Online Prediction of Stochastic Sequences with High Probability Regret Bounds
This work addresses the need for robust performance guarantees in online learning and prediction tasks, providing high-probability bounds that are incremental over prior expectation-based results.
The paper tackles the problem of universal prediction of stochastic sequences by deriving high-probability regret bounds that vanish over time, complementing existing expectation bounds. It achieves a convergence rate of O(T^{-1/2} δ^{-1/2}) with probability at least 1-δ for countable alphabets and proves an impossibility result showing this exponent in δ cannot be improved without extra assumptions.
We revisit the classical problem of universal prediction of stochastic sequences with a finite time horizon $T$ known to the learner. The question we investigate is whether it is possible to derive vanishing regret bounds that hold with high probability, complementing existing bounds from the literature that hold in expectation. We propose such high-probability bounds which have a very similar form as the prior expectation bounds. For the case of universal prediction of a stochastic process over a countable alphabet, our bound states a convergence rate of $\mathcal{O}(T^{-1/2} δ^{-1/2})$ with probability as least $1-δ$ compared to prior known in-expectation bounds of the order $\mathcal{O}(T^{-1/2})$. We also propose an impossibility result which proves that it is not possible to improve the exponent of $δ$ in a bound of the same form without making additional assumptions.