Bilevel optimization, the problem of minimizing a value function which involves the arg-minimum of another function, appears in many areas of machine learning. In a large scale empirical risk minimization setting where the number of samples is huge, it is crucial to develop stochastic methods, which only use a few samples at a time to progress. However, computing the gradient of the value function involves solving a linear system, which makes it difficult to derive unbiased stochastic estimates. To overcome this problem we introduce a novel framework, in which the solution of the inner problem, the solution of the linear system, and the main variable evolve at the same time. These directions are written as a sum, making it straightforward to derive unbiased estimates. The simplicity of our approach allows us to develop global variance reduction algorithms, where the dynamics of all variables is subject to variance reduction. We demonstrate that SABA, an adaptation of the celebrated SAGA algorithm in our framework, has $O(\frac1T)$ convergence rate, and that it achieves linear convergence under Polyak-Lojasciewicz assumption. This is the first stochastic algorithm for bilevel optimization that verifies either of these properties. Numerical experiments validate the usefulness of our method.
翻译:双层优化 双层优化, 将包含另一个函数最小化的值函数最小化的问题, 出现在机器学习的许多领域 。 在大规模实验风险最小化的大规模实验中, 样本数量巨大, 开发随机分析方法至关重要 。 然而, 计算值函数的梯度需要解决线性系统, 这使得难以得出公正的随机估计值。 要克服这个问题, 我们引入了一个新颖的框架, 解决内部问题、 线性系统解决方案和主要变量同时演变。 这些方向是写成一个总和, 直截了当地得出不偏直的估计数。 我们方法的简单性使我们能够开发全球差异减少算法, 所有的变量的动态都会降低差异。 我们证明, SABA, 是我们框架中值得庆祝的SAGA算法的调整, 具有$( orc1T) 的汇合率, 并在Polyak- Lojaciewicz假设下实现线性融合。 这是用于验证这些属性的双层优化方法的首个高级算法。