OpenNovelty.org

Second-Order Bounds for [0,1]-Valued Regression via Betting Loss

ICLR 2026 Conference SubmissionAnonymous Authors
OpenReview Score: 6.0Download Report PDF
RegressionSecond-Order Bounds
Abstract:

We consider the [0,1][0,1]-valued regression problem in the i.i.d. setting. In a related problem called cost-sensitive classification, Foster and Krishnamurthy (2021) have shown that the log loss minimizer achieves an improved generalization bound compared to that of the squared loss minimizer in the sense that the bound scales with the cost of the best classifier, which can be arbitrarily small depending on the problem at hand. Such a result is often called a first-order bound. For [0,1][0,1]-valued regression, we first show that the log loss minimizer leads to a similar first-order bound. We then ask if there exists a loss function that achieves a variance-dependent bound (also known as a second-order bound), which is a strict improvement upon first-order bounds. We answer this question in the affirmative by proposing a novel loss function called the betting loss. Our result is “variance-adaptive” in the sense that the bound is attained without any knowledge about the variance, which is in contrast to modeling label (or reward) variance or the label distribution itself explicitly as part of the function class, such as distributional reinforcement learning.

Disclaimer
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.
If you have any questions, please contact: mingzhang23@m.fudan.edu.cn

Overview

Overall Novelty Assessment

The paper contributes a novel betting loss function for variance-adaptive regression in bounded [0,1] settings, alongside first-order bounds via log loss minimization. It resides in the 'Betting-Based Approaches for Bounded Estimation' leaf, which contains only four papers total, including this work. This represents a relatively sparse research direction within the broader taxonomy of 26 papers across the field. The leaf focuses specifically on betting frameworks and composite martingales for deriving variance-adaptive confidence bounds, distinguishing it from neighboring approaches that use Gaussian processes, distributional learning, or structural constraints.

The taxonomy reveals that betting-based methods form one specialized branch under 'Variance-Adaptive Loss Functions and Bounds', alongside Gaussian process regression and theoretical minimax analysis. Neighboring branches include 'Adaptive Label Distribution Learning' (which models uncertainty through distributional predictions rather than point estimates) and 'Adaptive Regression Under Data Constraints' (handling noise and privacy). The betting-based leaf explicitly excludes non-betting approaches and methods not focused on confidence sequences, positioning this work within a game-theoretic framework distinct from the distributional or kernel-based methods prevalent in sibling branches.

Among six candidates examined across three contributions, none were found to clearly refute the paper's claims. The first-order log loss bound examined three candidates with zero refutations; the novel betting loss function similarly examined three candidates with zero refutations; the second-order bounds contribution examined zero candidates. This limited search scope—covering only top-K semantic matches plus citation expansion—suggests the analysis captures closely related work but cannot claim exhaustive coverage. The absence of refutations among examined candidates indicates the betting loss formulation and its variance-adaptive properties appear distinct within this small sample.

Based on the limited search of six candidates, the work appears to occupy a novel position within the sparse betting-based cluster. The taxonomy structure confirms this is an emerging rather than crowded direction, with only three sibling papers in the same leaf. However, the small search scope means potentially relevant work in adjacent branches (e.g., distributional learning, kernel methods) may not have been fully examined. The analysis covers semantic proximity but not exhaustive field-wide comparison.

Taxonomy

Core-task Taxonomy Papers
26
3
Claimed Contributions
6
Contribution Candidate Papers Compared
0
Refutable Paper

Research Landscape Overview

Core task: variance-adaptive regression with bounded labels. The field addresses regression problems where target values lie within known bounds and the model must adapt to heterogeneous or unknown variance structures. The taxonomy reveals several main branches: Variance-Adaptive Loss Functions and Bounds develops theoretical frameworks and loss designs that explicitly account for bounded domains and varying noise levels; Adaptive Label Distribution Learning focuses on modeling uncertainty through distributional predictions rather than point estimates; Regression with Localization and Bounding Box Uncertainty tackles spatial prediction tasks where outputs are inherently bounded (e.g., Bounding Box Uncertainty[4], Fetal Pose Estimation[5]); Adaptive Regression Under Data Constraints handles settings with limited or noisy supervision (e.g., Controlled Label Noise[11]); Adaptive Estimation and Learning Algorithms encompasses algorithmic innovations for variance adaptation (e.g., Adaptive Kernel Alignment[13]); and Specialized Regression Applications covers domain-specific methods like age estimation and crowd counting. These branches collectively span theoretical foundations, algorithmic strategies, and practical deployment scenarios, with some works bridging multiple categories through unified frameworks that combine bounded-domain constraints with adaptive variance modeling. Particularly active lines of work include betting-based approaches that leverage game-theoretic principles to construct adaptive confidence sequences under bounded observations, and distributional learning methods that predict full label distributions to capture uncertainty. Betting Loss Regression[0] sits squarely within the betting-based cluster alongside Estimating Means by Betting[1] and Second-Order Betting Bounds[20], emphasizing provable guarantees for bounded estimation through sequential betting strategies. Compared to Estimating Means by Betting[1], which establishes foundational betting frameworks, Betting Loss Regression[0] extends these ideas specifically to regression settings with bounded labels. The Owen Discussion Contribution[6] provides complementary theoretical perspectives on these betting constructions. A contrasting direction appears in works like Soft Equality Constraints[3] and Multivariate Convex Regression[2], which impose structural assumptions (convexity, equality constraints) rather than relying on betting mechanisms, illustrating the field's diversity in balancing statistical efficiency, computational tractability, and theoretical rigor under bounded-domain constraints.

Claimed Contributions

First-order bound for [0,1]-valued regression via log loss

The authors show that the log loss minimizer achieves a first-order generalization bound for [0,1]-valued regression, improving upon the standard squared loss bound by scaling with the variance proxy f*(x)(1-f*(x)) rather than worst-case constants.

3 retrieved papers
Novel betting loss function for variance-adaptive regression

The authors introduce a new loss function called betting loss that enables variance-adaptive learning without requiring knowledge of conditional variances, achieving second-order bounds that scale with the true conditional variance rather than worst-case upper bounds.

3 retrieved papers
Second-order generalization bounds for parametric function classes

The authors establish that their betting loss minimizer achieves second-order generalization bounds for parametric function classes (those with polynomial covering numbers), with explicit results for linear function classes matching minimax-optimal rates while adapting to conditional variance.

0 retrieved papers

Core Task Comparisons

Comparisons with papers in the same taxonomy category

Contribution Analysis

Detailed comparisons for each claimed contribution

Contribution

First-order bound for [0,1]-valued regression via log loss

The authors show that the log loss minimizer achieves a first-order generalization bound for [0,1]-valued regression, improving upon the standard squared loss bound by scaling with the variance proxy f*(x)(1-f*(x)) rather than worst-case constants.

Contribution

Novel betting loss function for variance-adaptive regression

The authors introduce a new loss function called betting loss that enables variance-adaptive learning without requiring knowledge of conditional variances, achieving second-order bounds that scale with the true conditional variance rather than worst-case upper bounds.

Contribution

Second-order generalization bounds for parametric function classes

The authors establish that their betting loss minimizer achieves second-order generalization bounds for parametric function classes (those with polynomial covering numbers), with explicit results for linear function classes matching minimax-optimal rates while adapting to conditional variance.

Table of Contents