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Discounted Online Convex Optimization: Uniform Regret Across a Continuous Interval

ICLR 2026 Conference SubmissionAnonymous Authors
OpenReview Score: 6.0Download Report PDF
Online Convex OptimizationDiscounted Online LearningAdaptive Algorithms
Abstract:

Reflecting the greater significance of recent history over the distant past in non-stationary environments, λ\lambda-discounted regret has been introduced in online convex optimization (OCO) to gracefully forget past data as new information arrives. When the discount factor λ\lambda is given, online gradient descent with an appropriate step size achieves an O(1/1−λ)O(1/\sqrt{1-\lambda}) discounted regret. However, the value of λ\lambda is often not predetermined in real-world scenarios. This gives rise to a significant \emph{open question}: is it possible to develop a discounted algorithm that adapts to an unknown discount factor. In this paper, we affirmatively answer this question by providing a novel analysis to demonstrate that smoothed OGD (SOGD) achieves a uniform O(log⁡T/1−λ)O(\sqrt{\log T/1-\lambda}) discounted regret, holding for all values of λ\lambda across a continuous interval simultaneously. The basic idea is to maintain multiple OGD instances to handle different discount factors, and aggregate their outputs sequentially by an online prediction algorithm named as Discounted-Normal-Predictor (DNP). Our analysis reveals that DNP can combine the decisions of two experts, even when they operate on discounted regret with different discount factors.

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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.
NOTE that some papers exist in multiple, slightly different versions (e.g., with different titles or URLs). The system may retrieve several versions of the same underlying work. The current automated pipeline does not reliably align or distinguish these cases, so human reviewers will need to disambiguate them manually.
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Overview

Overall Novelty Assessment

The paper addresses adapting online convex optimization to unknown discount factors by proposing a smoothed OGD algorithm that achieves uniform discounted regret bounds across a continuous interval of discount factors. According to the taxonomy, this work sits in the 'Uniform Regret Guarantees Across Discount Factor Intervals' leaf under 'Adaptive Discounting Mechanisms for Non-Stationary Environments'. Notably, this leaf contains only the original paper itself with no sibling papers, suggesting this specific research direction—achieving simultaneous guarantees for all discount factors in a range—is relatively unexplored within the examined literature.

The taxonomy reveals that the broader field divides into adaptive discounting mechanisms and problem-specific discounted regret analysis. The original paper's leaf sits alongside 'Instance-Specific Adaptive Discounting', which focuses on learning optimal discount factors for particular instances rather than providing uniform guarantees. The sibling branch 'Discounted Regret Analysis for Specific Problem Classes' contains work on linear regression and application-driven optimization, indicating that most prior research either targets specific problem structures or learns instance-specific parameters rather than pursuing the uniform guarantee approach taken here.

Among the three identified contributions, the literature search examined only five candidate papers total. The core contribution of uniform discounted regret bounds was examined against one candidate with no refutation found. The novel DNP analysis for combining experts was examined against four candidates, again with no refutations. The smoothed OGD framework itself was not examined against any candidates. Given this limited search scope of five papers, the analysis suggests these contributions appear distinct from the examined prior work, though the small candidate pool means substantial related work may exist beyond the top-K semantic matches retrieved.

Based on the limited search of five candidates and the sparse taxonomy leaf containing only the original paper, the work appears to occupy a relatively unexplored niche within discounted online learning. However, the small search scope and absence of sibling papers in the taxonomy leaf may reflect limitations in the literature retrieval process rather than definitive evidence of novelty. A more comprehensive search across online learning and non-stationary optimization literature would be needed to fully assess the contribution's originality.

Taxonomy

Core-task Taxonomy Papers
4
3
Claimed Contributions
5
Contribution Candidate Papers Compared
0
Refutable Paper

Research Landscape Overview

Core task: Adapting online convex optimization to unknown discount factors. The field addresses how online learning algorithms can handle discounted objectives when the discount factor is not known in advance. The taxonomy reveals two main branches: one focusing on adaptive discounting mechanisms for non-stationary environments, and another on discounted regret analysis for specific problem classes. The first branch develops methods that adjust to changing or unknown discount rates, enabling algorithms to remain robust across different temporal weighting schemes. The second branch examines how discounted regret bounds can be derived for particular settings such as linear regression or pricing problems, often leveraging problem structure to achieve tighter guarantees. Representative works like Discounted Linear Regression[1] and Discounted Adaptive Learning[2] illustrate how specialized problem classes benefit from tailored analysis, while applications such as Smart Grid Optimization[3] and Posted Pricing Valuations[4] demonstrate the practical relevance of discounting in dynamic decision-making. A central theme across these branches is the trade-off between generality and problem-specific efficiency: adaptive mechanisms aim for uniform performance across a range of discount factors, whereas specialized analyses exploit structure to improve bounds. Discounted Convex Optimization[0] sits within the adaptive discounting branch, specifically targeting uniform regret guarantees across discount factor intervals. This positions it closely to works like Discounted Adaptive Learning[2], which also emphasizes robustness to unknown parameters, but Discounted Convex Optimization[0] extends the scope to general convex settings rather than focusing on a narrower problem class. Compared to Discounted Linear Regression[1], which tailors its approach to linear models, the original paper pursues broader applicability at the cost of potentially looser constants. The main open question remains how to balance adaptivity with computational efficiency and whether problem-agnostic methods can match the performance of specialized algorithms in practice.

Claimed Contributions

Uniform discounted regret bound across continuous discount factor interval

The authors develop an algorithm that achieves O(√logT/(1−λ)) discounted regret uniformly for all discount factors λ in a continuous interval [1−1/τ, 1−1/T], without requiring prior knowledge of the specific discount factor value.

1 retrieved paper
Novel analysis of DNP for combining experts with different discount factors

The authors provide a novel theoretical analysis showing that Discounted-Normal-Predictor (DNP) can successfully aggregate decisions from experts operating under different discount factors, overcoming a key technical challenge in the meta-expert framework.

4 retrieved papers
Smoothed OGD algorithm with sequential aggregation framework

The authors propose a method that constructs multiple OGD instances with different step sizes corresponding to discretized discount factors, then uses DNP with conservative updating to sequentially aggregate their decisions, enabling adaptation to unknown discount factors.

0 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

Uniform discounted regret bound across continuous discount factor interval

The authors develop an algorithm that achieves O(√logT/(1−λ)) discounted regret uniformly for all discount factors λ in a continuous interval [1−1/τ, 1−1/T], without requiring prior knowledge of the specific discount factor value.

Contribution

Novel analysis of DNP for combining experts with different discount factors

The authors provide a novel theoretical analysis showing that Discounted-Normal-Predictor (DNP) can successfully aggregate decisions from experts operating under different discount factors, overcoming a key technical challenge in the meta-expert framework.

Contribution

Smoothed OGD algorithm with sequential aggregation framework

The authors propose a method that constructs multiple OGD instances with different step sizes corresponding to discretized discount factors, then uses DNP with conservative updating to sequentially aggregate their decisions, enabling adaptation to unknown discount factors.

Table of Contents

Discounted Online Convex Optimization: Uniform Regret Across a Continuous Interval | Novelty Validation