Ergodicity and excess switching rate of the Zig-Zag process

马尔可夫链蒙特卡罗（MCMC）已经成为贝叶斯推理的主要计算动力在过去的几十年里。由于计算标准的不断发展，以及互联网和全球化的出现，导致大量数据涌入并增加了贝叶斯模型的复杂性，MCMC文献在方法论和理论上都有了巨大的增长，专注于构建有效的算法来处理棘手的问题，高维和大数据。传统的MCMC算法的重点是构造一个可逆的，离散时间马尔可夫链承认所需的不变测度。然而，最近，连续时间，不可逆的方法引起了MCMC社区的极大兴趣，作为标准Metropolis-Hastings算法的有效和可扩展的替代方案（Hastings，1970）。特别地，Zig-Zag工艺（Bierkens等人，2019）为构建更高效的采样器提供了很大的希望。这部分是由于其不可逆的性质和被完全模拟的可能性不同于其他基于扩散的采样方法。部分原因是它能够通过允许子采样策略来减少大数据环境中的计算负担，并且仍然针对所需的不变度量，使其具有可扩展性。在某些情况下，使用Zig-Zag的子采样以及控制变量的想法已经被经验地证明可以实现超效率：以不随数据大小增加的计算成本生成基本上独立的样本。然而，需要注意的是，子采样通过引入过量的切换速率而在该过程中引起更多的扩散。这导致该过程更频繁地翻转其速度，减缓其混合。例如，一个典型的之字形开始时是尾部，在第一次翻转之前直接到达中心。然而，由于切换率过大，在到达中心之前，它将翻转不止一次，因此将花费更多的时间来探索目标测量。因此，算法需要运行更长的时间，即使每次迭代的成本降低。因此，总体效率，如通过算法效率和其实现成本的总和来精确测量的，可能会降低。这挑战了Zig-Zag阿萨标准MCMC方法的可扩展替代方案的承诺。该项目的目的是研究和量化子采样对处理速度的影响。这种效果会因不同的目标分布而异。并通过这样做，评论其对贝叶斯计算的有用性。目前的方向是研究Zig-Zag在大型数据集上的行为，并研究当数据大小达到无穷大时的限制过程。这个问题，据我们所知，还没有在理论上研究的文献。我们相信这是一个有趣的问题，将在贝叶斯社区有很多实际意义。对于由于子采样而导致的过程减慢被发现是微不足道的情况，Zig-Zag可以提供高度可扩展和超高效的采样算法。该项目的动机是其在贝叶斯计算的可能影响，并将使用应用概率的工具进行。总的来说，它福尔斯研究理事会的职权范围，涵盖工程和物理科学。参考文献：Bierkens，J.，Fearnhead，P.，和Roberts，G. O.（2019年）。用于大数据贝叶斯分析的锯齿形过程和超高效采样。统计年鉴，47（3）：1288- 1320。K.（1970年）。用马尔可夫链的蒙特卡罗抽样方法及其应用。Biometrika，57（1）：97-109.

Markov Chain Monte Carlo (MCMC) has emerged as the main computational powerhouse of Bayesian inference over the last few decades. There has been - fuelled by the constant evolution of computational standards, and necessitated by the advent of internet and globalization, which has lead to a massive influx of data and increased complexity of Bayesian models - a tremendous growth, both methodological and theoretical, in MCMC literature focused on constructing efficient algorithms to deal with problems of intractability, high dimensionality and big data. Traditional MCMC algorithms focus on constructing a reversible, discrete-time Markov chain admitting a desired invariant measure. However, more recently, continuous-time, non-reversible methods have aroused much interest in the MCMC community as efficient and scalable alternative to standard Metropolis-Hastings algorithms (Hastings, 1970). In particular, the Zig-Zag process (Bierkens et al., 2019) offers much promise for the construction of more efficient samplers. This is partly due to its non-reversible nature and the possibility of being simulated exactly unlike other diffusion based sampling methods. And partly because of its ability to reduce computational burden in big data settings by admitting a sub-sampling strategy and still target the desired invariant measure making it scalable. Under some situations, sub-sampling with Zig-Zag, together with a control variate idea, has been shown empirically to achieve super-efficiency: generating essentially independent samples at a computational cost that does not increase with the data-size. The caveat, however, is that sub-sampling induces more diffusivity in the process by introducing an excess switching rate. This causes the process to flip its velocity more frequently, slowing down its mixing. For example, a canonical Zig-Zag started out in tails comes straight to the centre before its first flip. However, with an excess switching rate, it would flip more than once before reaching the centre and consequently will take more time to explore the target measure. As a result, the algorithm needs to be run for a longer time, even though at a reduced cost per iteration. Thus the overall efficiency, as measured heuristically by the total of algorithmic efficiency and its cost of implementation might reduce. This challenges the promise of Zig-Zag asa scalable alternative to standard MCMC methods. The aim of this project is to study and quantify the effect of sub-sampling on the speed of the process. This effect would vary for different target distributions. And by doing so, comment on its usefulness for Bayesian computation. The current direction is to investigate the behaviour of Zig-Zag for large datasets and to look at the limiting process as the data-size goes to infinity. This problem, to our knowledge, has not been studied theoretically yet in the literature. We believe it to be an interesting problem that will have lots of practical implications in Bayesian community. For situations where the slowing down of the process due to sub-sampling is found to be insignificant, Zig-Zag can provide with a highly scalable and a super-efficient sampling algorithm. The project is motivated by its possible implications in Bayesian computation and will be carried out using tools of applied probability. Overall, it falls within the remit of Research council which covers engineering and physical sciences.References:Bierkens, J., Fearnhead, P., and Roberts, G. O. (2019). The zig-zag process and super-efficient sampling for Bayesian analysis of big data. The Annals of Statistics, 47(3):1288-1320.Hastings, W. K. (1970). Monte Carlo sampling methods using Markov chains and their applications. Biometrika, 57(1):97-109.