Huyên Pham

Huyên Pham, Xavier Warin

We develop a new policy gradient and actor-critic algorithm for solving mean-field control problems within a continuous time reinforcement learning setting. Our approach leverages a gradient-based representation of the value function, employing parametrized randomized policies. The learning for both the actor (policy) and critic (value function) is facilitated by a class of moment neural network functions on the Wasserstein space of probability measures, and the key feature is to sample directly trajectories of distributions. A central challenge addressed in this study pertains to the computational treatment of an operator specific to the mean-field framework. To illustrate the effectiveness of our methods, we provide a comprehensive set of numerical results. These encompass diverse examples, including multi-dimensional settings and nonlinear quadratic mean-field control problems with controlled volatility.

Achref Bachouch, Côme Huré, Nicolas Langrené et al.

This paper presents several numerical applications of deep learning-based algorithms that have been introduced in [HPBL18]. Numerical and comparative tests using TensorFlow illustrate the performance of our different algorithms, namely control learning by performance iteration (algorithms NNcontPI and ClassifPI), control learning by hybrid iteration (algorithms Hybrid-Now and Hybrid-LaterQ), on the 100-dimensional nonlinear PDEs examples from [EHJ17] and on quadratic backward stochastic differential equations as in [CR16]. We also performed tests on low-dimension control problems such as an option hedging problem in finance, as well as energy storage problems arising in the valuation of gas storage and in microgrid management. Numerical results and comparisons to quantization-type algorithms Qknn, as an efficient algorithm to numerically solve low-dimensional control problems, are also provided; and some corresponding codes are available on https://github.com/comeh/.

Alexandre Alouadi, Baptiste Barreau, Laurent Carlier et al.

We investigate the generative capabilities of the Schrödinger Bridge (SB) approach for time series. The SB framework formulates time series synthesis as an entropic optimal interpolation transport problem between a reference probability measure on path space and a target joint distribution. This results in a stochastic differential equation over a finite horizon that accurately captures the temporal dynamics of the target time series. While the SB approach has been largely explored in fields like image generation, there is a scarcity of studies for its application to time series. In this work, we bridge this gap by conducting a comprehensive evaluation of the SB method's robustness and generative performance. We benchmark it against state-of-the-art (SOTA) time series generation methods across diverse datasets, assessing its strengths, limitations, and capacity to model complex temporal dependencies. Our results offer valuable insights into the SB framework's potential as a versatile and robust tool for time series generation.

Côme Huré, Huyên Pham, Xavier Warin

We propose new machine learning schemes for solving high dimensional nonlinear partial differential equations (PDEs). Relying on the classical backward stochastic differential equation (BSDE) representation of PDEs, our algorithms estimate simultaneously the solution and its gradient by deep neural networks. These approximations are performed at each time step from the minimization of loss functions defined recursively by backward induction. The methodology is extended to variational inequalities arising in optimal stopping problems. We analyze the convergence of the deep learning schemes and provide error estimates in terms of the universal approximation of neural networks. Numerical results show that our algorithms give very good results till dimension 50 (and certainly above), for both PDEs and variational inequalities problems. For the PDEs resolution, our results are very similar to those obtained by the recent method in \cite{weinan2017deep} when the latter converges to the right solution or does not diverge. Numerical tests indicate that the proposed methods are not stuck in poor local minimaas it can be the case with the algorithm designed in \cite{weinan2017deep}, and no divergence is experienced. The only limitation seems to be due to the inability of the considered deep neural networks to represent a solution with a too complex structure in high dimension.

Côme Huré, Huyên Pham, Achref Bachouch et al.

This paper develops algorithms for high-dimensional stochastic control problems based on deep learning and dynamic programming. Unlike classical approximate dynamic programming approaches, we first approximate the optimal policy by means of neural networks in the spirit of deep reinforcement learning, and then the value function by Monte Carlo regression. This is achieved in the dynamic programming recursion by performance or hybrid iteration, and regress now methods from numerical probabilities. We provide a theoretical justification of these algorithms. Consistency and rate of convergence for the control and value function estimates are analyzed and expressed in terms of the universal approximation error of the neural networks, and of the statistical error when estimating network function, leaving aside the optimization error. Numerical results on various applications are presented in a companion paper (arxiv.org/abs/1812.05916) and illustrate the performance of the proposed algorithms.