The Gain from Ordering in Online Learning

Part of Advances in Neural Information Processing Systems 36 (NeurIPS 2023) Main Conference Track

Authors

Vasilis Kontonis, Mingchen Ma, Christos Tzamos

Abstract

We study fixed-design online learning where the learner is allowed to choose the order of the datapoints in order to minimize their regret (aka self-directed online learning). We focus on the fundamental task of online linear regression: the learner is given a dataset $X$ with $n$ examples in $d$ dimensions and at step $t$ they select a point $x_t \in X$, predict a value $\widetilde y_t$, and suffer loss $(\widetilde y_t - w^\ast \cdot x_t)^2$. The goal is to design algorithms that order the examples and achieve better regret than random- or worst-order online algorithms.For an arbitrary dataset $X$, we show that, under the Exponential Time Hypothesis, no efficient algorithm can approximate the optimal (best-order) regret within a factor of $d^{1/\poly(\log \log d)}$.We then show that, for structured datasets, we can bypass the above hardness result and achieve nearly optimal regret. When the examples of $X$ are drawn i.i.d.\ from the uniform distribution on the sphere, we present an algorithm based on the greedy heuristic of selecting easiest'' examples first that achieves a $\log d$-approximation of the optimal regret.