Uniform regret bounds over $R^d$ for the sequential linear regression problem with the square loss

TL;DR

This paper develops optimal uniform regret bounds for online linear regression with square loss, improving constants and handling sequential feature revelation, using variants of online ridge regression.

Contribution

It provides the first optimal constant regret bounds for both known and sequential feature settings in online linear regression.

Findings

Achieved regret bounds with constant 1 in front of $d B^2 \\ln T$ term.

Derived asymptotic regret bounds for sequentially revealed features.

Proposed variants of online ridge regression with data-dependent regularization.

Abstract

We consider the setting of online linear regression for arbitrary deterministic sequences, with the square loss. We are interested in the aim set by Bartlett et al. (2015): obtain regret bounds that hold uniformly over all competitor vectors. When the feature sequence is known at the beginning of the game, they provided closed-form regret bounds of $2d B^2 \ln T + \mathcal{O}_T(1)$, where $T$ is the number of rounds and $B$ is a bound on the observations. Instead, we derive bounds with an optimal constant of $1$ in front of the $d B^2 \ln T$ term. In the case of sequentially revealed features, we also derive an asymptotic regret bound of $d B^2 \ln T$ for any individual sequence of features and bounded observations. All our algorithms are variants of the online non-linear ridge regression forecaster, either with a data-dependent regularization or with almost no regularization.