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：我们提出了一种新的随机化方法，用于求解非线性方程组，能够找到稀疏解或满足特定简单约束的解。该方案仅需利用分量函数的梯度，并通过Bregman投影于牛顿方程的解空间。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，若分量函数非负，则问题转化为插值假设下的优化问题，该方法退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，我们的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，在凸性设定下，相比于标准Polyak步长，该方法每次迭代使Bregman距离更逼近精确解。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。收敛性在两类经典非线性框架下得到证明：针对凸非负函数的全局收敛，以及满足切向锥条件的局部收敛。最后，我们通过实例展示该方法在相同内存需求下优于同类方法的性能。