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Learning with dependent samples
Most of the results establishing asymptotic consistency and finite sample guarantees for learning algorithms assume that the observations used for training are realizations of an i.i.d. stochastic process. In this project, we will go beyond this classical framework, exploring the class of stationary and non-stationary mixing stochastic processes.
Keywords: probability theory, learning theory, dynamical systems, ergodic theory, control theory.
Recent years have witnessed an increasing enthusiasm towards data-driven methods, with researchers from disciplines such as computer science, electrical engineering and mathematics coming together to give rise to a new field called Data Science. The core aim of this multidisciplinary field is to understand and analyze actual phenomena with data.
While data-driven methods are having a tremendous impact on all scientific disciplines, a major shortcoming remains the assumption, shared by the vast majority of the results in the literature, that the data comes from an i.i.d. stochastic process. Recently, the possibility of extending statistical learning theory results to the case where the observations are non-i.i.d. has been explored in some papers. A common challenge in all these papers is establishing non-asymptotic sample complexity bounds.
An approach to the non-i.i.d. data problem is offered by the class of stationary and non-stationary mixing processes, for which it can be shown that similar statistical learning theory guarantees continue to hold. In this project the student will explore these results with the final aim of understanding the benefits and limitations of this approach to the important problem of learning linear and nonlinear dynamical systems.
This project investigates a timely and important problem, and good results will be published. Moreover, this project can be continued by the candidate and us in very interesting directions involving active learning and optimal control.
Some important (but not sufficient) literature:
1. V. Kuznetsov, M. Mohri, Generalization Bounds for Non-stationary Mixing Processes, Machine Learning, volume 106, pages 93–117(2017).
2. M. Simchowitz, H. Mania, S. Tu, M.I. Jordan, B. Recht, Learning Without Mixing: Towards A Sharp Analysis of Linear System Identification, Proceedings of the 31st Conference On Learning Theory, PMLR 75:439-473, 2018.
3. M.K.S. Faradonbeh, A. Tewari, G. Michailidis, Finite Time Identification in Unstable Linear Systems, Automatica, Volume 98, October 2018, Pages 342-353.
4. D.J. Foster, A. Rakhlin, T. Sarkar, Learning nonlinear dynamical systems from a single trajectory, Proceedings of the 2nd Conference on Learning for Dynamics and Control, PMLR 120:851-861, 2020.
5. Y. Sattar, S. Oymak, Non-asymptotic and Accurate Learning of Nonlinear Dynamical Systems, https://arxiv.org/pdf/2002.08538.pdf.
**Prerequisites:** This project is ideal for student with a strong background in probability theory (e.g., http://www.vvz.ethz.ch/lerneinheitPre.do?semkez=2020W&lerneinheitId=140197&lang=en) and an interest in statistical learning theory and dynamical systems theory. Knowledge in functional analysis and/or optimization theory is a plus.
Recent years have witnessed an increasing enthusiasm towards data-driven methods, with researchers from disciplines such as computer science, electrical engineering and mathematics coming together to give rise to a new field called Data Science. The core aim of this multidisciplinary field is to understand and analyze actual phenomena with data.
While data-driven methods are having a tremendous impact on all scientific disciplines, a major shortcoming remains the assumption, shared by the vast majority of the results in the literature, that the data comes from an i.i.d. stochastic process. Recently, the possibility of extending statistical learning theory results to the case where the observations are non-i.i.d. has been explored in some papers. A common challenge in all these papers is establishing non-asymptotic sample complexity bounds.
An approach to the non-i.i.d. data problem is offered by the class of stationary and non-stationary mixing processes, for which it can be shown that similar statistical learning theory guarantees continue to hold. In this project the student will explore these results with the final aim of understanding the benefits and limitations of this approach to the important problem of learning linear and nonlinear dynamical systems.
This project investigates a timely and important problem, and good results will be published. Moreover, this project can be continued by the candidate and us in very interesting directions involving active learning and optimal control.
Some important (but not sufficient) literature:
1. V. Kuznetsov, M. Mohri, Generalization Bounds for Non-stationary Mixing Processes, Machine Learning, volume 106, pages 93–117(2017). 2. M. Simchowitz, H. Mania, S. Tu, M.I. Jordan, B. Recht, Learning Without Mixing: Towards A Sharp Analysis of Linear System Identification, Proceedings of the 31st Conference On Learning Theory, PMLR 75:439-473, 2018. 3. M.K.S. Faradonbeh, A. Tewari, G. Michailidis, Finite Time Identification in Unstable Linear Systems, Automatica, Volume 98, October 2018, Pages 342-353. 4. D.J. Foster, A. Rakhlin, T. Sarkar, Learning nonlinear dynamical systems from a single trajectory, Proceedings of the 2nd Conference on Learning for Dynamics and Control, PMLR 120:851-861, 2020. 5. Y. Sattar, S. Oymak, Non-asymptotic and Accurate Learning of Nonlinear Dynamical Systems, https://arxiv.org/pdf/2002.08538.pdf.
**Prerequisites:** This project is ideal for student with a strong background in probability theory (e.g., http://www.vvz.ethz.ch/lerneinheitPre.do?semkez=2020W&lerneinheitId=140197&lang=en) and an interest in statistical learning theory and dynamical systems theory. Knowledge in functional analysis and/or optimization theory is a plus.
1. The student will familiarize himself/herself with the general concepts in statistical learning theory and mixing stochastic processes.
2. The student will read recent papers studying learning algorithms with stationary and non-stationary mixing processes.
3. The student will explore these results in the context of learning of linear and nonlinear dynamical systems.
4. The student will study the limitations of this framework in the context of learning of linear and nonlinear dynamical systems.
1. The student will familiarize himself/herself with the general concepts in statistical learning theory and mixing stochastic processes. 2. The student will read recent papers studying learning algorithms with stationary and non-stationary mixing processes. 3. The student will explore these results in the context of learning of linear and nonlinear dynamical systems. 4. The student will study the limitations of this framework in the context of learning of linear and nonlinear dynamical systems.