# Projection-Free Online Optimization with Stochastic Gradient: From Convexity to Submodularity

## TL;DR

This paper introduces the first online projection-free algorithms using stochastic gradients for convex and submodular optimization, achieving optimal regret bounds and enabling efficient large-scale online learning.

## Contribution

It presents Meta-Frank-Wolfe and One-Shot Frank-Wolfe algorithms that extend online optimization to non-convex submodular functions without projections, with theoretical regret guarantees.

## Key findings

- Meta-Frank-Wolfe achieves $O(\sqrt{T})$ regret bounds.
- One-Shot Frank-Wolfe achieves $O(T^{2/3})$ regret bounds.
- Algorithms outperform existing methods in experiments.

## Abstract

Online optimization has been a successful framework for solving large-scale problems under computational constraints and partial information. Current methods for online convex optimization require either a projection or exact gradient computation at each step, both of which can be prohibitively expensive for large-scale applications. At the same time, there is a growing trend of non-convex optimization in machine learning community and a need for online methods. Continuous DR-submodular functions, which exhibit a natural diminishing returns condition, have recently been proposed as a broad class of non-convex functions which may be efficiently optimized. Although online methods have been introduced, they suffer from similar problems. In this work, we propose Meta-Frank-Wolfe, the first online projection-free algorithm that uses stochastic gradient estimates. The algorithm relies on a careful sampling of gradients in each round and achieves the optimal $O( \sqrt{T})$ adversarial regret bounds for convex and continuous submodular optimization. We also propose One-Shot Frank-Wolfe, a simpler algorithm which requires only a single stochastic gradient estimate in each round and achieves an $O(T^{2/3})$ stochastic regret bound for convex and continuous submodular optimization. We apply our methods to develop a novel "lifting" framework for the online discrete submodular maximization and also see that they outperform current state-of-the-art techniques on various experiments.