论文标题
通过奇异扰动的深度放松受控的随机梯度下降
Deep Relaxation of Controlled Stochastic Gradient Descent via Singular Perturbations
论文作者
论文摘要
我们考虑了Chaudhari等人提出的随机微分方程的奇异扰动系统。 （res。Math。Sci。2018）通过均质化，在优化深神经网络中近似熵梯度下降。我们将其嵌入了一类更大的两尺度随机控制问题中，并依靠汉密尔顿 - 雅各比 - 贝尔曼方程的收敛结果，我们本人最近证明了无界数据（Esaim ControlOptim。Calc。var.var。2r.2r.2r.2r.2r.2r。2r.2r.2r.2r.2r.2r.2r.2r.2r.2r.2r.2r.2r.2r.2r.2r.2r.2r.2r.2r.2r.2r.2r.2r.var。2l.2r.2r。我们表明，值函数的限制本身就是通过扩展控制的有效控制问题的值函数，并且扰动系统的轨迹以适当的意义收敛到限制有效控制系统的轨迹。这些严格的结果提高了对Chaudhari等人使用的算法的收敛性的理解，以及将某些调谐参数建模为动态控制的可能的扩展。
We consider a singularly perturbed system of stochastic differential equations proposed by Chaudhari et al. (Res. Math. Sci. 2018) to approximate the Entropic Gradient Descent in the optimization of deep neural networks, via homogenisation. We embed it in a much larger class of two-scale stochastic control problems and rely on convergence results for Hamilton-Jacobi-Bellman equations with unbounded data proved recently by ourselves (ESAIM Control Optim. Calc. Var. 2023). We show that the limit of the value functions is itself the value function of an effective control problem with extended controls, and that the trajectories of the perturbed system converge in a suitable sense to the trajectories of the limiting effective control system. These rigorous results improve the understanding of the convergence of the algorithms used by Chaudhari et al., as well as of their possible extensions where some tuning parameters are modelled as dynamic controls.
