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.

翻译：我们重新审视了在有限时间范围$T$已知于学习者的条件下，对随机序列进行通用预测的经典问题。我们探究的问题是，是否能够推导出以高概率成立的渐近遗憾界，以补充现有文献中仅以期望形式成立的界。我们提出了此类高概率界，其形式与先前的期望界非常相似。对于在可数字母表上对随机过程进行通用预测的情况，我们的界表明，与先前已知的阶为$\mathcal{O}(T^{-1/2})$的期望界相比，以至少$1-δ$的概率，收敛速度为$\mathcal{O}(T^{-1/2} δ^{-1/2})$。我们还提出了一个不可能性结果，证明在不做出额外假设的情况下，无法改进相同形式界中$δ$的指数。