A Second-order Bound with Excess Losses
This work addresses the challenge of improving regret bounds in online learning for scenarios with expert confidence reporting and small excess losses, representing an incremental advancement in the field.
The paper tackles the problem of online aggregation of expert predictions by deriving new second-order regret bounds in terms of excess losses, achieved through variants of Prod and polynomially weighted average algorithms with multiple learning rates, and demonstrates applications including improved bounds for small excess losses and bounded regret against i.i.d. sequences.
We study online aggregation of the predictions of experts, and first show new second-order regret bounds in the standard setting, which are obtained via a version of the Prod algorithm (and also a version of the polynomially weighted average algorithm) with multiple learning rates. These bounds are in terms of excess losses, the differences between the instantaneous losses suffered by the algorithm and the ones of a given expert. We then demonstrate the interest of these bounds in the context of experts that report their confidences as a number in the interval [0,1] using a generic reduction to the standard setting. We conclude by two other applications in the standard setting, which improve the known bounds in case of small excess losses and show a bounded regret against i.i.d. sequences of losses.