Date/Time : 22 Nov 2023 15:00
Venue : S17 #04-06 (SR1)
Speaker : Anant Raj, University of Illinois Urbana-Champaign

In this talk, we delve into the intricate relationships between heavy-tailed distributions, generalization error, and algorithmic stability in the realm of noisy stochastic gradient descent. Recent research has illustrated the emergence of heavy tails in stochastic optimization and their intriguing links to generalization error. However, these studies often relied on challenging topological and statistical assumptions. Empirical evidence has further challenged existing theory, suggesting that the relationship between heavy tails and generalization is not always monotonic. In response, we introduce novel insights, exploring the relationship between tail behavior and generalization properties through the lens of algorithmic stability. Our analysis reveals that the stability of stochastic gradient descent (SGD) varies based on how we measure it, leading to interesting conclusions about its behavior.Expanding upon these findings, we extend the scope to a broader class of objective functions, including non-convex ones. Leveraging Wasserstein stability bounds for heavy-tailed stochastic processes, our research sheds light on the non-monotonic connection between generalization error and heavy tails, offering a more comprehensive perspective.Additionally, we introduce a unified approach for proving Wasserstein stability bounds in stochastic optimization, emphasizing time-uniform stability and its role in various scenarios, including convex and non-convex losses. Our approach is versatile and applicable to popular optimizers, highlighting the importance of ergodicity.

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Date/Time : 20 Nov 2023 14:00
Venue : S17 #04-06 (SR1)
Speaker : Julian Sester, National University of Singapore

In this talk I introduce a general framework for Markov decision problems (MDPs) under model uncertainty in a discrete-time infinite horizon setting. By providing a dynamic programming principle, we obtain a local-to-global paradigm, namely solving a local, that is, a one timestep robust optimization problem leads to an optimizer of the global (i.e., infinite time-steps) robust stochastic optimal control problem, as well as to a corresponding worst-case measure.

Based on the theoretical results, we are able to derive a novel Q-learning algorithm to solve numerically the associated robust MDPs where the corresponding ambiguity set of transition probabilities for the underlying  Markov decision process is a Wasserstein ball around a (possibly estimated) reference measure. Moreover, I will discuss two applications of the presented setting to problems in quantitative finance.

First, we apply this framework to portfolio optimization involving data of the S&P 500 where it turns out that in scenarios where the market is volatile or bearish, the optimal portfolio strategies from the corresponding robust optimization problem outperforms the ones without model uncertainty, showcasing the importance of taking model uncertainty into account.

Second, we study the optimal market-making problem in order-driven electronic markets while explicitly accounting for model uncertainty.

(The talk is based on joint works with Ying Chen, Ariel Neufeld, Mario Sikic, Hoang Hai Tran and Yijiong Zhang)

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Date/Time : 16 Nov 2023 10:00
Venue : via Zoom
Speaker : Yang Liu, University of California, Los Angeles

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https://nus-sg.zoom.us/j/83242625129?pwd=ZmRzME93QndWbTNZQUdPUVdIS080Zz09

Meeting ID: 832 4262 5129
Passcode: 157423

In this talk, I will first briefly introduce my work in Ph.D. years on physics-informed deep generative models. The general motivation is to exploit the flexibility and high-dimensional representability of deep generative models, and improve the generative models with physical and mathematical knowledge. I will present some numerical and theoretical results on PDE uncertainty quantification, optimal transport learning, particle dynamic inference, etc.

The main part of the talk will be on my recent work in UCLA. Realizing the limitations of task-specific models, I proposed a new paradigm for scientific machine learning, namely “In-Context Operator Networks” (ICON). A distinguishing feature of ICON is its ability to learn operators from data prompts during the inference phase, without weight adjustments. Much like how a single Large Language Model can solve a variety of natural language processing tasks specified by the language prompt, a single ICON model, trained as a generalist operator learner rather than being tuned to approximate a specific operator, can tackle a wide range of tasks involving different operators. I will show how a single ICON model manages 19 distinct problem types, encompassing forward and inverse ODE, PDE, and mean-field control (MFC) problems, each spanning a continuous spectrum of operators. I will also show the great accuracy of ICON on forward and backward predictions of conservation laws with discontinuities. The learning capability is generalized to operators beyond the training distribution without any fine-tuning.

Reference:
https://www.pnas.org/doi/10.1073/pnas.2310142120
https://arxiv.org/pdf/2308.05061.pdf

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Date/Time : 12 Sep 2023 10:00 - 12:00
Venue : I4-01-03 (Innovation 4.0 Building)
Speaker : Jean Bernard Lasserre, Laboratory for Analysis and Architecture of Systems

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Date/Time : 11 Sep 2023 10:00 - 12:00
Venue : I4-01-03 (Innovation 4.0 Building)
Speaker : Jean Bernard Lasserre, Laboratory for Analysis and Architecture of Systems

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Date/Time : 08 Sep 2023 10:00 - 12:00
Venue : I4-01-03 (Innovation 4.0 Building)
Speaker : Jean Bernard Lasserre, Laboratory for Analysis and Architecture of Systems

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Date/Time : 07 Sep 2023 10:00 - 12:00
Venue : I4-01-03 (Innovation 4.0 Building)
Speaker : Jean Bernard Lasserre, Laboratory for Analysis and Architecture of Systems

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Date/Time : 23 Mar 2023 16:00
Venue : S17 #04-06 (SR1)
Speaker : Tan Nguyen, University of California, Los Angeles

Self-attention is key to the remarkable success of transformers in sequence modeling tasks including many applications in natural language processing and computer vision. Like neural network layers, these attention mechanisms are often developed by heuristics and experience. In this talk, I present two novel mathematically principled frameworks to facilitate the process of designing efficient and robust transformer models with better expressivity.

In my first framework, I interpret self-attention in transformers as the Nadaraya-Watson estimator in a nonparametric regression setting. In the context of transformers, this Nadaraya-Watson estimator is constructed from the kernel density estimators (KDE) of the keys and queries in self-attention. However, such KDEs are not robust to contaminated samples. By viewing KDE as the solution to the kernel regression problem in a reproducing kernel Hilbert space, I adopt the state-of-the-art robust KDE methods based on, e.g., robust kernel regression and median-of-mean estimator, to design new robust self-attention mechanisms. The resulting family of robust self-attentions can be tailored to various transformer architectures.

In my second framework, I show that the self-attention corresponds to the support vector expansion derived from a support vector regression problem, whose primal formulation has the form of a neural network layer. Thus, I prove the primal-dual relationship between a self-attention layer and a neural network layer. Using my framework, I derive popular attention layers used in practice and propose two new attention mechanisms: 1) the Batch Normalized Attention (Attention-BN) derived from the batch normalization layer and 2) the Attention with Scaled Head (Attention-SH) derived from using less training data to fit the model. I empirically demonstrate the advantages of the Attention-BN and Attention-SH in reducing head redundancy, increasing the model’s accuracy, and improving the model’s efficiency in a variety of practical applications.

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Date/Time : 13 Jan 2022 09:00
Venue : via Zoom
Speaker : Holden Lee, Duke University

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https://nus-sg.zoom.us/j/81518116179?pwd=bGkvanY2aTlra1pLeEpKQzBSaVdBdz09

Meeting ID: : 815 1811 6179
Passcode: 461200

Abstract: Probabilistic methods in machine learning have made impressive progress in generating realistic images and videos, as well as building interpretable models with uncertainty quantification. In this talk, I will provide algorithms for two fundamental problems in probabilistic machine learning: sampling from multimodal distributions and estimating normalizing constants. I will show how to combine Langevin dynamics–an algorithm based on gradient descent with Gaussian noise–with tools from statistical physics and numerical analysis to design faster algorithms for these problems. Finally, I will touch on an application to understanding the representational power of normalizing flows, a deep generative model that provides explicit likelihood computation and produces high-resolution natural images.

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