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Tuesday, 04 June 2024, 13:30–15:00h, Room G309
Speaker: Prof. Dr. Lyudmila Grigoryeva
Title: Learning Dynamic Processes with Reservoir Computing
Host: TT. Prof. Dr. Tobias Sutter
Abstract: Many dynamic problems in engineering, control theory, signal processing, time series analysis, and forecasting can be described using input/output (IO) systems. Whenever a true functional IO relation cannot be derived from first principles, parsimonious and computationally efficient state-space systems can be used as universal approximants. We have shown that Reservoir Computing (RC) state-space systems with simple and easy-to-implement architectures enjoy universal approximation properties proved in different setups. The defining feature of RC systems is that some components (usually the state map) are randomly generated, and the observation equation is of a tractable form. From the machine learning perspective, RC systems can be seen as recurrent neural networks with random weights and a simple-to-train readout layer (often a linear map). RC systems serve as efficient, randomized, online computational tools for learning dynamic processes and enjoy generalization properties that can be explicitly derived. We will make a general introduction to up-to-date theoretical developments, discuss connections with research contributions in other fields, and address details of RC systems’ applications.
Tuesday, 25 June 2024, 13:30–15:00h, Room G309
Speaker: Dr. Matthias C. Caro
Title: (Un-)Decidability and (Un-)Computability in Machine Learning
Host: Dr. Lothar Sebastian Krapp
Abstract: While statistical learning theory has its origins in statistics and probability theory, algorithmic aspects, such as computational efficiency, of learning procedures have come into focus with the development of computational learning theory by computer scientists. Nevertheless, partially due to their statistical origins, fundamental results in learning theory are often phrased without taking requirements of (Turing) computability into account. In this talk, I will give an introduction to recent work exploring the intersection point between learning and computability theory. On the one hand, I will discuss results about the undecidability of (different notions of) learnability. On the other hand, I will outline current developments aiming towards a theory of computable learnability. Thereby, I hope to demonstrate the fruitfulness of revisiting our established understanding of learning theory with an emphasis on learning as an algorithmic process.
Tuesday, 02 July 2024, 13:30–15:00h, Room M628
Speaker: Prof. Dr. Gitta Kutyniok
Title: Reliable AI: Successes, Challenges, and Limitations
Host: Prof. Dr. Stefan Volkwein
Abstract: Artificial intelligence is currently leading to one breakthrough after the other, in industry, public life, and the sciences. However, one current major drawback is the lack of reliability of such methodologies.
In this talk we will take a mathematical viewpoint towards this problem, showing the power of such approaches to reliability. We will first provide an introduction into this vibrant research area, and also discuss the impact of the EU AI Act and the G7 Hiroshima Process. We will then survey recent advances, in particular, concerning generalization and explainability. This is followed by a discussion of fundamental limitations, which affect reliability of artificial intelligence, and show solutions to this serious obstacle in terms of an intriguing connection to next generation AI computing.
Tuesday, 16 July 2024, 13:30–15:00h, Room L602
Speaker: Asst. Prof. Dr. Ariel Neufeld
Title: Quantum Monte Carlo algorithm for solving Black-Scholes PDEs for high-dimensional option pricing in finance and its complexity analysis
Host: TT. Prof. Dr. Tobias Sutter
Abstract: In this talk we present a quantum Monte Carlo algorithm to solve high-dimensional Black-Scholes PDEs with correlation for high-dimensional option pricing. The payoff function of the option is of general form and is only required to be continuous and piece-wise affine (CPWA), which covers most of the relevant payoff functions used in finance. We provide a rigorous error analysis and complexity analysis of our algorithm. In particular, we prove that the computational complexity of our algorithm is bound polynomially in the space dimension d of the PDE and the reciprocal of the prescribed accuracy ε. Moreover, we show that for payoff functions which are bounded, our algorithm indeed has a speed-up compared to classical Monte Carlo methods. Furthermore, we present numerical simulations in one and two dimensions using our developed package within the Qiskit framework tailored to price CPWA options with respect to the Black-Scholes model, as well as discuss the potential extension of the numerical simulations to arbitrary space dimension. This talk is based on joint work with Jianjun Chen and Yongming Li.
Tuesday, 29 October 2024, 13:30–15:00h, Room M630
Speaker: Asst. Prof. Dr. Alexander Heinlein
Title: Domain decomposition for neural networks
Host: Prof. Dr. Stefan Volkwein
Abstract: Scientific machine learning (SciML) is a rapidly evolving research field that combines techniques from scientific computing and machine learning. This talk focuses on the application of domain decomposition methods to design neural network architectures and enhance neural network training, Specifically, it explores the use of domain decomposition techniques in neural network-based discretizations for solving partial differential equations with physics-informed neural networks (PINNs) and operator learning, as well as in classical machine learning tasks like semantic image segmentation using convolutional neural networks (CNNs). Computational results show that domain decomposition methods can improve efficiency — both in terms of time and memory — as well as enhance accuracy and robustness.
Tuesday, 05 November 2024, 13:30–15:00h, Room ZT1204
Speaker: Prof. Dr. Daniel Kuhn
Title: Wasserstein Distributionally Robust Optimization with Heterogeneous Data Sources
Host: TT. Prof. Dr. Tobias Sutter
Abstract: We study decision problems under uncertainty, where the decision-maker has access to K data sources that carry biased information about the underlying risk factors. The biases are measured by the mismatch between the risk factor distribution and the K data-generating distributions with respect to an optimal transport (OT) distance. In this situation the decision-maker can exploit the information contained in the biased samples by solving a distributionally robust optimization (DRO) problem, where the ambiguity set is defined as the intersection of K OT neighborhoods, each of which is centered at the empirical distribution on the samples generated by a biased data source. We show that if the decision-maker has a prior belief about the biases, then the out-of-sample performance of the DRO solution can improve with K - irrespective of the magnitude of the biases. We also show that, under standard convexity assumptions, the proposed DRO problem is computationally tractable if either K or the dimension of the risk factors is kept constant.
Tuesday, 10 December 2024, 13:30–15:00h, Room G227
Speaker: TT. Prof. Dr. Lukas Gonon
Title: Deep learning for pricing: from applications to theory and back
Host: TT. Prof. Dr. Tobias Sutter
Abstract: In the past years, deep learning algorithms have been applied to numerous classical problems from mathematical finance and financial services. For example, deep learning has been employed to numerically solve high-dimensional derivatives pricing and hedging tasks, to provide efficient volatility smoothing or to detect financial asset price bubbles. Theoretical foundations of deep learning for finance, however, are far less developed. In this talk, we start by revisiting some recently developed deep learning methods. We then present our recent results on theoretical foundations for approximating option prices, solutions to jump-diffusion PDEs and optimal stopping problems using (random) neural networks. We address neural network expressivity, highlight challenges in analysing optimization errors and show the potential of random neural networks for mitigating these difficulties. Our results allow to obtain more explicit convergence guarantees, thereby making employed neural network methods more trustable.
Thursday, 30 January 2025 17:00–18:30h, Room G201
Speaker: Asst. Prof. Dr. Mareike Dressler
Title: Perspectives on Signomial Programming for Nonlinear Dynamics: Theory and Applications
Host: Prof. Dr. Salma Kuhlmann
Abstract: Signomials generalize polynomials by allowing arbitrary real exponents, at the expense of restricting the resulting function to the positive orthant. In this talk, I present a novel convex relaxation hierarchy of lower bounds for signomial optimization and discuss how it can be used to study nonlinear dynamics. The hierarchy is derived through the newly-defined concept of signomial rings and a signomial Positivstellensatz based on conditional "sums of arithmetic-geometric exponentials" (SAGE). Numerical examples are provided to illustrate the performance of the hierarchy on problems in chemical engineering and reaction networks. To conclude, I provide an outlook on how any (hierarchical) inner-approximation of the signomial nonnegativity cone yields upper bounds for signomial optimization.
Tuesday, 08 April 2025 13:30–15:00h, Room G300
Speaker: Dr. Francis Bach
Title: Denoising diffusion models without diffusions
Host: Prof. Dr. Salma Kuhlmann
Abstract: Denoising diffusion models have led to impressive generative models in many domains. In this talk, I will present the framework of sampling by denoising with an alternative view that avoids continuous-time diffusions. This alternative view readily allows an extension to binary data. Joint work with Saeed Saremi (paper on arxiv).
Tuesday, 03 June 2025 13:30–15:00h, Room G300
Speaker: Jun.-Prof. Stephan Eckstein
Title: How knowledge of graphical structures helps in the estimation of probability measures
Host: TT. Prof. Tobias Sutter
Abstract: Estimating the distribution of a d-dimensional random variable (i.e., a d-dimensional probability measure) from data is a fundamental instance of the curse of dimensionality. In this talk, we explore whether structural knowledge about the data-generating process which gives rise to the distribution can help overcome this curse. More precisely, we work with the set of distributions of probabilistic graphical models for a known directed acyclic graph. We establish that this knowledge is only helpful if it holds in a suitably quantitative way, which is formalized via smoothness conditions on the stochastic kernels in the disintegration according to the graph. We provide explicit estimators which can leverage the graphical structure and we measure the estimation error in Wasserstein distance. We show that the rate of estimation becomes faster the fewer edges are in the graph and is sharp in many cases. The results are compared to existing works focusing on knowledge of smooth densities instead of graphical structures.