Optimized projection-free algorithms for online learning: construction and worst-case analysis

TL;DR

This paper develops and analyzes optimized projection-free algorithms for online learning, improving regret bounds and leveraging semidefinite programming for design and analysis.

Contribution

It introduces an improved online Frank-Wolfe algorithm with a simple potential-based proof and uses semidefinite programming for joint design and analysis.

Findings

The regret guarantee cannot be better than O(T^{3/4}) without extra assumptions.

Current algorithms do not have optimal constants.

Multiple linear optimization rounds do not improve regret bounds.

Abstract

This work studies and develop projection-free algorithms for online learning with linear optimization oracles (a.k.a. Frank-Wolfe) for handling the constraint set. More precisely, this work (i) provides an improved (optimized) variant of an online Frank-Wolfe algorithm along with its conceptually simple potential-based proof, and (ii) shows how to leverage semidefinite programming to jointly design and analyze online Frank-Wolfe-type algorithms numerically in a variety of settings-that include the design of the variant (i). Based on the semidefinite technique, we conclude with strong numerical evidence suggesting that no pure online Frank-Wolfe algorithm within our model class can have a regret guarantee better than O(T^3/4) (T is the time horizon) without additional assumptions, that the current algorithms do not have optimal constants, that the algorithm benefits from similar anytime properties O(t^3/4) not requiring to know T in advance, and that multiple linear optimization rounds do not generally help to obtain better regret bounds.