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Sample complexity bounds are a common performance metric in the reinforcement learning literature. In the discounted cost, infinite horizon setting, all of the known bounds can be arbitrarily large, as the discount factor approaches unity. These results seem to imply that a very large number of samples is required to achieve an epsilon-optimal policy. The objective of the present work is to introduce a new class of algorithms that have sample complexity uniformly bounded over all discount factors. One may argue that this is impossible, due to a recent min–max lower bound. The explanation is that these prior bounds concern value function approximation and not policy approximation. We show that the asymptotic covariance of the tabular Q-learning algorithm with an optimized step-size sequence is a quadratic function of a factor that goes to infinity, as discount factor approaches 1; an essentially known result. The new relative Q-learning algorithm proposed here is shown to have asymptotic covariance that is uniformly bounded over all discount factors. [ABSTRACT FROM AUTHOR]

Reinforcement learning aims to find policies that maximize an expected cumulative reward in Markov decision processes with unknown transition probabilities. Policy gradient (PG)-algorithms use stochastic gradients of the value function to update the policy. A major drawback of PG-algorithms is that they are limited to episodic tasks (multiple finite-horizon trajectories) unless stringent stationarity assumptions are imposed on the trajectories. Hence, they need restarts and cannot be fully implemented online, which is critical for systems need to adapt to new tasks and/or environments in deployment. Moreover, the standard stationary formulation ignores transient behaviors. This motivates our study of discounted MDPs of infinite horizon without restarts. However, it is unknown if in this case following stochastic PG-type estimates would improve the policy. The main result of this work is to establish that when policies belong to a reproducing kernel Hilbert space (RKHS), and the kernel is selected properly, then these PG-estimates are ascent directions for the value function conditioned to any arbitrary initial point. This allows us to prove convergence of our online algorithm to the local optima. A numerical example shows that an agent running our online algorithm learns to navigate and succeeds in a surveillance task that requires looping between two goal locations. This example corroborates our theoretical findings about the ascent directions of subsequent stochastic gradients. It also shows how our online algorithm guides the agent through a continuing cyclic trajectory that does not comply with the standard stationarity assumptions in the literature for non-episodic training. [ABSTRACT FROM AUTHOR]

Value iteration is a fixed point iteration technique utilized to obtain the optimal value function and policy in a discounted reward Markov decision process (MDP). Here, a contraction operator is constructed and applied repeatedly to arrive at the optimal solution. Value iteration is a first-order method and, therefore, it may take a large number of iterations to converge to the optimal solution. Successive relaxation is a popular technique that can be applied to solve a fixed point equation. It has been shown in the literature that under a special structure of the MDP, successive overrelaxation technique computes the optimal value function faster than standard value iteration. In this article, we propose a second-order value iteration procedure that is obtained by applying the Newton–Raphson method to the successive relaxation value iteration scheme. We prove the global convergence of our algorithm to the optimal solution asymptotically and show the second-order convergence. Through experiments, we demonstrate the effectiveness of our proposed approach. [ABSTRACT FROM AUTHOR]

In this article, we propose a symplectic algorithm for the stable manifolds of the Hamilton–Jacobi equations combined with an iterative procedure in [N. Sakamoto and A. J. van der Schaft, “Analytical approximation methods for the stabilizing solution of the Hamilton–Jacobi equation,” IEEE Trans. Autom. Control, 2008]. Our algorithm includes two key aspects. The first one is to prove a precise estimate for radius of convergence and the errors of local approximate stable manifolds. The second one is to extend the local approximate stable manifolds to larger ones by symplectic algorithms, which have better long-time behaviors than general-purpose schemes. Our approach avoids the case of divergence of the iterative sequence of approximate stable manifolds and reduces the computation cost. We illustrate the effectiveness of the algorithm by an optimal control problem with exponential nonlinearity. [ABSTRACT FROM AUTHOR]

This article develops a novel passive stochastic gradient algorithm. In passive stochastic approximation, the stochastic gradient algorithm does not have control over the location where noisy gradients of the cost function are evaluated. Classical passive stochastic gradient algorithms use a kernel that approximates a Dirac delta to weigh the gradients based on how far they are evaluated from the desired point. In this article, we construct a multikernel passive stochastic gradient algorithm. The algorithm performs substantially better in high dimensional problems and incorporates variance reduction. We analyze the weak convergence of the multikernel algorithm and its rate of convergence. In numerical examples, we study the multikernel version of the passive least mean squares algorithm for transfer learning to compare the performance with the classical passive version. [ABSTRACT FROM AUTHOR]

We propose a two-layer, semidecentralized algorithm to compute a local solution to the Stackelberg equilibrium problem in aggregative games with coupling constraints. Specifically, we focus on a single-leader, multiple-follower problem, and after equivalently recasting the Stackelberg game as a mathematical program with complementarity constraints (MPCC), we iteratively convexify a regularized version of the MPCC as the inner problem, whose solution generates a sequence of feasible descent directions for the original MPCC. Thus, by pursuing a descent direction at every outer iteration, we establish convergence to a local Stackelberg equilibrium. Finally, the proposed algorithm is tested on a numerical case study, a hierarchical instance of the charging coordination problem of plug-in electric vehicles. [ABSTRACT FROM AUTHOR]

This article is concerned with the multiagent optimization problem. A distributed randomized gradient-free mirror descent (DRGFMD) method is developed by introducing a randomized gradient-free oracle in the mirror descent scheme where the non-Euclidean Bregman divergence is used. The classical gradient descent method is generalized without using subgradient information of objective functions. The proposed algorithms are the first distributed non-Euclidean zeroth-order methods, which achieve an approximate $O(\frac{1}{\sqrt{T}})$ $T$ -rate of convergence, recovering the best known optimal rate of distributed nonsmooth constrained convex optimization. Moreover, a decentralized reciprocal weighted averaging (RWA) approximating sequence is first investigated, the convergence for RWA sequence is shown to hold over time-varying graph. Rates of convergence are comprehensively explored for the algorithm with RWA (DRGFMD-RWA). The technique on constructing the decentralized RWA sequence provides new insight in searching for minimizers in distributed algorithms. [ABSTRACT FROM AUTHOR]

This article compiles several aspects of the dynamics of stochastic approximation algorithms with Markov iterate-dependent noise when the iterates are not known to be stable beforehand. We achieve the same by extending the lock-in probability (i.e., the probability of convergence of the iterates to a specific attractor of the limiting ordinary differential equation (o.d.e.) given that the iterates are in its domain of attraction after a sufficiently large number of iterations (say) $n_0$) framework to such recursions. Specifically, with the more restrictive assumption of Markov iterate-dependent noise supported on a bounded subset of the Euclidean space, we give a lower bound for the lock-in probability. We use these results to prove almost sure convergence of the iterates to the specified attractor when the iterates satisfy an asymptotic tightness condition. The novelty of our approach is that if the state space of the Markov process is compact, we prove almost sure convergence under much weaker assumptions compared to the work by Andrieu et al., which solves the general state-space case under much restrictive assumptions by providing sufficient conditions for stability of the iterates. We also extend our single-timescale results to the case where there are two separate recursions over two different timescales. This, in turn, is shown to be useful in analyzing the tracking ability of general adaptive algorithms. Additionally, we show that our results can be used to derive a sample complexity estimate of such recursions, which then can be used for step-size selection. [ABSTRACT FROM AUTHOR]

We consider the stochastic generalized Nash equilibrium problem (SGNEP) with expected-value cost functions. Inspired by Yi and Pavel (2019), we propose a distributed generalized Nash equilibrium seeking algorithm based on the preconditioned forward–backward operator splitting for SGNEPs, where, at each iteration, the expected value of the pseudogradient is approximated via a number of random samples. Our main contribution is to show almost sure convergence of the proposed algorithm if the pseudogradient mapping is restricted (monotone and) cocoercive. [ABSTRACT FROM AUTHOR]

Value functions derived from Markov decision processes arise as a central component of algorithms as well as performance metrics in many statistics and engineering applications of machine learning. Computation of the solution to the associated Bellman equations is challenging in most practical cases of interest. A popular class of approximation techniques, known as temporal difference (TD) learning algorithms, are an important subclass of general reinforcement learning methods. The algorithms introduced in this article are intended to resolve two well-known issues with TD-learning algorithms. Their slow convergence due to very high central limit theorem variance, and the fact that, for the problem of computing the relative value function, consistent algorithms exist only in special cases. First we show that the gradients of these value functions admit a representation that lends itself to algorithm design. Based on this result, a new class of differential TD-learning algorithms is introduced. For Markovian models on Euclidean space with smooth dynamics, the algorithms are shown to be consistent under general conditions. Numerical results show dramatic variance reduction in comparison to standard methods. [ABSTRACT FROM AUTHOR]

We study gradient methods for solving distributed convex optimization problems over a network when the communication bandwidth between the nodes is limited, and so information that is exchanged between the nodes must be quantized. This imperfect communication poses a fundamental challenge, and if not properly accounted for may prevent the convergence of these algorithms. Our first contribution in this article is to propose a modified consensus-based gradient method for solving such problems using random (dithered) quantization. This algorithm can be interpreted as a distributed variant of a two-time-scale stochastic approximation algorithm. We then study the convergence and derive upper bounds on the rates of convergence of the proposed method as a function of the bandwidth available between the nodes and the underlying network topology for both convex and strongly convex objective functions. Our results complement existing literature, where such convergence and explicit formulas of the convergence rates are missing. Finally, we provide numerical simulations to compare the convergence properties of the distributed gradient methods with and without quantization for solving the well-known regression problems over networks, for both quadratic and absolute loss functions. [ABSTRACT FROM AUTHOR]

Economic model predictive control (EMPC) has recently become popular because of its ability to control constrained nonlinear systems while explicitly optimizing a prescribed performance criterion. Large performance gains have been reported for many applications and closed-loop stability has been recently investigated. However, computational performance still remains an open issue and only few contributions have proposed real-time algorithms tailored to EMPC. We perform a step towards computationally cheap algorithms for EMPC by proposing a new positive-definite Hessian approximation which does not hinder fast convergence and is suitable for being used within the real-time iteration (RTI) scheme. We provide two simulation examples to demonstrate the effectiveness of RTI-based EMPC relying on the proposed Hessian approximation. [ABSTRACT FROM AUTHOR]

In this article, we introduce a class of novel distributed algorithms for solving stochastic big-data convex optimization problems over directed graphs. In the addressed set-up, the dimension of the decision variable can be extremely high and the objective function can be nonsmooth. The general algorithm consists of two main steps: a consensus step and an update on a single block of the optimization variable, which is then broadcast to neighbors. Three special instances of the proposed method, involving particular problem structures, are then presented. In the general case, the convergence of a dynamic consensus algorithm over random row stochastic matrices is shown. Then, the convergence of the proposed algorithm to the optimal cost is proven in expected value. Exact convergence is achieved when using diminishing (local) stepsizes, whereas approximate convergence is attained when constant stepsizes are employed. The convergence rate is shown to be sublinear and an explicit rate is provided in the case of constant stepsizes. Finally, the algorithm is tested on a distributed classification problem, first on synthetic data and, then, on a real, high-dimensional, text dataset. [ABSTRACT FROM AUTHOR]

Asynchronous stochastic approximations (SAs) are an important class of model-free algorithms, tools, and techniques that are popular in multiagent and distributed control scenarios. To counter Bellman's curse of dimensionality, such algorithms are coupled with function approximations. Although the learning/control problem becomes more tractable, function approximations affect stability and convergence. In this article, we present verifiable sufficient conditions for stability and convergence of asynchronous SAs with biased approximation errors. The theory developed herein is used to analyze policy gradient methods and noisy value iteration schemes. Specifically, we analyze the asynchronous approximate counterparts of the policy gradient (A2PG) and value iteration (A2VI) schemes. It is shown that the stability of these algorithms is unaffected by biased approximation errors, provided that they are asymptotically bounded. With respect to convergence (of A2VI and A2PG), a relationship between the limiting set and the approximation errors is established. Finally, experimental results that support the theory are presented. [ABSTRACT FROM AUTHOR]

Controlled interacting particle systems such as the ensemble Kalman filter (EnKF) and the feedback particle filter (FPF) are numerical algorithms to approximate the solution of the nonlinear filtering problem in continuous time. The distinguishing feature of these algorithms is that the Bayesian update step is implemented using a feedback control law. It has been noted in the literature that the control law is not unique. This is the main problem addressed in this article. To obtain a unique control law, the filtering problem is formulated here as an optimal transportation problem. An explicit formula for the (mean-field type) optimal control law is derived in the linear Gaussian setting. Comparisons are made with the control laws for different types of EnKF algorithms described in the literature. Via empirical approximation of the mean-field control law, a finite- $N$ controlled interacting particle algorithm is obtained. For this algorithm, the equations for empirical mean and covariance are derived and shown to be identical to the Kalman filter. This allows strong conclusions on convergence and error properties based on the classical filter stability theory for the Kalman filter. It is shown that, under certain technical conditions, the mean squared error converges to zero even with a finite number of particles. A detailed propagation of chaos analysis is carried out for the finite- $N$ algorithm. The analysis is used to prove weak convergence of the empirical distribution as $N\rightarrow \infty$. For a certain simplified filtering problem, analytical comparison of the mse with the importance sampling-based algorithms is described. The analysis helps explain the favorable scaling properties of the control-based algorithms reported in several numerical studies in recent literature. [ABSTRACT FROM AUTHOR]

This article studies a class of nonsmooth decentralized multiagent optimization problems where the agents aim at minimizing a sum of local strongly-convex smooth components plus a common nonsmooth term. We propose a general primal-dual algorithmic framework that unifies many existing state-of-the-art algorithms. We establish linear convergence of the proposed method to the exact minimizer in the presence of the nonsmooth term. Moreover, for the more general class of problems with agent specific nonsmooth terms, we show that linear convergence cannot be achieved (in the worst case) for the class of algorithms that uses the gradients and the proximal mappings of the smooth and nonsmooth parts, respectively. We further provide a numerical counterexample that shows how some state-of-the-art algorithms fail to converge linearly for strongly convex objectives and different local non smooth terms. [ABSTRACT FROM AUTHOR]

We study distributed big-data nonconvex optimization in multiagent networks. We consider the (constrained) minimization of the sum of a smooth (possibly) nonconvex function, i.e., the agents’ sum-utility, plus a convex (possibly) nonsmooth regularizer. Our interest is on big-data problems in which there is a large number of variables to optimize. If treated by means of standard distributed optimization algorithms, these large-scale problems may be intractable due to the prohibitive local computation and communication burden at each node. We propose a novel distributed solution method where, at each iteration, agents update in an uncoordinated fashion only one block of the entire decision vector. To deal with the nonconvexity of the cost function, the novel scheme hinges on successive convex approximation techniques combined with a novel blockwise perturbed push-sum consensus protocol, which is instrumental to perform local block-averaging operations and tracking of gradient averages. Asymptotic convergence to stationary solutions of the nonconvex problem is established. Finally, numerical results show the effectiveness of the proposed algorithm and highlight how the block dimension impacts on the communication overhead and practical convergence speed. [ABSTRACT FROM AUTHOR]

This article addresses the problem of seeking a common fixed point for a finite collection of nonexpansive operators over time-varying multi-agent networks in real Hilbert spaces. Each operator is assumed to be only privately and approximately known to each individual agent, and all agents need to cooperate to solve this problem by local communications over time-varying networks. To handle this problem, inspired by the centralized inexact Krasnoselski –Mann iteration, we propose a distributed algorithm, called distributed inexact averaged operator algorithm (DIO). It is shown that the DIO can converge weakly to a common fixed point of the family of nonexpansive operators. Moreover, under the assumption that all operators and their own fixed point sets are (boundedly) linearly regular, it is proved that the distributed averaged operator algorithm converges with a rate O(γlog16(4k)) for some constant γ ∈ (0,1), where k is the iteration number. To reduce computational complexity, a scenario, where only a random part of coordinates of each operator is computed at each iteration, is further considered. In this case, a corresponding algorithm, named distributed block-coordinate inexact averaged operator algorithm, is developed. The algorithm is proved to be weakly convergent to a common fixed point of the group of considered operators almost surely, and, with the extra assumption of (bounded) linear regularity, the distributed block-coordinate averaged operator algorithm is convergent in the mean square sense with a rate O(ηlog4(4k)) for some constant η ∈ (0,1). Furthermore, it is shown that the same convergence rates can still be guaranteed under a more relaxed (bounded) power regularity condition. A couple of examples are finally presented to illustrate the effectiveness of the proposed algorithms. [ABSTRACT FROM AUTHOR]

It is well known that the nonlinear filtering (NLF) problem has important applications in both military and civil industries. The central question is to solve the posterior conditional density function of the states, which satisfies the Kushner or the Duncan–Mortensen–Zakai (DMZ) equation after suitable change of probability measure. In this article, we shall follow the so-called Yau–Yau's algorithm to split the solution of the DMZ equation into on- and off-line part, where the off-line part is to solve the forward Kolmogorov equation (FKE) with the initial conditions to be the orthonormal bases in some suitable function space. Instead of the generalized Hermite function investigated by the second and the third author of this article, we shall explore the generalized Legendre polynomials. The Legendre spectral method (LSM) is used to numerically solve the FKE. Under certain conditions, the convergence rate of LSM is twice faster than that of the Hermite spectral method. Two two-dimensional numerical experiments of NLF problems (time-invariant and time-varying cases) have been numerically solved to illustrate the feasibility of our algorithm. Our algorithm outperforms the extended Kalman Filter and particle filter in both real-time manner and accuracy. [ABSTRACT FROM AUTHOR]

This article develops a fully decentralized multiagent algorithm for policy evaluation. The proposed scheme can be applied to two distinct scenarios. In the first scenario, a collection of agents have distinct datasets gathered by following different behavior policies (none of which is required to explore the full state space) in different instances of the same environment and they all collaborate to evaluate a common target policy. The network approach allows for efficient exploration of the state space and allows all agents to converge to the optimal solution even in situations where neither agent can converge on its own without cooperation. The second scenario is that of multiagent games, in which the state is global and rewards are local. In this scenario, agents collaborate to estimate the value function of a target team policy. The proposed algorithm combines off-policy learning, eligibility traces, and linear function approximation. The proposed algorithm is of the variance-reduced kind and achieves linear convergence with O(1) memory requirements. The linear convergence of the algorithm is established analytically and simulations are used to illustrate the effectiveness of the method. [ABSTRACT FROM AUTHOR]

This article considers a distributed Nash equilibrium seeking problem, where the players only have partial access to other players’ actions, such as their neighbors’ actions. Thus, the players are supposed to communicate with each other to estimate other players’ actions. To solve the problem, a leader–following consensus gradient-free distributed Nash equilibrium seeking algorithm is proposed. This algorithm utilizes only the measurements of the player’ local cost function without the knowledge of its explicit expression or the requirement on its smoothness. Hence, the algorithm is gradient-free during the entire updating process. Moreover, the analysis on the convergence of the Nash equilibrium is studied for the algorithm with both diminishing and constant step-sizes, respectively. Specifically, in the case of diminishing step-size, it is shown that the players’ actions converge to the Nash equilibrium almost surely, while in the case of fixed step-size, the convergence to the neighborhood of the Nash equilibrium is achieved. The performance of the proposed algorithm is verified through numerical simulations. [ABSTRACT FROM AUTHOR]

We develop a stochastic approximation-type algorithm to solve finite state/action, infinite-horizon, risk-aware Markov decision processes. Our algorithm has two loops. The inner loop computes the risk by solving a stochastic saddle-point problem. The outer loop performs Q-learning to compute an optimal risk-aware policy. Several widely investigated risk measures (e.g., conditional value-at-risk, optimized certainty equivalent, and absolute semideviation) are covered by our algorithm. Almost sure convergence and the convergence rate of the algorithm are established. For an error tolerance ε > 0 for optimal Q-value estimation gap and learning rate k ∈ (1/2,1], the overall convergence rate of our algorithm is Ω((ln(1/δε)/ε2)1/k + (ln(1/ε))1/(1−k)) with probability at least 1−ε. [ABSTRACT FROM AUTHOR]

In this article, we examine a novel generic network cost minimization problem, in which every node has a local decision vector to optimize. Each node incurs a cost associated with its decision vector, while each link incurs a cost related to the decision vectors of its two end nodes. All nodes collaborate to minimize the overall network cost. The formulated network cost minimization problem has broad applications in distributed signal processing and control, in which the notion of link costs often arises. To solve this problem in a decentralized manner, we develop a distributed variant of Newton's method, which possesses faster convergence than alternative first-order optimization methods such as gradient descent and alternating direction method of multipliers. The proposed method is based on an appropriate splitting of the Hessian matrix and an approximation of its inverse, which is used to determine the Newton step. Global linear convergence of the proposed algorithm is established under several standard technical assumptions on the local cost functions. Furthermore, analogous to classical centralized Newton's method, a quadratic convergence phase of the algorithm over a certain time interval is identified. Finally, numerical simulations are conducted to validate the effectiveness of the proposed algorithm and its superiority over other first-order methods, especially when the cost functions are ill-conditioned. Complexity issues of the proposed distributed Newton's method and alternative first-order methods are also discussed. [ABSTRACT FROM AUTHOR]

In this article, we consider an online distributed composite optimization problem over a time-varying multiagent network that consists of multiple interacting nodes, where the objective function of each node consists of two parts: a loss function that changes over time and a regularization function. This problem naturally arises in many real-world applications ranging from wireless sensor networks to signal processing. We propose a class of online distributed optimization algorithms that are based on approximate mirror descent, which utilizes the Bregman divergence as a distance-measuring function that includes the Euclidean distances as a special case. We consider two standard information feedback models when designing the algorithms, that is, full-information feedback and bandit feedback. For the full-information feedback model, the first algorithm attains an average regularized regret of order O(1/√T) with the total number of rounds T. The second algorithm, which only requires the information of the values of the loss function at two predicted points instead of the gradient information, achieves the same average regularized regret as that of the first algorithm. Simulation results of a distributed online regularized linear regression problem are provided to illustrate the performance of the proposed algorithms. [ABSTRACT FROM AUTHOR]

This article presents a multicommodity, discrete-time, distributed, and noncooperative routing algorithm, which is proved to converge to an equilibrium in the presence of heterogeneous, unknown, time-varying but bounded delays. Under mild assumptions on the latency functions, which describe the cost associated with the network paths, two algorithms are proposed: The former assumes that each commodity relies only on measurements of the latencies associated with its own paths; the latter assumes that each commodity has (at least indirectly) access to the measures of the latencies of all the network paths. Both algorithms are proven to drive the system state to an invariant set that approximates and contains the Wardrop equilibrium, defined as a network state in which no traffic flow over the network paths can improve its routing unilaterally, with the latter achieving a better reconstruction of the Wardrop equilibrium. Numerical simulations show the effectiveness of the proposed approach. [ABSTRACT FROM AUTHOR]

In this article, we propose a novel family of higher order sliding mode (HOSM) controllers with discontinuous integral term for a class of single-input–single-output (SISO) nonlinear systems. They belong to a recently introduced class of controllers, named continuous HOSM since they are able to compensate exactly and in finite-time perturbations/uncertainties using a continuous control signal. They are obtained by introducing a discontinuous integral action at the input of the plant. Here, continuously differentiable Lyapunov functions are used to rigorously prove the stability of the closed-loop system. We also investigate an approximation of this strategy, where a continuous integral term replaces the discontinuous one. The robustness properties of the closed-loop controller with continuous and discontinuous integral term are studied. [ABSTRACT FROM AUTHOR]

One of the most important problems in the field of distributed optimization is the problem of minimizing a sum of local convex objective functions over a networked system. Most of the existing work in this area focuses on developing distributed algorithms in a synchronous setting under the presence of a central clock, where the agents need to wait for the slowest one to finish the update, before proceeding to the next iterate. Asynchronous distributed algorithms remove the need for a central coordinator, reduce the synchronization wait, and allow some agents to compute faster and execute more iterations. In the asynchronous setting, the only known algorithms for solving this problem could achieve an either linear or sublinear rate of convergence. In this paper, we build upon the existing literature to develop and analyze an asynchronous Newton-based method to solve a penalized version of the problem. We show that this algorithm guarantees almost sure convergence with a global linear and local quadratic rate in expectation. Numerical studies confirm the superior performance of our algorithm against other asynchronous methods. [ABSTRACT FROM AUTHOR]

This paper studies consensus problems of multi-agent systems with binary-valued communications. Different from most existing works, the agents considered in this paper can only get binary-valued observations of its neighbors’ states with random noises. A consensus algorithm is proposed: first, each agent estimates its neighbors’ states by the recursive projection algorithm; then, each agent designs the control timely based on the estimates. It is proved that the estimates of the states can converge to the true states with a faster convergence rate than that in the parameter estimation. Moreover, the states of the agents can achieve mean-square consensus, and the corresponding consensus speed can achieve $O(1/t)$ under certain conditions. Finally, simulations are given to demonstrate the theoretical results. [ABSTRACT FROM AUTHOR]

We introduce deterministic perturbation (DP) schemes for the recently proposed random directions stochastic approximation, and propose new first-order and second-order algorithms. In the latter case, these are the first second-order algorithms to incorporate DPs. We show that the gradient and/or Hessian estimates in the resulting algorithms with DPs are asymptotically unbiased, so that the algorithms are provably convergent. Furthermore, we derive convergence rates to establish the superiority of the first-order and second-order algorithms, for the special case of a convex and quadratic optimization problem, respectively. Numerical experiments are used to validate the theoretical results. [ABSTRACT FROM AUTHOR]

Many problems of interest for cyber-physical network systems can be formulated as mixed-integer linear programs in which the constraints are distributed among the agents. In this paper, we propose a distributed algorithmic framework to solve this class of optimization problems in a peer-to-peer network with no coordinator and with limited computation and communication capabilities. At each communication round, agents locally solve a small linear program, generate suitable cutting planes, and communicate a fixed number of active constraints. Within the distributed framework, we first propose an algorithm that, under the assumption of integer-valued optimal cost, guarantees finite-time convergence to an optimal solution. Second, we propose an algorithm for general problems that provides a suboptimal solution up to a given tolerance in a finite number of communication rounds. Both algorithms work under asynchronous, directed, unreliable networks. Finally, through numerical computations, we analyze the algorithm scalability in terms of the network size. Moreover, for a multi-agent multi-task assignment problem, we show, consistently with the theory, its robustness to packet loss. [ABSTRACT FROM AUTHOR]

This paper proposes an iterative learning control (ILC) algorithm with gain adaptation for discrete-time stochastic systems. The algorithm is based on Kesten's accelerated stochastic approximation (SA) algorithm. The gain adaptation uses only tracking error information, and, hence, is a data-driven adaptation approach. If stochastic noises account for a small proportion of the tracking error, the learning gain matrix remains constant with a high probability. If stochastic noises dominate the tracking error, the learning gain matrix is decreasing. Therefore, the new ILC algorithm converges more quickly than existing SA-based algorithms. In addition, the classic P-type ILC law for noise-free systems is a special case of the new ILC algorithm. The behaviors of the proposed ILC algorithm are demonstrated through examples. [ABSTRACT FROM AUTHOR]

When using standard learning techniques for uncertainty approximation, persistently exciting inputs are necessary in order to achieve good parameter estimation. It has been shown in recent years that, through concurrent learning (CL), it is possible to achieve better learning without requiring persistency of excitation. We define good learning here by how well an uncertainty can be reconstructed given the estimated parameters. Most studies concerning CL have however been done in the continuous-time (CT) framework. While working with discrete-time (DT) structured uncertainties, we have shown in an earlier study that the concept of CL could be used to solve the parameter identification problem granted, much like in the CT domain, a less restrictive condition compared to that of persistency of excitation is verified. This paper furthers our fundamental study of CL in the DT framework while drawing comparisons with the traditional but fundamental gradient descent technique. As a main contribution, via formal derivations, we present a generalized gradient-based CL motivated DT algorithm for online approximation of both DT structured and unstructured uncertainties. Numerical simulations are provided to show how well the designed algorithm leverages memory usage to achieve better learning. [ABSTRACT FROM AUTHOR]

In this paper, a novel distributed stochastic approximation algorithm (DSAA) is proposed to seek roots of the sum of local functions, each of which is associated with an agent from multiple agents connected over a network. At each iteration, each agent updates its estimate for the root utilizing the noisy observations of its local function and the information derived from the neighboring agents. The key difference of the proposed algorithm from the existing ones consists in the expanding truncations (so it is called the DSAAWET), by which the boundedness of the estimates can be guaranteed without imposing the growth-rate constraints on the local functions. The estimates generated by the DSAAWET are shown to converge almost surely to a consensus set, which belongs to a connected subset of the root set of the sum function. In comparison with the existing results, we impose weaker conditions on the local functions and on the observation noise. We then apply the proposed algorithm to two applications, one from signal processing and the other one from distributed optimization, and establish the almost sure convergence. Numerical simulation results are also included. [ABSTRACT FROM AUTHOR]

We propose universal randomized function approximation-based empirical value learning (EVL) algorithms for Markov decision processes. The “empirical” nature comes from each iteration being done empirically from samples available from simulations of the next state. This makes the Bellman operator a random operator. A parametric and a nonparametric method for function approximation using a parametric function space and a reproducing kernel Hilbert space respectively are then combined with EVL. Both function spaces have the universal function approximation property. Basis functions are picked randomly. Convergence analysis is performed using a random operator framework with techniques from the theory of stochastic dominance. Finite time sample complexity bounds are derived for both universal approximate dynamic programming algorithms. Numerical experiments support the versatility and computational tractability of this approach. [ABSTRACT FROM AUTHOR]

Q-learning is a sample-based model-free algorithm that solves Markov decision problems asymptotically, but in finite time, it can perform poorly when random rewards and transitions result in large variance of value estimates. We pinpoint its cause to be the estimation bias due to the maximum operator in Q-learning algorithm, and present the evidence of max-operator bias in its Q value estimates. We then present an asymptotically optimal bias-correction strategy and construct an extension to bias-corrected Q-learning algorithm to multistate Markov decision processes, with asymptotic convergence properties as strong as those from Q-learning. We report the empirical performance of the bias-corrected Q-learning algorithm with multistate extension in two model problems: A multiarmed bandit version of Roulette and an electricity storage control simulation. The bias-corrected Q-learning algorithm with multistate extension is shown to control max-operator bias effectively, where the bias-resistance can be tuned predictably by adjusting a correction parameter. [ABSTRACT FROM AUTHOR]

In this paper, we propose a distributed Newton method for decenteralized optimization of large sums of convex functions. Our proposed method is based on creating a set of separable finite sum minimization problems by utilizing a decomposition technique known as Global Consensus that distributes the computation across nodes of a graph and enforces a consensus constraint among the separated variables. The key idea is to exploit the sparsity of the dual Hessian and recast the computation of the Newton step as one of the efficiently solving symmetric diagonally dominant linear equations. We show that our method outperforms the state-of-the-art algorithms, including ADMM. We validate our algorithm both theoretically and empirically. On the theory side, we demonstrate that our algorithm exhibits superlinear convergence within a neighborhood of optimality. Empirically, we show the superiority of this new method on a variety of large-scale optimization problems. The proposed approach is scalable to large problems and has a low communication overhead. [ABSTRACT FROM AUTHOR]

This paper investigates the interaction between control and communications in networked systems by studying a class of stochastic approximation algorithms that accommodate random network topology switching processes, time-varying functions, nonlinear dynamics, additive and nonadditive noises, and other uncertainties. Interaction among control strategy and the multiple stochastic processes introduces critical challenges in such problems. By modeling the random switching as a discrete-time Markov chain and studying multiple stochastic uncertainties in a unified framework, it is shown that under broad conditions, the algorithms are convergent. The performance of the algorithms is further analyzed by establishing their rate of convergence and asymptotic characterizations. Simulation case studies are conducted to evaluate the performance of the procedures in various aspects. [ABSTRACT FROM AUTHOR]

We are interested in understanding stability (almost sure boundedness) of stochastic approximation algorithms (SAs) driven by a “controlled Markov” process. Analyzing this class of algorithms is important, since many reinforcement learning (RL) algorithms can be cast as SAs driven by a “controlled Markov” process. In this paper, we present easily verifiable sufficient conditions for stability and convergence of SAs driven by a “controlled Markov” process. Many RL applications involve continuous state spaces. While our analysis readily ensures stability for such continuous state applications, traditional analyses do not. As compared to literature, our analysis presents a two-fold generalization: 1) the Markov process may evolve in a continuous state space and 2) the process need not be ergodic under any given stationary policy. Temporal difference (TD) learning is an important policy evaluation method in RL. The theory developed herein, is used to analyze generalized $\text{TD}(0)$ , an important variant of TD. Our theory is also used to analyze a TD formulation of supervised learning for forecasting problems. [ABSTRACT FROM AUTHOR]

This paper studies linear stochastic approximation (SA) algorithms and their application to multiagent systems in engineering and sociology. As main contribution, we provide necessary and sufficient conditions for convergence of linear SA algorithms to a deterministic or random final vector. We also characterize the system convergence rate, when the system is convergent. Moreover, differing from non-negative gain functions in traditional SA algorithms, this paper considers also the case when the gain functions are allowed to take arbitrary real numbers. Using our general treatment, we provide necessary and sufficient conditions to reach consensus and group consensus for first-order discrete-time multiagent system over random signed networks and with state-dependent noise. Finally, we extend our results to the setting of multidimensional linear SA algorithms and characterize the behavior of the multidimensional Friedkin–Johnsen model over random interaction networks. [ABSTRACT FROM AUTHOR]

This paper considers the problem of a leader that seeks to optimally influence the opinions of agents in a directed network through connecting with a limited number of the agents (“direct followers”), possibly in the presence of a fixed competing leader. The settings involving a single leader and two competing leaders are unified into a general combinatoric optimization problem, for which two heuristic approaches are developed. The first approach is based on a convex relaxation scheme, possibly in combination with the $\ell _1$ -norm regularization technique, and the second is based on a greedy selection strategy. The main technical novelties of this work are in the establishment of supermodularity of the objective function and convexity of its continuous relaxation. The greedy approach is guaranteed to have a lower bound on the approximation ratio sharper than $(1-1/e)$ , while the convex approach can benefit from efficient (customized) numerical solvers to have practically comparable solutions possibly with faster computation times. The two approaches can be combined to provide improved results. In numerical examples, the approximation ratio can be made to reach ${\text{90}\%}$ or higher depending on the number of direct followers. [ABSTRACT FROM AUTHOR]

This paper presents a discrete-time distributed and noncooperative routing algorithm, which is proved, via Lyapunov arguments, to asymptotically converge to a specific equilibrium condition among the traffic flows over the network paths, known as Wardrop equilibrium. This convergence result improves the discrete-time algorithms in the literature, which achieve approximate convergence to the Wardrop equilibrium. Numerical simulations show the effectiveness of the proposed approach. [ABSTRACT FROM AUTHOR]

In this paper, we provide a detailed analysis of the global convergence properties of an extensively studied and extremely effective fixed-point algorithm for the Kullback–Leibler approximation of spectral densities, proposed by Pavon and Ferrante in their paper “On the Georgiou–Lindquist approach to constrained Kullback–Leibler approximation of spectral densities.” Our main result states that the algorithm globally converges to one of its fixed points. [ABSTRACT FROM AUTHOR]

Cumulative prospect theory (CPT) is a popular approach for modeling human preferences. It is based on probabilistic distortions and generalizes the expected utility theory. We bring the CPT to a stochastic optimization framework and propose algorithms for both estimation and optimization of CPT-value objectives. We propose an empirical distribution function-based scheme to estimate the CPT value, and then, use this scheme in the inner loop of a CPT-value optimization procedure. We propose both gradient based as well as gradient-free CPT-value optimization algorithms that are based on two well-known simulation optimization ideas: simultaneous perturbation stochastic approximation and model-based parameter search, respectively. We provide theoretical convergence guarantees for all the proposed algorithms and also illustrate the potential of CPT-based criteria in a traffic signal control application. [ABSTRACT FROM AUTHOR]

We consider large-scale Markov decision processes (MDPs) with a time-consistent risk measure of variability in cost under the risk-aware MDP paradigm. Previous studies showed that risk-aware MDPs, based on a minimax approach to handling risk, can be solved using dynamic programming for small- to medium-sized problems. However, due to the “curse of dimensionality,” MDPs that model real-life problems are typically prohibitively large for such approaches. In this technical note, we employ an approximate dynamic programming approach and develop a family of simulation-based algorithms to approximately solve large-scale risk-aware MDPs with time-consistent risk measures. In parallel, we develop a unified convergence analysis technique to derive sample complexity bounds for this new family of algorithms. [ABSTRACT FROM AUTHOR]

Consensus conditions and convergence speeds are crucial for distributed consensus algorithms of networked systems. Based on a basic first-order average-consensus protocol with time-varying topologies and additive noises, this paper first investigates its critical consensus condition on network topology by stochastic approximation frameworks. A new joint-connectivity condition called extensible joint-connectivity that contains a parameter $\delta$ (termed the extensible exponent) is proposed. With this and a balanced topology condition, we show that a critical value of \delta$ for consensus is 1/2$ . Optimization on convergence rate of this protocol is further investigated. It is proved that the fastest convergence rate, which is the theoretic optimal rate among all controls, is of the order 1/t for the worst topologies, which are balanced and satisfy the extensible joint-connectivity condition. For practical implementation, certain open-loop control strategies are introduced to achieve consensus with a convergence rate of the same order as the fastest convergence rate. Furthermore, a consensus condition is derived for nonstationary and strongly correlated random topologies. The algorithms and consensus conditions are applied to distributed consensus computation of mobile ad-hoc networks; and their related critical exponents are derived from relative velocities of mobile agents for guaranteeing consensus. [ABSTRACT FROM PUBLISHER]

Search algorithms for optimizing a complex problem are mainly categorized as gradient-driven and stochastic search, each with its advantages and shortcomings. A newly developed algorithm, GO-POLARS, is proposed with a hyperspherical coordinate framework, which could perturb a given direction with well-controlled variation. It designs a steerable stochastic search algorithm that explores toward a promising direction, such as the gradient, at any desired levels. In this note, we provide an analytical study on the hyperspherical coordinates and the corresponding random distributions and, thus, prove the local convergence property of the GO-POLARS. Extensive numerical experiments are illustrated to show its advantages compared to conventional search algorithms. [ABSTRACT FROM PUBLISHER]

We develop algorithms that find and track the optimal solution trajectory of time-varying convex optimization problems that consist of local and network-related objectives. The algorithms are derived from the prediction-correction methodology, which corresponds to a strategy where the time-varying problem is sampled at discrete time instances, and then, a sequence is generated via alternatively executing predictions on how the optimizers at the next time sample are changing and corrections on how they actually have changed. Prediction is based on how the optimality conditions evolve in time, while correction is based on a gradient or Newton method, leading to decentralized prediction-correction gradient and decentralized prediction-correction Newton. We extend these methods to cases where the knowledge on how the optimization programs are changing in time is only approximate and propose decentralized approximate prediction-correction gradient and decentralized approximate prediction-correction Newton. Convergence properties of all the proposed methods are studied and empirical performance is shown on an application of a resource allocation problem in a wireless network. We observe that the proposed methods outperform existing running algorithms by orders of magnitude. The numerical results showcase a tradeoff between convergence accuracy, sampling period, and network communications. [ABSTRACT FROM PUBLISHER]

Decentralized optimization algorithms have received much attention due to the recent advances in network information processing. However, conventional decentralized algorithms based on projected gradient descent are incapable of handling high-dimensional constrained problems, as the projection step becomes computationally prohibitive. To address this problem, this paper adopts a projection-free optimization approach, a.k.a. the Frank–Wolfe (FW) or conditional gradient algorithm. We first develop a decentralized FW (DeFW) algorithm from the classical FW algorithm. The convergence of the proposed algorithm is studied by viewing the decentralized algorithm as an inexact FW algorithm. Using a diminishing step size rule and letting t be the iteration number, we show that the DeFW algorithm's convergence rate is {\mathcal O}(1/t) for convex objectives; is {\mathcal O}(1/t^2)$ for strongly convex objectives with the optimal solution in the interior of the constraint set; and is {\mathcal O}(1/\sqrt{t}) toward a stationary point for smooth but nonconvex objectives. We then show that a consensus-based DeFW algorithm meets the above guarantees with two communication rounds per iteration. We demonstrate the advantages of the proposed DeFW algorithm on low-complexity robust matrix completion and communication efficient sparse learning. Numerical results on synthetic and real data are presented to support our findings. [ABSTRACT FROM PUBLISHER]

In this paper, we propose a distributed algorithm for optimization problems that involve a separable, possibly nonconvex objective function subject to convex local constraints and linear coupling constraints. The method is based on the accelerated distributed augmented Lagrangians (ADAL) algorithm that was recently developed by the authors to address convex problems. Here, we extend this line of work in two ways. First, we establish convergence of the method to a local minimum of the problem, using assumptions that are common in the analysis of nonconvex optimization methods. To the best of our knowledge, this is the first work that shows convergence to local minima specifically for a distributed augmented Lagrangian (AL) method applied to nonconvex optimization problems; distributed AL methods are known to perform very well when used to solve convex problems. Second, we propose a more general and decentralized rule to select the stepsizes of the method. This improves on the authors’ original ADAL method, where the stepsize selection used global information at initialization. Numerical results are included to verify the correctness and efficiency of the proposed distributed method. [ABSTRACT FROM AUTHOR]