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Register a new userFinite-Time Analysis of Asynchronous Stochastic Approximation and $Q$-Learning
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Guannan Qu
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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We consider a general asynchronous Stochastic Approximation (SA) scheme featuring a weighted infinity-norm contractive operator, and prove a bound on its finite-time convergence rate on a single trajectory. Additionally, we specialize the result to asynchronous $Q$-learning. The resulting bound matches the sharpest available bound for synchronous $Q$-learning, and improves over previous known bounds for asynchronous $Q$-learning.
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Motivated by broad applications in reinforcement learning and machine learning, this paper considers the popular stochastic gradient descent (SGD) when the gradients of the underlying objective function are sampled from Markov processes. This Markov sampling leads to the gradient samples being biased and not independent. The existing results for the convergence of SGD under Markov randomness are often established under the assumptions on the boundedness of either the iterates or the gradient samples. Our main focus is to study the finite-time convergence of SGD for different types of objective functions, without requiring these assumptions. We show that SGD converges nearly at the same rate with Markovian gradient samples as with independent gradient samples. The only difference is a logarithmic factor that accounts for the mixing time of the Markov chain.
Although Q-learning is one of the most successful algorithms for finding the best action-value function (and thus the optimal policy) in reinforcement learning, its implementation often suffers from large overestimation of Q-function values incurred by random sampling. The double Q-learning algorithm proposed in~citet{hasselt2010double} overcomes such an overestimation issue by randomly switching the update between two Q-estimators, and has thus gained significant popularity in practice. However, the theoretical understanding of double Q-learning is rather limited. So far only the asymptotic convergence has been established, which does not characterize how fast the algorithm converges. In this paper, we provide the first non-asymptotic (i.e., finite-time) analysis for double Q-learning. We show that both synchronous and asynchronous double Q-learning are guaranteed to converge to an $epsilon$-accurate neighborhood of the global optimum by taking $tilde{Omega}left(left( frac{1}{(1-gamma)^6epsilon^2}right)^{frac{1}{omega}} +left(frac{1}{1-gamma}right)^{frac{1}{1-omega}}right)$ iterations, where $omegain(0,1)$ is the decay parameter of the learning rate, and $gamma$ is the discount factor. Our analysis develops novel techniques to derive finite-time bounds on the difference between two inter-connected stochastic processes, which is new to the literature of stochastic approximation.
Stochastic approximation, a data-driven approach for finding the fixed point of an unknown operator, provides a unified framework for treating many problems in stochastic optimization and reinforcement learning. Motivated by a growing interest in multi-agent and multi-task learning, we consider in this paper a decentralized variant of stochastic approximation. A network of agents, each with their own unknown operator and data observations, cooperatively find the fixed point of the aggregate operator. The agents work by running a local stochastic approximation algorithm using noisy samples from their operators while averaging their iterates with their neighbors on a decentralized communication graph. Our main contribution provides a finite-time analysis of this decentralized stochastic approximation algorithm and characterizes the impacts of the underlying communication topology between agents. Our model for the data observed at each agent is that it is sampled from a Markov processes; this lack of independence makes the iterates biased and (potentially) unbounded. Under mild assumptions on the Markov processes, we show that the convergence rate of the proposed methods is essentially the same as if the samples were independent, differing only by a log factor that represents the mixing time of the Markov process. We also present applications of the proposed method on a number of interesting learning problems in multi-agent systems, including a decentralized variant of Q-learning for solving multi-task reinforcement learning.
In machine learning and optimization community there are two main approaches for convex risk minimization problem, namely, the Stochastic Approximation (SA) and the Sample Average Approximation (SAA). In terms of oracle complexity (required number of stochastic gradient evaluations), both approaches are considered equivalent on average (up to a logarithmic factor). The total complexity depends on the specific problem, however, starting from work cite{nemirovski2009robust} it was generally accepted that the SA is better than the SAA. Nevertheless, in case of large-scale problems SA may run out of memory as storing all data on one machine and organizing online access to it can be impossible without communications with other machines. SAA in contradistinction to SA allows parallel/distributed calculations. In this paper, we shed new light on the comparison of SA and SAA for particular problem of calculating the population (regularized) Wasserstein barycenter of discrete measures. The conclusion is valid even for non-parallel (non-decentralized) setup.
Asynchronous stochastic approximations (SAs) are an important class of model-free algorithms, tools and techniques that are popular in multi-agent and distributed control scenarios. To counter Bellmans 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 paper, 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 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 are presented that support the theory.
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