[Paper Review] Generalization Analysis for Game-Theoretic Machine Learning

Haifang Li, Fei Tian+3 more|arXiv (Cornell University)|Oct 9, 2014

Advanced Bandit Algorithms Research11 references3 citations

TL;DR

This paper presents the first generalization analysis for game-theoretic machine learning (GTML), a framework that optimizes mechanisms in dynamic systems with self-interested agents by modeling their behavior as Markov processes. It decomposes generalization error into behavior learning and mechanism learning errors, deriving non-asymptotic bounds using Markov chain stability and a novel nested covering number concept, with a concrete error bound established for GSP auctions with reserve prices.

ABSTRACT

For Internet applications like sponsored search, cautions need to be taken when using machine learning to optimize their mechanisms (e.g., auction) since self-interested agents in these applications may change their behaviors (and thus the data distribution) in response to the mechanisms. To tackle this problem, a framework called game-theoretic machine learning (GTML) was recently proposed, which first learns a Markov behavior model to characterize agents' behaviors, and then learns the optimal mechanism by simulating agents' behavior changes in response to the mechanism. While GTML has demonstrated practical success, its generalization analysis is challenging because the behavior data are non-i.i.d. and dependent on the mechanism. To address this challenge, first, we decompose the generalization error for GTML into the behavior learning error and the mechanism learning error; second, for the behavior learning error, we obtain novel non-asymptotic error bounds for both parametric and non-parametric behavior learning methods; third, for the mechanism learning error, we derive a uniform convergence bound based on a new concept called nested covering number of the mechanism space and the generalization analysis techniques developed for mixing sequences. To the best of our knowledge, this is the first work on the generalization analysis of GTML, and we believe it has general implications to the theoretical analysis of other complicated machine learning problems.

Motivation & Objective

• To address the lack of generalization theory for GTML, where behavior data are non-i.i.d. and dependent on the mechanism.
• To formally analyze the generalization error in GTML by decomposing it into behavior learning and mechanism learning components.
• To develop new theoretical tools—specifically, the nested covering number—for handling dependent data in mechanism learning.
• To provide the first non-asymptotic generalization error bounds for GTML in real-world applications like sponsored search.

Proposed method

• Decomposes GTML's generalization error into behavior learning error and mechanism learning error using the stability of Markov chain stationary distributions.
• Applies Hoeffding's inequality for Markov chains to derive non-asymptotic bounds for parametric and non-parametric behavior learning methods.
• Introduces the concept of nested covering number to measure complexity of the mechanism space under dependent data.
• Uses uniform convergence techniques for mixing sequences to bound the mechanism learning error.
• Establishes a total error bound by combining behavior and mechanism learning error bounds, with explicit dependence on sample size and function class complexity.
• Applies the framework to GSP auctions with reserve prices, deriving a concrete generalization error bound using pseudo-dimension of the reserve price function class.

Experimental results

Research questions

• RQ1 How can generalization error be formally decomposed in GTML when behavior data are non-i.i.d. and dependent on the mechanism?
• RQ2 What non-asymptotic error bounds can be established for behavior learning under Markovian assumptions in GTML?
• RQ3 How can the complexity of the mechanism space be measured in the presence of dependent data to enable generalization analysis?
• RQ4 What is the generalization error bound for GTML in the context of GSP auctions with reserve prices?
• RQ5 Can the proposed theoretical framework be applied to real-world dynamic systems involving strategic agents?

Key findings

• The paper establishes a non-asymptotic generalization error bound for behavior learning using Hoeffding-type inequalities for Markov chains, valid for both parametric and non-parametric models.
• It introduces the novel concept of nested covering number to analyze the complexity of the mechanism space under dependent data, enabling uniform convergence bounds.
• For GSP auctions with reserve prices, the second-layer covering number is bounded using the pseudo-dimension of the reserve price function class.
• A total generalization error bound is derived for GTML in GSP auctions, showing exponential decay in the number of behavior samples and sub-exponential decay in mechanism samples.
• The bound is of the form $ O(e^{-T_1}) + Oig( ext{covering terms} imes e^{-T_2^{s/(1+s)}}ig) $, indicating strong convergence under appropriate sample sizes.
• The result provides the first formal generalization guarantee for GTML in sponsored search, validating its theoretical robustness in dynamic, strategic environments.