Adaptive Minimax Regret against Smooth Logarithmic Losses over High-Dimensional ℓ_1-Balls via Envelope Complexity
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We develop a new theoretical framework, the envelope complexity, to analyze the minimax regret with logarithmic loss functions and derive a Bayesian predictor that achieves the adaptive minimax regret over high-dimensional ℓ_1-balls up to the major term. The prior is newly derived for achieving the minimax regret and called the spike-and-tails (ST) prior as it looks like. The resulting regret bound is so simple that it is completely determined with the smoothness of the loss function and the radius of the balls except with logarithmic factors, and it has a generalized form of existing regret/risk bounds. In the preliminary experiment, we confirm that the ST prior outperforms the conventional minimax-regret prior under non-high-dimensional asymptotics.
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