Institute for Computational and Experimental Research in Mathematics (ICERM)

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Abstract

With the wide adoption of machine learning algorithms across various application domains, there is a growing interest in the fairness properties of such algorithms. The vast majority of the activity in the field of group fairness addresses disparities between predefined groups based on protected features such as gender, age, and race, which need to be available at train, and often also at test, time. These approaches are static and retrospective, since algorithms designed to protect groups identified a priori cannot anticipate and protect the needs of different at-risk groups in the future. In this work we analyze the space of solutions for worst-case fairness beyond demographics, and propose Blind Pareto Fairness (BPF), a method that leverages no-regret dynamics to recover a fair minimax classifier that reduces worst-case risk of any potential subgroup of sufficient size, and guarantees that the remaining population receives the best possible level of service. BPF addresses fairness beyond demographics, that is, it does not rely on predefined notions of at-risk groups, neither at train nor at test time. Our experimental results show that the proposed framework improves worst-case risk in multiple standard datasets, while simultaneously providing better levels of service for the remaining population, in comparison to competing methods.

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Path signature has unique advantages on extracting high order differential features of sequential data. Our team has been studying the path signature theory and actively applied it to various applications, including infant cognitive score prediction, human motion recognition, hand-written character recognition, hand-written text line recognition and writer identification etc. In this talk, I will share our most recent works on infant cognitive score prediction using learnable path signature features and simple deep learning models. The cognitive score can reveal individual’s abilities on intelligence, motion, language abilities. Recent research discovered that the cognitive ability is closely related with individual’s cortical structure and its development. We have proposed two frameworks to predict the cognitive score with different path signature features. For the first framework, we construct the temporal path signature along the age growth and extract signature features from longitudinal structural MRI data. By incorporating the cortical temporal path signature into the multi-stream deep learning model, the individual cognitive score can be predicted, even with missing data issues. For the second framework, we propose the learnable path signature algorithm to compute the developmental feature. Further, we obtain the brain region-wise development graph for the first two-year infant. Then we have employed the graph convolutional network for the score prediction. These two frameworks have been tested on two in-house cognitive data sets and reached state-of-the-art results.

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Distribution regression on sequential data describes the task of learning a function from a group of time series to a single scalar target. I will present a generic framework, based on the expected signature, which enables to compactly summarise a cloud of time series and make decisions on it. I will then demonstrate empirically how this framework achieves state-of-the-art performance on both synthetic and real-world examples from thermodynamics, mathematical finance and agricultural science.

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To recover the underlying paths from a given signature representation can not only give confidence in using signature as features in machine learning tasks but also be useful for data augmentation or key-frame extraction. The "Signatory" python package allows the signature transformation to act as a layer in a trainable neural network. Inspired by it, we tried to recover different types of underlying paths from a given signature-based representation. By visualizing the results, we discussed the effects of different hyper-parameters for our signature feature set.

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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.

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I will discuss methods for using machine learning to speed up solutions of nonlinear partial differential equations, focusing on learning discretizations for coarse graining the numerical solutions of PDEs. I will start with examples in 1d, and then move on to the Navier Stokes equation.

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Techniques that address sequential data have been a central theme in machine learning research in the past years. More recently, such considerations have entered the field of finance-related ML applications in several areas where we face inherently path dependent problems: from (deep) pricing and hedging (of path-dependent options) to generative modeling of synthetic market data, which we refer to as market generation.
We revisit Deep Hedging from the perspective of the role of the data streams used for training and highlight how this perspective motivates the use of highly accurate generative models for synthetic data generation. From this, we draw conclusions regarding the implications for risk management and model governance of these applications, in contrast torisk-management in classical quantitative finance approaches.
Indeed, financial ML applications and their risk-management heavily rely on a solid means of measuring and efficiently computing (smilarity-)metrics between datasets consisting of sample paths of stochastic processes. Stochastic processes are at their core random variables with values on path space. However, while the distance between two (finite dimensional) distributions was historically well understood, the extension of this notion to the level of stochastic processes remained a challenge until recently. We discuss the effect of different choices of such metrics while revisiting some topics that are central to ML-augmented quantitative finance applications (such as the synthetic generation and the evaluation of similarity of data streams) from a regulatory (and model governance) perpective. Finally, we discuss the effect of considering refined metrics which respect and preserve the information structure (the filtration) of the marketand the implications and relevance of such metrics on financial results.

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In this tutorial, I will introduce two transforms that work with signature transforms, one is designed to handle missing data, and the other one is designed to embed the effect of the absolute position of the data stream into signature features in a unified and efficient way.

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I will begin by discussing tensor rank, and the equations that arise when decomposing tensors into rank one terms. I will then consider decompositions of signature tensors, and their systems of equations. Along the way, I will mention joint work with Max Pfeffer and Bernd Sturmfels, and with Terry Lyons and Cris Salvi.

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Consider a stochastic differential equation dY(t)=f(X(t))dX(t), where X(t) is a pure jump Levy process with finite p-variation, 1<= p < 2, and f is alpha-Lipschitz for some alpha>p. Following the geometric solution construction of Levy-driven stochastic differential equations, we develop a class of epsilon-strong simulation algorithms that allows us to construct a probability space, supporting both Y and a fully simulatable process Y_epsilon, such that Y_epsilon is within epsilon distance from Y under the Skorokhod J1 topology on compact time intervals with probability 1. Moreover, the user can adaptively refine the accuracy levels. This tolerance-enforcement feature allows us to easily combine our algorithm with multilevel Monte Carlo method for efficient estimation of expectations, and adding as a benefit a straightforward analysis of the rate of convergence.

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There are two types of convergence results for stochastic gradient descent: (1) SGD finds minimizers of convex objective functions and (2) SGD finds critical points of smooth objective functions. We show that, if the objective landscape and noise possess certain properties which are reminiscent of deep learning problems, then we can obtain global convergence guarantees of first type under second type assumptions for a fixed (small, but positive) learning rate. The convergence is exponential, but with a large random coefficient. If the learning rate exceeds a certain threshold, we discuss minimum selection by studying the invariant distribution of a continuous time SGD model. We show that at a critical threshold, SGD prefers minimizers where the objective function is 'flat' in a precise sense.

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In this talk, I will discuss some of the details needed for the theory of controlled rough paths in the infinite dimensional Banach space setting. A key point will be that one may use truncated signatures of smooth curves as ``generating functions'' of the algebraic identities needed to make the theory work. This is a report on work in progress.

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We consider the problem of generating exact samples at finitely many locations from the solution of a generic multidimensional stochastic differential equation (SDE) driven by Brownian motion at a given time. If the SDE can be transformed into one with a constant diffusion coefficient and gradient drift exact samples can be obtained by sequential acceptance / rejection. In general, in this talk, we will explain how to use the theory of rough paths to obtain such exact samples. This is the first generic algorithm for exact samples of generic multivariate diffusions.

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