电流涨落在不可逆采样器中的作用
Role of current fluctuations in nonreversible samplers
摘要：
已知的是，与具有相同平稳分布的可逆过程相比，打破细致平衡条件的不可逆马尔可夫过程的分布更快地收敛到平稳分布。这在实践中用于加速马尔可夫链蒙特卡罗算法，该算法通过添加不可逆跃迁或非梯度漂移项来对吉布斯分布进行采样。细致平衡的打破也加速了经验估计在长时间极限下向其遍历期望的收敛。在这里，我们给出了一个物理解释的第二种形式的加速度与经验估计的波动，使用水平2.5的大偏差，其特点的可能性密度和当前的波动在马尔可夫过程中的电流。专注于扩散过程中，我们表明，有加速收敛，因为估计波动出现在一般与电流波动，导致增加了大的偏差成本相比，可逆的情况下，这表明没有电流。我们研究了当前的波动最有可能出现在一个给定的估计波动，并提供加速度上的界限，根据近似的电流。我们用二维的Ornstein-Uhlenbeck过程和圆上的布朗运动来说明这些结果。
英文摘要：
It is known that the distribution of nonreversible Markov processes breaking the detailed balance condition converges faster to the stationary distribution compared to reversible processes having the same stationary distribution. This is used in practice to accelerate Markov chain Monte Carlo algorithms that sample the Gibbs distribution by adding nonreversible transitions or non-gradient drift terms. The breaking of detailed balance also accelerates the convergence of empirical estimators to their ergodic expectation in the long-time limit. Here, we give a physical interpretation of this second form of acceleration in terms of currents associated with the fluctuations of empirical estimators using the level 2.5 of large deviations, which characterises the likelihood of density and current fluctuations in Markov processes. Focusing on diffusion processes, we show that there is accelerated convergence because estimator fluctuations arise in general with current fluctuations, leading to an added large deviation cost compared to the reversible case, which shows no current. We study the current fluctuation most likely to arise in conjunction with a given estimator fluctuation and provide bounds on the acceleration, based on approximations of this current. We illustrate these results for the Ornstein-Uhlenbeck process in two dimensions and the Brownian motion on the circle.
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