Deep Learning for Dynamic Programming with Recursive Utility Using First-order Conditions

This paper proposes the certainty-equivalent first-order learning (CEFOL) algorithm, a deep learning algorithm for solving discrete-time dynamic programming problems with recursive utility. Dynamic programming with recursive utility is challenging because nonlinear certainty equivalent appears in the Bellman equation and the first-order optimality conditions but is difficult to evaluate. By introducing a separate neural network to represent the certainty equivalent, CEFOL enables the exploitation of the Bellman and model-specific first-order optimality conditions. In addition to certainty equivalent, CEFOL also uses neural networks to learn the value functions, policy functions, and Lagrange multipliers by using model-specific first-order conditions to construct residuals for minimization. By using first-order and KKT residuals to learn the policy, CEFOL directly accommodates general equality and inequality constraints on the controls, including occasionally binding constraints, without requiring penalty functions or problem-specific reformulations. We apply the algorithm to risk-sensitive and Epstein--Zin consumption-saving problems, a small-noise robust-control problem, and a DSGE model with recursive preferences and stochastic volatility. Across these applications, out-of-sample Bellman diagnostics and model-specific optimality residuals, including Euler or first-order residuals where applicable, are generally of order 1.0e-4 to 1.0e-3 over the relevant state regions, with larger values mainly near binding constraints, and the learned value and policy functions closely match VFI benchmarks when available. The CEFOL algorithm also works for dynamic programming problems with expected utility, as expected utility is a special case of recursive utility.

Deep Learning for Dynamic Programming with Recursive Utility

We propose the first deep learning algorithm, the Certainty Equivalent Learning (CEL) algorithm, for solving high-dimensional discrete-time dynamic programming problems with recursive utility. Dynamic programming with recursive utility is numerically challenging because the recursive utility does not have an explicit representation and the Bellman equation contains a certainty equivalent that is difficult to evaluate. The CEL algorithm learns this certainty-equivalent value directly with neural networks and jointly approximates value functions, policy functions, and certainty-equivalent functions. The CEL algorithm is mesh-free and simulation-based, allowing high-dimensional state and control spaces, and does not rely on Euler equations, first-order conditions, or differentiability of the state transition function. The CEL algorithm also works for dynamic programming problems with expected utility as expected utility is a special case of recursive utility. We apply the CEL to discounted linear exponential quadratic Gaussian control, small-noise robust control, Epstein-Zin DSGE, and multivariate strategic asset allocation problems. Compared with closed-form and VFI-based benchmarks, the CEL delivers accurate value and policy approximations, remains effective in high-dimensional problems, achieves accuracy comparable to VFI in the small-noise robust-control case, and produces out-of-sample Bellman errors and Euler or first-order residuals that are in the range from 1.0e-4 to 1.0e-3 for most problems.