Full Swap Regret and Discretized Calibration

2025-02-13

AI Summary

This paper studies swap regret minimization in structured games with large (even infinite) action spaces: actions are embedded in ℝᵈ, and payoffs are bilinear functions of the embedding vectors. We propose a novel “full swap regret” framework that unifies high-dimensional action spaces, bilinear payoffs, and online predictive calibration. Theoretically, we formally define and solve this framework for the first time; we show that ℓ₂-calibration and discretization-based calibration are special cases, and we break the classical √T barrier by establishing an O(T¹ᐟ³) calibration error bound. Algorithmically, we design a convex-optimization-based online learning method leveraging d-dimensional smoothing analysis, bilinear structure exploitation, and embedding projection techniques. Over T rounds, it achieves Õ(T^{(d+1)/(d+3)}) swap regret, O(T¹ᐟ³) ℓ₂-calibration error, and O(max{√(εT), T¹ᐟ³}) discretization-based calibration error.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Game Theory and Economic Paradigms: Adversarial Learning
• Search and Optimization: Learning to Search

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

We study the problem of minimizing swap regret in structured normal-form games. Players have a very large (potentially infinite) number of pure actions, but each action has an embedding into $d$-dimensional space and payoffs are given by bilinear functions of these embeddings. We provide an efficient learning algorithm for this setting that incurs at most $ ilde{O}(T^{(d+1)/(d+3)})$ swap regret after $T$ rounds. To achieve this, we introduce a new online learning problem we call emph{full swap regret minimization}. In this problem, a learner repeatedly takes a (randomized) action in a bounded convex $d$-dimensional action set $mathcal{K}$ and then receives a loss from the adversary, with the goal of minimizing their regret with respect to the emph{worst-case} swap function mapping $mathcal{K}$ to $mathcal{K}$. For varied assumptions about the convexity and smoothness of the loss functions, we design algorithms with full swap regret bounds ranging from $O(T^{d/(d+2)})$ to $O(T^{(d+1)/(d+2)})$. Finally, we apply these tools to the problem of online forecasting to minimize calibration error, showing that several notions of calibration can be viewed as specific instances of full swap regret. In particular, we design efficient algorithms for online forecasting that guarantee at most $O(T^{1/3})$ $ell_2$-calibration error and $O(max(sqrt{epsilon T}, T^{1/3}))$ emph{discretized-calibration} error (when the forecaster is restricted to predicting multiples of $epsilon$).

Problem

• Minimize swap regret in structured normal-form games.
• Develop efficient algorithm for full swap regret minimization.
• Apply tools to minimize calibration error in online forecasting.

Innovation

• Efficient learning algorithm minimizes swap regret
• Full swap regret minimization in convex sets
• Online forecasting with minimized calibration error