A Mean-Field Optimal Control Formulation of Deep Learning (1807.01083v1)

Abstract: Recent work linking deep neural networks and dynamical systems opened up new avenues to analyze deep learning. In particular, it is observed that new insights can be obtained by recasting deep learning as an optimal control problem on difference or differential equations. However, the mathematical aspects of such a formulation have not been systematically explored. This paper introduces the mathematical formulation of the population risk minimization problem in deep learning as a mean-field optimal control problem. Mirroring the development of classical optimal control, we state and prove optimality conditions of both the Hamilton-Jacobi-BeLLMan type and the Pontryagin type. These mean-field results reflect the probabilistic nature of the learning problem. In addition, by appealing to the mean-field Pontryagin's maximum principle, we establish some quantitative relationships between population and empirical learning problems. This serves to establish a mathematical foundation for investigating the algorithmic and theoretical connections between optimal control and deep learning.

• The paper introduces a mean-field formulation that reinterprets deep learning as an optimal control problem over differential equations.
• It derives an infinite-dimensional HJB equation and employs viscosity solutions to ensure uniqueness and optimality in the learning framework.
• It establishes a mean-field Pontryagin’s maximum principle to provide necessary conditions, enhancing insights into neural network generalization.

A Mean-Field Optimal Control Formulation of Deep Learning

This paper explores a mathematical formulation of the population risk minimization problem in deep learning as a mean-field optimal control problem. By leveraging concepts from dynamical systems and control theory, it offers a framework for analyzing deep neural networks that recasts learning as an optimal control problem over differential equations. The formulation investigates the role of probabilistic assumptions by modeling deep learning as a mean-field control problem, where the optimal control parameters depend on the distribution of input-target pairs at the population level. The authors present essential mathematical conditions for optimality, including those derived from the Hamilton-Jacobi-BeLLMan (HJB) equation and Pontryagin's maximum principle (PMP).

Key Contributions and Findings

• Mean-Field Optimal Control: The paper provides a comprehensive view of deep learning as a control problem embedded in continuous dynamical systems. This conceptual linkage allows the examination of neural networks through the lens traditionally reserved for classical control systems.
• Hamilton-Jacobi-BeLLMan Equation: The authors derive an infinite-dimensional HJB equation for the value function associated with the learning process. This equation incorporates feedback mechanisms into the learning framework, serving as a tool to iteratively optimize the control policy.
• Viscosity Solutions: The paper leverages the concept of viscosity solutions to provide existence and uniqueness for the HJB equation given the probabilistic formulation of the learning task. This approach accommodates the complexity of probability distributions inherent in neural network applications.
• Pontryagin's Maximum Principle: A mean-field version of this computational principle is established to provide necessary conditions for optimal control in the probabilistic setting. This principle enables a localized view of optimal trajectories compared to the global view provided by the HJB equation.
• Small-Time Unique Solutions: Under certain conditions, such as small-time horizon and strong concavity of the problem's Hamiltonian, the authors prove uniqueness in the solutions of the PMP. This aligns with practical cases in deep learning where model capacity might be restricted.
• Error Analysis and Generalization: The paper rigorously relates solutions of the mean-field model to empirical learning scenarios by analyzing the sampled PMP. It provides insights into the generalization capacity of neural networks, offering avenues to explain real-world phenomena regarding model overfitting and underperformance.

Implications for Deep Learning and AI

The authors provide a novel perspective by pairing deep learning with extrinsic mathematical formalizations rooted in optimal control theory. The mean-field approach creates avenues for understanding complexities unique to continuous, high-dimensional systems found in neural network architectures. Additionally, the framework provides depth to discussions on generalization capabilities, emphasizing the potential for enhancing machine learning models' capacity without explicit reliance on parameter count. The inclusion of batch normalization through mean-field dynamics further underscores the adaptability and robustness of this mathematical paradigm in addressing contemporary challenges in AI.

Future Directions

This work opens numerous potential explorations within AI and deep learning. Future research could dive deeper into practical aspects such as efficient numerical solutions for the derived PDEs and ODE systems. Additionally, investigating the role of more complex neural structures, such as attention mechanisms, under similar control-theoretic frameworks could reveal significant insights into optimizing neural networks' design and training. Extending the mean-field dynamics theory to dissect mechanisms underlying batch normalization and other regularization methods provides another intriguing path. Overall, this formulation stands as a cornerstone for integrating control theory insights into the evolving landscape of artificial intelligence.