Certified Residual Quasi-Newton Methods for Distributed Variational Inequalities

Second-order methods for smooth monotone variational inequalities reach the optimal rate $O(T^{-3\mathrm{/}2})$, but a distributed exact Jacobian costs $d$ times more communication than an operator value. We show that similarity does part of the work for free: if the server’s Jacobian differs from the global one by at most $\beta$, using it gives $O(L_1{D}^3{T}^{-3\mathrm{/}2}+\beta D^2{T}^{-1})$ at first-order communication cost. A quasi-Newton approximation of the residual Jacobian $\mathrm{\nabla }F-\mathrm{\nabla }{F}_1$, built from secants already communicated, improves the model but cannot remove the $T^{-1}$ term, because any uniform bound on the Jacobian error leaves it in the rate. We therefore certify the surrogate only along the candidate step: one Jacobian-vector product tests it, and a failed test is reused as an exact correction. This attains the exact rate $O(L_1{D}^3{T}^{-3\mathrm{/}2})$ while transmitting only vectors. Experiments on LIBSVM and synthetic instances measure accuracy against communication.