We introduce a clipping strategy for Stochastic Gradient Descent (SGD) which uses quantiles of the gradient norm as clipping thresholds. We prove that this new strategy provides a robust and efficient optimization algorithm for smooth objectives (convex or non-convex), that tolerates heavy-tailed samples (including infinite variance) and a fraction of outliers in the data stream akin to Huber contamination. Our mathematical analysis leverages the connection between constant step size SGD and Markov chains and handles the bias introduced by clipping in an original way. For strongly convex objectives, we prove that the iteration converges to a concentrated distribution and derive high probability bounds on the final estimation error. In the non-convex case, we prove that the limit distribution is localized on a neighborhood with low gradient. We propose an implementation of this algorithm using rolling quantiles which leads to a highly efficient optimization procedure with strong robustness properties, as confirmed by our numerical experiments.

引入了一种剪裁策略，使用梯度范数的分位数作为剪裁阈值，为平滑目标（凸或非凸）提供鲁棒且高效的优化算法，容忍重尾样本和数据中的异常值，数学分析说明了其收敛性质以及对初始估计误差的高概率界限，并通过实验证实了其高效性和鲁棒性。

基于梯度分位数截断的鲁棒随机优化