Prediction with Advice of Unknown Number of Experts

In the framework of prediction with expert advice, we consider a recently introduced kind of regret bounds: the bounds that depend on the effective instead of nominal number of experts. In contrast to the NormalHedge bound, which mainly depends on the effective number of experts and also weakly depends on the nominal one, we obtain a bound that does not contain the nominal number of experts at all. We use the defensive forecasting method and introduce an application of defensive forecasting to multivalued supermartingales.

💡 Research Summary

The paper addresses the classic problem of prediction with expert advice in an online setting, focusing on a newer class of regret bounds that depend on the effective number of experts rather than the nominal total. Traditional bounds, such as those derived from the Hedge or Weighted Majority algorithms, contain a term proportional to (\sqrt{T\ln N}), where (N) is the total number of experts. This dependence can be overly pessimistic when many experts are redundant or irrelevant. Recent work introduced the notion of an effective number of experts (K), defined as the size of the subset of experts that actually achieve the minimal cumulative loss. The NormalHedge algorithm, for instance, yields a bound of order (\sqrt{T\ln K}) but still retains a weak (\ln N) term, meaning the bound is not completely independent of the nominal expert count.

The authors propose a fundamentally different approach that eliminates any appearance of (N) from the regret bound. Their method is built on defensive forecasting, a meta‑algorithmic technique that frames the learner’s predictions as bets and forces the environment (adversary) to respect a supermartingale condition. While earlier defensive‑forecasting constructions used a single‑valued supermartingale—essentially a scalar potential that must not increase in expectation—the present work extends the concept to multivalued supermartingales. In this generalized setting, the potential is a function of the entire loss vector across all experts, and the supermartingale condition must hold simultaneously for every admissible weight vector the learner might choose.

The construction proceeds as follows. At each round (t) the learner maintains a weight vector (\mathbf{w}_t) over the experts. After observing the loss vector (\mathbf{l}t) (the losses of all experts at that round), the learner updates the weights to (\mathbf{w}{t+1}) in a way that guarantees \