In this article, we consider an optimization problem where the objective function is evaluated at the fixed-point of a contraction mapping parameterized by a control variable, and optimization takes place over this control variable. Since the derivative of the fixed-point with respect to the parameter can usually not be evaluated exactly, an adjoint dynamical system can be used to estimate gradients. Using this estimation procedure, the optimization algorithm alternates between derivative estimation and an approximate gradient descent step. We analyze a variant of this approach involving dynamic time-scaling, where after each parameter update the adjoint system is iterated until a convergence threshold is passed. We prove that, under certain conditions, the algorithm can find approximate stationary points of the objective function. We demonstrate the approach in the settings of an inverse problem in chemical kinetics, and learning in attractor networks.

在本文中，我们考虑了一个优化问题，其中在控制变量参数化的收缩映射的固定点上评估目标函数，并对该控制变量进行优化。由于定点相对于参数的导数通常无法精确评估，因此可以使用辅助动力学系统来估计梯度。使用此估计程序，优化算法会在导数估计和近似梯度下降步骤之间交替。我们分析了这种方法的一种变体，其中涉及动态时标，其中在每个参数更新之后，对伴随系统进行迭代，直到通过收敛阈值为止。我们证明，在一定条件下，该算法可以找到目标函数的近似平稳点。