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Stability of stochastic approximation under verifiable conditions

机译：可验证条件下的随机逼近稳定性

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In this paper we address the problem of the stability and convergence of the stochastic approximation proceduretheta(n+1) = theta(n) + gamma(n+1)[h(theta(n)) + xi(n+1)].The stability of such sequences {theta(n)} is known to heavily rely on the behavior of the mean field h at the boundary of the parameter set and the magnitude of the stepsizes used. The conditions typically required to ensure convergence, and in particular the boundedness or stability of {theta(n)}, are either too difficult to check in practice or not satisfied at all. This is the case even for very simple models. The most popular technique for circumventing the stability problem consists of constraining {theta(n)} to a compact subset K in the parameter space. This is obviously not a satisfactory solution, as the choice of K is a delicate one. In this paper we first prove a "deterministic" stability result, which relies on simple conditions on the sequences {xi(n)} and {gamma(n)}. We then propose and analyze an algorithm based on projections on adaptive truncation sets, which ensures that the aforementioned conditions required for stability are satisfied. We focus in particular on the case where {xi(n)} is a so-called Markov state-dependent noise. We establish both the stability and convergence with probability 1 (w.p.1) of the algorithm under a set of simple and verifiable assumptions. We illustrate our results with an example related to adaptive Markov chain Monte Carlo algorithms.
机译：在本文中，我们解决了随机逼近过程theta（n + 1）= theta（n）+ gamma（n + 1）[h（theta（n））+ xi（n + 1）]的稳定性和收敛性的问题已知这样的序列{theta（n）}的稳定性在很大程度上取决于在参数集边界处的平均场h的行为以及所用步长的大小。确保收敛通常需要的条件，尤其是{theta（n）}的有界性或稳定性，在实践中太难检查或根本无法满足。即使是非常简单的模型也是如此。避免稳定性问题的最流行技术是将{theta（n）}约束到参数空间中的紧凑子集K。显然，这不是令人满意的解决方案，因为选择K是一个微妙的选择。在本文中，我们首先证明了“确定性”稳定性结果，该结果取决于序列{xi（n）}和{gamma（n）}上的简单条件。然后，我们提出并分析了基于自适应截断集投影的算法，该算法可确保满足稳定性所需的上述条件。我们特别关注{xi（n）}是所谓的依赖于马尔可夫状态的噪声的情况。在一组简单且可验证的假设下，我们以算法的概率1（w.p.1）建立稳定性和收敛性。我们以与自适应马尔可夫链蒙特卡洛算法相关的示例说明我们的结果。

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