论文标题
一类无需无功能功能优化算法的复杂性
Complexity of a Class of First-Order Objective-Function-Free Optimization Algorithms
论文作者
论文摘要
储层计算是预测湍流的有力工具，其简单的架构具有处理大型系统的计算效率。然而，其实现通常需要完整的状态向量测量和系统非线性知识。我们使用非线性投影函数将系统测量扩展到高维空间，然后将其输入到储层中以获得预测。我们展示了这种储层计算网络在时空混沌系统上的应用，该系统模拟了湍流的若干特征。我们表明，使用径向基函数作为非线性投影器，即使只有部分观测并且不知道控制方程，也能稳健地捕捉复杂的系统非线性。最后，我们表明，当测量稀疏、不完整且带有噪声，甚至控制方程变得不准确时，我们的网络仍然可以产生相当准确的预测，从而为实际湍流系统的无模型预测铺平了道路。
A parametric class of trust-region algorithms for unconstrained nonconvex optimization is considered where the value of the objective function is never computed. The class contains a deterministic version of the first-order Adagrad method typically used for minimization of noisy function, but also allows the use of (possibly approximate) second-order information when available. The rate of convergence of methods in the class is analyzed and is shown to be identical to that known for first-order optimization methods using both function and gradients values, recovering existing results for purely-first order variants and improving the explicit dependence on problem dimension. This rate is shown to be essentially sharp. A new class of methods is also presented, for which a slightly worse and essentially sharp complexity result holds. Limited numerical experiments show that the new methods' performance may be comparable to that of standard steepest descent, despite using significantly less information, and that this performance is relatively insensitive to noise.
