Isotuning With Applications To Scale-Free Online Learning

TL;DR

This paper introduces isotuning, a versatile framework for designing fast, adaptive, and scale-free online learning algorithms that automatically adjust to data difficulty and domain size, with strong theoretical guarantees.

Contribution

It develops a general isotuning method for adaptive learning rates, combined with online correction and null updates, to enhance existing algorithms with scale-free and adaptive regret bounds.

Findings

Restores adaptivity to small losses in FTRL for unbounded domains

Provides scale-free adaptive guarantees for Mirror Descent variants

Extends Adapt-ML-Prod and improves several algorithms like Prod, AdaHedge, BOA, Soft-Bayes

Abstract

We extend and combine several tools of the literature to design fast, adaptive, anytime and scale-free online learning algorithms. Scale-free regret bounds must scale linearly with the maximum loss, both toward large losses and toward very small losses. Adaptive regret bounds demonstrate that an algorithm can take advantage of easy data and potentially have constant regret. We seek to develop fast algorithms that depend on as few parameters as possible, in particular they should be anytime and thus not depend on the time horizon. Our first and main tool, isotuning, is a generalization of the idea of designing adaptive learning rates that balance the trade-off of the regret. We provide a simple and versatile theorem that can be applied to a wide range of settings, and competes with the best balancing in hindsight within a factor 2. The second tool is an online correction, which allows us to obtain centered bounds for many algorithms, to prevent the regret bounds from being vacuous when the domain is overly large or only partially constrained. The last tool, null updates, prevents the algorithm from performing overly large updates, which could result in unbounded regret, or even invalid updates. We develop a general theory to combine all these tools and apply it to several standard algorithms. In particular, we (almost entirely) restore the adaptivity to small losses of FTRL for unbounded domains, design and prove scale-free adaptive guarantees for a variant of Mirror Descent (at least when the Bregman divergence is convex in its second argument), extend Adapt-ML-Prod to scale-free guarantees, and provide several additional contributions about Prod, AdaHedge, BOA and Soft-Bayes.