arXiv subjects

Konrad Kleinberg

Publications and source records attributed to Konrad Kleinberg.

4 recordsLinked to original sources

arXiv2026

Walk-on-Spheres Monte Carlo and deep neural network approximations of elliptic PDEs with drift and killing

In this paper we provide Monte Carlo and deep neural network approximations for stochastic representations of solutions to linear elliptic partial differential equations with constant diffusion, drift and killing. Building on the modified Walk-on-Spheres algorithm of Beznea et al. (arXiv:2209.01432), we introduce Monte Carlo estimators that explicitly incorporate sampled random times arising in the analyzed stochastic representations. We establish uniform error bounds for these estimators and show that, under suitable assumptions, a prescribed approximation accuracy is achieved with sample complexities growing at most polynomially in both the inverse accuracy and the problem dimension. Furthermore, we prove a deep neural network approximation result for the stochastic representations. Assuming suitable neural network representations of the boundary data and the distance function to the boundary, we use the constructed Monte Carlo to design deep neural networks that approximate the representation uniformly with a number of parameters growing at most polynomially in the inverse accuracy and the problem dimension. These results extend previous complexity analyses to a broader class of elliptic equations involving drift and killing.

math.NA↗

arXiv2025

Convexity and strict convexity for compositional neural networks in high-dimensional optimal control

Neural networks (NNs) have emerged as powerful tools for solving high-dimensional optimal control problems. In particular, their compositional structure has been shown to enable efficient approximation of high-dimensional functions, helping to mitigate the curse of dimensionality in optimal control problems. In this work, we build upon the theoretical framework developed by Kang & Gong (SIAM J. Control Optim. 60(2):786-813, 2022), particularly their results on NN approximations for compositional functions in optimal control. Theorem 6.2 in Kang & Gong (SIAM J. Control Optim. 60(2):786-813, 2022) establishes that, under suitable assumptions on the compositional structure and its associated features, optimal control problems with strictly convex cost functionals admit a curse-of-dimensionality-free approximation of the optimal control by NNs. We extend this result in two directions. First, we analyze the strict convexity requirement on the cost functional and demonstrate that reformulating a discrete-time optimal control problem with linear transitions and stage costs as a terminal cost problem ensures the necessary strict convexity. Second, we establish a generalization of Theorem 6.2 in Kang & Gong (SIAM J. Control Optim. 60(2):786-813, 2022) which provides weak error bounds for optimal control approximations by NNs when the cost functional is only convex rather than strictly convex.

math.OC↗

arXiv2025

Deep neural networks can provably solve Bellman equations for Markov decision processes without the curse of dimensionality

Discrete time stochastic optimal control problems and Markov decision processes (MDPs) are fundamental models for sequential decision-making under uncertainty and as such provide the mathematical framework underlying reinforcement learning theory. A central tool for solving MDPs is the Bellman equation and its solution, the so-called $Q$-function. In this article, we construct deep neural network (DNN) approximations for $Q$-functions associated to MDPs with infinite time horizon and finite control set $A$. More specifically, we show that if the the payoff function and the random transition dynamics of the MDP can be suitably approximated by DNNs with leaky rectified linear unit (ReLU) activation, then the solutions $Q_d\colon \mathbb R^d\to \mathbb R^{|A|}$, $d\in \mathbb{N}$, of the associated Bellman equations can also be approximated in the $L^2$-sense by DNNs with leaky ReLU activation whose numbers of parameters grow at most polynomially in both the dimension $d\in \mathbb{N}$ of the state space and the reciprocal $1/\varepsilon$ of the prescribed error $\varepsilon\in (0,1)$. Our proof relies on the recently introduced full-history recursive multilevel fixed-point (MLFP) approximation scheme.

math.OC↗

arXiv2023

Nonlinear Monte Carlo methods with polynomial runtime for Bellman equations of discrete time high-dimensional stochastic optimal control problems

Discrete time stochastic optimal control problems and Markov decision processes (MDPs), respectively, serve as fundamental models for problems that involve sequential decision making under uncertainty and as such constitute the theoretical foundation of reinforcement learning. In this article we study the numerical approximation of MDPs with infinite time horizon, finite control set, and general state spaces. Our set-up in particular covers infinite-horizon optimal stopping problems of discrete time Markov processes. A key tool to solve MDPs are Bellman equations which characterize the value functions of the MDPs and determine the optimal control strategies. By combining ideas from the full-history recursive multilevel Picard approximation method, which was recently introduced to solve certain nonlinear partial differential equations, and ideas from $Q$-learning we introduce a class of suitable nonlinear Monte Carlo methods and prove that the proposed methods do overcome the curse of dimensionality in the numerical approximation of the solutions of Bellman equations and the associated discrete time stochastic optimal control problems.

math.OC↗