We study multiproposal Markov chain Monte Carlo algorithms, such as Multiple-try or generalised Metropolis-Hastings schemes, which have recently received renewed attention due to their amenability to parallel computing. First, we prove that no multiproposal scheme can speed-up convergence relative to the corresponding single proposal scheme by more than a factor of $K$, where $K$ denotes the number of proposals at each iteration. This result applies to arbitrary target distributions and it implies that serial multiproposal implementations are always less efficient than single proposal ones. Secondly, we consider log-concave distributions over Euclidean spaces, proving that, in this case, the speed-up is at most logarithmic in $K$, which implies that even parallel multiproposal implementations are fundamentally limited in the computational gain they can offer. Crucially, our results apply to arbitrary multiproposal schemes and purely rely on the two-step structure of the associated kernels (i.e. first generate $K$ candidate points, then select one among those). Our theoretical findings are validated through numerical simulations.

翻译：我们研究多提案马尔可夫链蒙特卡洛算法，例如多重尝试或广义Metropolis-Hastings方案，这些算法由于适合并行计算而近期重新受到关注。首先，我们证明任何多提案方案相对于相应的单提案方案，其收敛速度的提升不可能超过因子$K$，其中$K$表示每次迭代的提案数量。该结果适用于任意目标分布，且意味着串行多提案实现的效率始终低于单提案实现。其次，我们考虑欧几里得空间上的对数凹分布，证明在此情况下，加速效果至多为$K$的对数级，这表明即使并行多提案实现所能提供的计算增益也存在根本性限制。关键的是，我们的结果适用于任意多提案方案，且仅依赖于相关核的两步结构（即首先生成$K$个候选点，随后从中选择一个）。我们的理论发现通过数值模拟得到了验证。