On the necessity of adaptive regularisation:Optimal anytime online learning on $\boldsymbol{\ell_p}$-balls
This work addresses the necessity of adaptive regularization for optimal online learning, with implications for algorithm design in machine learning, though it is incremental in refining theoretical understanding.
The paper tackles the problem of online convex optimization on ℓp-balls for p>2, showing that Follow-the-Regularised-Leader with adaptive regularization is anytime optimal across dimension regimes, while proving that fixed regularization is suboptimal in at least one regime and providing lower bounds for linear bandits in high dimensions.
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 $p \geq 1$.