We study the necessity of predictive information in a class of queueing admission control problems, where a system manager is allowed to divert incoming jobs up to a fixed rate, in order to minimize the queueing delay experienced by the admitted jobs. (2014) show that the system's delay performance can be significantly improved by having access to future information in the form of a lookahead window, during which the times of future arrivals and services are revealed. They prove that, while delay under an optimal online policy diverges to infinity in the heavy-traffic regime, it can stay bounded by making use of future information. However, the diversion polices of (2014) require the length of the lookahead window to grow to infinity at a non-trivial rate in the heavy-traffic regime, and it remained open whether substantial performance improvement could still be achieved with less future information. We resolve this question to a large extent by establishing an asymptotically tight lower bound on how much future information is necessary to achieve superior performance, which matches the upper bound of (2014) up to a constant multiplicative factor. Our result hence demonstrates that the system's heavy-traffic delay performance is highly sensitive to the amount of future information available. Our proof is based on analyzing certain excursion probabilities of the input sample paths, and exploiting a connection between a policy's diversion decisions and subsequent server idling, which may be of independent interest for related dynamic resource allocation problems.
Unfortunately, these arguments employed in the Markov setting do not seem to carry over easily when the lookahead window is taken into account. While our setting can still be cast as an MDP by incorporating the lookahead window into the state space, the structure of the state space is now considerably more complex (and increasingly so, as @math ), and it is not so clear as to whether any monotonicity property continues to hold. Our proof techniques circumvent this complexity by focusing on the macroscopic'' sample-path characteristics of the system, instead of the more refined details of the Bellman equations. As a trade-off, our analysis is more coarse'' by nature, and it provides neither a characterization of the multiplicative in the delay scaling, nor a concrete diversion policy that achieves the lower bound of the necessary amount of future information (which, fortunately, has already been given in @cite_5 ).
@cite_5
2104050194
Consider a Markov decision process (MDP) that admits a set of state-action features, which can linearly express the process's probabilistic transition model. We propose a parametric Q-learning algorithm that finds an approximate-optimal policy using a sample size proportional to the feature dimension @math and invariant with respect to the size of the state space. To further improve its sample efficiency, we exploit the monotonicity property and intrinsic noise structure of the Bellman operator, provided the existence of anchor state-actions that imply implicit non-negativity in the feature space. We augment the algorithm using techniques of variance reduction, monotonicity preservation, and confidence bounds. It is proved to find a policy which is @math -optimal from any initial state with high probability using @math sample transitions for arbitrarily large-scale MDP with a discount factor @math . A matching information-theoretical lower bound is proved, confirming the sample optimality of the proposed method with respect to all parameters (up to polylog factors). Consider the problem of approximating the optimal policy of a Markov decision process (MDP) by sampling state transitions. In contrast to existing reinforcement learning methods that are based on successive approximations to the nonlinear Bellman equation, we propose a Primal-Dual @math Learning method in light of the linear duality between the value and policy. The @math learning method is model-free and makes primal-dual updates to the policy and value vectors as new data are revealed. For infinite-horizon undiscounted Markov decision process with finite state space @math and finite action space @math , the @math learning method finds an @math -optimal policy using the following number of sample transitions @math where @math is an upper bound of mixing times across all policies and @math is a parameter characterizing the range of stationary distributions across policies. The @math learning method also applies to the computational problem of MDP where the transition probabilities and rewards are explicitly given as the input. In the case where each state transition can be sampled in @math time, the @math learning method gives a sublinear-time algorithm for solving the averaged-reward MDP. For undiscounted reinforcement learning in Markov decision processes (MDPs) we consider the total regret of a learning algorithm with respect to an optimal policy. In order to describe the transition structure of an MDP we propose a new parameter: An MDP has diameter D if for any pair of states s,s' there is a policy which moves from s to s' in at most D steps (on average). We present a reinforcement learning algorithm with total regret O(DS√AT) after T steps for any unknown MDP with S states, A actions per state, and diameter D. A corresponding lower bound of Ω(√DSAT) on the total regret of any learning algorithm is given as well. These results are complemented by a sample complexity bound on the number of suboptimal steps taken by our algorithm. This bound can be used to achieve a (gap-dependent) regret bound that is logarithmic in T. Finally, we also consider a setting where the MDP is allowed to change a fixed number of l times. We present a modification of our algorithm that is able to deal with this setting and show a regret bound of O(l1 3T2 3DS√A). We study an admissions control problem, where a queue with service rate @math receives incoming jobs at rate @math , and the decision maker is allowed to redirect away jobs up to a rate of @math , with the objective of minimizing the time-average queue length. We show that the amount of information about the future has a significant impact on system performance, in the heavy-traffic regime. When the future is unknown, the optimal average queue length diverges at rate @math , as @math . In sharp contrast, when all future arrival and service times are revealed beforehand, the optimal average queue length converges to a finite constant, @math , as @math . We further show that the finite limit of @math can be achieved using only a finite lookahead window starting from the current time frame, whose length scales as @math , as @math . This leads to the conjecture of an interesting duality between queuing delay and the amount of information about the future. We propose a new approach to the problem of searching a space of policies for a Markov decision process (MDP) or a partially observable Markov decision process (POMDP), given a model. Our approach is based on the following observation: Any (PO)MDP can be transformed into an "equivalent" POMDP in which all state transitions (given the current state and action) are deterministic. This reduces the general problem of policy search to one in which we need only consider POMDPs with deterministic transitions. We give a natural way of estimating the value of all policies in these transformed POMDPs. Policy search is then simply performed by searching for a policy with high estimated value. We also establish conditions under which our value estimates will be good, recovering theoretical results similar to those of Kearns, Mansour and Ng (1999), but with "sample complexity" bounds that have only a polynomial rather than exponential dependence on the horizon time. Our method applies to arbitrary POMDPs, including ones with infinite state and action spaces. We also present empirical results for our approach on a small discrete problem, and on a complex continuous state continuous action problem involving learning to ride a bicycle. A critical issue for the application of Markov decision processes (MDPs) to realistic problems is how the complexity of planning scales with the size of the MDP. In stochastic environments with very large or infinite state spaces, traditional planning and reinforcement learning algorithms may be inapplicable, since their running time typically grows linearly with the state space size in the worst case. In this paper we present a new algorithm that, given only a generative model (a natural and common type of simulator) for an arbitrary MDP, performs on-line, near-optimal planning with a per-state running time that has no dependence on the number of states. The running time is exponential in the horizon time (which depends only on the discount factor γ and the desired degree of approximation to the optimal policy). Our algorithm thus provides a different complexity trade-off than classical algorithms such as value iteration—rather than scaling linearly in both horizon time and state space size, our running time trades an exponential dependence on the former in exchange for no dependence on the latter. Our algorithm is based on the idea of sparse sampling. We prove that a randomly sampled look-ahead tree that covers only a vanishing fraction of the full look-ahead tree nevertheless suffices to compute near-optimal actions from any state of an MDP. Practical implementations of the algorithm are discussed, and we draw ties to our related recent results on finding a near-best strategy from a given class of strategies in very large partially observable MDPs (Kearns, Mansour, & Ng. Neural information processing systems 13, to appear). We first consider a multi-user, single-hop wireless network with arbitrarily varying (and possibly non-ergodic) arrivals and channels. We design an opportunistic scheduling algorithm that guarantees all sessions have a bounded worst case delay. The algorithm has no knowledge of the future, but yields throughput-utility that is close to (or better than) that of a T-slot lookahead policy that makes “ideal” decisions based on perfect knowledge up to T slots into the future. We then extend the algorithm to treat worst case delay guarantees in multi-hop networks. Our analysis uses a sample-path version of Lyapunov optimization together with a novel virtual queue structure. We introduce a class of MPDs which greatly simplify Reinforcement Learning. They have discrete state spaces and continuous control spaces. The controls have the effect of rescaling the transition probabilities of an underlying Markov chain. A control cost penalizing KL divergence between controlled and uncontrolled transition probabilities makes the minimization problem convex, and allows analytical computation of the optimal controls given the optimal value function. An exponential transformation of the optimal value function makes the minimized Bellman equation linear. Apart from their theoretical significance, the new MDPs enable efficient approximations to traditional MDPs. Shortest path problems are approximated to arbitrary precision with largest eigenvalue problems, yielding an O (n) algorithm. Accurate approximations to generic MDPs are obtained via continuous embedding reminiscent of LP relaxation in integer programming. Off-policy learning of the optimal value function is possible without need for state-action values; the new algorithm (Z-learning) outperforms Q-learning. In this paper we consider the problem of computing an ϵ-optimal policy of a discounted Markov Decision Process (DMDP) provided we can only access its transition function through a generative sampling model that given any state-action pair samples from the transition function in O(1) time. Given such a DMDP with states , actions , discount factor γ∈(0,1), and rewards in range [0,1] we provide an algorithm which computes an ϵ-optimal policy with probability 1−δ where both the run time spent and number of sample taken is upper bounded by O[ | || | (1−γ)3ϵ2 log( | || | (1−γ)δϵ )log( 1 (1−γ)ϵ )] . For fixed values of ϵ, this improves upon the previous best known bounds by a factor of (1−γ)−1 and matches the sample complexity lower bounds proved in azar2013minimax up to logarithmic factors. We also extend our method to computing ϵ-optimal policies for finite-horizon MDP with a generative model and provide a nearly matching sample complexity lower bound. We present a reinforcement learning game player that can interact with a General Game Playing system and transfer knowledge learned in one game to expedite learning in many other games. We use the technique of value-function transfer where general features are extracted from the state space of a previous game and matched with the completely different state space of a new game. To capture the underlying similarity of vastly disparate state spaces arising from different games, we use a game-tree lookahead structure for features. We show that such feature-based value function transfer learns superior policies faster than a reinforcement learning agent that does not use knowledge transfer. Furthermore, knowledge transfer using lookahead features can capture opponent-specific value-functions, i.e. can exploit an opponent's weaknesses to learn faster than a reinforcement learner that uses lookahead with minimax (pessimistic) search against the same opponent. Lookahead search is perhaps the most natural and widely used game playing strategy. Given the practical importance of the method, the aim of this paper is to provide a theoretical performance examination of lookahead search in a wide variety of applications. To determine a strategy play using lookahead search, each agent predicts multiple levels of possible re-actions to her move (via the use of a search tree), and then chooses the play that optimizes her future payoff accounting for these re-actions. There are several choices of optimization function the agents can choose, where the most appropriate choice of function will depend on the specifics of the actual game - we illustrate this in our examples. Furthermore, the type of search tree chosen by computationally-constrained agent can vary. We focus on the case where agents can evaluate only a bounded number, k, of moves into the future. That is, we use depth k search trees and call this approach k-lookahead search. We apply our method in five well-known settings: industrial organization (Cournot's model); AdWord auctions; congestion games; valid-utility games and basic-utility games; cost-sharing network design games. We consider two questions. First, what is the expected social quality of outcome when agents apply lookahead search? Second, what interactive behaviours can be exhibited when players use lookahead search? We demonstrate how the answer depends on the game played. Although many real-world stochastic planning problems are more naturally formulated by hybrid models with both discrete and continuous variables, current state-of-the-art methods cannot adequately address these problems. We present the first framework that can exploit problem structure for modeling and solving hybrid problems efficiently. We formulate these problems as hybrid Markov decision processes (MDPs with continuous and discrete state and action variables), which we assume can be represented in a factored way using a hybrid dynamic Bayesian network (hybrid DBN). This formulation also allows us to apply our methods to collaborative multiagent settings. We present a new linear program approximation method that exploits the structure of the hybrid MDP and lets us compute approximate value functions more efficiently. In particular, we describe a new factored discretization of continuous variables that avoids the exponential blow-up of traditional approaches. We provide theoretical bounds on the quality of such an approximation and on its scale-up potential. We support our theoretical arguments with experiments on a set of control problems with up to 28-dimensional continuous state space and 22-dimensional action space. Data analysis often concerns not only the space where data come from, but also various types of maps attached to data. In recent years, several related structures have been used to study maps on data, including Reeb spaces, mappers and multiscale mappers. The construction of these structures also relies on the so-called of a cover of the domain. In this paper, we aim to analyze the topological information encoded in these structures in order to provide better understanding of these structures and facilitate their practical usage. More specifically, we show that the one-dimensional homology of the nerve complex @math of a path-connected cover @math of a domain @math cannot be richer than that of the domain @math itself. Intuitively, this result means that no new @math -homology class can be "created" under a natural map from @math to the nerve complex @math . Equipping @math with a pseudometric @math , we further refine this result and characterize the classes of @math that may survive in the nerve complex using the notion of of the covering elements in @math . These fundamental results about nerve complexes then lead to an analysis of the @math -homology of Reeb spaces, mappers and multiscale mappers. The analysis of @math -homology groups unfortunately does not extend to higher dimensions. Nevertheless, by using a map-induced metric, establishing a Gromov-Hausdorff convergence result between mappers and the domain, and interleaving relevant modules, we can still analyze the persistent homology groups of (multiscale) mappers to establish a connection to Reeb spaces. We initiate the study of numerical linear algebra in the sliding window model, where only the most recent @math updates in the data stream form the underlying set. Although most existing work in the sliding window model uses the smooth histogram framework, most interesting linear-algebraic problems are not smooth; we show that the spectral norm, vector induced matrix norms, generalized regression, and low-rank approximation are not amenable for the smooth histogram framework. To overcome this challenge, we first give a deterministic algorithm that achieves spectral approximation in the sliding window model that can be viewed as a generalization of smooth histograms, using the Loewner ordering of PSD matrices. We then give algorithms for both spectral approximation and low-rank approximation that are space-optimal up to polylogarithmic factors. Our algorithms are based on a new notion of "reverse online" leverage scores that account for both how unique and how recent a row is, while preserving sparsity so that both our algorithms run in input sparsity runtime, up to lower order factors. We show that our techniques have applications to linear-algebraic problems in other settings. Specifically, we show that our analysis immediately implies an algorithm for low-rank approximation in the online setting that is space-optimal up to logarithmic factors, as well as nearly input sparsity time. We show our deterministic spectral approximation algorithm can be used to handle @math spectral approximation in the sliding window model under a certain assumption on the bit complexity of the entries. Finally, we show that our downsampling framework can be applied to the problem of approximate matrix multiplication and provide upper and lower bounds that are tight up to @math factors. We study the problem of learning near-optimal behavior in finite Markov Decision Processes (MDPs) with a polynomial number of samples. These "PAC-MDP" algorithms include the well-known E3 and R-MAX algorithms as well as the more recent Delayed Q-learning algorithm. We summarize the current state-of-the-art by presenting bounds for the problem in a unified theoretical framework. A more refined analysis for upper and lower bounds is presented to yield insight into the differences between the model-free Delayed Q-learning and the model-based R-MAX. Approximate linear programming (ALP) represents one of the major algorithmic families to solve large-scale Markov decision processes (MDP). In this work, we study a primal-dual formulation of the ALP, and develop a scalable, model-free algorithm called bilinear @math learning for reinforcement learning when a sampling oracle is provided. This algorithm enjoys a number of advantages. First, it adopts (bi)linear models to represent the high-dimensional value function and state-action distributions, using given state and action features. Its run-time complexity depends on the number of features, not the size of the underlying MDPs. Second, it operates in a fully online fashion without having to store any sample, thus having minimal memory footprint. Third, we prove that it is sample-efficient, solving for the optimal policy to high precision with a sample complexity linear in the dimension of the parameter space. We consider the problem of learning an unknown Markov Decision Process (MDP) that is weakly communicating in the infinite horizon setting. We propose a Thompson Sampling-based reinforcement learning algorithm with dynamic episodes (TSDE). At the beginning of each episode, the algorithm generates a sample from the posterior distribution over the unknown model parameters. It then follows the optimal stationary policy for the sampled model for the rest of the episode. The duration of each episode is dynamically determined by two stopping criteria. The first stopping criterion controls the growth rate of episode length. The second stopping criterion happens when the number of visits to any state-action pair is doubled. We establish @math bounds on expected regret under a Bayesian setting, where @math and @math are the sizes of the state and action spaces, @math is time, and @math is the bound of the span. This regret bound matches the best available bound for weakly communicating MDPs. Numerical results show it to perform better than existing algorithms for infinite horizon MDPs. In the context of standard Markov decision processes (MDPs), the connection between Dynamic Program (DP) and Linear Program (LP) is well understood and is well established under sufficiently general conditions. LP based approach facilitates solving the constrained MDPs. Multiplicative or Risk sensitive MDPs, introduced to control the fluctuations variations around the expected value, are relatively less studied objects. DP equations are considerably well understood even in the context of Risk MDPs, however the LP connection is not known. We consider a finite horizon risk MDP problem and establish the connections between the DP and LP approaches. We augment the state space with a suitable component, to obtain the optimal policies for constrained risk MDPs. We apply this results to a server selection problem in Ber M K K queues, with a constraint on the utilization of the fast server. We discuss some interesting structural properties of the risk optimal policies. Partially Observable Markov Decision Processes (POMDPs) provide a rich framework for sequential decision-making under uncertainty in stochastic domains. However, solving a POMDP is often intractable except for small problems due to their complexity. Here, we focus on online approaches that alleviate the computational complexity by computing good local policies at each decision step during the execution. Online algorithms generally consist of a lookahead search to find the best action to execute at each time step in an environment. Our objectives here are to survey the various existing online POMDP methods, analyze their properties and discuss their advantages and disadvantages; and to thoroughly evaluate these online approaches in different environments under various metrics (return, error bound reduction, lower bound improvement). Our experimental results indicate that state-of-the-art online heuristic search methods can handle large POMDP domains efficiently. We present a general framework for applying machine-learning algorithms to the verification of Markov decision processes (MDPs). The primary goal of these techniques is to improve performance by avoiding an exhaustive exploration of the state space. Our framework focuses on probabilistic reachability, which is a core property for verification, and is illustrated through two distinct instantiations. The first assumes that full knowledge of the MDP is available, and performs a heuristic-driven partial exploration of the model, yielding precise lower and upper bounds on the required probability. The second tackles the case where we may only sample the MDP, and yields probabilistic guarantees, again in terms of both the lower and upper bounds, which provides efficient stopping criteria for the approximation. The latter is the first extension of statistical model checking for unbounded properties in MDPs. In contrast with other related techniques, our approach is not restricted to time-bounded (finite-horizon) or discounted properties, nor does it assume any particular properties of the MDP. We also show how our methods extend to LTL objectives. We present experimental results showing the performance of our framework on several examples.