On the necessity of adaptive regularisation:Optimal anytime online learning on $\boldsymbol{\ell_p}$-balls

TL;DR

This paper investigates online convex optimization on $oldsymbol{ extit{ ext{l}_p}}$-balls for $p > 2$, demonstrating that adaptive regularisation is essential for optimal regret across different dimension regimes.

Contribution

It proves that adaptive regularisation in FTRL is necessary for anytime optimality and shows fixed regularisers are sub-optimal in certain regimes.

Findings

FTRL with time-varying regularisation achieves anytime optimal regret.

Fixed regularisers are sub-optimal in high or low-dimensional regimes.

Lower bounds indicate sub-linear regret is impossible in high dimensions for certain $ extit{ ext{l}_p}$-balls.

Abstract

We study online convex optimization on $\ell_p$-balls in $\mathbb{R}^d$ for $p > 2$. While always sub-linear, the optimal regret exhibits a shift between the high-dimensional setting ($d > T$), when the dimension $d$ is greater than the time horizon $T$ and the low-dimensional setting ($d \leq T$). We show that Follow-the-Regularised-Leader (FTRL) with time-varying regularisation which is adaptive to the dimension regime is anytime optimal for all dimension regimes. Motivated by this, we ask whether it is possible to obtain anytime optimality of FTRL with fixed non-adaptive regularisation. Our main result establishes that for separable regularisers, adaptivity in the regulariser is necessary, and that any fixed regulariser will be sub-optimal in one of the two dimension regimes. Finally, we provide lower bounds which rule out sub-linear regret bounds for the linear bandit problem in sufficiently high-dimension for all $\ell_p$-balls with $p \geq 1$.