Functional Bregman Divergences

Understanding the principles behind Shampoo has recently guided the development of more effective neural network optimizers. These methods learn a preconditioner by optimizing the Frobenius or Kullback-Leibler (KL) divergence against the gradient second moment. In this work, we investigate how the choice of divergence shapes preconditioning, which remains unclear and blocks further improvements. To do so, we develop a unified Bregman divergence framework that connects all popular divergences, allowing us to study them jointly. Through empirical spectral analysis of gradient second moments, we examine how divergence choice shapes Kronecker approximation and interacts with finite-sample error in preconditioning. We find that some divergences can better compensate for finite-sample underestimation of the empirical second moment, helping explain the differing behavior of their corresponding Shampoo variants. We further validate this explanation through GPT-2 pretraining experiments. By connecting divergence choice to practical training behavior, we believe our framework provides principled guidance for understanding the foundations of, and further improving, Shampoo.

Consider a community of agents who are seeking consensus on a set of parameters. The agents agree to use the same Bregman-type divergence to quantify disagreement between their individual estimates of the parameters but have varying confidence in each other's abilities. Each agent is happy to revise their estimate by moving to the weighted barycenter of all individual estimates with higher weights applied to more trusted agents. We show that such revisions naturally lead to an iterative algorithm which converges to a unique consensus estimate of the parameters. Furthermore, since the consensus estimate is itself a barycenter with computable weights, the group emerges as a collective super-agent with a well-formed opinion regarding the ability of each individual agent.

Many representation learning problems involve directed relations, such as lexical entailment, sentence entailment, ontology hierarchy, and citation links. Standard Euclidean, cosine, and Mahalanobis heads are symmetric, while generic neural scorers can model directionality but provide limited geometric structure. This paper proposes a role-aware neural convex divergence head for asymmetric representation learning. The head applies source- and target-role projections before evaluating an input-convex neural Bregman divergence, yielding a nonnegative structured score in the role-projected space. We characterize its projected-space identity, source-role convexity, directional-gap decomposition, and Hessian-based local curvature. Experiments on lexical, sentence, ontology, and directed graph benchmarks compare symmetric distances, unstructured asymmetric scorers, order/hyperbolic baselines, plain ICNN-Bregman heads, and the proposed role-aware variant. Across ten random seeds on the main semantic and ontology benchmarks, role-aware projections consistently improve directional accuracy over plain ICNN-Bregman heads while preserving zero observed negative divergence rate. The results also identify a boundary case: on large fixed-feature citation prediction, specialized symmetric or hyperbolic baselines remain stronger in ranking accuracy. Overall, the proposed head is best understood as a structured and interpretable plug-in distance module for tasks where directional relations matter.

Classification and Regression Trees (CART) constitute one of the most influential paradigms in statistical learning. Although a variety of impurity measures have been proposed for different statistical models, these criteria are typically introduced on a case-by-case basis and analyzed separately. In this paper, we study CART through the lens of Bregman divergences. This perspective places the classical least-squares criterion, Poisson deviance, Kullback-Leibler-type losses, and other impurity measures associated with exponential-family models within a common framework. As a result, key ingredients of the CART methodology -- including node representatives, impurity measures, and split selection rules -- can be expressed and analyzed through general properties of convex functions rather than through separate model-specific constructions. Beyond the algorithmic formulation, we investigate theoretical properties of Bregman-based CART procedures. In particular, we analyze how geometric properties of the generating convex function influence impurity reductions and stability of recursive partitions. We also establish consistency results within the proposed framework, providing a unified theoretical treatment for a broad family of CART type procedures. Our results provide a geometric interpretation of impurity-based tree construction and show that many classical CART impurity criteria admit a common interpretation within a Bregman framework.

This work introduces a general framework for calibeating based on regret minimization. As compared to Foster and Hart's seminal calibeating work which had specialized treatments of Brier score (squared loss) and log loss, we consider a large family of proper losses that includes $α$-Tsallis losses (for $α\in [1, 2]$) and Lipschitz losses. Our results for Tsallis losses also hold for an unscaled version of Tsallis loss that recovers log loss. Our analysis is oriented around the Bregman divergence view of a proper loss. Technically, our results for the family of Tsallis losses that we consider are U-calibration results, simultaneously obtaining logarithmic regret for all losses in this family while having a weaker dependence on the dimension compared to previous results. Of potential independent interest, we also show a new regret equality for the regret of Be The Regularized Leader. This regret equality holds for general proper losses and itself is based on two results related to online updating formulas for the generalized variance, the latter being a previously introduced generalization of variance based on Bregman divergences.

Aligning language models for both helpfulness and safety typically requires complex pipelines-separate reward and cost models, online reinforcement learning, and primal-dual updates. Recent direct preference optimization approaches simplify training but incorporate safety through ad-hoc modifications such as multi-stage procedures or heuristic margin terms, lacking a principled derivation. We show that the likelihood ratio of the optimal safe policy admits a closed-form decomposition that reduces safety alignment to a density ratio matching problem. Minimizing Bregman divergences between the data and model ratios yields Bregman Safety Optimization (BSO), a family of single-stage loss functions, each induced by a convex generator, that provably recover the optimal safe policy. BSO is both general and simple: it requires no auxiliary models, introduces only one hyperparameter beyond standard preference optimization, and recovers existing safety-aware methods as special cases. Experiments across safety alignment benchmarks show that BSO consistently improves the safety-helpfulness trade-off.

The Distributional Alignment Game framework provides a powerful variational perspective on Answer-Level Fine-Tuning (ALFT). However, standard algorithms for these games rely on estimating logarithmic rewards from small batches, introducing a systematic bias due to Jensen's inequality that can destabilize training. In this paper, we systematically resolve this structural estimation bias. First, we generalize the alignment game to arbitrary Bregman divergences, showing that for a family of geometries inducing polynomial rewards, we can construct provably exact and unbiased estimators using U-statistics. Second, for the canonical KL divergence game where an exact solution is impossible, we derive a globally robust minimax polynomial estimator that is provably optimal, achieving the fundamental statistical error limit of $Θ(1/K^2)$, which we establish via the Ditzian-Totik theorem. Finally, we synthesize these two approaches to propose a novel Variance-Optimal Augmented Polynomial Optimization Program (AQP) Estimator, proving that by systematically reducing variance, our method achieves not only optimal bias but also provably accelerated game convergence, leading to more efficient and stable training with zero online computational overhead.

Bregman divergences play a pivotal role in statistics, machine learning and computational information geometry. Particularly in the context of machine learning, they are central to clustering, exponential families, parameter estimation and optimisation, among other things. Despite this, the full toolkit of Hilbert spaces and in particular reproducing kernel Hilbert spaces have not been systematically developed and applied to functional Bregman divergences, where points are functions rather than finite-dimensional parameter vectors. While other types of functional Bregman divergences have been studied, these are typically in a Banach space rather than more directly aligned with kernel methods and Hilbert-space geometry commonly used in machine learning. We consider functional Bregman divergences on a Hilbert space, where the self-dual pairing and Riesz representer afford us particularly convenient calculus. Further specialising Bregman generators as a composition involving a kernel mean embedding makes such divergences easy to estimate. We discuss applications in clustering, universal estimation, robust estimation and generative modelling, and contrast our approach with other types of Bregman divergences.