Exponential Weights Algorithms for Selective Learning
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We study the selective learning problem introduced by Qiao and Valiant (2019), in which the learner observes n labeled data points one at a time. At a time of its choosing, the learner selects a window length w and a model ℓ̂ from the model class ℒ, and then labels the next w data points using ℓ̂. The excess risk incurred by the learner is defined as the difference between the average loss of ℓ̂ over those w data points and the smallest possible average loss among all models in ℒ over those w data points. We give an improved algorithm, termed the hybrid exponential weights algorithm, that achieves an expected excess risk of O((loglog|ℒ| + loglog n)/log n). This result gives a doubly exponential improvement in the dependence on |ℒ| over the best known bound of O(√(|ℒ|/log n)). We complement the positive result with an almost matching lower bound, which suggests the worst-case optimality of the algorithm. We also study a more restrictive family of learning algorithms that are bounded-recall in the sense that when a prediction window of length w is chosen, the learner's decision only depends on the most recent w data points. We analyze an exponential weights variant of the ERM algorithm in Qiao and Valiant (2019). This new algorithm achieves an expected excess risk of O(√(log |ℒ|/log n)), which is shown to be nearly optimal among all bounded-recall learners. Our analysis builds on a generalized version of the selective mean prediction problem in Drucker (2013); Qiao and Valiant (2019), which may be of independent interest.
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