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Abstract:
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We study the minimax regret for the classical problem of sequential probabilistic forecasting under logarithmic loss. At each time step, the player observes some covariates, forecasts a probability distribution, and incurs the negative log-likelihood of the observed outcome under the forecasted distribution. We allow the observations to be chosen adversarially rather than from a fixed distribution, so it is standard to study the regret; i.e., the difference between the player’s total loss and the total loss of the single best forecaster in some arbitrary, potentially nonparametric class of “experts”. This problem requires more refined techniques than, say, square loss or classification loss, as both the loss values and their gradient may now be unbounded.
Exploiting the curvature of logarithmic loss in a novel way, we obtain upper bounds that depend on the complexity of the expert class through its (sequential) metric entropy, and that cannot be improved without additional assumptions on the expert class. As an application of our techniques, we resolve the minimax regret for nonparametric Lipschitz classes of experts.
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