Implicit Parameter-free Online Learning with Truncated Linear Models

TL;DR

This paper introduces efficient, parameter-free online learning algorithms that leverage truncated linear models for improved loss approximation, maintaining optimal regret without requiring parameter tuning.

Contribution

The paper proposes a novel implicit update method enabling parameter-free algorithms to utilize truncated linear models efficiently, reducing computational cost and avoiding overshooting.

Findings

Algorithms achieve optimal regret bounds.

Empirical results show practical utility.

Update requires only one gradient per step.

Abstract

Parameter-free algorithms are online learning algorithms that do not require setting learning rates. They achieve optimal regret with respect to the distance between the initial point and any competitor. Yet, parameter-free algorithms do not take into account the geometry of the losses. Recently, in the stochastic optimization literature, it has been proposed to instead use truncated linear lower bounds, which produce better performance by more closely modeling the losses. In particular, truncated linear models greatly reduce the problem of overshooting the minimum of the loss function. Unfortunately, truncated linear models cannot be used with parameter-free algorithms because the updates become very expensive to compute. In this paper, we propose new parameter-free algorithms that can take advantage of truncated linear models through a new update that has an "implicit" flavor. Based on a novel decomposition of the regret, the new update is efficient, requires only one gradient at each step, never overshoots the minimum of the truncated model, and retains the favorable parameter-free properties. We also conduct an empirical study demonstrating the practical utility of our algorithms.