Second-order Quantile Methods for Experts and Combinatorial Games

TL;DR

This paper introduces a novel adaptive method for sequential decision making that combines second-order regret bounds and quantile bounds, improving performance on easy data in expert advice and combinatorial tasks.

Contribution

The paper presents a new potential-based algorithm that adaptively combines learning rates to achieve both second-order and quantile regret bounds, a novel combination in the field.

Findings

First bounds that are both second-order and quantile-based

Efficient algorithms constructed using a tailored prior

Demonstrated improved performance on easy data scenarios

Abstract

We aim to design strategies for sequential decision making that adjust to the difficulty of the learning problem. We study this question both in the setting of prediction with expert advice, and for more general combinatorial decision tasks. We are not satisfied with just guaranteeing minimax regret rates, but we want our algorithms to perform significantly better on easy data. Two popular ways to formalize such adaptivity are second-order regret bounds and quantile bounds. The underlying notions of 'easy data', which may be paraphrased as "the learning problem has small variance" and "multiple decisions are useful", are synergetic. But even though there are sophisticated algorithms that exploit one of the two, no existing algorithm is able to adapt to both. In this paper we outline a new method for obtaining such adaptive algorithms, based on a potential function that aggregates a range of learning rates (which are essential tuning parameters). By choosing the right prior we construct efficient algorithms and show that they reap both benefits by proving the first bounds that are both second-order and incorporate quantiles.