2026/07/20 by Lisha Chen
#math.OC
For smooth nonconvex stochastic multi-objective problems, stochastic multi-gradient descent (SMG) computes an approximate steepest common descent direction of the objectives from stochastic gradients. With unbiased, variance-bounded stochastic gradients, this note establishes a new convergence rate for SMG in terms of the squared Pareto-stationarity (PS) measure. With a constant stepsize and linearly growing mini-batches, this measure at the algorithm's output is \widetilde O(T-1) after T iterations. This improves on the \widetilde O(T-1/4) bound obtained by Chen et al. (2024) under the same setting, where \widetilde O(⋅) suppresses logarithmic factors. The key to the rate improvement is to exploit the Lipschitz continuity of the PS measure, defined by the norm of the multi-gradient descent algorithm (MGDA) direction, rather than the (1/2)-Hölder continuity of the MGDA direction used by Chen et al. (2024). The proof was discovered while the author was preparing homework for a graduate course: ChatGPT 5.4 Thinking Extended generated the initial proof strategy in response to an author-written homework-solution prompt; the author then verified and reorganized the resulting argument. The appendices document the prompt and summarize the student submissions.