TCS+ Talk

Abstract: I will present a new approach to randomized lower bounds, particularly in the setting where we wish to give a fine-grained analysis of randomized algorithms that achieve small bias. The approach is as follows: instead of considering ordinary randomized algorithms which give an output in {0,1} and may err, we switch models to look at "forecasting" randomized algorithms which output a confidence in [0,1] for whether they think the answer is 1. When scored by a proper scoring rule, the performance of the best forecasting algorithm is closely related to the bias of the best (ordinary) randomized algorithm, but is more amenable to analysis.

As an application, I'll present a new minimax theorem for randomized algorithms, which can be viewed as a strengthening of Yao's minimax theorem. Yao's minimax theorem guarantees the existence of a hard distribution for a function f such that solving f against this distribution (to a desired error level) is as hard as solving f in the worst case (to that same error level). However, the hard distribution provided by Yao's theorem depends on the chosen error level. Our minimax theorem removes this dependence, giving a distribution which certifies the hardness of f against all bias levels at once. In recent work, we used this minimax theorem to give a tight composition theorem for randomized query complexity.

Based on joint work with Eric Blais.

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