Exploiting Symmetry in High-Dimensional Dynamic Programming - Mahdi Ebrahimi Kahou; Jesús Fernández-Villaverde; Jesse Perla; Arnav Sood (Center for Economic Studies and Ifo Institute (CESifo), 2021)
We propose a new method for solving high-dimensional dynamic programming problems and recursive competitive equilibria with a large (but finite) number of heterogeneous agents using deep learning. The ,,curse of dimensionality“ is avoided due to four complementary techniques: (1) exploiting symmetry in the approximate law of motion and the value function; (2) constructing a concentration of measure to calculate high-dimensional expectations using a single Monte Carlo draw from the distribution of idiosyncratic shocks; (3) sampling methods to ensure the model fits along manifolds of interest; and (4) selecting the most generalizable over-parameterized deep learning approximation without calculating the stationary distribution or applying a transversality condition. As an application, we solve a global solution of a multi-firm version of the classic Lucas and Prescott (1971) model of ,,investment under uncertainty.“ First, we compare the solution against a linear-quadratic Gaussian version for validation and benchmarking. Next, we solve nonlinear versions with aggregate shocks. Finally, we describe how our approach applies to a large class of models in economics.
Keywords: C45; C60; C63; 330; dynamic programming; deep learning; breaking the curse of dimensionality