We propose a new randomized method for solving systems of nonlinear equations, which can find sparse solutions or solutions under certain simple constraints. The scheme only takes gradients of component functions and uses Bregman projections onto the solution space of a Newton equation. In the special case of euclidean projections, the method is known as nonlinear Kaczmarz method. Furthermore, if the component functions are nonnegative, we are in the setting of optimization under the interpolation assumption and the method reduces to SGD with the recently proposed stochastic Polyak step size. For general Bregman projections, our method is a stochastic mirror descent with a novel adaptive step size. We prove that in the convex setting each iteration of our method results in a smaller Bregman distance to exact solutions as compared to the standard Polyak step. Our generalization to Bregman projections comes with the price that a convex one-dimensional optimization problem needs to be solved in each iteration. This can typically be done with globalized Newton iterations. Convergence is proved in two classical settings of nonlinearity: for convex nonnegative functions and locally for functions which fulfill the tangential cone condition. Finally, we show examples in which the proposed method outperforms similar methods with the same memory requirements.

翻译：我们提出了一种求解非线性方程组的新型随机化方法，该方法能够找到稀疏解或满足特定简单约束条件的解。该方案仅需利用分量函数的梯度，并通过在牛顿方程解空间上执行Bregman投影来实现。在欧几里得投影的特殊情形下，该方法即为非线性Kaczmarz方法。此外，当分量函数非负时，我们处于插值假设下的优化场景，该方法可简化为采用近期提出的随机Polyak步长的SGD算法。对于一般Bregman投影，本方法是一种具有新型自适应步长的随机镜像下降算法。我们证明：在凸函数设定下，相较于标准Polyak步长，本文方法的每次迭代都能使Bregman距离更接近精确解。向Bregman投影的推广需以每次迭代求解一个凸一维优化问题为代价，该问题通常可通过全局化牛顿迭代完成。我们分别在两种经典非线性场景中证明收敛性：对于凸非负函数，以及满足切向锥条件的局部函数。最后，通过实例展示该方法在同等内存需求下优于同类方法的表现。