A framework for bilevel optimization that enables stochastic and global variance reduction algorithms
This work addresses a bottleneck in bilevel optimization for machine learning applications with huge datasets by enabling efficient stochastic methods, representing a novel method for a known problem rather than a foundational breakthrough.
The authors tackled the challenge of deriving unbiased stochastic gradient estimates in large-scale bilevel optimization by introducing a novel framework that simultaneously evolves the inner problem solution, linear system solution, and main variable, enabling the development of stochastic algorithms with variance reduction. They demonstrated that their SABA algorithm achieves an O(1/T) convergence rate and linear convergence under specific assumptions, marking the first stochastic method for bilevel optimization with these properties.
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.