Adaptation and learning over networks for nonlinear system modeling

In this chapter, we analyze nonlinear filtering problems in distributed environments, e.g., sensor networks or peer-to-peer protocols. In these scenarios, the agents in the environment receive measurements in a streaming fashion, and they are required to estimate a common (nonlinear) model by alternating local computations and communications with their neighbors. We focus on the important distinction between single-task problems, where the underlying model is common to all agents, and multitask problems, where each agent might converge to a different model due to, e.g., spatial dependencies or other factors. Currently, most of the literature on distributed learning in the nonlinear case has focused on the single-task case, which may be a strong limitation in real-world scenarios. After introducing the problem and reviewing the existing approaches, we describe a simple kernel-based algorithm tailored for the multitask case. We evaluate the proposal on a simulated benchmark task, and we conclude by detailing currently open problems and lines of research.Comment: To be published as a chapter in `Adaptive Learning Methods for Nonlinear System Modeling', Elsevier Publishing, Eds. D. Comminiello and J.C. Principe (2018

Stochastic Optimization For Multi-Agent Statistical Learning And Control

The goal of this thesis is to develop a mathematical framework for optimal, accurate, and affordable complexity statistical learning among networks of autonomous agents. We begin by noting the connection between statistical inference and stochastic programming, and consider extensions of this setup to settings in which a network of agents each observes a local data stream and would like to make decisions that are good with respect to information aggregated across the entire network. There is an open-ended degree of freedom in this problem formulation, however: the selection of the estimator function class which defines the feasible set of the stochastic program. Our central contribution is the design of stochastic optimization tools in reproducing kernel Hilbert spaces that yield optimal, accurate, and affordable complexity statistical learning for a multi-agent network. To obtain this result, we first explore the relative merits and drawbacks of different function class selections. In Part I, we consider multi-agent expected risk minimization this problem setting for the case that each agent seems to learn a common globally optimal generalized linear models (GLMs) by developing a stochastic variant of Arrow-Hurwicz primal-dual method. We establish convergence to the primal-dual optimal pair when either consensus or ``proximity constraints encode the fact that we want all agents\u27 to agree, or nearby agents to make decisions that are close to one another. Empirically, we observe that these convergence results are substantiated but that convergence may not translate into statistical accuracy. More broadly, optimality within a given estimator function class is not the same as one that makes minimal inference errors. The optimality-accuracy tradeoff of GLMs motivates subsequent efforts to learn more sophisticated estimators based upon learned feature encodings of the data that is fed into the statistical model. The specific tool we turn to in Part II is dictionary learning, where we optimize both over regression weights and an encoding of the data, which yields a non-convex problem. We investigate the use of stochastic methods for online task-driven dictionary learning, and obtain promising performance for the task of a ground robot learning to anticipate control uncertainty based on its past experience. Heartened by this implementation, we then consider extensions of this framework for a multi-agent network to each learn globally optimal task-driven dictionaries based on stochastic primal-dual methods. However, it is here the non-convexity of the optimization problem causes problems: stringent conditions on stochastic errors and the duality gap limit the applicability of the convergence guarantees, and impractically small learning rates are required for convergence in practice. Thus, we seek to learn nonlinear statistical models while preserving convexity, which is possible through kernel methods ( Part III). However, the increased descriptive power of nonparametric estimation comes at the cost of infinite complexity. Thus, we develop a stochastic approximation algorithm in reproducing kernel Hilbert spaces (RKHS) that ameliorates this complexity issue while preserving optimality: we combine the functional generalization of stochastic gradient method (FSGD) with greedily constructed low-dimensional subspace projections based on matching pursuit. We establish that the proposed method yields a controllable trade-off between optimality and memory, and yields highly accurate parsimonious statistical models in practice. % Then, we develop a multi-agent extension of this method by proposing a new node-separable penalty function and applying FSGD together with low-dimensional subspace projections. This extension allows a network of autonomous agents to learn a memory-efficient approximation to the globally optimal regression function based only on their local data stream and message passing with neighbors. In practice, we observe agents are able to stably learn highly accurate and memory-efficient nonlinear statistical models from streaming data. From here, we shift focus to a more challenging class of problems, motivated by the fact that true learning is not just revising predictions based upon data but augmenting behavior over time based on temporal incentives. This goal may be described by Markov Decision Processes (MDPs): at each point, an agent is in some state of the world, takes an action and then receives a reward while randomly transitioning to a new state. The goal of the agent is to select the action sequence to maximize its long-term sum of rewards, but determining how to select this action sequence when both the state and action spaces are infinite has eluded researchers for decades. As a precursor to this feat, we consider the problem of policy evaluation in infinite MDPs, in which we seek to determine the long-term sum of rewards when starting in a given state when actions are chosen according to a fixed distribution called a policy. We reformulate this problem as a RKHS-valued compositional stochastic program and we develop a functional extension of stochastic quasi-gradient algorithm operating in tandem with the greedy subspace projections mentioned above. We prove convergence with probability 1 to the Bellman fixed point restricted to this function class, and we observe a state of the art trade off in memory versus Bellman error for the proposed method on the Mountain Car driving task, which bodes well for incorporating policy evaluation into more sophisticated, provably stable reinforcement learning techniques, and in time, developing optimal collaborative multi-agent learning-based control systems

Random Inverse Problems Over Graphs: Decentralized Online Learning

We establish a framework of random inverse problems with real-time observations over graphs, and present a decentralized online learning algorithm based on online data streams, which unifies the distributed parameter estimation in Hilbert space and the least mean square problem in reproducing kernel Hilbert space (RKHS-LMS). We transform the algorithm convergence into the asymptotic stability of randomly time-varying difference equations in Hilbert space with L2-bounded martingale difference terms and develop the L2 -asymptotic stability theory. It is shown that if the network graph is connected and the sequence of forward operators satisfies the infinitedimensional spatio-temporal persistence of excitation condition, then the estimates of all nodes are mean square and almost surely strongly consistent. By equivalently transferring the distributed learning problem in RKHS to the random inverse problem over graphs, we propose a decentralized online learning algorithm in RKHS based on non-stationary and non-independent online data streams, and prove that the algorithm is mean square and almost surely strongly consistent if the operators induced by the random input data satisfy the infinite-dimensional spatio-temporal persistence of excitation condition

Learning Theory and Approximation

The main goal of this workshop – the third one of this type at the MFO – has been to blend mathematical results from statistical learning theory and approximation theory to strengthen both disciplines and use synergistic effects to work on current research questions. Learning theory aims at modeling unknown function relations and data structures from samples in an automatic manner. Approximation theory is naturally used for the advancement and closely connected to the further development of learning theory, in particular for the exploration of new useful algorithms, and for the theoretical understanding of existing methods. Conversely, the study of learning theory also gives rise to interesting theoretical problems for approximation theory such as the approximation and sparse representation of functions or the construction of rich kernel reproducing Hilbert spaces on general metric spaces. This workshop has concentrated on the following recent topics: Pitchfork bifurcation of dynamical systems arising from mathematical foundations of cell development; regularized kernel based learning in the Big Data situation; deep learning; convergence rates of learning and online learning algorithms; numerical refinement algorithms to learning; statistical robustness of regularized kernel based learning

Optimal Statistical Rates for Decentralised Non-Parametric Regression with Linear Speed-Up

We analyse the learning performance of Distributed Gradient Descent in the context of multi-agent decentralised non-parametric regression with the square loss function when i.i.d. samples are assigned to agents. We show that if agents hold sufficiently many samples with respect to the network size, then Distributed Gradient Descent achieves optimal statistical rates with a number of iterations that scales, up to a threshold, with the inverse of the spectral gap of the gossip matrix divided by the number of samples owned by each agent raised to a problem-dependent power. The presence of the threshold comes from statistics. It encodes the existence of a "big data" regime where the number of required iterations does not depend on the network topology. In this regime, Distributed Gradient Descent achieves optimal statistical rates with the same order of iterations as gradient descent run with all the samples in the network. Provided the communication delay is sufficiently small, the distributed protocol yields a linear speed-up in runtime compared to the single-machine protocol. This is in contrast to decentralised optimisation algorithms that do not exploit statistics and only yield a linear speed-up in graphs where the spectral gap is bounded away from zero. Our results exploit the statistical concentration of quantities held by agents and shed new light on the interplay between statistics and communication in decentralised methods. Bounds are given in the standard non-parametric setting with source/capacity assumptions