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：我们提出一种新的随机化方法用于求解非线性方程组，该方法能够找到稀疏解或在简单约束条件下的解。本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进行Bregman投影实现。在欧几里得投影的特殊情况下，该方法即为非线性Kaczmarz方法。进一步地，若分量函数非负，则对应插值假设下的优化场景，此时该方法退化为采用近期提出的随机Polyak步长的随机梯度下降法。对于一般的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，相较于标准Polyak步长，本方法每次迭代都能使Bregman距离更接近精确解。向Bregman投影的推广需要付出代价：每次迭代需求解一个一维凸优化问题。这通常可通过全局化牛顿迭代完成。我们在两种经典非线性场景下证明了收敛性：凸非负函数情形及局部满足切锥条件的函数情形。最后，我们通过算例表明，在同等内存需求条件下，所提方法优于同类方法。