Entropic Fictitious Play for Mean Field Optimization Problem
Authors: Fan Chen, Zhenjie Ren, Songbo Wang
JMLR 2023 | Venue PDF | Archive PDF | Plain Text | LLM Run Details
| Reproducibility Variable | Result | LLM Response |
|---|---|---|
| Research Type | Experimental | Exponential convergence is proved in this paper and we also verify our theoretical results by simple numerical examples. Keywords: mean field optimization, neural network, fictitious play, relative entropy, free energy function. In Section 4 we provide a toy numerical experiment to showcase the feasibility of the algorithm for the training two-layer neural networks. We present our numerical results. We plot the learned approximative functions for different training epochs (t/ t = 10, 20, 50, 100, 200, 600) and compare them to the objective in Figure 1(a). We find that in the last training epoch the sine function is well approximated. We also investigate the validation error, calculated from 1000 evenly distributed points in the interval [0, 1], and plot its evolution in Figure 1(b). |
| Researcher Affiliation | Academia | School of Mathematical Sciences, Shanghai Jiao Tong University Shanghai, China. CEREMADE, Université Paris-Dauphine, PSL Paris, France. CMAP, CNRS, École polytechnique, Institut Polytechnique de Paris Palaiseau, France. |
| Pseudocode | Yes | Algorithm 1: Entropic fictitious play algorithm. Algorithm 2: EFP with Langevin inner iterations. |
| Open Source Code | No | The paper does not contain an explicit statement about the release of source code for the methodology described, nor does it provide a direct link to a code repository. |
| Open Datasets | No | As a toy example, we approximate the 1-periodic sine function z 7 sin(2πz) defined on [0, 1] by a two-layer neural network. We pick K = 101 samples evenly distributed on the interval [0, 1], i.e. zk = k 1/101, and set yk = sin 2πzk for k = 1, . . . , 101. |
| Dataset Splits | Yes | As a toy example, we approximate the 1-periodic sine function z 7 sin(2πz) defined on [0, 1] by a two-layer neural network. We pick K = 101 samples evenly distributed on the interval [0, 1], i.e. zk = k 1/101, and set yk = sin 2πzk for k = 1, . . . , 101. We also investigate the validation error, calculated from 1000 evenly distributed points in the interval [0, 1], and plot its evolution in Figure 1(b). |
| Hardware Specification | Yes | The whole training process consumes 63.02 seconds on the laptop (CPU model: i7-9750H). |
| Software Dependencies | No | The paper discusses various algorithms (e.g., Unadjusted Langevin Algorithm, Markov chain Monte Carlo) and theoretical concepts but does not specify any software dependencies with version numbers for implementing or running the experiments. |
| Experiment Setup | Yes | The parameters for the outer iteration are time step t = 0.2, horizon T = 120.0, learning rate α = 1, the number of neurons N = 1000, the initial distribution of neurons m0 = N(0, 15^2). For each t, we calculate the inner iteration (19) with the parameters: regularization σ^2/2 = 0.0005, time step s = 0.1, time horizon for the first step Sfirst = 100.0, and the remaining Sother = 5.0, the number of particles for simulating the Langevin dynamics M = N = 1000. |