In this work, we revisit the problem of solving large-scale semidefinite programs using randomized first-order methods and stochastic smoothing. We introduce two oblivious stochastic mirror descent algorithms based on a complementary composite setting. One algorithm is designed for non-smooth objectives, while an accelerated version is tailored for smooth objectives. Remarkably, both algorithms work without prior knowledge of the Lipschitz constant or smoothness of the objective function. For the non-smooth case with $\mathcal{M}-$bounded oracles, we prove a convergence rate of $ O( {\mathcal{M}}/{\sqrt{T}} ) $. For the $L$-smooth case with a feasible set bounded by $D$, we derive a convergence rate of $ O( {L^2 D^2}/{(T^{2}\sqrt{T})} + {(D_0^2+\sigma^2)}/{\sqrt{T}} )$, where $D_0$ is the starting distance to an optimal solution, and $ \sigma^2$ is the stochastic oracle variance. These rates had only been obtained so far by either assuming prior knowledge of the Lipschitz constant or the starting distance to an optimal solution. We further show how to extend our framework to relative scale and demonstrate the efficiency and robustness of our methods on large scale semidefinite programs.

翻译：本文重新审视了利用随机一阶方法与随机平滑技术求解大规模半定规划的问题。基于互补复合设置，我们提出了两种无感知随机镜像下降算法：一种面向非光滑目标函数，另一种加速版本则针对光滑目标函数设计。值得注意的是，这两种算法均无需事先获知目标函数的Lipschitz常数或光滑性参数。对于具有$\mathcal{M}-$有界预言的非光滑情形，我们证明了$O( {\mathcal{M}}/{\sqrt{T}} )$的收敛速率；对于可行集有界（界为$D$）的$L$-光滑情形，我们推导出$O( {L^2 D^2}/{(T^{2}\sqrt{T})} + {(D_0^2+\sigma^2)}/{\sqrt{T}} )$的收敛速率，其中$D_0$为初始点到最优解的距离，$\sigma^2$为随机预言方差。此前，这些收敛速率仅在假设已知Lipschitz常数或初始最优解距离的情况下才能实现。我们进一步展示了如何将该框架扩展至相对尺度，并通过大规模半定规划实验验证了所提方法的效率与鲁棒性。