This paper considers a class of reinforcement-based learning (namely, perturbed learning automata) and provides a stochastic-stability analysis in repeatedly-played, positive-utility, finite strategic-form games. Prior work in this class of learning dynamics primarily analyzes asymptotic convergence through stochastic approximations, where convergence can be associated with the limit points of an ordinary-differential equation (ODE). However, analyzing global convergence through an ODE-approximation requires the existence of a Lyapunov or a potential function, which naturally restricts the analysis to a fine class of games. To overcome these limitations, this paper introduces an alternative framework for analyzing asymptotic convergence that is based upon an explicit characterization of the invariant probability measure of the induced Markov chain. We further provide a methodology for computing the invariant probability measure in positive-utility games, together with an illustration in the context of coordination games.
The case of noisy utility measurements, which are present in many engineering applications, has not currently been addressed through aspiration-based learning. The only exception is reference @cite_19 , under benchmark-based dynamics, where (synchronized) are introduced through which each agent plays a fixed action for the duration of the exploration phase. If such exploration phases are large in duration (as required by the results in @cite_19 ), this may reduce the robustness of the dynamics to dynamic changes in the environment (e.g., changes in the utility function). One reason that such robustness analysis is currently not possible in this class of dynamics is the fact that decisions are taken directly based on the measured performances (e.g., by comparing the currently measured performance with the benchmark performance in @cite_19 ).
@cite_19
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Abstract Recent advances in convex optimization have led to new strides in the phase retrieval problem over finite-dimensional vector spaces. However, certain fundamental questions remain: What sorts of measurement vectors uniquely determine every signal up to a global phase factor, and how many are needed to do so? Furthermore, which measurement ensembles yield stability? This paper presents several results that address each of these questions. We begin by characterizing injectivity, and we identify that the complement property is indeed a necessary condition in the complex case. We then pose a conjecture that 4 M − 4 generic measurement vectors are both necessary and sufficient for injectivity in M dimensions, and we prove this conjecture in the special cases where M = 2 , 3 . Next, we shift our attention to stability, both in the worst and average cases. Here, we characterize worst-case stability in the real case by introducing a numerical version of the complement property. This new property bears some resemblance to the restricted isometry property of compressed sensing and can be used to derive a sharp lower Lipschitz bound on the intensity measurement mapping. Localized frames are shown to lack this property (suggesting instability), whereas Gaussian random measurements are shown to satisfy this property with high probability. We conclude by presenting results that use a stochastic noise model in both the real and complex cases, and we leverage Cramer–Rao lower bounds to identify stability with stronger versions of the injectivity characterizations. In this paper, we consider a class of restless multiarmed bandit processes (RMABs) that arises in dynamic multichannel access, user server scheduling, and optimal activation in multiagent systems. For this class of RMABs, we establish the indexability and obtain Whittle index in closed form for both discounted and average reward criteria. These results lead to a direct implementation of Whittle index policy with remarkably low complexity. When arms are stochastically identical, we show that Whittle index policy is optimal under certain conditions. Furthermore, it has a semiuniversal structure that obviates the need to know the Markov transition probabilities. The optimality and the semiuniversal structure result from the equivalence between Whittle index policy and the myopic policy established in this work. For nonidentical arms, we develop efficient algorithms for computing a performance upper bound given by Lagrangian relaxation. The tightness of the upper bound and the near-optimal performance of Whittle index policy are illustrated with simulation examples. In this paper, the problem of safe exploration in the active learning context is considered. Safe exploration is especially important for data sampling from technical and industrial systems, e.g. combustion engines and gas turbines, where critical and unsafe measurements need to be avoided. The objective is to learn data-based regression models from such technical systems using a limited budget of measured, i.e. labelled, points while ensuring that critical regions of the considered systems are avoided during measurements. We propose an approach for learning such models and exploring new data regions based on Gaussian processes GP's. In particular, we employ a problem specific GP classifier to identify safe and unsafe regions, while using a differential entropy criterion for exploring relevant data regions. A theoretical analysis is shown for the proposed algorithm, where we provide an upper bound for the probability of failure. To demonstrate the efficiency and robustness of our safe exploration scheme in the active learning setting, we test the approach on a policy exploration task for the inverse pendulum hold up problem. Applications of compressed sensing motivate the possibility of using different operators to encode and decode a signal of interest. Since it is clear that the operators cannot be too different, we can view the discrepancy between the two matrices as a perturbation. The stability of l 1 -minimization and greedy algorithms to recover the signal in the presence of additive noise is by now well-known. Recently however, work has been done to analyze these methods with noise in the measurement matrix, which generates a multiplicative noise term. This new framework of generalized perturbations (i.e., both additive and multiplicative noise) extends the prior work on stable signal recovery from incomplete and inaccurate measurements of Candes, Romberg and Tao using Basis Pursuit (BP), and of Needell and Tropp using Compressive Sampling Matching Pursuit (CoSaMP). We show, under reasonable assumptions, that the stability of the reconstructed signal by both BP and CoSaMP is limited by the noise level in the observation. Our analysis extends easily to arbitrary greedy methods.