Fast rates for prediction with limited expert advice

TL;DR

This paper studies the impact of limited expert advice on prediction accuracy, showing that observing multiple experts can achieve faster convergence rates, and introduces algorithms to optimize expert advice usage under constraints.

Contribution

It introduces novel algorithms that attain fast convergence rates with limited expert advice and provides instance-dependent bounds for advice queries.

Findings

Single expert advice per round yields a slow 1/√T rate.

Observing at least two experts per round achieves a 1/T fast rate.

Algorithms are designed for both training and testing phases under advice constraints.

Abstract

We investigate the problem of minimizing the excess generalization error with respect to the best expert prediction in a finite family in the stochastic setting, under limited access to information. We assume that the learner only has access to a limited number of expert advices per training round, as well as for prediction. Assuming that the loss function is Lipschitz and strongly convex, we show that if we are allowed to see the advice of only one expert per round for T rounds in the training phase, or to use the advice of only one expert for prediction in the test phase, the worst-case excess risk is $\Omega$(1/ $\sqrt$ T) with probability lower bounded by a constant. However, if we are allowed to see at least two actively chosen expert advices per training round and use at least two experts for prediction, the fast rate O(1/T) can be achieved. We design novel algorithms achieving this rate in this setting, and in the setting where the learner has a budget constraint on the total number of observed expert advices, and give precise instance-dependent bounds on the number of training rounds and queries needed to achieve a given generalization error precision.