arXiv subjects

Samuel Cohen

Publications and source records attributed to Samuel Cohen.

18 recordsLinked to original sources

arXiv2026

Global Convergence of DGM and PINN Algorithms for Solving Nonlinear PDEs

The Deep Galerkin Method (DGM) and Physics Informed Neural Networks (PINNs) have become widely-used methods for solving partial differential equations (PDEs) in the rapidly growing field of scientific machine learning. In these methods, a neural network is trained to approximate the PDE solution by using (stochastic) gradient descent to minimize the PDE residual of the neural network. Due to the non-convexity of the PDE residual objective function, the trained neural network may, in principle, only converge to a local minimizer of the objective function (which would not be a solution of the PDE). Therefore, there is a longstanding question regarding the mathematical foundations of these algorithms, and it is highly valuable to establish that the trained neural network will converge to the PDE solution. In this paper, we consider a class of semilinear PDEs with nonlinearities in the solution and its first derivative. For this class of PDEs, we prove that neural networks trained with gradient descent to minimize the PDE residual objective function will converge to the PDE solution as the network width and training time $\rightarrow \infty$.

cs.LG↗

arXiv2025

Asynchronous Nonlinear Sheaf Diffusion for Multi-Agent Coordination

Cellular sheaves and sheaf Laplacians provide a far-reaching generalization of graphs and graph Laplacians, resulting in a wide array of applications ranging from machine learning to multi-agent control. In the context of multi-agent systems, so called coordination sheaves provide a unifying formalism that models heterogeneous agents and coordination goals over undirected communication topologies, and applying sheaf diffusion drives agents to achieve their coordination goals. Existing literature on sheaf diffusion assumes that agents can communicate and compute updates synchronously, which is an unrealistic assumption in many scenarios where communication delays or heterogeneous agents with different compute capabilities cause disagreement among agents. To address these challenges, we introduce asynchronous nonlinear sheaf diffusion. Specifically, we show that under mild assumptions on the coordination sheaf and bounded delays in communication and computation, nonlinear sheaf diffusion converges to a minimizer of the Dirichlet energy of the coordination sheaf at a linear rate proportional to the delay bound. We further show that this linear convergence is attained from arbitrary initial conditions and the analysis depends on the spectrum of the sheaf Laplacian in a manner that generalizes the standard graph Laplacian case. We provide several numerical simulations to validate our theoretical results.

math.OC↗

arXiv2025

Distributed Multi-agent Coordination over Cellular Sheaves

Techniques for coordination of multi-agent systems are vast and varied, often utilizing purpose-built solvers or controllers with tight coupling to the types of systems involved or the coordination goal. In this paper, we introduce a general unified framework for heterogeneous multi-agent coordination using the language of cellular sheaves and nonlinear sheaf Laplacians, which are generalizations of graphs and graph Laplacians. Specifically, we introduce the concept of a nonlinear homological program encompassing a choice of cellular sheaf on an undirected graph, nonlinear edge potential functions, and constrained convex node objectives, which constitutes a standard form for a wide class of coordination problems. We use the alternating direction method of multipliers to derive a distributed optimization algorithm for solving these nonlinear homological programs. To demonstrate the applicability of this framework, we show how heterogeneous coordination goals including combinations of consensus, formation, and flocking can be formulated as nonlinear homological programs and provide numerical simulations showing the efficacy of our distributed solution algorithm.

math.OC↗

arXiv2023

On Combining Expert Demonstrations in Imitation Learning via Optimal Transport

Imitation learning (IL) seeks to teach agents specific tasks through expert demonstrations. One of the key approaches to IL is to define a distance between agent and expert and to find an agent policy that minimizes that distance. Optimal transport methods have been widely used in imitation learning as they provide ways to measure meaningful distances between agent and expert trajectories. However, the problem of how to optimally combine multiple expert demonstrations has not been widely studied. The standard method is to simply concatenate state (-action) trajectories, which is problematic when trajectories are multi-modal. We propose an alternative method that uses a multi-marginal optimal transport distance and enables the combination of multiple and diverse state-trajectories in the OT sense, providing a more sensible geometric average of the demonstrations. Our approach enables an agent to learn from several experts, and its efficiency is analyzed on OpenAI Gym control environments and demonstrates that the standard method is not always optimal.

cs.LG↗

arXiv2023

Optimal Transport for Offline Imitation Learning

With the advent of large datasets, offline reinforcement learning (RL) is a promising framework for learning good decision-making policies without the need to interact with the real environment. However, offline RL requires the dataset to be reward-annotated, which presents practical challenges when reward engineering is difficult or when obtaining reward annotations is labor-intensive. In this paper, we introduce Optimal Transport Reward labeling (OTR), an algorithm that assigns rewards to offline trajectories, with a few high-quality demonstrations. OTR's key idea is to use optimal transport to compute an optimal alignment between an unlabeled trajectory in the dataset and an expert demonstration to obtain a similarity measure that can be interpreted as a reward, which can then be used by an offline RL algorithm to learn the policy. OTR is easy to implement and computationally efficient. On D4RL benchmarks, we show that OTR with a single demonstration can consistently match the performance of offline RL with ground-truth rewards.

cs.LG↗

arXiv2023

Inefficiency of CFMs: hedging perspective and agent-based simulations

We investigate whether the fee income from trades on the CFM is sufficient for the liquidity providers to hedge away the exposure to market risk. We first analyse this problem through the lens of continuous-time financial mathematics and derive an upper bound for not-arbitrage fee income that would make CFM efficient and liquidity provision fair. We then evaluate our findings by performing multi-agent simulations by varying CFM fees, market volatility, and rate of arrival of liquidity takers. We observe that, on average, fee income generated from liquidity provision is insufficient to compensate for market risk.

q-fin.MF↗

arXiv2022

Analysis and mitigation of pulse-pile-up tail artifacts in warm-plasma pulse-height X-ray spectra

Pulse pile-up in pulse-height energy analyzers increases when the incident rate of pulses increases relative to the inverse of the dead time per pulse of the detection system. Changes in the observed energy distributions with incident rate and detector-electronics-formed pulse shape then occur. We focus on weak high energy tails in X-ray spectra, important for measurements on partially ionized, warm, pure-hydrogen plasma. A first-principles two-photon pulse-pile-up model is derived specific to trapezoidal-shaped pulses; quantitative agreement is found between the measurements and the model predictions. The modeling is then used to diagnose pulse-pile-up tail artifacts and mitigate them in relatively low count-rate spectra.

physics.plasm-ph↗

arXiv2022

Matching Normalizing Flows and Probability Paths on Manifolds

Continuous Normalizing Flows (CNFs) are a class of generative models that transform a prior distribution to a model distribution by solving an ordinary differential equation (ODE). We propose to train CNFs on manifolds by minimizing probability path divergence (PPD), a novel family of divergences between the probability density path generated by the CNF and a target probability density path. PPD is formulated using a logarithmic mass conservation formula which is a linear first order partial differential equation relating the log target probabilities and the CNF's defining vector field. PPD has several key benefits over existing methods: it sidesteps the need to solve an ODE per iteration, readily applies to manifold data, scales to high dimensions, and is compatible with a large family of target paths interpolating pure noise and data in finite time. Theoretically, PPD is shown to bound classical probability divergences. Empirically, we show that CNFs learned by minimizing PPD achieve state-of-the-art results in likelihoods and sample quality on existing low-dimensional manifold benchmarks, and is the first example of a generative model to scale to moderately high dimensional manifolds.

stat.ML↗

arXiv2022

Meta Optimal Transport

We study the use of amortized optimization to predict optimal transport (OT) maps from the input measures, which we call Meta OT. This helps repeatedly solve similar OT problems between different measures by leveraging the knowledge and information present from past problems to rapidly predict and solve new problems. Otherwise, standard methods ignore the knowledge of the past solutions and suboptimally re-solve each problem from scratch. We instantiate Meta OT models in discrete and continuous settings between grayscale images, spherical data, classification labels, and color palettes and use them to improve the computational time of standard OT solvers. Our source code is available at http://github.com/facebookresearch/meta-ot

cs.LG↗

arXiv2021

Cross-Domain Imitation Learning via Optimal Transport

Cross-domain imitation learning studies how to leverage expert demonstrations of one agent to train an imitation agent with a different embodiment or morphology. Comparing trajectories and stationary distributions between the expert and imitation agents is challenging because they live on different systems that may not even have the same dimensionality. We propose Gromov-Wasserstein Imitation Learning (GWIL), a method for cross-domain imitation that uses the Gromov-Wasserstein distance to align and compare states between the different spaces of the agents. Our theory formally characterizes the scenarios where GWIL preserves optimality, revealing its possibilities and limitations. We demonstrate the effectiveness of GWIL in non-trivial continuous control domains ranging from simple rigid transformation of the expert domain to arbitrary transformation of the state-action space.

cs.LG↗

arXiv2021

Riemannian Convex Potential Maps

Modeling distributions on Riemannian manifolds is a crucial component in understanding non-Euclidean data that arises, e.g., in physics and geology. The budding approaches in this space are limited by representational and computational tradeoffs. We propose and study a class of flows that uses convex potentials from Riemannian optimal transport. These are universal and can model distributions on any compact Riemannian manifold without requiring domain knowledge of the manifold to be integrated into the architecture. We demonstrate that these flows can model standard distributions on spheres, and tori, on synthetic and geological data. Our source code is freely available online at http://github.com/facebookresearch/rcpm

cs.LG↗

arXiv2021

Healing Products of Gaussian Processes

Gaussian processes (GPs) are nonparametric Bayesian models that have been applied to regression and classification problems. One of the approaches to alleviate their cubic training cost is the use of local GP experts trained on subsets of the data. In particular, product-of-expert models combine the predictive distributions of local experts through a tractable product operation. While these expert models allow for massively distributed computation, their predictions typically suffer from erratic behaviour of the mean or uncalibrated uncertainty quantification. By calibrating predictions via a tempered softmax weighting, we provide a solution to these problems for multiple product-of-expert models, including the generalised product of experts and the robust Bayesian committee machine. Furthermore, we leverage the optimal transport literature and propose a new product-of-expert model that combines predictions of local experts by computing their Wasserstein barycenter, which can be applied to both regression and classification.

stat.ML↗

arXiv2021

Sliced Multi-Marginal Optimal Transport

Multi-marginal optimal transport enables one to compare multiple probability measures, which increasingly finds application in multi-task learning problems. One practical limitation of multi-marginal transport is computational scalability in the number of measures, samples and dimensionality. In this work, we propose a multi-marginal optimal transport paradigm based on random one-dimensional projections, whose (generalized) distance we term the sliced multi-marginal Wasserstein distance. To construct this distance, we introduce a characterization of the one-dimensional multi-marginal Kantorovich problem and use it to highlight a number of properties of the sliced multi-marginal Wasserstein distance. In particular, we show that (i) the sliced multi-marginal Wasserstein distance is a (generalized) metric that induces the same topology as the standard Wasserstein distance, (ii) it admits a dimension-free sample complexity, (iii) it is tightly connected with the problem of barycentric averaging under the sliced-Wasserstein metric. We conclude by illustrating the sliced multi-marginal Wasserstein on multi-task density estimation and multi-dynamics reinforcement learning problems.

stat.ML↗

arXiv2021

Generalised correlated batched bandits via the ARC algorithm with application to dynamic pricing

The Asymptotic Randomised Control (ARC) algorithm provides a rigorous approximation to the optimal strategy for a wide class of Bayesian bandits, while retaining low computational complexity. In particular, the ARC approach provides nearly optimal choices even when the payoffs are correlated or more than the reward is observed. The algorithm is guaranteed to asymptotically optimise the expected discounted payoff, with error depending on the initial uncertainty of the bandit. In this paper, we extend the ARC framework to consider a batched bandit problem where observations arrive from a generalised linear model. In particular, we develop a large sample approximation to allow correlated and generally distributed observation. We apply this to a classic dynamic pricing problem based on a Bayesian hierarchical model and demonstrate that the ARC algorithm outperforms alternative approaches.

math.OC↗

arXiv2020

Estimating Barycenters of Measures in High Dimensions

Barycentric averaging is a principled way of summarizing populations of measures. Existing algorithms for estimating barycenters typically parametrize them as weighted sums of Diracs and optimize their weights and/or locations. However, these approaches do not scale to high-dimensional settings due to the curse of dimensionality. In this paper, we propose a scalable and general algorithm for estimating barycenters of measures in high dimensions. The key idea is to turn the optimization over measures into an optimization over generative models, introducing inductive biases that allow the method to scale while still accurately estimating barycenters. We prove local convergence under mild assumptions on the discrepancy showing that the approach is well-posed. We demonstrate that our method is fast, achieves good performance on low-dimensional problems, and scales to high-dimensional settings. In particular, our approach is the first to be used to estimate barycenters in thousands of dimensions.

stat.ML↗

arXiv2020

Aligning Time Series on Incomparable Spaces

Dynamic time warping (DTW) is a useful method for aligning, comparing and combining time series, but it requires them to live in comparable spaces. In this work, we consider a setting in which time series live on different spaces without a sensible ground metric, causing DTW to become ill-defined. To alleviate this, we propose Gromov dynamic time warping (GDTW), a distance between time series on potentially incomparable spaces that avoids the comparability requirement by instead considering intra-relational geometry. We demonstrate its effectiveness at aligning, combining and comparing time series living on incomparable spaces. We further propose a smoothed version of GDTW as a differentiable loss and assess its properties in a variety of settings, including barycentric averaging, generative modeling and imitation learning.

cs.LG↗

arXiv2019

Multi-Graph Decoding for Code-Switching ASR

In the FAME! Project, a code-switching (CS) automatic speech recognition (ASR) system for Frisian-Dutch speech is developed that can accurately transcribe the local broadcaster's bilingual archives with CS speech. This archive contains recordings with monolingual Frisian and Dutch speech segments as well as Frisian-Dutch CS speech, hence the recognition performance on monolingual segments is also vital for accurate transcriptions. In this work, we propose a multi-graph decoding and rescoring strategy using bilingual and monolingual graphs together with a unified acoustic model for CS ASR. The proposed decoding scheme gives the freedom to design and employ alternative search spaces for each (monolingual or bilingual) recognition task and enables the effective use of monolingual resources of the high-resourced mixed language in low-resourced CS scenarios. In our scenario, Dutch is the high-resourced and Frisian is the low-resourced language. We therefore use additional monolingual Dutch text resources to improve the Dutch language model (LM) and compare the performance of single- and multi-graph CS ASR systems on Dutch segments using larger Dutch LMs. The ASR results show that the proposed approach outperforms baseline single-graph CS ASR systems, providing better performance on the monolingual Dutch segments without any accuracy loss on monolingual Frisian and code-mixed segments.

cs.CL↗

arXiv2011

Sublinear Expectations and Martingales in discrete time

We give a theory of sublinear expectations and martingales in discrete time. Without assuming the existence of a dominating probability measure, we derive the extensions of classical results on uniform integrability, optional stopping of martingales, and martingale convergence. We also give a theory of BSDEs in the context of sublinear expectations and a finite-state space, including general existence and comparison results.

math.PR↗