# A Geometry-Aware Efficient Algorithm for Compositional Entropic Risk Minimization arXiv: https://arxiv.org/abs/2602.02877 · HTML: https://mlsanity.com/paper/2602.02877/ ## Key facts - Cited by 1 papers in the map (data through 14 Oct 2026). - Builds on 39 of its 56 references that are in the map. - First posted on arXiv 2 Feb 2026. ## Builds on (most cited first, 30 of 39) - [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015) - [Learning Transferable Visual Models From Natural Language Supervision](https://mlsanity.com/paper/2103.00020/) (2021) - [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020) - [A Simple Framework for Contrastive Learning of Visual Representations](https://mlsanity.com/paper/2002.05709/) (2020) - [The Many Faces of Robustness: A Critical Analysis of Out-of-Distribution Generalization](https://mlsanity.com/paper/2006.16241/) (2020) - [Do ImageNet Classifiers Generalize to ImageNet?](https://mlsanity.com/paper/1902.10811/) (2019) - [Microsoft COCO Captions: Data Collection and Evaluation Server](https://mlsanity.com/paper/1504.00325/) (2015) - [Natural Adversarial Examples](https://mlsanity.com/paper/1907.07174/) (2019) - [Understanding Contrastive Representation Learning through Alignment and Uniformity on the Hypersphere](https://mlsanity.com/paper/2005.10242/) (2020) - [Reproducible scaling laws for contrastive language-image learning](https://mlsanity.com/paper/2212.07143/) (2022) - [Learning Robust Global Representations by Penalizing Local Predictive Power](https://mlsanity.com/paper/1905.13549/) (2019) - [Approximate Nearest Neighbor Negative Contrastive Learning for Dense Text Retrieval](https://mlsanity.com/paper/2007.00808/) (2020) - [DataComp: In search of the next generation of multimodal datasets](https://mlsanity.com/paper/2304.14108/) (2023) - [Data Filtering Networks](https://mlsanity.com/paper/2309.17425/) (2023) - [Large-Scale Methods for Distributionally Robust Optimization](https://mlsanity.com/paper/2010.05893/) (2020) - [Partial FC: Training 10 Million Identities on a Single Machine](https://mlsanity.com/paper/2010.05222/) (2020) - [BioCLIP: A Vision Foundation Model for the Tree of Life](https://mlsanity.com/paper/2311.18803/) (2023) - [Tilted Empirical Risk Minimization](https://mlsanity.com/paper/2007.01162/) (2020) - [Stochastic Compositional Gradient Descent: Algorithms for Minimizing Compositions of Expected-Value Functions](https://mlsanity.com/paper/1411.3803/) (2014) - [Biased Stochastic First-Order Methods for Conditional Stochastic Optimization and Applications in Meta Learning](https://mlsanity.com/paper/2002.10790/) (2020) - [Provable Stochastic Optimization for Global Contrastive Learning: Small Batch Does Not Harm Performance](https://mlsanity.com/paper/2202.12387/) (2022) - [An Online Method for A Class of Distributionally Robust Optimization with Non-Convex Objectives](https://mlsanity.com/paper/2006.10138/) (2020) - [NGAME: Negative Mining-aware Mini-batching for Extreme Classification](https://mlsanity.com/paper/2207.04452/) (2022) - [When AUC meets DRO: Optimizing Partial AUC for Deep Learning with Non-Convex Convergence Guarantee](https://mlsanity.com/paper/2203.00176/) (2022) - [Finite-Sum Coupled Compositional Stochastic Optimization: Theory and Applications](https://mlsanity.com/paper/2202.12396/) (2022) - [Not All Semantics are Created Equal: Contrastive Self-supervised Learning with Automatic Temperature Individualization](https://mlsanity.com/paper/2305.11965/) (2023) - [Large-scale Stochastic Optimization of NDCG Surrogates for Deep Learning with Provable Convergence](https://mlsanity.com/paper/2202.12183/) (2022) - [Optimal Algorithms for Convex Nested Stochastic Composite Optimization](https://mlsanity.com/paper/2011.10076/) (2020) - [Multi-block-Single-probe Variance Reduced Estimator for Coupled Compositional Optimization](https://mlsanity.com/paper/2207.08540/) (2022) - [Stochastic Constrained DRO with a Complexity Independent of Sample Size](https://mlsanity.com/paper/2210.05740/) (2022) ## Built on by (most cited first, 1 of 1) - [Scalable Cox Regression via Grouped Risk Sets and Sharper LogSumExp Rates](https://mlsanity.com/paper/2609.40120/) (2026) ## Abstract (arXiv) This paper studies optimization for a family of problems termed compositional entropic risk minimization, in which each data's loss is formulated as a Log-Expectation-Exponential (Log-E-Exp) function. The Log-E-Exp formulation serves as an abstraction of the Log-Sum-Exponential (LogSumExp) function when the explicit summation inside the logarithm is taken over a gigantic number of items and is therefore expensive to evaluate. While entropic risk objectives of this form arise in many machine learning problems, existing optimization algorithms suffer from several fundamental limitations including non-convergence, numerical instability, and slow convergence rates. To address these limitations, we propose a geometry-aware stochastic algorithm, termed SCENT, for the dual formulation of entropic risk minimization cast as a min--min optimization problem. The key to our design is a stochastic proximal mirror descent (SPMD) update for the dual variable, equipped with a Bregman divergence induced by a negative exponential function that faithfully captures the geometry of the objective. Our main contributions are threefold: (i) we establish an O(1/√T) convergence rate of the proposed SCENT algorithm for convex problems; (ii) we theoretically characterize the advantages of SPMD over standard SGD update for optimizing the dual variable; and (iii) we demonstrate the empirical effectiveness of SCENT on extreme classification, partial AUC maximization, contrastive learning and distributionally robust optimization, where it consistently outperforms existing baselines. Code is available at https://github.com/Optimization-AI/SCENT.