A Geometry-Aware Efficient Algorithm for Compositional Entropic Risk Minimization

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 $O(1/\sqrt{T})$ 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.

We consider the problem of training Machine Learning models to discriminate one data from all other data, such as recognizing one face among hundreds of thousands. In implementation, this involves reading a large amount of data and doing computation on them at the same time, which is very slow given the resources we have. Existing works then leverage a strategy called Entropic Risk Minimization to avoid the prohibitive amount of computation, but these methods do not always guide the model in the right direction, and the training could be numerically unstable and slow. We designed a new training method, named SCENT, that leverages the Entropic Risk Minimization strategy in a more careful way. The key idea of SCENT is to adapt the model update to the (mathematical) "shape" of the problem, which is to update the model in a more correct direction calculated based on the property of the problem. Theoretically, we prove that SCENT has a faster training speed and is more stable than existing methods. Empirically, we show that models trained by SCENT achieve better performance on tasks such as classifying data into hundreds of thousands classes (like recognizing faces and classifying species).