Adeline Fermanian

Adeline Fermanian, Terry Lyons, James Morrill et al. · oxford

This article provides a concise overview of some of the recent advances in the application of rough path theory to machine learning. Controlled differential equations (CDEs) are discussed as the key mathematical model to describe the interaction of a stream with a physical control system. A collection of iterated integrals known as the signature naturally arises in the description of the response produced by such interactions. The signature comes equipped with a variety of powerful properties rendering it an ideal feature map for streamed data. We summarise recent advances in the symbiosis between deep learning and CDEs, studying the link with RNNs and culminating with the Neural CDE model. We concluded with a discussion on signature kernel methods.

Pierre Marion, Adeline Fermanian, Gérard Biau et al.

Deep ResNets are recognized for achieving state-of-the-art results in complex machine learning tasks. However, the remarkable performance of these architectures relies on a training procedure that needs to be carefully crafted to avoid vanishing or exploding gradients, particularly as the depth $L$ increases. No consensus has been reached on how to mitigate this issue, although a widely discussed strategy consists in scaling the output of each layer by a factor $α_L$. We show in a probabilistic setting that with standard i.i.d.~initializations, the only non-trivial dynamics is for $α_L = \frac{1}{\sqrt{L}}$; other choices lead either to explosion or to identity mapping. This scaling factor corresponds in the continuous-time limit to a neural stochastic differential equation, contrarily to a widespread interpretation that deep ResNets are discretizations of neural ordinary differential equations. By contrast, in the latter regime, stability is obtained with specific correlated initializations and $α_L = \frac{1}{L}$. Our analysis suggests a strong interplay between scaling and regularity of the weights as a function of the layer index. Finally, in a series of experiments, we exhibit a continuous range of regimes driven by these two parameters, which jointly impact performance before and after training.

Linus Bleistein, Adeline Fermanian, Anne-Sophie Jannot et al.

We address the problem of learning the dynamics of an unknown non-parametric system linking a target and a feature time series. The feature time series is measured on a sparse and irregular grid, while we have access to only a few points of the target time series. Once learned, we can use these dynamics to predict values of the target from the previous values of the feature time series. We frame this task as learning the solution map of a controlled differential equation (CDE). By leveraging the rich theory of signatures, we are able to cast this non-linear problem as a high-dimensional linear regression. We provide an oracle bound on the prediction error which exhibits explicit dependencies on the individual-specific sampling schemes. Our theoretical results are illustrated by simulations which show that our method outperforms existing algorithms for recovering the full time series while being computationally cheap. We conclude by demonstrating its potential on real-world epidemiological data.

James Morrill, Adeline Fermanian, Patrick Kidger et al.

The 'signature method' refers to a collection of feature extraction techniques for multivariate time series, derived from the theory of controlled differential equations. There is a great deal of flexibility as to how this method can be applied. On the one hand, this flexibility allows the method to be tailored to specific problems, but on the other hand, can make precise application challenging. This paper makes two contributions. First, the variations on the signature method are unified into a general approach, the \emph{generalised signature method}, of which previous variations are special cases. A primary aim of this unifying framework is to make the signature method more accessible to any machine learning practitioner, whereas it is now mostly used by specialists. Second, and within this framework, we derive a canonical collection of choices that provide a domain-agnostic starting point. We derive these choices as a result of an extensive empirical study on 26 datasets and go on to show competitive performance against current benchmarks for multivariate time series classification. Finally, to ease practical application, we make our techniques available as part of the open-source [redacted] project.

Sobihan Surendran, Antoine Godichon-Baggioni, Adeline Fermanian et al.

Stochastic Gradient Descent (SGD) with adaptive steps is widely used to train deep neural networks and generative models. Most theoretical results assume that it is possible to obtain unbiased gradient estimators, which is not the case in several recent deep learning and reinforcement learning applications that use Monte Carlo methods. This paper provides a comprehensive non-asymptotic analysis of SGD with biased gradients and adaptive steps for non-convex smooth functions. Our study incorporates time-dependent bias and emphasizes the importance of controlling the bias of the gradient estimator. In particular, we establish that Adagrad, RMSProp, and AMSGRAD, an exponential moving average variant of Adam, with biased gradients, converge to critical points for smooth non-convex functions at a rate similar to existing results in the literature for the unbiased case. Finally, we provide experimental results using Variational Autoenconders (VAE) and applications to several learning frameworks that illustrate our convergence results and show how the effect of bias can be reduced by appropriate hyperparameter tuning.

Linus Bleistein, Van-Tuan Nguyen, Adeline Fermanian et al.

We consider the task of learning individual-specific intensities of counting processes from a set of static variables and irregularly sampled time series. We introduce a novel modelization approach in which the intensity is the solution to a controlled differential equation. We first design a neural estimator by building on neural controlled differential equations. In a second time, we show that our model can be linearized in the signature space under sufficient regularity conditions, yielding a signature-based estimator which we call CoxSig. We provide theoretical learning guarantees for both estimators, before showcasing the performance of our models on a vast array of simulated and real-world datasets from finance, predictive maintenance and food supply chain management.

Sobihan Surendran, Adeline Fermanian, Sylvain Le Corff

Combinatorial Optimization problems are widespread in domains such as logistics, manufacturing, and drug discovery, yet their NP-hard nature makes them computationally challenging. Recent Neural Combinatorial Optimization methods leverage deep learning to learn solution strategies, trained via Supervised or Reinforcement Learning (RL). While promising, these approaches often rely on task-specific augmentations, perform poorly on out-of-distribution instances, and lack robust inference mechanisms. Moreover, existing latent space models either require labeled data or rely on pre-trained policies. In this work, we propose LGS-Net, a novel latent space model that conditions on problem instances, and introduce an efficient inference method, Latent Guided Sampling (LGS), based on Markov Chain Monte Carlo and Stochastic Approximation. We show that the iterations of our method form a time-inhomogeneous Markov Chain and provide rigorous theoretical convergence guarantees. Empirical results on benchmark routing tasks show that our method achieves state-of-the-art performance among RL-based approaches.

Massil Hihat, Guillaume Garrigos, Adeline Fermanian et al.

In this paper, we consider a deterministic online linear regression model where we allow the responses to be multivariate. To address this problem, we introduce MultiVAW, a method that extends the well-known Vovk-Azoury-Warmuth algorithm to the multivariate setting, and show that it also enjoys logarithmic regret in time. We apply our results to the online hierarchical forecasting problem and recover an algorithm from this literature as a special case, allowing us to relax the hypotheses usually made for its analysis.

Adeline Fermanian, Pierre Marion, Jean-Philippe Vert et al.

Building on the interpretation of a recurrent neural network (RNN) as a continuous-time neural differential equation, we show, under appropriate conditions, that the solution of a RNN can be viewed as a linear function of a specific feature set of the input sequence, known as the signature. This connection allows us to frame a RNN as a kernel method in a suitable reproducing kernel Hilbert space. As a consequence, we obtain theoretical guarantees on generalization and stability for a large class of recurrent networks. Our results are illustrated on simulated datasets.

Adeline Fermanian

Sequential and temporal data arise in many fields of research, such as quantitative finance, medicine, or computer vision. A novel approach for sequential learning, called the signature method and rooted in rough path theory, is considered. Its basic principle is to represent multidimensional paths by a graded feature set of their iterated integrals, called the signature. This approach relies critically on an embedding principle, which consists in representing discretely sampled data as paths, i.e., functions from $[0,1]$ to $\mathbb{R}^d$. After a survey of machine learning methodologies for signatures, the influence of embeddings on prediction accuracy is investigated with an in-depth study of three recent and challenging datasets. It is shown that a specific embedding, called lead-lag, is systematically the strongest performer across all datasets and algorithms considered. Moreover, an empirical study reveals that computing signatures over the whole path domain does not lead to a loss of local information. It is concluded that, with a good embedding, combining signatures with other simple algorithms achieves results competitive with state-of-the-art, domain-specific approaches.