Replica exchange stochastic gradient Langevin dynamics (reSGLD) is an effective sampler for non-convex learning in large-scale datasets. However, the simulation may encounter stagnation issues when the high-temperature chain delves too deeply into the distribution tails. To tackle this issue, we propose reflected reSGLD (r2SGLD): an algorithm tailored for constrained non-convex exploration by utilizing reflection steps within a bounded domain. Theoretically, we observe that reducing the diameter of the domain enhances mixing rates, exhibiting a \emph{quadratic} behavior. Empirically, we test its performance through extensive experiments, including identifying dynamical systems with physical constraints, simulations of constrained multi-modal distributions, and image classification tasks. The theoretical and empirical findings highlight the crucial role of constrained exploration in improving the simulation efficiency.

翻译：副本交换随机梯度朗之万动力学（reSGLD）是一种在大规模数据集中进行非凸学习的有效采样方法。然而，当高温链过度深入分布尾部时，模拟可能会遇到停滞问题。为解决这一问题，我们提出了反射reSGLD（r2SGLD）：一种通过在有界域内利用反射步骤专为约束非凸探索设计的算法。理论上，我们观察到减小域直径可提高混合速率，展现出二次方行为。实验上，我们通过大量实验测试其性能，包括具有物理约束的动力系统识别、约束多模态分布模拟以及图像分类任务。理论和实验发现强调了约束探索在提高模拟效率中的关键作用。