最优控制论神经网络优化器：从反向传播到动态规划

深度神经网络(DNN)的优化一直是现代机器学习和人工智能领域的驱动力。由于 DNN 以持续的非线性传播序列为特征，给定目标函数确定其最优参数自然符合最优控制编程的框架。将 DNN 解读为动力系统的这一观点已 proven crucial in offering a theoretical foundation for principled analysis from numerical equations to physics。 In parallel to these theoretical pursuits, this paper focuses on an algorithmic perspective. Our motivated observation is 's striking algorithmic resemblance between the Backpropagation algorithm for computing gradients in DNNs and the optimality conditions for dynamical systems, expressed through another backward process known as dynamic programming. Consolidating this connection, where Backpropagation admits a variational structure, solving an approximate dynamic programming up to the first-order expansion leads to a new class of optimization methods exploring higher-order expansions of the Bellman equation. The resulting optimizer, termed Optimal Control Theoretic Neural Optimizer (OCNOpt), enables rich algorithmic opportunities, including layer-wise feedback policies, game-theoretic applications, and higher-order training of continuous-time models such as Neural ODEs. Extensive experiments demonstrate that OCNOpt improves upon existing methods in robustness and efficiency while maintaining manageable computational complexity, paving new avenues for principled algorithmic design grounded in dynamical systems and optimal control theory。