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Online Prediction of Stochastic Sequences with High Probability Regret Bounds

ICLR 2026 Conference SubmissionAnonymous Authors
OpenReview Score: 6.0Download Report PDF
online predictionlearning theoryhigh-probability boundregretstochastic sequences
Abstract:

We revisit the classical problem of universal prediction of stochastic sequences with a finite time horizon TT 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 O(T−1/2δ−1/2)\mathcal{O}(T^{-1/2} \delta^{-1/2}) with probability as least 1−δ1-\delta compared to prior known in-expectation bounds of the order O(T−1/2)\mathcal{O}(T^{-1/2}). We also propose an impossibility result which proves that it is not possible to improve the exponent of δ\delta in a bound of the same form without making additional assumptions.

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Overview

Taxonomy

Core-task Taxonomy Papers
37
3
Claimed Contributions
22
Contribution Candidate Papers Compared
1
Refutable Paper

Research Landscape Overview

Core task: universal prediction of stochastic sequences with finite time horizon. This field addresses the challenge of making sequential predictions under uncertainty when the forecasting window is bounded, blending algorithmic theory with statistical guarantees. The taxonomy reveals several major branches: Universal Prediction Theory and Algorithmic Foundations explores fundamental regret bounds and learning-theoretic principles, often drawing on compression-based methods and context-tree techniques; Confidence Sequences and Statistical Guarantees focuses on time-uniform inference and anytime-valid bounds that remain valid across stopping times; Probabilistic Forecasting Methods emphasizes distributional predictions and uncertainty quantification; Stochastic Approximation and Optimization tackles iterative algorithms under noisy feedback; Sequence Prediction with Neural Networks and Learning Models leverages deep architectures for pattern recognition; Domain-Specific Prediction Applications targets concrete settings such as traffic forecasting or activity recognition; and Pattern Matching and Compression-Based Prediction exploits data compression principles for prediction. These branches intersect in their treatment of finite horizons, yet differ in whether they prioritize worst-case guarantees, probabilistic modeling, or application-driven performance. A particularly active line of work centers on high-probability regret bounds for stochastic sequences, where researchers seek tight finite-time guarantees that hold with specified confidence. Online Prediction of Stochastic[0] sits squarely within this branch, emphasizing rigorous probabilistic bounds over finite horizons. Nearby efforts such as On Confidence Sequences for[3] and Gambling-Based Confidence Sequences for[4] develop anytime-valid inference tools that complement prediction algorithms, while Finite-time High-probability Bounds for[19] and O1k Finite-Time Bound for[8] refine concentration inequalities for sequential settings. In contrast, works like Probabilistic Forecasting with Stochastic[2] lean toward distributional forecasting rather than worst-case regret, and classical contributions such as Learning to predict by[9] anchor the field in universal coding and compression. The central tension across these directions is balancing computational tractability, statistical efficiency, and the strength of assumptions on the data-generating process, with Online Prediction of Stochastic[0] contributing to the algorithmic foundations by addressing high-probability guarantees under minimal stochastic assumptions.

Claimed Contributions

High-probability regret bounds for universal prediction of stochastic sequences

The authors derive high-probability regret bounds for universal prediction of stochastic sequences that achieve a convergence rate of O(T^(-1/2)δ^(-1/2)) with probability at least 1-δ, complementing prior expectation bounds of order O(T^(-1/2)). These bounds hold under general bounded loss functions and do not require finite underlying spaces.

4 retrieved papers
Impossibility result for improving the error probability dependence

The authors prove that the dependence on the error probability δ in their high-probability bounds cannot be improved beyond the polynomial factors (1/δ or 1/√δ) without additional assumptions. This establishes fundamental limits on achievable high-probability regret bounds for this problem.

9 retrieved papers
Relaxation of technical assumptions compared to prior work

The authors extend the theoretical framework beyond finite spaces by using weaker technical assumptions than previous work, allowing their results to apply to more general measurable spaces while maintaining similar convergence guarantees.

9 retrieved papers
Can Refute

Core Task Comparisons

Comparisons with papers in the same taxonomy category

Within the taxonomy built over the current TopK core-task papers, the original paper is assigned to a leaf with no direct siblings and no cousin branches under the same grandparent topic. In this retrieved landscape, it appears structurally isolated, which is one partial signal of novelty, but still constrained by search coverage and taxonomy granularity.

Contribution Analysis

Detailed comparisons for each claimed contribution

Contribution

High-probability regret bounds for universal prediction of stochastic sequences

The authors derive high-probability regret bounds for universal prediction of stochastic sequences that achieve a convergence rate of O(T^(-1/2)δ^(-1/2)) with probability at least 1-δ, complementing prior expectation bounds of order O(T^(-1/2)). These bounds hold under general bounded loss functions and do not require finite underlying spaces.

Contribution

Impossibility result for improving the error probability dependence

The authors prove that the dependence on the error probability δ in their high-probability bounds cannot be improved beyond the polynomial factors (1/δ or 1/√δ) without additional assumptions. This establishes fundamental limits on achievable high-probability regret bounds for this problem.

Contribution

Relaxation of technical assumptions compared to prior work

The authors extend the theoretical framework beyond finite spaces by using weaker technical assumptions than previous work, allowing their results to apply to more general measurable spaces while maintaining similar convergence guarantees.

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