Randomized linear solvers randomly compress and solve a linear system with compelling theoretical convergence rates and computational complexities. However, such solvers suffer a substantial disconnect between their theoretical rates and actual efficiency in practice. Fortunately, these solvers are quite flexible and can be adapted to specific problems and computing environments to ensure high efficiency in practice, even at the cost of lower effectiveness (i.e., having a slower theoretical rate of convergence). While highly efficient adapted solvers can be readily designed by application experts, will such solvers still converge and at what rate? To answer this, we distill three general criteria for randomized adaptive solvers, which, as we show, will guarantee a worst-case exponential rate of convergence of the solver applied to consistent and inconsistent linear systems irrespective of whether such systems are over-determined, under-determined or rank-deficient. As a result, we enable application experts to design randomized adaptive solvers that achieve efficiency and can be verified for effectiveness using our theory. We demonstrate our theory on twenty-six solvers, nine of which are novel or novel block extensions of existing methods to the best of our knowledge.

翻译：随机线性求解器通过随机压缩与求解线性系统，在理论上具有令人信服的收敛速度和计算复杂度。然而，此类求解器的理论收敛速度与实际计算效率之间存在显著差距。幸运的是，这类求解器具有高度灵活性，可针对特定问题和计算环境进行自适应调整，从而在实际应用中确保高效率——即便这种调整可能以降低有效性（即理论收敛速度变慢）为代价。当领域专家能够轻易设计出高效的自适应求解器时，我们不禁要问：此类求解器是否仍能保持收敛？收敛速度又将如何？为回答这一问题，我们提炼出随机自适应求解器的三项通用准则。研究表明，这些准则能确保求解器在相容与不相容线性系统（无论超定、欠定还是秩亏）中均具有最坏情况下的指数级收敛速度。由此，我们使领域专家能够设计出兼具效率性且可通过理论验证有效性的随机自适应求解器。我们已在26种求解器上验证了该理论，其中9种据我们所知为全新方法或现有方法的全新块扩展。