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Lipschitz Bandits with Stochastic Delayed Feedback

ICLR 2026 Conference SubmissionAnonymous Authors
OpenReview Score: 6.5Download Report PDF
banditLipschitz bandit
Abstract:

The Lipschitz bandit problem extends stochastic bandits to a continuous action set defined over a metric space, where the expected reward function satisfies a Lipschitz condition. In this work, we introduce a new problem of Lipschitz bandit in the presence of stochastic delayed feedback, where the rewards are not observed immediately but after a random delay. We consider both bounded and unbounded stochastic delays, and design algorithms that attain sublinear regret guarantees in each setting. For bounded delays, we propose a delay-aware zooming algorithm that retains the optimal performance of the delay-free setting up to an additional term that scales with the maximum delay τmax⁡\tau_{\max}. For unbounded delays, we propose a novel phased learning strategy that accumulates reliable feedback over carefully scheduled intervals, and establish a regret lower bound showing that our method is nearly optimal up to logarithmic factors. Finally, we present experimental results to demonstrate the efficiency of our algorithms under various delay scenarios.

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This report is AI-GENERATED using Large Language Models and WisPaper (A scholar search engine). It analyzes academic papers' tasks and contributions against retrieved prior work. While this system identifies POTENTIAL overlaps and novel directions, ITS COVERAGE IS NOT EXHAUSTIVE AND JUDGMENTS ARE APPROXIMATE. These results are intended to assist human reviewers and SHOULD NOT be relied upon as a definitive verdict on novelty.
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Overview

Overall Novelty Assessment

The paper addresses Lipschitz bandits with stochastic delayed feedback, proposing algorithms for both bounded and unbounded delay settings. According to the taxonomy, this work resides in the 'Lipschitz Bandits with Bounded Stochastic Delays' leaf, which currently contains only this paper as a sibling. This positioning indicates a relatively sparse research direction within the broader continuous action space bandits literature, suggesting the specific combination of Lipschitz continuity and stochastic delays has received limited prior attention in the examined literature.

The taxonomy reveals neighboring work in adjacent leaves: 'Bandit Convex Optimization with Delayed Feedback' addresses convex loss minimization under delays, while the 'Multi-Agent and Game-Theoretic Learning with Delays' branch explores strategic interactions with delayed rewards. The paper's focus on single-agent Lipschitz bandits distinguishes it from game-theoretic settings and from convex optimization approaches that may not exploit metric space structure. The taxonomy's scope notes explicitly exclude non-Lipschitz reward structures and adversarial delay models, clarifying that this work occupies a distinct methodological niche emphasizing stochastic delay distributions over continuous metric spaces.

Among the eight candidates examined through semantic search, none were found to refute the paper's three main contributions. The delay-aware zooming algorithm examined two candidates with zero refutations; the phased learning strategy for unbounded delays examined three candidates with zero refutations; and the instance-dependent lower bound examined three candidates with zero refutations. This limited search scope suggests that within the top-ranked semantically similar papers, no direct prior work addressing the same algorithmic framework for Lipschitz bandits with stochastic delays was identified, though the small candidate pool limits the strength of this signal.

Based on the constrained literature search of eight candidates and the sparse taxonomy leaf containing only this paper, the work appears to occupy a relatively unexplored intersection of Lipschitz continuity and stochastic delays. However, the limited search scope means potentially relevant work outside the top semantic matches may exist. The analysis covers immediate neighbors in the taxonomy but does not exhaustively survey all bandit or delay-related literature.

Taxonomy

Core-task Taxonomy Papers
6
3
Claimed Contributions
8
Contribution Candidate Papers Compared
0
Refutable Paper

Research Landscape Overview

Core task: Lipschitz bandits with stochastic delayed feedback. This field addresses sequential decision-making problems where an agent selects actions from a continuous space under Lipschitz smoothness assumptions, but observes rewards only after random delays. The taxonomy organizes research into three main branches. The first, Continuous Action Space Bandits with Delays, focuses on algorithms that exploit smoothness or structure in infinite action sets while handling stochastic or adversarial delays; representative works include Lipschitz Bandits Delayed[0] and Bandit Convex Delayed[2], which develop regret bounds under bounded delay distributions. The second branch, Multi-Agent and Game-Theoretic Learning with Delays, examines settings where multiple learners interact, possibly in competitive or cooperative scenarios, and feedback arrives asynchronously; examples are Gradient-free Delayed Games[3], Cooperative Multi-player Bandit[5], and Distributed Delayed Learning[4]. The third branch, General Sequential Decision Making with Delayed Feedback, captures broader frameworks that apply to various bandit and reinforcement learning problems with delayed observations, such as Sequential Delayed Framework[1] and Network Adversarial Bandit[6]. A particularly active line of work contrasts single-agent continuous bandits with multi-agent or distributed settings. In the former, the main challenge is balancing exploration of a large action space with the uncertainty introduced by delayed rewards, often requiring careful discretization or tree-based search strategies. In the latter, coordination and communication constraints add complexity, as seen in Cooperative Multi-player Bandit[5] and Distributed Delayed Learning[4]. Lipschitz Bandits Delayed[0] sits squarely within the Continuous Action Space branch, emphasizing bounded stochastic delays and Lipschitz continuity to derive tight regret guarantees. Its focus on stochastic delay models distinguishes it from works like Bandit Convex Delayed[2], which may consider adversarial delays or convex loss functions, and from the game-theoretic emphasis of Gradient-free Delayed Games[3]. Overall, the field remains open regarding optimal delay-dependent regret rates and the interplay between smoothness assumptions and delay distributions.

Claimed Contributions

Delay-aware zooming algorithm for bounded delays

The authors extend the classic zooming algorithm to handle bounded stochastic delays in Lipschitz bandits. Their Delayed Zooming algorithm achieves a regret bound of Õ(T^(dz+1)/(dz+2) + τmax·T^(dz/(dz+2))), matching the optimal delay-free rate with an additive delay-dependent term.

2 retrieved papers
Phased learning algorithm for unbounded delays

The authors introduce Delayed Lipschitz Phased Pruning (DLPP), a phased elimination-based algorithm that handles unbounded and potentially infinite delays. The algorithm achieves near-optimal regret with an additive dependence on delay distribution quantiles, supported by matching lower bounds.

3 retrieved papers
Instance-dependent lower bound for delayed Lipschitz bandits

The authors prove an instance-dependent regret lower bound for Lipschitz bandits with stochastic delays, demonstrating that their upper bounds are nearly tight. The lower bound shows that the regret must scale with both the zooming dimension and delay quantiles.

3 retrieved papers

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

Delay-aware zooming algorithm for bounded delays

The authors extend the classic zooming algorithm to handle bounded stochastic delays in Lipschitz bandits. Their Delayed Zooming algorithm achieves a regret bound of Õ(T^(dz+1)/(dz+2) + τmax·T^(dz/(dz+2))), matching the optimal delay-free rate with an additive delay-dependent term.

Contribution

Phased learning algorithm for unbounded delays

The authors introduce Delayed Lipschitz Phased Pruning (DLPP), a phased elimination-based algorithm that handles unbounded and potentially infinite delays. The algorithm achieves near-optimal regret with an additive dependence on delay distribution quantiles, supported by matching lower bounds.

Contribution

Instance-dependent lower bound for delayed Lipschitz bandits

The authors prove an instance-dependent regret lower bound for Lipschitz bandits with stochastic delays, demonstrating that their upper bounds are nearly tight. The lower bound shows that the regret must scale with both the zooming dimension and delay quantiles.

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