Many real-world classification tasks require predicting multiple labels per instance, necessitating the optimization of complex evaluation metrics such as the $F$-measure and Jaccard index. While the Empirical Utility Maximization (EUM) framework is natural for these population-level metrics, existing theoretical results are largely limited to asymptotic Bayes-consistency. In this paper, we develop principled learning algorithms for optimizing a broad class of generalized metrics within the EUM framework, grounded in the stronger notion of $H$-consistency. Our key contribution is the design of novel surrogate loss functions for multi-label learning that admit provable $H$-consistency bounds, enabling optimization with non-asymptotic guarantees tailored to the hypothesis class and finite samples. Crucially, we prove these combinatorially formulated surrogates decompose exactly, operating in strictly $O(l)$ time without approximations. Building on this foundation, we introduce MMO (Multi-Label Metric Optimization), a new family of algorithms for optimizing generalized linear-fractional metrics. We validate our approach through extensive experiments, demonstrating robust scalability and superior performance over state-of-the-art continuous baselines on large-scale datasets (MS-COCO, Reuters-21578) in high-sparsity, deep learning regimes. Our results offer both theoretical rigor and practical effectiveness for general multi-label metric optimization.

Learning algorithms can be significantly improved by routing complex or uncertain inputs to specialized experts, balancing accuracy with computational cost. This approach, known as learning to defer, is essential in domains like natural language generation, medical diagnosis, and computer vision, where an effective deferral can reduce errors at low extra resource consumption. However, the two-stage learning to defer setting, which leverages existing predictors such as a collection of LLMs or other classifiers, often faces challenges due to an expert imbalance problem. This imbalance can lead to suboptimal performance, with deferral algorithms favoring the majority expert. We present a comprehensive study of two-stage learning to defer in expert imbalance settings. We cast the deferral loss optimization as a novel cost-sensitive learning problem over the input-expert domain. We derive new margin-based loss functions and guarantees tailored to this setting, and develop novel algorithms for cost-sensitive learning. Leveraging these results, we design principled deferral algorithms, MILD (Margin-based Imbalanced Learning to Defer), specifically suited for expert imbalance settings. Extensive experiments demonstrate the effectiveness of our approach, showing clear improvements over existing baselines on both image classification and real-world Large Language Model (LLM) routing tasks.

Abstractsurrogate loss (e.g., the logistic loss) as a proxy for the true objective: the non-convex, discontinuous 0-1 ranking Preference learning has become the foundationloss. This reliance raises a fundamental theoretical question of aligning Large Language Models (LLMs) withthat remains largely unanswered for deep networks: Does human intent. Popular methods, such as Direct Preference Optimization (DPO), minimize surrominimizing these surrogate losses actually guarantee the minimization of the true ranking error? However, we demonstrate that for In this work, we investigate this question through the lens the equicontinuous hypothesis sets typical of neu-of H-consistency (Mao, Mohri, and Zhong, 2023e). We ral networks, these standard surrogates are theo-formulate LLM preference learning as a pairwise ranking retically inconsistent, yielding vacuous general-problem and derive a series of results that bridge the gap between learning theory and practical fine-tuning. To resolve this, we formulate LLM alignment within a margin-shifted rankingwe identify a fundamental theoretical deficiency in standard framework. We demonstrate that for equicontinuous hypothbounds that depend on enforcing a separationesis sets, a property satisfied by neural networks, standard margin γ. Crucially, we extend this to Structure-surrogate minimization yields vacuous consistency guaranAware H-consistency, introducing a novel ob-tees. Specifically, without explicit constraints, a model can achieve arbitrarily low surrogate risk while maintaining ajective (SA-DPO) that adapts the margin based on the semantic distance between responses tohigh ranking error, effectively "cheating" the objective by handle synonyms and hard pairs. Finally, weshrinking score differences rather than learning the correct analyze the trade-off between consistency andordering. We prove that enforcing a confidence the Polynomial Hinge family) offer superior con-gap γ is not merely a heuristic, but a strict requirement for sistency guarantees for capacity-bounded models H-consistency in the deep learning regime. However, while compared to the standard logistic loss used in DPO. a uniform margin restores consistency, it is a blunt instrument. We show that demanding a large, fixed margin on semantically identical pairs (synonyms) forces the model to hallucinate differences where none exist, introducing bias 1. Introductionand instability. To address this, we propose Structure-Aware H-consistency and a corresponding objective, StructureThe alignment of Large Language Models (LLMs) has shifted from explicit Reward Modeling (Stiennon et al., Aware DPO (SA-DPO).

The choice of loss function in classification involves a fundamental trade-off: smooth losses (like Cross-Entropy) enable fast optimization rates but yield slow square-root consistency bounds, while piecewise-linear losses (like Hinge) offer fast linear consistency rates but suffer from non-differentiability. We propose Linear-Core (LC) Surrogates, a new family of convex loss functions that resolve this tension by stitching a linear core to a smooth tail. We prove that these surrogates are differentiable everywhere while retaining strict linear $H$-consistency bounds, effectively combining the optimization benefits of smoothness with the statistical efficiency of margin-based losses. In the structured prediction setting, we show that this smoothness unlocks a massive computational and energy advantage: it allows for an unbiased stochastic gradient estimator that bypasses the quadratic complexity $O(|\mathscr{Y}|^2)$ of exact inference (e.g., Viterbi). Empirically, our method achieves a 23$\times$ speedup over Structured SVMs on large-vocabulary sequence tagging tasks and demonstrates superior robustness to instance-dependent label noise, outperforming Cross-Entropy by 2.6% on corrupted CIFAR-10.

A series of recent publications by Awasthi, Mao, Mohri, and Zhong [2022b] have introduced the key notion of H-consistency bounds for surrogate loss functions. These are upper bounds on the zero-one estimation error of any predictor in a hypothesis set, expressed in terms of its surrogate loss estimation error. They are both non-asymptotic and hypothesis set-specific and thus stronger and more informative than Bayes-consistency. However, determining if they hold and deriving these bounds have required a specific proof and analysis for each surrogate loss. Can we derive more general tools and characterizations?

We present a comprehensive study of surrogate loss functions for learning to defer.

Many of these algorithms have been successfully used with specific loss functions such as the Hamming loss. Their use has been also extended to multivariate performance measures such as Precision/Recall orF1-score (Joachims,2005),which depend onpredictions onalltraining points.

We first examine how H -consistency bounds vary across surrogates based on the number of classes.