Efficient Opportunistic Approachability

2026-02-24

AI Summary

This work addresses the problem of opportunistic approachability: how a learner can efficiently approach a smaller target set when the adversary is constrained to a low-dimensional action space. Building on Blackwell’s approachability theory, we propose the first computationally efficient algorithm that eliminates the need for online calibration subroutines, thereby overcoming the exponential complexity barrier of traditional methods in high-dimensional settings. Our algorithm achieves a convergence rate of \(O(T^{-1/4})\) in general cases and attains the optimal rate of \(O(T^{-1/2})\) when the adversary’s action space has dimension at most two, significantly improving upon prior suboptimal results that only guaranteed rates of \(T^{-O(1/d)}\).

Technology Category

• Search and Optimization: Adversarial Search
• Game Theory and Economic Paradigms: Adversarial Learning
• Machine Learning: Online Learning & Bandits

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• User Modeling, Personalization and Recommendation: Attacks and countermeasures in recommendation systems

Abstract

We study the problem of opportunistic approachability: a generalization of Blackwell approachability where the learner would like to obtain stronger guarantees (i.e., approach a smaller set) when their adversary limits themselves to a subset of their possible action space. Bernstein et al. (2014) introduced this problem in 2014 and presented an algorithm that guarantees sublinear approachability rates for opportunistic approachability. However, this algorithm requires the ability to produce calibrated online predictions of the adversary's actions, a problem whose standard implementations require time exponential in the ambient dimension and result in approachability rates that scale as $T^{-O(1/d)}$. In this paper, we present an efficient algorithm for opportunistic approachability that achieves a rate of $O(T^{-1/4})$ (and an inefficient one that achieves a rate of $O(T^{-1/3})$), bypassing the need for an online calibration subroutine. Moreover, in the case where the dimension of the adversary's action set is at most two, we show it is possible to obtain the optimal rate of $O(T^{-1/2})$.

Problem

• opportunistic approachability
• Blackwell approachability
• adversarial action space
• sublinear convergence
• online learning

Innovation

• opportunistic approachability
• efficient algorithm
• convergence rate
• online calibration
• Blackwell approachability