Marc Sabate‐Vidales

Hao Ni, Lukasz Szpruch, Marc Sabate-Vidales et al.

Synthetic data is an emerging technology that can significantly accelerate the development and deployment of AI machine learning pipelines. In this work, we develop high-fidelity time-series generators, the SigWGAN, by combining continuous-time stochastic models with the newly proposed signature $W_1$ metric. The former are the Logsig-RNN models based on the stochastic differential equations, whereas the latter originates from the universal and principled mathematical features to characterize the measure induced by time series. SigWGAN allows turning computationally challenging GAN min-max problem into supervised learning while generating high fidelity samples. We validate the proposed model on both synthetic data generated by popular quantitative risk models and empirical financial data. Codes are available at https://github.com/SigCGANs/Sig-Wasserstein-GANs.git.

Fabian Germ, Marc Sabate-Vidales

We consider the filtering and prediction problem for a diffusion process. The signal and observation are modeled by stochastic differential equations (SDEs) driven by correlated Wiener processes. In classical estimation theory, measure-valued stochastic partial differential equations (SPDEs) are derived for the filtering and prediction measures. These equations can be hard to solve numerically. We provide an approximation algorithm using conditional generative adversarial networks (GANs) in combination with signatures, an object from rough path theory. The signature of a sufficiently smooth path determines the path completely. As a result, in some cases, GANs based on signatures have been shown to efficiently approximate the law of a stochastic process. For our algorithm we extend this method to sample from the conditional law, given noisy, partial observation. Our generator is constructed using neural differential equations (NDEs), relying on their universal approximator property. We show well-posedness in providing a rigorous mathematical framework. Numerical results show the efficiency of our algorithm.

Mateusz B. Majka, Marc Sabate-Vidales, Łukasz Szpruch

Stochastic Gradient Algorithms (SGAs) are ubiquitous in computational statistics, machine learning and optimisation. Recent years have brought an influx of interest in SGAs, and the non-asymptotic analysis of their bias is by now well-developed. However, relatively little is known about the optimal choice of the random approximation (e.g mini-batching) of the gradient in SGAs as this relies on the analysis of the variance and is problem specific. While there have been numerous attempts to reduce the variance of SGAs, these typically exploit a particular structure of the sampled distribution by requiring a priori knowledge of its density's mode. It is thus unclear how to adapt such algorithms to non-log-concave settings. In this paper, we construct a Multi-index Antithetic Stochastic Gradient Algorithm (MASGA) whose implementation is independent of the structure of the target measure and which achieves performance on par with Monte Carlo estimators that have access to unbiased samples from the distribution of interest. In other words, MASGA is an optimal estimator from the mean square error-computational cost perspective within the class of Monte Carlo estimators. We prove this fact rigorously for log-concave settings and verify it numerically for some examples where the log-concavity assumption is not satisfied.