A Short Note on a Variant of the Squint Algorithm

Prediction with expert advice is a fundamental problem in online learning. When the time horizon $T$ is known in advance, the minimax cumulative regret over $n$ experts is asymptotically $\sqrt{\frac{T \ln n}{2}}$. This is achieved by the Multiplicative Weights Update algorithm with a learning rate tuned to $T$, and is known to be tight. If instead the regret bound is required to hold simultaneously at every time $t$, the best known guarantee has been $\sqrt{t \ln n}$---a factor of $\sqrt{2}$ worse---and it has remained unknown whether this factor of $\sqrt{2}$ is necessary. We show that it is not. We give an algorithm, requiring no knowledge of the horizon, whose cumulative regret satisfies $R_t \le \bigl(1 + O(\sqrt{\ln \ln n / \ln n})\bigr)\sqrt{t \ln n / 2}$ simultaneously for every $t \ge 1$.

In Orabona and Pál [2016], we introduced the shifted KT potentials, to remove the $\ln \ln T$ factor in the parameter-free learning with expert bound. In this short technical note, I show that this is equivalent to changing the prior in the Krichevsky--Trofimov algorithm. Then, I show how to use the same idea to remove the $\ln \ln T$ factor in the data-independent bound for the Squint algorithm.

For the fundamental problem of prediction from expert advice, Freund asked whether one could compete with a comparator that switches $k$ times among only $m$ experts. For a discrete set of experts, this problem was resolved by Bousquet and Warmuth (2020), who gave an efficient algorithm with optimal regret matching the information-theoretic limit. We study the analogous problem in online convex optimization, where the action space is continuous. An inefficient algorithm can be derived by discretizing the action space and applying the Bousquet--Warmuth algorithm, giving regret of $\widetilde{O}(md+k)$, while the best known efficient algorithm for this problem has regret $\widetilde{O}(dk)$. We show that this computational inefficiency is inherent, establishing a separation between the information-theoretic and polynomial-time regret achievable for this problem. Specifically, under the hypergraphic planted-clique conjecture, we show that for every polynomial-time algorithm there exist instances for which it incurs at least $(k\sqrt d)^{1-o(1)}$ regret.