Lecture notes on rough paths and applications to machine learning

These notes expound the recent use of the signature transform and rough path theory in data science and machine learning. We develop the core theory of the signature from first principles and then survey some recent popular applications of this approach, including signature-based kernel methods and neural rough differential equations. The notes are based on a course given by the two authors at Imperial College London.

Theoretical Foundations of Deep Selective State-Space Models

Structured state-space models (SSMs) such as S4, stemming from the seminal work of Gu et al., are gaining popularity as effective approaches for modeling sequential data. Deep SSMs demonstrate outstanding performance across a diverse set of domains, at a reduced training and inference cost compared to attention-based transformers. Recent developments show that if the linear recurrence powering SSMs allows for multiplicative interactions between inputs and hidden states (e.g. GateLoop, Mamba, GLA), then the resulting architecture can surpass in both in accuracy and efficiency attention-powered foundation models trained on text, at scales of billion parameters. In this paper, we give theoretical grounding to this recent finding using tools from Rough Path Theory: we show that when random linear recurrences are equipped with simple input-controlled transitions (selectivity mechanism), then the hidden state is provably a low-dimensional projection of a powerful mathematical object called the signature of the input -- capturing non-linear interactions between tokens at distinct timescales. Our theory not only motivates the success of modern selective state-space models such as Mamba but also provides a solid framework to understand the expressive power of future SSM variants.

Signature Kernel Conditional Independence Tests in Causal Discovery for Stochastic Processes

Inferring the causal structure underlying stochastic dynamical systems from observational data holds great promise in domains ranging from science and health to finance. Such processes can often be accurately modeled via stochastic differential equations (SDEs), which naturally imply causal relationships via "which variables enter the differential of which other variables". In this paper, we develop a kernel-based test of conditional independence (CI) on "path-space" -- solutions to SDEs -- by leveraging recent advances in signature kernels. We demonstrate strictly superior performance of our proposed CI test compared to existing approaches on path-space. Then, we develop constraint-based causal discovery algorithms for acyclic stochastic dynamical systems (allowing for loops) that leverage temporal information to recover the entire directed graph. Assuming faithfulness and a CI oracle, our algorithm is sound and complete. We empirically verify that our developed CI test in conjunction with the causal discovery algorithm reliably outperforms baselines across a range of settings.

A Neural RDE approach for continuous-time non-Markovian stochastic control problems

We propose a novel framework for solving continuous-time non-Markovian stochastic control problems by means of neural rough differential equations (Neural RDEs) introduced in Morrill et al. (2021). Non-Markovianity naturally arises in control problems due to the time delay effects in the system coefficients or the driving noises, which leads to optimal control strategies depending explicitly on the historical trajectories of the system state. By modelling the control process as the solution of a Neural RDE driven by the state process, we show that the control-state joint dynamics are governed by an uncontrolled, augmented Neural RDE, allowing for fast Monte-Carlo estimation of the value function via trajectories simulation and memory-efficient backpropagation. We provide theoretical underpinnings for the proposed algorithmic framework by demonstrating that Neural RDEs serve as universal approximators for functions of random rough paths. Exhaustive numerical experiments on non-Markovian stochastic control problems are presented, which reveal that the proposed framework is time-resolution-invariant and achieves higher accuracy and better stability in irregular sampling compared to existing RNN-based approaches.

Non-adversarial training of Neural SDEs with signature kernel scores

Neural SDEs are continuous-time generative models for sequential data. State-of-the-art performance for irregular time series generation has been previously obtained by training these models adversarially as GANs. However, as typical for GAN architectures, training is notoriously unstable, often suffers from mode collapse, and requires specialised techniques such as weight clipping and gradient penalty to mitigate these issues. In this paper, we introduce a novel class of scoring rules on pathspace based on signature kernels and use them as objective for training Neural SDEs non-adversarially. By showing strict properness of such kernel scores and consistency of the corresponding estimators, we provide existence and uniqueness guarantees for the minimiser. With this formulation, evaluating the generator-discriminator pair amounts to solving a system of linear path-dependent PDEs which allows for memory-efficient adjoint-based backpropagation. Moreover, because the proposed kernel scores are well-defined for paths with values in infinite dimensional spaces of functions, our framework can be easily extended to generate spatiotemporal data. Our procedure permits conditioning on a rich variety of market conditions and significantly outperforms alternative ways of training Neural SDEs on a variety of tasks including the simulation of rough volatility models, the conditional probabilistic forecasts of real-world forex pairs where the conditioning variable is an observed past trajectory, and the mesh-free generation of limit order book dynamics.

Neural signature kernels as infinite-width-depth-limits of controlled ResNets

Motivated by the paradigm of reservoir computing, we consider randomly initialized controlled ResNets defined as Euler-discretizations of neural controlled differential equations (Neural CDEs). We show that in the infinite-width-then-depth limit and under proper scaling, these architectures converge weakly to Gaussian processes indexed on some spaces of continuous paths and with kernels satisfying certain partial differential equations (PDEs) varying according to the choice of activation function. In the special case where the activation is the identity, we show that the equation reduces to a linear PDE and the limiting kernel agrees with the signature kernel of Salvi et al. (2021). In this setting, we also show that the width-depth limits commute. We name this new family of limiting kernels neural signature kernels. Finally, we show that in the infinite-depth regime, finite-width controlled ResNets converge in distribution to Neural CDEs with random vector fields which, depending on whether the weights are shared across layers, are either time-independent and Gaussian or behave like a matrix-valued Brownian motion.

New directions in the applications of rough path theory

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.

Neural Stochastic Partial Differential Equations

Stochastic partial differential equations (SPDEs) are the mathematical tool of choice to model complex spatio-temporal dynamics of systems subject to the influence of randomness. We introduce the Neural SPDE model providing an extension to two important classes of physics-inspired neural architectures. On the one hand, it extends all the popular neural -- ordinary, controlled, stochastic, rough -- differential equation models in that it is capable of processing incoming information even when the latter evolves in an infinite dimensional state space. On the other hand, it extends Neural Operators -- recent generalizations of neural networks modelling mappings between functional spaces -- in that it can be used to learn complex SPDE solution operators $(u_0,\xi) \mapsto u$ depending simultaneously on an initial condition $u_0$ and on a stochastic forcing term $\xi$, while remaining resolution-invariant and equation-agnostic. A Neural SPDE is constrained to respect real physical dynamics and consequently requires only a modest amount of data to train, depends on a significantly smaller amount of parameters and has better generalization properties compared to Neural Operators. Through various experiments on semilinear SPDEs with additive and multiplicative noise (including the stochastic Navier-Stokes equations) we demonstrate how Neural SPDEs can flexibly be used in a supervised learning setting as well as conditional generative models to sample solutions of SPDEs conditioned on prior knowledge, systematically achieving in both cases better performance than all alternative models.

Higher Order Kernel Mean Embeddings to Capture Filtrations of Stochastic Processes

Stochastic processes are random variables with values in some space of paths. However, reducing a stochastic process to a path-valued random variable ignores its filtration, i.e. the flow of information carried by the process through time. By conditioning the process on its filtration, we introduce a family of higher order kernel mean embeddings (KMEs) that generalizes the notion of KME and captures additional information related to the filtration. We derive empirical estimators for the associated higher order maximum mean discrepancies (MMDs) and prove consistency. We then construct a filtration-sensitive kernel two-sample test able to pick up information that gets missed by the standard MMD test. In addition, leveraging our higher order MMDs we construct a family of universal kernels on stochastic processes that allows to solve real-world calibration and optimal stopping problems in quantitative finance (such as the pricing of American options) via classical kernel-based regression methods. Finally, adapting existing tests for conditional independence to the case of stochastic processes, we design a causal-discovery algorithm to recover the causal graph of structural dependencies among interacting bodies solely from observations of their multidimensional trajectories.