论文标题
误差界限和随机近似的应用，并具有未偿还增益
Error Bounds and Applications for Stochastic Approximation with Non-Decaying Gain
论文作者
论文摘要
这项工作分析了随机变化问题的随机近似算法。设置是在采样时间$τ_k$下最小化标量值损耗函数$ f_k（\ cdot）$，或在$τ_k$ in r^p $ in r^p $的情况下，定位vector-valued函数$ g_k（\ cdot）$的序列的根。可用的信息是仅在一个或两个设计点上评估的$ f_k（\ cdot）$或$ g_k（\ cdot）$的噪声浪费的观察结果。鉴于随机变化的随机近似设置，我们将随机近似算法应用于非销售收益，以便表示为$ \hatθ_k$表示的递归估计值可以保持其在跟踪时变的最佳最佳最佳表示为$θ_K^*$。 第3章提供了针对根平方错误$ \ sqrt {总体而言，边界适用于对时间变化的温和假设，对观察噪声和偏置项的适度限制。在第3章中建立了跟踪能力之后，我们还讨论了第4章中$ \hatθ_k$的浓度行为。$ \hatθ_k$连续插值的弱收敛限制显示出遵循非自动普通微分方程的轨迹。第3章和第4章都是概率论点，并且可能对对单个实验运行有用的增益调整策略提供了太多指导。因此，第5章讨论了基于估计HESSIAN信息和噪声水平的数据依赖于数据的增益策略。总体而言，这项工作回答了“动态系统的估计是多少，$θ_k^*$”和“我们可以信任$ \hatθ_k$作为$θ_k^*$的估计。”
This work analyzes the stochastic approximation algorithm with non-decaying gains as applied in time-varying problems. The setting is to minimize a sequence of scalar-valued loss functions $f_k(\cdot)$ at sampling times $τ_k$ or to locate the root of a sequence of vector-valued functions $g_k(\cdot)$ at $τ_k$ with respect to a parameter $θ\in R^p$. The available information is the noise-corrupted observation(s) of either $f_k(\cdot)$ or $g_k(\cdot)$ evaluated at one or two design points only. Given the time-varying stochastic approximation setup, we apply stochastic approximation algorithms with non-decaying gains, so that the recursive estimate denoted as $\hatθ_k$ can maintain its momentum in tracking the time-varying optimum denoted as $θ_k^*$. Chapter 3 provides a bound for the root-mean-squared error $ \sqrt{E(\|\hatθ_k-θ_k^*\|^2})$. Overall, the bounds are applicable under a mild assumption on the time-varying drift and a modest restriction on the observation noise and the bias term. After establishing the tracking capability in Chapter 3, we also discuss the concentration behavior of $\hatθ_k $ in Chapter 4. The weak convergence limit of the continuous interpolation of $\hatθ_k$ is shown to follow the trajectory of a non-autonomous ordinary differential equation. Both Chapter 3 and Chapter 4 are probabilistic arguments and may not provide much guidance on the gain-tuning strategies useful for one single experiment run. Therefore, Chapter 5 discusses a data-dependent gain-tuning strategy based on estimating the Hessian information and the noise level. Overall, this work answers the questions "what is the estimate for the dynamical system $θ_k^*$" and "how much we can trust $\hatθ_k $ as an estimate for $θ_k^*$."
