Logarithmic Regret from Sublinear Hints

TL;DR

This paper demonstrates that in online linear optimization, a logarithmic regret can be achieved with significantly fewer hints than previously thought, using only O(√T) hints instead of at every step.

Contribution

It shows that logarithmic regret is possible with only O(√T) hints, challenging the necessity of hints at every step and establishing lower bounds for hints needed.

Findings

O(log T) regret with O(√T) hints

Lower bound of Ω(√T) hints for better regret

Applications to optimistic regret bounds and abstention

Abstract

We consider the online linear optimization problem, where at every step the algorithm plays a point $x_t$ in the unit ball, and suffers loss $\langle c_t, x_t\rangle$ for some cost vector $c_t$ that is then revealed to the algorithm. Recent work showed that if an algorithm receives a hint $h_t$ that has non-trivial correlation with $c_t$ before it plays $x_t$, then it can achieve a regret guarantee of $O(\log T)$, improving on the bound of $\Theta(\sqrt{T})$ in the standard setting. In this work, we study the question of whether an algorithm really requires a hint at every time step. Somewhat surprisingly, we show that an algorithm can obtain $O(\log T)$ regret with just $O(\sqrt{T})$ hints under a natural query model; in contrast, we also show that $o(\sqrt{T})$ hints cannot guarantee better than $\Omega(\sqrt{T})$ regret. We give two applications of our result, to the well-studied setting of optimistic regret bounds and to the problem of online learning with abstention.