Parameter-free Regret in High Probability with Heavy Tails

TL;DR

This paper introduces algorithms for online convex optimization over unbounded domains that achieve high-probability regret bounds with heavy-tailed subgradients, advancing beyond expectation-based results.

Contribution

It develops new regularization techniques to attain parameter-free, high-probability regret bounds in unbounded domains with heavy-tailed subgradients, a significant improvement over prior expectation-based methods.

Findings

Achieves regret of ilde{O}(||u|| T^{1/p} log(1/δ)) with high probability.

Handles subgradients with bounded p-th moments for p in (1, 2].

Overcomes challenges of unbounded iterates without relying on standard martingale concentration.

Abstract

We present new algorithms for online convex optimization over unbounded domains that obtain parameter-free regret in high-probability given access only to potentially heavy-tailed subgradient estimates. Previous work in unbounded domains considers only in-expectation results for sub-exponential subgradients. Unlike in the bounded domain case, we cannot rely on straight-forward martingale concentration due to exponentially large iterates produced by the algorithm. We develop new regularization techniques to overcome these problems. Overall, with probability at most $\delta$, for all comparators $\mathbf{u}$ our algorithm achieves regret $\tilde{O}(\| \mathbf{u} \| T^{1/\mathfrak{p}} \log (1/\delta))$ for subgradients with bounded $\mathfrak{p}^{th}$ moments for some $\mathfrak{p} \in (1, 2]$.