Beatrice Acciaio

Beatrice Acciaio, Stephan Eckstein, Songyan Hou

We build a time-causal variational autoencoder (TC-VAE) for robust generation of financial time series data. Our approach imposes a causality constraint on the encoder and decoder networks, ensuring a causal transport from the real market time series to the fake generated time series. Specifically, we prove that the TC-VAE loss provides an upper bound on the causal Wasserstein distance between market distributions and generated distributions. Consequently, the TC-VAE loss controls the discrepancy between optimal values of various dynamic stochastic optimization problems under real and generated distributions. To further enhance the model's ability to approximate the latent representation of the real market distribution, we integrate a RealNVP prior into the TC-VAE framework. Finally, extensive numerical experiments show that TC-VAE achieves promising results on both synthetic and real market data. This is done by comparing real and generated distributions according to various statistical distances, demonstrating the effectiveness of the generated data for downstream financial optimization tasks, as well as showcasing that the generated data reproduces stylized facts of real financial market data.

Beatrice Acciaio, Anastasis Kratsios, Gudmund Pammer

Several problems in stochastic analysis are defined through their geometry, and preserving that geometric structure is essential to generating meaningful predictions. Nevertheless, how to design principled deep learning (DL) models capable of encoding these geometric structures remains largely unknown. We address this open problem by introducing a universal causal geometric DL framework in which the user specifies a suitable pair of metric spaces $\mathscr{X}$ and $\mathscr{Y}$ and our framework returns a DL model capable of causally approximating any ``regular'' map sending time series in $\mathscr{X}^{\mathbb{Z}}$ to time series in $\mathscr{Y}^{\mathbb{Z}}$ while respecting their forward flow of information throughout time. Suitable geometries on $\mathscr{Y}$ include various (adapted) Wasserstein spaces arising in optimal stopping problems, a variety of statistical manifolds describing the conditional distribution of continuous-time finite state Markov chains, and all Fréchet spaces admitting a Schauder basis, e.g. as in classical finance. Suitable spaces $\mathscr{X}$ are compact subsets of any Euclidean space. Our results all quantitatively express the number of parameters needed for our DL model to achieve a given approximation error as a function of the target map's regularity and the geometric structure both of $\mathscr{X}$ and of $\mathscr{Y}$. Even when omitting any temporal structure, our universal approximation theorems are the first guarantees that Hölder functions, defined between such $\mathscr{X}$ and $\mathscr{Y}$ can be approximated by DL models.

Konstantin Klemmer, Tianlin Xu, Beatrice Acciaio et al.

From ecology to atmospheric sciences, many academic disciplines deal with data characterized by intricate spatio-temporal complexities, the modeling of which often requires specialized approaches. Generative models of these data are of particular interest, as they enable a range of impactful downstream applications like simulation or creating synthetic training data. Recent work has highlighted the potential of generative adversarial nets (GANs) for generating spatio-temporal data. A new GAN algorithm COT-GAN, inspired by the theory of causal optimal transport (COT), was proposed in an attempt to better tackle this challenge. However, the task of learning more complex spatio-temporal patterns requires additional knowledge of their specific data structures. In this study, we propose a novel loss objective combined with COT-GAN based on an autoregressive embedding to reinforce the learning of spatio-temporal dynamics. We devise SPATE (spatio-temporal association), a new metric measuring spatio-temporal autocorrelation by using the deviance of observations from their expected values. We compute SPATE for real and synthetic data samples and use it to compute an embedding loss that considers space-time interactions, nudging the GAN to learn outputs that are faithful to the observed dynamics. We test this new objective on a diverse set of complex spatio-temporal patterns: turbulent flows, log-Gaussian Cox processes and global weather data. We show that our novel embedding loss improves performance without any changes to the architecture of the COT-GAN backbone, highlighting our model's increased capacity for capturing autoregressive structures. We also contextualize our work with respect to recent advances in physics-informed deep learning and interdisciplinary work connecting neural networks with geographic and geophysical sciences.

Tianlin Xu, Beatrice Acciaio

Causal Optimal Transport (COT) results from imposing a temporal causality constraint on classic optimal transport problems, which naturally generates a new concept of distances between distributions on path spaces. The first application of the COT theory for sequential learning was given in Xu et al. (2020), where COT-GAN was introduced as an adversarial algorithm to train implicit generative models optimized for producing sequential data. Relying on (Xu et al., 2020), the contribution of the present paper is twofold. First, we develop a conditional version of COT-GAN suitable for sequence prediction. This means that the dataset is now used in order to learn how a sequence will evolve given the observation of its past evolution. Second, we improve on the convergence results by working with modifications of the empirical measures via kernel smoothing due to (Pflug and Pichler (2016)). The resulting kernel conditional COT-GAN algorithm is illustrated with an application for video prediction.

Tianlin Xu, Li K. Wenliang, Michael Munn et al.

We introduce COT-GAN, an adversarial algorithm to train implicit generative models optimized for producing sequential data. The loss function of this algorithm is formulated using ideas from Causal Optimal Transport (COT), which combines classic optimal transport methods with an additional temporal causality constraint. Remarkably, we find that this causality condition provides a natural framework to parameterize the cost function that is learned by the discriminator as a robust (worst-case) distance, and an ideal mechanism for learning time dependent data distributions. Following Genevay et al.\ (2018), we also include an entropic penalization term which allows for the use of the Sinkhorn algorithm when computing the optimal transport cost. Our experiments show effectiveness and stability of COT-GAN when generating both low- and high-dimensional time series data. The success of the algorithm also relies on a new, improved version of the Sinkhorn divergence which demonstrates less bias in learning.