We develop a framework for the average-case analysis of random quadratic problems and derive algorithms that are optimal under this analysis. This yields a new class of methods that achieve acceleration given a model of the Hessian's eigenvalue distribution. We develop explicit algorithms for the uniform, Marchenko-Pastur, and exponential distributions. These methods are momentum-based algorithms, whose hyper-parameters can be estimated without knowledge of the Hessian's smallest singular value, in contrast with classical accelerated methods like Nesterov acceleration and Polyak momentum. Through empirical benchmarks on quadratic and logistic regression problems, we identify regimes in which the the proposed methods improve over classical (worst-case) accelerated methods.

翻译：我们开发了随机二次问题平均分析框架, 并得出了根据本分析最理想的算法。 这产生了一种新的方法, 有了赫西安的二元值分布模型, 加速了 。 我们为制服、 Marchenko- Pastur 和指数分布开发了明确的算法 。 这些方法都是基于动力的算法, 在不了解赫西安人最小单值的情况下, 其超参数可以被估计, 与古典加速法( Nesterov 加速法 ) 和 Polyak 动力法( Polyak 动力法) 不同。 通过关于二次回归和物流回归问题的经验基准, 我们找出了拟议方法比古典( 微量) 加速法改进的制度 。