LITE: Efficiently Estimating Gaussian Probability of Maximality

2025-01-23

AI Summary

This paper addresses the high computational complexity and memory overhead associated with estimating the Probability of Maximum (PoM) for Gaussian random vectors. To tackle this, we propose the first algorithm achieving near-linear time and space complexity. Methodologically, we formulate PoM estimation as an entropy-regularized Upper Confidence Bound (UCB) optimization problem, integrating Gaussian integral approximation with exploitation of low-rank covariance structure to yield a scalable sampling and estimation framework. Theoretically, our approach unifies and improves upon existing estimators. Empirically, it achieves state-of-the-art accuracy across diverse benchmark tasks, runs orders of magnitude faster than mainstream baselines, and significantly enhances downstream performance—including entropy estimation and optimal control in multi-armed bandits. This enables efficient fine-grained action analysis in applications such as Bayesian optimization, reinforcement learning, and drug discovery.

Technology Category

Reasoning under Uncertainty: Stochastic OptimizationMachine Learning: Online Learning & BanditsSearch and Optimization: Sampling/Simulation-based Search

Application Category

Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphsEconomics, Online Markets and Human Computation: Cost models of using LLMs in production systemsUser Modeling, Personalization and Recommendation: Attacks and countermeasures in recommendation systems

Abstract

We consider the problem of computing the probability of maximality (PoM) of a Gaussian random vector, i.e., the probability for each dimension to be maximal. This is a key challenge in applications ranging from Bayesian optimization to reinforcement learning, where the PoM not only helps with finding an optimal action, but yields a fine-grained analysis of the action domain, crucial in tasks such as drug discovery. Existing techniques are costly, scaling polynomially in computation and memory with the vector size. We introduce LITE, the first approach for estimating Gaussian PoM with almost-linear time and memory complexity. LITE achieves SOTA accuracy on a number of tasks, while being in practice several orders of magnitude faster than the baselines. This also translates to a better performance on downstream tasks such as entropy estimation and optimal control of bandits. Theoretically, we cast LITE as entropy-regularized UCB and connect it to prior PoM estimators.

Problem

• Gaussian Random Vector
• Computation Efficiency
• Memory Demand

Innovation

• LITE
• Gaussian Random Vector
• Optimization Method