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arXiv 1712.09652cs.LGmath.OC
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离策略学习的几种基于时间差的学习算法的收敛性

On Convergence of some Gradient-based Temporal-Differences Algorithms for Off-Policy Learning

University of Alberta(阿尔伯塔大学)

Huizhen Yu

摘要:

我们认为关闭政策的时间差(TD)学习方法的政策评估有限空间和折扣奖励标准的马尔可夫决策过程中,我们提出了一系列的收敛结果的几个基于梯度的TD算法与线性函数逼近。我们分析的算法包括:(i)两种基本形式的双时间尺度的基于梯度的TD算法,我们称之为GTD和最小化均方投影贝尔曼误差使用随机梯度下降;(ii)他们的“鲁棒”有偏的变种;(iii)他们的镜像下降版本结合镜像下降的想法与TD学习;和(iv)一个单一的时间尺度版本的GTD解决极大极小问题制定近似的政策评估。我们得出收敛结果的三种类型的步长:恒定的步长,缓慢减少步长,以及标准类型的平方可和的条件下减少步长。对于前两种类型的步长,我们应用随机逼近理论的弱收敛方法来描述算法的渐近行为,对于标准类型的步长,我们分析了算法的行为相对于一个更强的模式的收敛,几乎必然收敛。我们的收敛结果是上述TD算法与三个一般的方式设置其$λ$参数:(i)状态依赖的$λ$,(ii)最近提出的计划,使用历史依赖的$λ$,以保持有界的资格迹的算法,同时允许相对较大的值的$λ$ ;以及(iii)设置$λ$参数的复合方案,其结合了前述两种方案,并允许更广泛的广义Bellman算子类用于TD方法的近似策略评估。

英文摘要:

We consider off-policy temporal-difference (TD) learning methods for policy evaluation in Markov decision processes with finite spaces and discounted reward criteria, and we present a collection of convergence results for several gradient-based TD algorithms with linear function approximation. The algorithms we analyze include: (i) two basic forms of two-time-scale gradient-based TD algorithms, which we call GTD and which minimize the mean squared projected Bellman error using stochastic gradient-descent; (ii) their "robustified" biased variants; (iii) their mirror-descent versions which combine the mirror-descent idea with TD learning; and (iv) a single-time-scale version of GTD that solves minimax problems formulated for approximate policy evaluation. We derive convergence results for three types of stepsizes: constant stepsize, slowly diminishing stepsize, as well as the standard type of diminishing stepsize with a square-summable condition. For the first two types of stepsizes, we apply the weak convergence method from stochastic approximation theory to characterize the asymptotic behavior of the algorithms, and for the standard type of stepsize, we analyze the algorithmic behavior with respect to a stronger mode of convergence, almost sure convergence. Our convergence results are for the aforementioned TD algorithms with three general ways of setting their $λ$-parameters: (i) state-dependent $λ$; (ii) a recently proposed scheme of using history-dependent $λ$ to keep the eligibility traces of the algorithms bounded while allowing for relatively large values of $λ$; and (iii) a composite scheme of setting the $λ$-parameters that combines the preceding two schemes and allows a broader class of generalized Bellman operators to be used for approximate policy evaluation with TD methods.

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