Bilevel Optimization: Convergence Analysis and Enhanced Design (2010.07962v3)

Abstract: Bilevel optimization has arisen as a powerful tool for many machine learning problems such as meta-learning, hyperparameter optimization, and reinforcement learning. In this paper, we investigate the nonconvex-strongly-convex bilevel optimization problem. For deterministic bilevel optimization, we provide a comprehensive convergence rate analysis for two popular algorithms respectively based on approximate implicit differentiation (AID) and iterative differentiation (ITD). For the AID-based method, we orderwisely improve the previous convergence rate analysis due to a more practical parameter selection as well as a warm start strategy, and for the ITD-based method we establish the first theoretical convergence rate. Our analysis also provides a quantitative comparison between ITD and AID based approaches. For stochastic bilevel optimization, we propose a novel algorithm named stocBiO, which features a sample-efficient hypergradient estimator using efficient Jacobian- and Hessian-vector product computations. We provide the convergence rate guarantee for stocBiO, and show that stocBiO outperforms the best known computational complexities orderwisely with respect to the condition number $\kappa$ and the target accuracy $\epsilon$. We further validate our theoretical results and demonstrate the efficiency of bilevel optimization algorithms by the experiments on meta-learning and hyperparameter optimization.

• The paper refines deterministic bilevel optimization by enhancing AID and pioneering convergence rate analysis for ITD with practical parameter selection and warm-start strategies.
• The paper introduces stocBiO, a novel algorithm that improves sample efficiency using optimized Jacobian and Hessian-vector computations for stochastic bilevel problems.
• The results demonstrate significant computational complexity reductions, offering faster convergence and practical scalability for large-scale machine learning applications.

Bilevel Optimization: Convergence Analysis and Enhanced Design

The paper offers a comprehensive exploration of bilevel optimization, a framework gaining substantial traction across various machine learning domains such as meta-learning, hyperparameter optimization, and reinforcement learning. The focus is on a particular class of this problem, namely the nonconvex-strongly-convex bilevel optimization, where the upper-level objective function is nonconvex, and the lower-level function is strongly convex. This setting is prevalent in many scenarios where bilevel optimization is applied, such as in meta-learning where the lower-level function often includes a strongly-convex regularizer.

Summary of Contributions

1. Deterministic Bilevel Optimization: The paper refines the theoretical underpinning of two existing optimization algorithms—Approximate Implicit Differentiation (AID) and Iterative Differentiation (ITD). The authors enhance the convergence rate analysis for AID by employing a more practical parameter selection and a warm-start strategy, which introduces a significantly improved computational complexity. Notably, the paper pioneers the first theoretical convergence rate analysis for the ITD-based method, comparing it quantitatively against the AID-based approach.
2. Stochastic Bilevel Optimization: A novel algorithm, stocBiO, is introduced for dealing with stochastic bilevel optimization problems. This algorithm leverages a sample-efficient hypergradient estimator, improved by efficient Jacobian- and Hessian-vector product computations. The provided convergence rate guarantee indicates that stocBiO outshines existing solutions in terms of computational complexities with respect to the condition number and target accuracy.

Numerical and Theoretical Implications

The paper asserts significant advancements in both deterministic and stochastic bilevel optimization. For deterministic cases, the convergence rate analysis for AID-BiO reveals marked improvements over previous work, with a reduction in complexity by the condition number order. This result is pivotal as it translates into faster computational times and reduced resource usage, potentially influencing how large-scale machine learning problems are tackled.

In stochastic settings, stocBiO's introduction is particularly noteworthy. The new algorithm demonstrates enhanced efficiency with complexities improved by orders of magnitude compared to established methods such as BSA and TTSA. These improvements not only advance the theoretical framework but also position stocBiO as a preferable option in practical applications like reinforcement learning and hyperparameter tuning on massive datasets.

Future Directions

From a theoretical perspective, this research lays groundwork that could be expanded by exploring the generalization of bilevel optimization algorithms to other convexity structures, or by incorporating more sophisticated learning rate schedules. Practically, the implications for scaling machine learning models to high-dimensional data are significant, warranting further investigation into the synergy between bilevel optimization and emerging architectures in AI.

In conclusion, the paper substantially contributes to the body of knowledge on bilevel optimization, enhancing both our theoretical understanding and practical toolkit. By methodically addressing the convergence and efficiency issues with novel approaches, it sets the stage for significant advancements in how complex machine learning models are trained and optimized.