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Register a new userTowards Scalable Koopman Operator Learning: Convergence Rates and A Distributed Learning Algorithm
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We propose an alternating optimization algorithm to the nonconvex Koopman operator learning problem for nonlinear dynamic systems. We show that the proposed algorithm will converge to a critical point with rate $O(1/T)$ and $O(frac{1}{log T})$ for the constant and diminishing learning rates, respectively, under some mild conditions. To cope with the high dimensional nonlinear dynamical systems, we present the first-ever distributed Koopman operator learning algorithm. We show that the distributed Koopman operator learning has the same convergence properties as the centralized Koopman operator learning, in the absence of optimal tracker, so long as the basis functions satisfy a set of state-based decomposition conditions. Numerical experiments are provided to complement our theoretical results.
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We provide a framework for learning of dynamical systems rooted in the concept of representations and Koopman operators. The interplay between the two leads to the full description of systems that can be represented linearly in a finite dimension, based on the properties of the Koopman operator spectrum. The geometry of state space is connected to the notion of representation, both in the linear case - where it is related to joint level sets of eigenfunctions - and in the nonlinear representation case. As shown here, even nonlinear finite-dimensional representations can be learned using the Koopman operator framework, leading to a new class of representation eigenproblems. The connection to learning using neural networks is given. An extension of the Koopman operator theory to static maps between different spaces is provided. The effect of the Koopman operator spectrum on Mori-Zwanzig type representations is discussed.
The Koopman operator allows for handling nonlinear systems through a (globally) linear representation. In general, the operator is infinite-dimensional - necessitating finite approximations - for which there is no overarching framework. Although there are principled ways of learning such finite approximations, they are in many instances overlooked in favor of, often ill-posed and unstructured methods. Also, Koopman operator theory has long-standing connections to known system-theoretic and dynamical system notions that are not universally recognized. Given the former and latter realities, this work aims to bridge the gap between various concepts regarding both theory and tractable realizations. Firstly, we review data-driven representations (both unstructured and structured) for Koopman operator dynamical models, categorizing various existing methodologies and highlighting their differences. Furthermore, we provide concise insight into the paradigms relation to system-theoretic notions and analyze the prospect of using the paradigm for modeling control systems. Additionally, we outline the current challenges and comment on future perspectives.
The Large Intelligent Surface (LIS) is a promising technology in the areas of wireless communication, remote sensing and positioning. It consists of a continuous radiating surface located in the proximity of the users, with the capability to communicate by transmission and reception (replacing base stations). Despite of its potential, there are numerous challenges from implementation point of view, being the interconnection data-rate, computational complexity, and storage the most relevant ones. In order to address those challenges, hierarchical architectures with distributed processing techniques are envisioned to to be relevant for this task, while ensuring scalability. In this work we perform algorithm-architecture codesign to propose two distributed interference cancellation algorithms, and a tree-based interconnection topology for uplink processing. We also analyze the performance, hardware requirements, and architecture tradeoffs for a discrete LIS, in order to provide concrete case studies and guidelines for efficient implementation of LIS systems.
In this paper, we address the problem of privacy-preserving distributed learning and the evaluation of machine-learning models by analyzing it in the widespread MapReduce abstraction that we extend with privacy constraints. We design SPINDLE (Scalable Privacy-preservINg Distributed LEarning), the first distributed and privacy-preserving system that covers the complete ML workflow by enabling the execution of a cooperative gradient-descent and the evaluation of the obtained model and by preserving data and model confidentiality in a passive-adversary model with up to N-1 colluding parties. SPINDLE uses multiparty homomorphic encryption to execute parallel high-depth computations on encrypted data without significant overhead. We instantiate SPINDLE for the training and evaluation of generalized linear models on distributed datasets and show that it is able to accurately (on par with non-secure centrally-trained models) and efficiently (due to a multi-level parallelization of the computations) train models that require a high number of iterations on large input data with thousands of features, distributed among hundreds of data providers. For instance, it trains a logistic-regression model on a dataset of one million samples with 32 features distributed among 160 data providers in less than three minutes.
We study the Bayesian inverse problem of learning a linear operator on a Hilbert space from its noisy pointwise evaluations on random input data. Our framework assumes that this target operator is self-adjoint and diagonal in a basis shared with the Gaussian prior and noise covariance operators arising from the imposed statistical model and is able to handle target operators that are compact, bounded, or even unbounded. We establish posterior contraction rates with respect to a family of Bochner norms as the number of data tend to infinity and derive related lower bounds on the estimation error. In the large data limit, we also provide asymptotic convergence rates of suitably defined excess risk and generalization gap functionals associated with the posterior mean point estimator. In doing so, we connect the posterior consistency results to nonparametric learning theory. Furthermore, these convergence rates highlight and quantify the difficulty of learning unbounded linear operators in comparison with the learning of bounded or compact ones. Numerical experiments confirm the theory and demonstrate that similar conclusions may be expected in more general problem settings.
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