradient descent in RKHSs without regularization or strong convexity.

Findings

Explicit convergence rates with high probability for averaged iterates.

Explicit convergence rates with high probability for the last iterate.

Convergence results obtained without boundedness assumptions on iterates.

Abstract

In this paper we study the convergence of online gradient descent algorithms in reproducing kernel Hilbert spaces (RKHSs) without regularization. We establish a sufficient condition and a necessary condition for the convergence of excess generalization errors in expectation. A sufficient condition for the almost sure convergence is also given. With high probability, we provide explicit convergence rates of the excess generalization errors for both averaged iterates and the last iterate, which in turn also imply convergence rates with probability one. To our best knowledge, this is the first high-probability convergence rate for the last iterate of online gradient descent algorithms without strong convexity. Without any boundedness assumptions on iterates, our results are derived by a novel use of two measures of the algorithm's one-step progress, respectively by generalization errors and by distances in RKHSs, where the variances of the involved martingales are cancelled out by the descent property of the algorithm.