The celebrated minimax principle of Yao (1977) says that for any Boolean-valued function $f$ with finite domain, there is a distribution $mu$ over the domain of $f$ such that computing $f$ to error $epsilon$ against inputs from $mu$ is just as hard as computing $f$ to error $epsilon$ on worst-case inputs. Notably, however, the distribution $mu$ depends on the target error level $epsilon$: the hard distribution which is tight for bounded error might be trivial to solve to small bias, and the hard distribution which is tight for a small bias level might be far from tight for bounded error levels. In this work, we introduce a new type of minimax theorem which can provide a hard distribution $mu$ that works for all bias levels at once. We show that this works for randomized query complexity, randomized communication complexity, some randomized circuit models, quantum query and communication complexities, approximate polynomial degree, and approximate logrank. We also prove an improved version of Impagliazzos hardcore lemma. Our proofs rely on two innovations over the classical approach of using Von Neumanns minimax theorem or linear programming duality. First, we use Sions minimax theorem to prove a minimax theorem for ratios of bilinear functions representing the cost and score of algorithms. Second, we introduce a new way to analyze low-bias randomized algorithms by viewing them as forecasting algorithms evaluated by a proper scoring rule. The expected score of the forecasting version of a randomized algorithm appears to be a more fine-grained way of analyzing the bias of the algorithm. We show that such expected scores have many elegant mathematical properties: for example, they can be amplified linearly instead of quadratically. We anticipate forecasting algorithms will find use in future work in which a fine-grained analysis of small-bias algorithms is required.
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