Improved Balanced Classification with Theoretically Grounded Loss Functions

The balanced loss is a widely adopted objective for multi-class classification under class imbalance. By assigning equal importance to all classes, regardless of their frequency, it promotes fairness and ensures that minority classes are not overlooked. However, directly minimizing the balanced classification loss is typically intractable, which makes the design of effective surrogate losses a central question. This paper introduces and studies two advanced surrogate loss families: Generalized Logit-Adjusted (GLA) loss functions and Generalized Class-Aware weighted (GCA) losses. GLA losses generalize Logit-Adjusted losses, which shift logits based on class priors, to the broader general cross-entropy loss family. GCA loss functions extend the standard class-weighted losses, which scale losses inversely by class frequency, by incorporating class-dependent confidence margins and extending them to the general cross-entropy family. We present a comprehensive theoretical analysis of consistency for both loss families. We show that GLA losses are Bayes-consistent, but only $H$-consistent for complete (i.e., unbounded) hypothesis sets. Moreover, their $H$-consistency bounds depend inversely on the minimum class probability, scaling at least as $1/\mathsf p_{\min}$. In contrast, GCA losses are $H$-consistent for any hypothesis set that is bounded or complete, with $H$-consistency bounds that scale more favorably as $1/\sqrt{\mathsf p_{\min}}$, offering significantly stronger theoretical guarantees in imbalanced settings. We report the results of experiments demonstrating that, empirically, both the GCA losses with calibrated class-dependent confidence margins and GLA losses can greatly outperform straightforward class-weighted losses as well as the LA losses. GLA generally performs slightly better in common benchmarks, whereas GCA exhibits a slight edge in highly imbalanced settings.

Principled Algorithms for Optimizing Generalized Metrics in Binary Classification

In applications with significant class imbalance or asymmetric costs, metrics such as the $F_β$-measure, AM measure, Jaccard similarity coefficient, and weighted accuracy offer more suitable evaluation criteria than standard binary classification loss. However, optimizing these metrics present significant computational and statistical challenges. Existing approaches often rely on the characterization of the Bayes-optimal classifier, and use threshold-based methods that first estimate class probabilities and then seek an optimal threshold. This leads to algorithms that are not tailored to restricted hypothesis sets and lack finite-sample performance guarantees. In this work, we introduce principled algorithms for optimizing generalized metrics, supported by $H$-consistency and finite-sample generalization bounds. Our approach reformulates metric optimization as a generalized cost-sensitive learning problem, enabling the design of novel surrogate loss functions with provable $H$-consistency guarantees. Leveraging this framework, we develop new algorithms, METRO (Metric Optimization), with strong theoretical performance guarantees. We report the results of experiments demonstrating the effectiveness of our methods compared to prior baselines.

Coherence Mechanisms for Provable Self-Improvement

Self-improvement is a critical capability for large language models and other intelligent systems, enabling them to refine their behavior and internal consistency without external supervision. Despite its importance, prior approaches largely rely on empirical heuristics and lack formal guarantees. In this paper, we propose a principled framework for self-improvement based on the concept of \emph{coherence}, which requires that a model's outputs remain consistent under task-preserving transformations of the input. We formalize this concept using projection-based mechanisms that update a baseline model to be coherent while remaining as close as possible to its original behavior. We provide rigorous theoretical guarantees that these mechanisms achieve \emph{monotonic improvement}, measured by a reduction in expected Bregman divergence. Our analysis is comprehensive, covering both \emph{direct} and \emph{two-step} projection methods, and robustly extends these guarantees to non-realizable settings, empirical (finite-sample) distributions, and relaxed coherence constraints. Furthermore, we establish a general \emph{characterization theorem}, showing that any mechanism with similar provable improvement guarantees must inherently conform to a coherence-based structure. This culminates in rigidity results under the demand for universal improvement, establishing coherence as a fundamental and, in a formal sense, necessary principle for provable self-improvement.

Budgeted Multiple-Expert Deferral

Learning to defer uncertain predictions to costly experts offers a powerful strategy for improving the accuracy and efficiency of machine learning systems. However, standard training procedures for deferral algorithms typically require querying all experts for every training instance, an approach that becomes prohibitively expensive when expert queries incur significant computational or resource costs. This undermines the core goal of deferral: to limit unnecessary expert usage. To overcome this challenge, we introduce the budgeted deferral framework, which aims to train effective deferral algorithms while minimizing expert query costs during training. We propose new algorithms for both two-stage and single-stage multiple-expert deferral settings that selectively query only a subset of experts per training example. While inspired by active learning, our setting is fundamentally different: labels are already known, and the core challenge is to decide which experts to query in order to balance cost and predictive performance. We establish theoretical guarantees for both of our algorithms, including generalization bounds and label complexity analyses. Empirical results across several domains show that our algorithms substantially reduce training costs without sacrificing prediction accuracy, demonstrating the practical value of our budget-aware deferral algorithms.

Mastering Multiple-Expert Routing: Realizable $H$-Consistency and Strong Guarantees for Learning to Defer

The problem of learning to defer with multiple experts consists of optimally assigning input instances to experts, balancing the trade-off between their accuracy and computational cost. This is a critical challenge in natural language generation, but also in other fields such as image processing, and medical diagnostics. Recent studies have proposed surrogate loss functions to optimize deferral, but challenges remain in ensuring their consistency properties. This paper introduces novel surrogate loss functions and efficient algorithms with strong theoretical learning guarantees. We address open questions regarding realizable $H$-consistency, $H$-consistency bounds, and Bayes-consistency for both single-stage (jointly learning predictor and deferral function) and two-stage (learning only the deferral function with a fixed expert) learning scenarios. For single-stage deferral, we introduce a family of new realizable $H$-consistent surrogate losses and further prove $H$-consistency for a selected member. For two-stage deferral, we derive new surrogate losses that achieve realizable $H$-consistency, $H$-consistency bounds, and Bayes-consistency for the two-expert scenario and, under natural assumptions, multiple-expert scenario. Additionally, we provide enhanced theoretical guarantees under low-noise assumptions for both scenarios. Finally, we report the results of experiments using our proposed surrogate losses, comparing their performance against existing baselines.

High-Dimensional Calibration from Swap Regret

We study the online calibration of multi-dimensional forecasts over an arbitrary convex set $\mathcal{P} \subset \mathbb{R}^d$ relative to an arbitrary norm $\Vert\cdot\Vert$. We connect this with the problem of external regret minimization for online linear optimization, showing that if it is possible to guarantee $O(\sqrt{\rho T})$ worst-case regret after $T$ rounds when actions are drawn from $\mathcal{P}$ and losses are drawn from the dual $\Vert \cdot \Vert_*$ unit norm ball, then it is also possible to obtain $\epsilon$-calibrated forecasts after $T = \exp(O(\rho /\epsilon^2))$ rounds. When $\mathcal{P}$ is the $d$-dimensional simplex and $\Vert \cdot \Vert$ is the $\ell_1$-norm, the existence of $O(\sqrt{T\log d})$-regret algorithms for learning with experts implies that it is possible to obtain $\epsilon$-calibrated forecasts after $T = \exp(O(\log{d}/\epsilon^2)) = d^{O(1/\epsilon^2)}$ rounds, recovering a recent result of Peng (2025). Interestingly, our algorithm obtains this guarantee without requiring access to any online linear optimization subroutine or knowledge of the optimal rate $\rho$ -- in fact, our algorithm is identical for every setting of $\mathcal{P}$ and $\Vert \cdot \Vert$. Instead, we show that the optimal regularizer for the above OLO problem can be used to upper bound the above calibration error by a swap regret, which we then minimize by running the recent TreeSwap algorithm with Follow-The-Leader as a subroutine. Finally, we prove that any online calibration algorithm that guarantees $\epsilon T$ $\ell_1$-calibration error over the $d$-dimensional simplex requires $T \geq \exp(\mathrm{poly}(1/\epsilon))$ (assuming $d \geq \mathrm{poly}(1/\epsilon)$). This strengthens the corresponding $d^{\Omega(\log{1/\epsilon})}$ lower bound of Peng, and shows that an exponential dependence on $1/\epsilon$ is necessary.

Swap Regret and Correlated Equilibria Beyond Normal-Form Games

Swap regret is a notion that has proven itself to be central to the study of general-sum normal-form games, with swap-regret minimization leading to convergence to the set of correlated equilibria and guaranteeing non-manipulability against a self-interested opponent. However, the situation for more general classes of games -- such as Bayesian games and extensive-form games -- is less clear-cut, with multiple candidate definitions for swap-regret but no known efficiently minimizable variant of swap regret that implies analogous non-manipulability guarantees. In this paper, we present a new variant of swap regret for polytope games that we call ``profile swap regret'', with the property that obtaining sublinear profile swap regret is both necessary and sufficient for any learning algorithm to be non-manipulable by an opponent (resolving an open problem of Mansour et al., 2022). Although we show profile swap regret is NP-hard to compute given a transcript of play, we show it is nonetheless possible to design efficient learning algorithms that guarantee at most $O(\sqrt{T})$ profile swap regret. Finally, we explore the correlated equilibrium notion induced by low-profile-swap-regret play, and demonstrate a gap between the set of outcomes that can be implemented by this learning process and the set of outcomes that can be implemented by a third-party mediator (in contrast to the situation in normal-form games).

Multi-Label Learning with Stronger Consistency Guarantees

We present a detailed study of surrogate losses and algorithms for multi-label learning, supported by $H$-consistency bounds. We first show that, for the simplest form of multi-label loss (the popular Hamming loss), the well-known consistent binary relevance surrogate suffers from a sub-optimal dependency on the number of labels in terms of $H$-consistency bounds, when using smooth losses such as logistic losses. Furthermore, this loss function fails to account for label correlations. To address these drawbacks, we introduce a novel surrogate loss, multi-label logistic loss, that accounts for label correlations and benefits from label-independent $H$-consistency bounds. We then broaden our analysis to cover a more extensive family of multi-label losses, including all common ones and a new extension defined based on linear-fractional functions with respect to the confusion matrix. We also extend our multi-label logistic losses to more comprehensive multi-label comp-sum losses, adapting comp-sum losses from standard classification to the multi-label learning. We prove that this family of surrogate losses benefits from $H$-consistency bounds, and thus Bayes-consistency, across any general multi-label loss. Our work thus proposes a unified surrogate loss framework benefiting from strong consistency guarantees for any multi-label loss, significantly expanding upon previous work which only established Bayes-consistency and for specific loss functions. Additionally, we adapt constrained losses from standard classification to multi-label constrained losses in a similar way, which also benefit from $H$-consistency bounds and thus Bayes-consistency for any multi-label loss. We further describe efficient gradient computation algorithms for minimizing the multi-label logistic loss.

Enhanced $H$-Consistency Bounds

Recent research has introduced a key notion of $H$-consistency bounds for surrogate losses. These bounds offer finite-sample guarantees, quantifying the relationship between the zero-one estimation error (or other target loss) and the surrogate loss estimation error for a specific hypothesis set. However, previous bounds were derived under the condition that a lower bound of the surrogate loss conditional regret is given as a convex function of the target conditional regret, without non-constant factors depending on the predictor or input instance. Can we derive finer and more favorable $H$-consistency bounds? In this work, we relax this condition and present a general framework for establishing enhanced $H$-consistency bounds based on more general inequalities relating conditional regrets. Our theorems not only subsume existing results as special cases but also enable the derivation of more favorable bounds in various scenarios. These include standard multi-class classification, binary and multi-class classification under Tsybakov noise conditions, and bipartite ranking.

Realizable $H$-Consistent and Bayes-Consistent Loss Functions for Learning to Defer

We present a comprehensive study of surrogate loss functions for learning to defer. We introduce a broad family of surrogate losses, parameterized by a non-increasing function $\Psi$, and establish their realizable $H$-consistency under mild conditions. For cost functions based on classification error, we further show that these losses admit $H$-consistency bounds when the hypothesis set is symmetric and complete, a property satisfied by common neural network and linear function hypothesis sets. Our results also resolve an open question raised in previous work (Mozannar et al., 2023) by proving the realizable $H$-consistency and Bayes-consistency of a specific surrogate loss. Furthermore, we identify choices of $\Psi$ that lead to $H$-consistent surrogate losses for any general cost function, thus achieving Bayes-consistency, realizable $H$-consistency, and $H$-consistency bounds simultaneously. We also investigate the relationship between $H$-consistency bounds and realizable $H$-consistency in learning to defer, highlighting key differences from standard classification. Finally, we empirically evaluate our proposed surrogate losses and compare them with existing baselines.