We present an optimization framework that exhibits dimension-independent convergence on a broad class of semidefinite programs (SDPs). Our approach first regularizes the primal problem with the von Neumann entropy, then solve the regularized problem using dual gradient ascent with respect to a problem-adapted norm. In particular, we show that the dual gradient norm converges to zero at a rate independent of the ambient dimension and, via rounding arguments, construct primal-feasible solutions in certain special cases. We also derive explicit convergence rates for the objective. In order to achieve optimal computational scaling, we must accommodate the use of stochastic gradients constructed via randomized trace estimators. Throughout we illustrate the generality of our framework via three important special cases -- the Goemans-Williamson SDP relaxation of the Max-Cut problem, the optimal transport linear program, and several SDP relaxations of the permutation synchronization problem. Numerical experiments confirm that our methods achieve dimension-independent convergence in practice.

翻译：我们提出了一种优化框架，该框架在一类广泛的半定规划问题上展现出与维度无关的收敛性。我们的方法首先使用冯·诺依曼熵对原始问题进行正则化，然后利用问题自适应范数下的对偶梯度上升法求解正则化问题。特别地，我们证明了对偶梯度范数以与问题维度无关的速率收敛至零，并通过舍入论证在特定特殊情况下构造出原始可行解。我们还推导了目标函数的显式收敛速率。为实现最优计算复杂度，我们采用了基于随机迹估计器构建的随机梯度。全文通过三个重要特例——Max-Cut问题的Goemans-Williamson半定规划松弛、最优传输线性规划以及置换同步问题的若干半定规划松弛——阐释了我们框架的普适性。数值实验证实，我们的方法在实践中实现了与维度无关的收敛。