arXiv subjects

Samuel N. Cohen

Publications and source records attributed to Samuel N. Cohen.

At least 19 recordsLinked to original sources

arXiv2026

Neural Actor-Critic Methods for Hamilton-Jacobi-Bellman PDEs: Asymptotic Analysis and Numerical Studies

We mathematically analyze and numerically study an actor-critic machine learning algorithm for solving high-dimensional Hamilton-Jacobi-Bellman (HJB) partial differential equations from stochastic control theory. The architecture of the critic (the estimator for the value function) is structured so that the boundary condition is always perfectly satisfied (rather than being included in the training loss) and utilizes a biased gradient which reduces computational cost. The actor (the estimator for the optimal control) is trained by minimizing the integral of the Hamiltonian over the domain, where the Hamiltonian is estimated using the critic. We show that the training dynamics of the actor and critic neural networks converge in a Sobolev-type space to a certain infinite-dimensional ordinary differential equation (ODE) as the number of hidden units in the actor and critic $\rightarrow \infty$. Further, under a convexity-like assumption on the Hamiltonian, we prove that any fixed point of this limit ODE is a solution of the original stochastic control problem. This provides an important guarantee for the algorithm's performance in light of the fact that finite-width neural networks may only converge to a local minimizers (and not optimal solutions) due to the non-convexity of their loss functions. In our numerical studies, we demonstrate that the algorithm can solve stochastic control problems accurately in up to 200 dimensions. In particular, we construct a series of increasingly complex stochastic control problems with known analytic solutions and study the algorithm's numerical performance on them. These problems range from a linear-quadratic regulator equation to highly challenging equations with non-convex Hamiltonians, allowing us to identify and analyze the strengths and limitations of this neural actor-critic method for solving HJB equations.

math.OC↗

arXiv2026

The Exponentially Weighted Signature

The signature is a canonical representation of a multidimensional path over an interval. However, it treats all historical information uniformly, offering no intrinsic mechanism for contextualising the relevance of the past. To address this, we introduce the Exponentially Weighted Signature (EWS), generalising the Exponentially Fading Memory (EFM) signature from diagonal to general bounded linear operators. These operators enable cross-channel coupling at the level of temporal weighting together with richer memory dynamics including oscillatory, growth, and regime-dependent behaviour, while preserving the algebraic strengths of the classical signature. We show that the EWS is the unique solution to a linear controlled differential equation on the tensor algebra, and that it generalises both state-space models and the Laplace and Fourier transforms of the path. The group-like structure of the EWS enables efficient computation and makes the framework amenable to gradient-based learning, with the full semigroup action parametrised by and learned through its generator. We use this framework to empirically demonstrate the expressivity gap between the EWS and both the signature and EFM on two SDE-based regression tasks.

stat.ML↗

arXiv2026

Deep Hilbert--Galerkin Methods for Infinite-Dimensional PDEs and Optimal Control

We develop deep learning-based approximation methods for fully nonlinear second-order PDEs on separable Hilbert spaces, such as HJB equations for infinite-dimensional control, by parameterizing solutions via Hilbert--Galerkin Neural Operators (HGNOs). We prove the first Universal Approximation Theorems (UATs) which are sufficiently powerful to address these problems, based on novel topologies for Hessian terms and corresponding novel continuity assumptions on the fully nonlinear operator. These topologies are non-sequential and non-metrizable, making the problem delicate. In particular, we prove UATs for functions on Hilbert spaces, together with their Fréchet derivatives up to second order, and for unbounded operators applied to the first derivative, ensuring that HGNOs are able to approximate all the PDE terms. For control problems, we further prove UATs for optimal feedback controls in terms of our approximating value function HGNO. We develop numerical training methods, which we call Deep Hilbert--Galerkin and Hilbert Actor-Critic (reinforcement learning) Methods, for these problems by minimizing the $L^2_μ(H)$-norm of the residual of the PDE on the whole Hilbert space, not just a projected PDE to finite dimensions. This is the first paper to propose such an approach. The models considered arise in many applied sciences, such as functional differential equations in physics and Kolmogorov and HJB PDEs related to controlled PDEs, SPDEs, path-dependent systems, partially observed stochastic systems, and mean-field SDEs. We numerically solve examples of Kolmogorov and HJB PDEs related to the optimal control of deterministic and stochastic heat and Burgers' equations, demonstrating the promise of our deep learning-based approach.

cs.LG↗

arXiv2025

Nowcasting using regression on signatures

We introduce a new method of nowcasting using regression on path signatures. Path signatures capture the geometric properties of sequential data. Because signatures embed observations in continuous time, they naturally handle mixed frequencies and missing data. We prove theoretically, and with simulations, that regression on signatures subsumes the linear Kalman filter and retains desirable consistency properties. Nowcasting with signatures is more robust to disruptions in data series than previous methods, making it useful in stressed times (for example, during COVID-19). This approach is performant in nowcasting US GDP growth, and in nowcasting UK unemployment.

econ.EM↗

arXiv2025

Linking Path-Dependent and Stochastic Volatility Models

We explore a link between stochastic volatility (SV) and path-dependent volatility (PDV) models. Using assumed density filtering, we map a given SV model into a corresponding PDV representation. The resulting specification is lightweight, improves in-sample fit, and delivers robust out-of-sample forecasts. We also introduce a calibration procedure for both SV and PDV models that produces standard errors for parameter estimates and supports joint calibration of SPX/VIX smile.

q-fin.MF↗

arXiv2025

Mean-Field Generalisation Bounds for Learning Controls in Stochastic Environments

We consider a data-driven formulation of the classical discrete-time stochastic control problem. Our approach exploits the natural structure of many such problems, in which significant portions of the system are uncontrolled. Employing the dynamic programming principle and the mean-field interpretation of single-hidden layer neural networks, we formulate the control problem as a series of infinite-dimensional minimisation problems. When regularised carefully, we provide practically verifiable assumptions for non-asymptotic bounds on the generalisation error achieved by the minimisers to this problem, thus ensuring stability in overparametrised settings, for controls learned using finitely many observations. We explore connections to the traditional noisy stochastic gradient descent algorithm, and subsequently show promising numerical results for some classic control problems.

math.OC↗

arXiv2025

Generalization and Robustness of the Tilted Empirical Risk

The generalization error (risk) of a supervised statistical learning algorithm quantifies its prediction ability on previously unseen data. Inspired by exponential tilting, \citet{li2020tilted} proposed the {\it tilted empirical risk} (TER) as a non-linear risk metric for machine learning applications such as classification and regression problems. In this work, we examine the generalization error of the tilted empirical risk in the robustness regime under \textit{negative tilt}. Our first contribution is to provide uniform and information-theoretic bounds on the {\it tilted generalization error}, defined as the difference between the population risk and the tilted empirical risk, under negative tilt for unbounded loss function under bounded $(1+ε)$-th moment of loss function for some $ε\in(0,1]$ with a convergence rate of $O(n^{-ε/(1+ε)})$ where $n$ is the number of training samples, revealing a novel application for TER under no distribution shift. Secondly, we study the robustness of the tilted empirical risk with respect to noisy outliers at training time and provide theoretical guarantees under distribution shift for the tilted empirical risk. We empirically corroborate our findings in simple experimental setups where we evaluate our bounds to select the value of tilt in a data-driven manner.

stat.ML↗

arXiv2024

Hyperbolic contractivity and the Hilbert metric on probability measures

This paper gives a self-contained introduction to the Hilbert projective metric $\mathcal{H}$ and its fundamental properties, with a particular focus on the space of probability measures. We start by defining the Hilbert pseudo-metric on convex cones, focusing mainly on dual formulations of $\mathcal{H}$ . We show that linear operators on convex cones contract in the distance given by the hyperbolic tangent of $\mathcal{H}$, which in particular implies Birkhoff's classical contraction result for $\mathcal{H}$. Turning to spaces of probability measures, where $\mathcal{H}$ is a metric, we analyse the dual formulation of $\mathcal{H}$ in the general setting, and explore the geometry of the probability simplex under $\mathcal{H}$ in the special case of discrete probability measures. Throughout, we compare $\mathcal{H}$ with other distances between probability measures. In particular, we show how convergence in $\mathcal{H}$ implies convergence in total variation, $p$-Wasserstein distance, and any $f$-divergence. Furthermore, we derive a novel sharp bound for the total variation between two probability measures in terms of their Hilbert distance.

math.PR↗

arXiv2024

Understanding Transfer Learning via Mean-field Analysis

We propose a novel framework for exploring generalization errors of transfer learning through the lens of differential calculus on the space of probability measures. In particular, we consider two main transfer learning scenarios, $α$-ERM and fine-tuning with the KL-regularized empirical risk minimization and establish generic conditions under which the generalization error and the population risk convergence rates for these scenarios are studied. Based on our theoretical results, we show the benefits of transfer learning with a one-hidden-layer neural network in the mean-field regime under some suitable integrability and regularity assumptions on the loss and activation functions.

stat.ML↗

arXiv2024

Generalization Error of Graph Neural Networks in the Mean-field Regime

This work provides a theoretical framework for assessing the generalization error of graph neural networks in the over-parameterized regime, where the number of parameters surpasses the quantity of data points. We explore two widely utilized types of graph neural networks: graph convolutional neural networks and message passing graph neural networks. Prior to this study, existing bounds on the generalization error in the over-parametrized regime were uninformative, limiting our understanding of over-parameterized network performance. Our novel approach involves deriving upper bounds within the mean-field regime for evaluating the generalization error of these graph neural networks. We establish upper bounds with a convergence rate of $O(1/n)$, where $n$ is the number of graph samples. These upper bounds offer a theoretical assurance of the networks' performance on unseen data in the challenging over-parameterized regime and overall contribute to our understanding of their performance.

stat.ML↗

arXiv2023

Optimal adaptive control with separable drift uncertainty

We consider a problem of stochastic optimal control with separable drift uncertainty in strong formulation on a finite horizon. The drift coefficient of the state $Y^{u}$ is multiplicatively influenced by an unknown random variable $λ$, while admissible controls $u$ are required to be adapted to the observation filtration. Choosing a control actively influences the state and information acquisition simultaneously and comes with a learning effect. The problem, initially non-Markovian, is embedded into a higher-dimensional Markovian, full information control problem with control-dependent filtration and noise. To that problem, we apply the stochastic Perron method to characterize the value function as the unique viscosity solution to the HJB equation, explicitly construct $\varepsilon$-optimal controls and show that the values of strong and weak formulations agree. Numerical illustrations show a significant difference between the adaptive control and the certainty equivalence control.

math.OC↗

arXiv2023

Neural Q-learning for solving PDEs

Solving high-dimensional partial differential equations (PDEs) is a major challenge in scientific computing. We develop a new numerical method for solving elliptic-type PDEs by adapting the Q-learning algorithm in reinforcement learning. Our "Q-PDE" algorithm is mesh-free and therefore has the potential to overcome the curse of dimensionality. Using a neural tangent kernel (NTK) approach, we prove that the neural network approximator for the PDE solution, trained with the Q-PDE algorithm, converges to the trajectory of an infinite-dimensional ordinary differential equation (ODE) as the number of hidden units $\rightarrow \infty$. For monotone PDE (i.e. those given by monotone operators, which may be nonlinear), despite the lack of a spectral gap in the NTK, we then prove that the limit neural network, which satisfies the infinite-dimensional ODE, converges in $L^2$ to the PDE solution as the training time $\rightarrow \infty$. More generally, we can prove that any fixed point of the wide-network limit for the Q-PDE algorithm is a solution of the PDE (not necessarily under the monotone condition). The numerical performance of the Q-PDE algorithm is studied for several elliptic PDEs.

math.NA↗

arXiv2023

Mean-field Analysis of Generalization Errors

We propose a novel framework for exploring weak and $L_2$ generalization errors of algorithms through the lens of differential calculus on the space of probability measures. Specifically, we consider the KL-regularized empirical risk minimization problem and establish generic conditions under which the generalization error convergence rate, when training on a sample of size $n$, is $\mathcal{O}(1/n)$. In the context of supervised learning with a one-hidden layer neural network in the mean-field regime, these conditions are reflected in suitable integrability and regularity assumptions on the loss and activation functions.

stat.ML↗

arXiv2023

Global Convergence of Deep Galerkin and PINNs Methods for Solving Partial Differential Equations

Numerically solving high-dimensional partial differential equations (PDEs) is a major challenge. Conventional methods, such as finite difference methods, are unable to solve high-dimensional PDEs due to the curse-of-dimensionality. A variety of deep learning methods have been recently developed to try and solve high-dimensional PDEs by approximating the solution using a neural network. In this paper, we prove global convergence for one of the commonly-used deep learning algorithms for solving PDEs, the Deep Galerkin Method (DGM). DGM trains a neural network approximator to solve the PDE using stochastic gradient descent. We prove that, as the number of hidden units in the single-layer network goes to infinity (i.e., in the ``wide network limit"), the trained neural network converges to the solution of an infinite-dimensional linear ordinary differential equation (ODE). The PDE residual of the limiting approximator converges to zero as the training time $\rightarrow \infty$. Under mild assumptions, this convergence also implies that the neural network approximator converges to the solution of the PDE. A closely related class of deep learning methods for PDEs is Physics Informed Neural Networks (PINNs). Using the same mathematical techniques, we can prove a similar global convergence result for the PINN neural network approximators. Both proofs require analyzing a kernel function in the limit ODE governing the evolution of the limit neural network approximator. A key technical challenge is that the kernel function, which is a composition of the PDE operator and the neural tangent kernel (NTK) operator, lacks a spectral gap, therefore requiring a careful analysis of its properties.

math.NA↗

arXiv2023

Exponential contractions and robustness for approximate Wonham filters

We consider the problem of estimating the state of a continuous-time Markov chain from noisy observations. We show that the corresponding optimal filter is strictly contracting pathwise, when considered in the Hilbert projective space, and give explicit deterministic and pathwise rates of convergence. Using this, we provide alternative proofs of the robustness of optimal filters, improving on known error estimates, and derive rigorous and computable error bounds for approximate filters.

math.ST↗

arXiv2022

A Framework for Auditable Synthetic Data Generation

Synthetic data has gained significant momentum thanks to sophisticated machine learning tools that enable the synthesis of high-dimensional datasets. However, many generation techniques do not give the data controller control over what statistical patterns are captured, leading to concerns over privacy protection. While synthetic records are not linked to a particular real-world individual, they can reveal information about users indirectly which may be unacceptable for data owners. There is thus a need to empirically verify the privacy of synthetic data -- a particularly challenging task in high-dimensional data. In this paper we present a general framework for synthetic data generation that gives data controllers full control over which statistical properties the synthetic data ought to preserve, what exact information loss is acceptable, and how to quantify it. The benefits of the approach are that (1) one can generate synthetic data that results in high utility for a given task, while (2) empirically validating that only statistics considered safe by the data curator are used to generate the data. We thus show the potential for synthetic data to be an effective means of releasing confidential data safely, while retaining useful information for analysts.

cs.CR↗

arXiv2022

TAPAS: a Toolbox for Adversarial Privacy Auditing of Synthetic Data

Personal data collected at scale promises to improve decision-making and accelerate innovation. However, sharing and using such data raises serious privacy concerns. A promising solution is to produce synthetic data, artificial records to share instead of real data. Since synthetic records are not linked to real persons, this intuitively prevents classical re-identification attacks. However, this is insufficient to protect privacy. We here present TAPAS, a toolbox of attacks to evaluate synthetic data privacy under a wide range of scenarios. These attacks include generalizations of prior works and novel attacks. We also introduce a general framework for reasoning about privacy threats to synthetic data and showcase TAPAS on several examples.

cs.CR↗

arXiv2022

Asymptotic Randomised Control with applications to bandits

We consider a general multi-armed bandit problem with correlated (and simple contextual and restless) elements, as a relaxed control problem. By introducing an entropy regularisation, we obtain a smooth asymptotic approximation to the value function. This yields a novel semi-index approximation of the optimal decision process. This semi-index can be interpreted as explicitly balancing an exploration-exploitation trade-off as in the optimistic (UCB) principle where the learning premium explicitly describes asymmetry of information available in the environment and non-linearity in the reward function. Performance of the resulting Asymptotic Randomised Control (ARC) algorithm compares favourably well with other approaches to correlated multi-armed bandits.

math.OC↗