arXiv subjects

Jihao Long

Publications and source records attributed to Jihao Long.

21 records · Page 2Linked to original sources

arXiv2022

A spectral-based analysis of the separation between two-layer neural networks and linear methods

We propose a spectral-based approach to analyze how two-layer neural networks separate from linear methods in terms of approximating high-dimensional functions. We show that quantifying this separation can be reduced to estimating the Kolmogorov width of two-layer neural networks, and the latter can be further characterized by using the spectrum of an associated kernel. Different from previous work, our approach allows obtaining upper bounds, lower bounds, and identifying explicit hard functions in a united manner. We provide a systematic study of how the choice of activation functions affects the separation, in particular the dependence on the input dimension. Specifically, for nonsmooth activation functions, we extend known results to more activation functions with sharper bounds. As concrete examples, we prove that any single neuron can instantiate the separation between neural networks and random feature models. For smooth activation functions, one surprising finding is that the separation is negligible unless the norms of inner-layer weights are polynomially large with respect to the input dimension. By contrast, the separation for nonsmooth activation functions is independent of the norms of inner-layer weights.

stat.ML↗

arXiv2022

An $L^2$ Analysis of Reinforcement Learning in High Dimensions with Kernel and Neural Network Approximation

Reinforcement learning (RL) algorithms based on high-dimensional function approximation have achieved tremendous empirical success in large-scale problems with an enormous number of states. However, most analysis of such algorithms gives rise to error bounds that involve either the number of states or the number of features. This paper considers the situation where the function approximation is made either using the kernel method or the two-layer neural network model, in the context of a fitted Q-iteration algorithm with explicit regularization. We establish an $\tilde{O}(H^3|\mathcal {A}|^{\frac14}n^{-\frac14})$ bound for the optimal policy with $Hn$ samples, where $H$ is the length of each episode and $|\mathcal {A}|$ is the size of action space. Our analysis hinges on analyzing the $L^2$ error of the approximated Q-function using $n$ data points. Even though this result still requires a finite-sized action space, the error bound is independent of the dimensionality of the state space.

cs.LG↗

arXiv2021

Convergence of Deep Fictitious Play for Stochastic Differential Games

Stochastic differential games have been used extensively to model agents' competitions in Finance, for instance, in P2P lending platforms from the Fintech industry, the banking system for systemic risk, and insurance markets. The recently proposed machine learning algorithm, deep fictitious play, provides a novel efficient tool for finding Markovian Nash equilibrium of large $N$-player asymmetric stochastic differential games [J. Han and R. Hu, Mathematical and Scientific Machine Learning Conference, pages 221-245, PMLR, 2020]. By incorporating the idea of fictitious play, the algorithm decouples the game into $N$ sub-optimization problems, and identifies each player's optimal strategy with the deep backward stochastic differential equation (BSDE) method parallelly and repeatedly. In this paper, we prove the convergence of deep fictitious play (DFP) to the true Nash equilibrium. We can also show that the strategy based on DFP forms an $\eps$-Nash equilibrium. We generalize the algorithm by proposing a new approach to decouple the games, and present numerical results of large population games showing the empirical convergence of the algorithm beyond the technical assumptions in the theorems.

math.OC↗