We propose a Langevin diffusion-based algorithm for non-convex optimization and sampling on a product manifold of spheres. Under a logarithmic Sobolev inequality, we establish a guarantee for finite iteration convergence to the Gibbs distribution in terms of Kullback--Leibler divergence. We show that with an appropriate temperature choice, the suboptimality gap to the global minimum is guaranteed to be arbitrarily small with high probability. As an application, we consider the Burer--Monteiro approach for solving a semidefinite program (SDP) with diagonal constraints, and analyze the proposed Langevin algorithm for optimizing the non-convex objective. In particular, we establish a logarithmic Sobolev inequality for the Burer--Monteiro problem when there are no spurious local minima, but under the presence saddle points. Combining the results, we then provide a global optimality guarantee for the SDP and the Max-Cut problem. More precisely, we show that the Langevin algorithm achieves $\epsilon$ accuracy with high probability in $\widetilde{\Omega}( \epsilon^{-5} )$ iterations.

翻译：我们提出一种基于朗之万扩散的算法，用于在球面乘积流形上进行非凸优化与采样。在满足对数索博列夫不等式的前提下，我们建立了有限迭代收敛至吉布斯分布的库尔贝克-莱布勒散度保证。研究表明，通过恰当的温度选择，算法能以高概率保证与全局最优解的次优性间隙任意小。作为应用，我们考虑采用Burer-Monteiro方法求解对角约束半定规划，并分析用于优化非凸目标函数的所提朗之万算法。特别地，当不存在虚假局部极小值但存在鞍点时，我们建立了Burer-Monteiro问题的对数索博列夫不等式。结合这些结果，我们进而提供了半定规划及最大割问题的全局最优性保证。具体而言，我们证明朗之万算法在$\widetilde{\Omega}( \epsilon^{-5} )$次迭代内以高概率达到$\epsilon$精度。