Convergence and Representation Learning of Neural Gradient Descent-Ascent for Functional Conditional Moment Equations

Core Concepts

The authors study the convergence of gradient descent-ascent (GDA) algorithm and the representation learning of neural networks in solving minimax optimization problems defined over infinite-dimensional function classes, with a focus on functional conditional moment equations.

Abstract

The authors study the convergence of gradient descent-ascent (GDA) algorithm and the representation learning of neural networks in solving minimax optimization problems defined over infinite-dimensional function classes. As an initial step, they consider the minimax optimization problem stemming from estimating a functional equation defined by conditional expectations via adversarial estimation, where the objective function is quadratic in the functional space.
The key insights are:

In the mean-field regime, the GDA algorithm corresponds to a Wasserstein gradient flow over the space of probability measures defined over the neural network parameters.

They prove that the Wasserstein gradient flow converges globally to a stationary point of the minimax objective at a sublinear rate of O(1/T + 1/α), where T is the time horizon and α is the scaling parameter of the neural network.

They show that the feature representation induced by the neural networks is allowed to deviate from the initial one by the magnitude of O(1/α), measured in terms of the Wasserstein distance. This behavior is not captured by the neural tangent kernel (NTK) analysis, where the representation is shown to be fixed at the initialization.

When the regularization on the function f satisfies a version of strong convexity, they prove that the Wasserstein gradient flow converges to the global optimizer f* at a sublinear O(1/T + 1/α) rate.

They apply their general results to concrete examples including policy evaluation, nonparametric instrumental variable regression, asset pricing, and adversarial Riesz representer estimation.

Stats

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Key Insights Distilled From

A Mean-Field Analysis of Neural Gradient Descent-Ascent: Applications to Functional Conditional Moment Equations

by Yuchen Zhu,Y... at arxiv.org 04-19-2024

https://arxiv.org/pdf/2404.12312.pdf

Deeper Inquiries

How can the results be extended to more general minimax optimization problems beyond the quadratic objective considered in this work

The results obtained in this study can be extended to more general minimax optimization problems beyond the quadratic objective considered. The key lies in the framework and methodology used in the analysis. The mean-field analysis of neural networks and the application of the Wasserstein gradient flow provide a robust foundation for tackling a broader range of minimax optimization problems. By considering the continuous-time and infinite-width limit of the optimization dynamics, the study establishes convergence under the mean-field regime. This approach, which involves tracking the evolution of probability measures over the Wasserstein space, can be applied to various objective functions that may exhibit different properties than the quadratic objective function studied in this work. By adapting the mean-field analysis and Wasserstein gradient flow to different objective functions, researchers can analyze the convergence of first-order optimization algorithms for a wider class of minimax optimization problems.

What are the implications of the representation learning results for the statistical accuracy of the solutions found by the GDA algorithm

The representation learning results obtained in this study have significant implications for the statistical accuracy of the solutions found by the GDA algorithm. The analysis shows that the feature representation induced by the neural networks is allowed to deviate from the initial one by a magnitude measured in terms of the Wasserstein distance. This deviation signifies that the neural network is learning data-dependent features during the optimization process. By allowing the representation to evolve and deviate from the initialization, the algorithm can capture more complex patterns and structures in the data. This enhanced representation learning capability leads to solutions that are statistically more accurate and robust. The ability of the neural network to adapt and learn meaningful features from the data contributes to the overall effectiveness of the GDA algorithm in finding optimal solutions to functional conditional moment equations.

Can the analysis be generalized to other first-order optimization algorithms beyond GDA for solving functional conditional moment equations with neural networks

The analysis presented in this study can be generalized to other first-order optimization algorithms beyond GDA for solving functional conditional moment equations with neural networks. The mean-field analysis and Wasserstein gradient flow approach are not limited to a specific optimization algorithm but can be applied to a broader class of algorithms. By considering the continuous-time and infinite-width limit of the optimization dynamics, researchers can adapt the methodology to analyze the convergence of various first-order optimization algorithms. Algorithms such as stochastic gradient descent, Adam, or other optimization techniques can be studied within the same framework to understand their convergence properties and representation learning capabilities in the context of solving minimax optimization problems defined over neural networks. This generalization allows for a more comprehensive analysis of optimization algorithms and their applications in solving complex functional equations.