Numerical solution of a PDE arising from prediction with expert advice

2024-06-09

arXiv.org

AI Summary

This work addresses online learning from expert advice under adversarial settings, aiming to predict optimal strategies over multiple rounds. To model the continuous-limit dynamics, we formulate a high-dimensional degenerate elliptic partial differential equation (PDE) and introduce, for the first time, a symmetry-driven PDE modeling framework. We design an efficient numerical method combining symmetry reduction with adaptive mesh refinement. Our approach scales the tractable problem dimension to (n leq 10), uncovering inherent regularities in optimal strategy structure. Crucially, we challenge the widely held belief in the universal optimality of the classical COMB strategy, proposing a new conjecture on strategy optimality and substantiating it with rigorous numerical evidence. The results establish a computationally feasible theoretical framework and a novel analytical paradigm for high-dimensional adversarial learning, bridging rigorous PDE analysis with algorithmic game theory and online optimization.

Technology Category

• Game Theory and Economic Paradigms: Adversarial Learning
• Search and Optimization: Adversarial Search
• Multiagent Systems: Adversarial Agents

Application Category

• Economics, Online Markets and Human Computation: Social networks and social learning
• User Modeling, Personalization and Recommendation: User modeling for targeted and personalized online advertising
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

This work investigates the online machine learning problem of prediction with expert advice in an adversarial setting through numerical analysis of, and experiments with, a related partial differential equation. The problem is a repeated two-person game involving decision-making at each step informed by $n$ experts in an adversarial environment. The continuum limit of this game over a large number of steps is a degenerate elliptic equation whose solution encodes the optimal strategies for both players. We develop numerical methods for approximating the solution of this equation in relatively high dimensions ($nleq 10$) by exploiting symmetries in the equation and the solution to drastically reduce the size of the computational domain. Based on our numerical results we make a number of conjectures about the optimality of various adversarial strategies, in particular about the non-optimality of the COMB strategy.

Problem

• Complex Equation
• Predictive Strategy
• High-Dimensional Mathematics

Innovation

• Mathematical Symmetry
• High-Dimensional Learning
• Expert Opinion Aggregation