654 research outputs found
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.
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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
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Probabilistic Programming for Deep Learning
We propose the idea of deep probabilistic programming, a synthesis of advances for systems at the intersection of probabilistic modeling and deep learning. Such systems enable the development of new probabilistic models and inference algorithms that would otherwise be impossible: enabling unprecedented scales to billions of parameters, distributed and mixed precision environments, and AI accelerators; integration with neural architectures for modeling massive and high-dimensional datasets; and the use of computation graphs for automatic differentiation and arbitrary manipulation of probabilistic programs for flexible inference and model criticism.
After describing deep probabilistic programming, we discuss applications in novel variational inference algorithms and deep probabilistic models. First, we introduce the variational Gaussian process (VGP), a Bayesian nonparametric variational family, which adapts its shape to match complex posterior distributions. The VGP generates approximate posterior samples by generating latent inputs and warping them through random non-linear mappings; the distribution over random mappings is learned during inference, enabling the transformed outputs to adapt to varying complexity of the true posterior. Second, we introduce hierarchical implicit models (HIMs). HIMs combine the idea of implicit densities with hierarchical Bayesian modeling, thereby defining models via simulators of data with rich hidden structure
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Stochastic Methods in Optimization and Machine Learning
Stochastic methods are indispensable to the modeling, analysis and design of complex systems involving randomness. In this thesis, we show how simulation techniques and simulation-based computational methods can be applied to a wide spectrum of applied domains including engineering, optimization and machine learning. Moreover, we show how analytical tools in statistics and computer science including empirical processes, probably approximately correct learning, and hypothesis testing can be used in these contexts to provide new theoretical results. In particular, we apply these techniques and present how our results can create new methodologies or improve upon existing state-of-the-art in three areas: decision making under uncertainty (chance-constrained programming, stochastic programming), machine learning (covariate shift, reinforcement learning) and estimation problems arising from optimization (gradient estimate of composite functions) or stochastic systems (solution of stochastic PDE).
The work in the above three areas will be organized into six chapters, where each area contains two chapters. In Chapter 2, we study how to obtain feasible solutions for chance-constrained programming using data-driven, sampling-based scenario optimization (SO) approach. When the data size is insufficient to statistically support a desired level of feasibility guarantee, we explore how to leverage parametric information, distributionally robust optimization and Monte Carlo simulation to obtain a feasible solution of chance-constrained programming in small-sample situations.
In Chapter 3, We investigate the feasibility of sample average approximation (SAA) for general stochastic optimization problems, including two-stage stochastic programming without the relatively complete recourse. We utilize results from the Vapnik-Chervonenkis (VC) dimension and Probably Approximately Correct learning to provide a general framework.
In Chapter 4, we design a robust importance re-weighting method for estimation/learning problem in the covariate shift setting that improves the best-know rate. In Chapter 5, we develop a model-free reinforcement learning approach to solve constrained Markov decision processes (MDP). We propose a two-stage procedure that generates policies with simultaneous guarantees on near-optimality and feasibility.
In Chapter 6, we use multilevel Monte Carlo to construct unbiased estimators for expectations of random parabolic PDE. We obtain estimators with finite variance and finite expected computational cost, but bypassing the curse of dimensionality. In Chapter 7, we introduce unbiased gradient simulation algorithms for solving stochastic composition optimization (SCO) problems. We show that the unbiased gradients generated by our algorithms have finite variance and finite expected computational cost
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