Non-Linear Counterfactual Aggregate Optimization

2025-09-03

AI Summary

This paper addresses the direct optimization of nonlinear aggregate objective functions—specifically, when the objective is a sum of many small contributions and the aggregation function is nonlinear (precluding reduction to expectation maximization), rendering standard methods ineffective. We propose a scalable counterfactual gradient descent algorithm that leverages concentration properties of the aggregate sum to optimize nonlinear objectives—such as success probability or uplift threshold attainment rate—without linear approximations or distributional assumptions. Our core innovation integrates counterfactual reasoning and concentration analysis directly into a gradient-based optimization framework, enabling end-to-end optimization of the global objective. Experiments demonstrate that our method significantly improves the probability of achieving target outcomes in success-driven tasks (e.g., A/B testing), outperforming expectation-maximization–based baselines.

Technology Category

• Search and Optimization: Non-convex Optimization
• Machine Learning: Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• User Modeling, Personalization and Recommendation: Attacks and countermeasures in recommendation systems

Abstract

We consider the problem of directly optimizing a non-linear function of an outcome, where this outcome itself is the sum of many small contributions. The non-linearity of the function means that the problem is not equivalent to the maximization of the expectation of the individual contribution. By leveraging the concentration properties of the sum of individual outcomes, we derive a scalable descent algorithm that directly optimizes for our stated objective. This allows for instance to maximize the probability of successful A/B test, for which it can be wiser to target a success criterion, such as exceeding a given uplift, rather than chasing the highest expected payoff.

Problem

• Optimizing non-linear functions of aggregate outcomes
• Maximizing probability of exceeding success thresholds
• Addressing non-equivalence to individual expectation maximization

Innovation

• Non-linear counterfactual aggregate optimization algorithm
• Scalable descent method for objective optimization
• Concentration properties leverage individual outcomes