Bilevel Optimization for Machine Learning: Algorithm Design and Convergence Analysis

Abstract

Bilevel optimization has become a powerful framework in various machine learning applications including meta-learning, hyperparameter optimization, and network architecture search. There are generally two classes of bilevel optimization formulations for machine learning: 1) problem-based bilevel optimization, whose inner-level problem is formulated as finding a minimizer of a given loss function; and 2) algorithm-based bilevel optimization, whose inner-level solution is an output of a fixed algorithm. For the first class, two popular types of gradient-based algorithms have been proposed for hypergradient estimation via approximate implicit differentiation (AID) and iterative differentiation (ITD). Algorithms for the second class include the popular model-agnostic meta-learning (MAML) and almost no inner loop (ANIL). However, the convergence rate and fundamental limitations of bilevel optimization algorithms have not been well explored. This thesis provides a comprehensive convergence rate analysis for bilevel algorithms in the aforementioned two classes. We further propose principled algorithm designs for bilevel optimization with higher efficiency and scalability. For the problem-based formulation, we provide a convergence rate analysis for AID- and ITD-based bilevel algorithms. We then develop acceleration bilevel algorithms, for which we provide shaper convergence analysis with relaxed assumptions. We also provide the first lower bounds for bilevel optimization, and establish the optimality by providing matching upper bounds under certain conditions. We finally propose new stochastic bilevel optimization algorithms with lower complexity and higher efficiency in practice. For the algorithm-based formulation, we develop a theoretical convergence for general multi-step MAML and ANIL, and characterize the impact of parameter selections and loss geometries on the their complexities.

Ph.D. dissertation, 253 pages, 7 tables and 8 figures. All comments are welcome

References in corpus (12)

• Theoretical Models of Learning to Learn
• Meta-SGD: Learning to Learn Quickly for Few-Shot Learning
• On Differentiating Parameterized Argmin and Argmax Problems with Application to Bi-level Optimization
• Online Meta-Learning
• Learning-to-Learn Stochastic Gradient Descent with Biased Regularization
• Self-Tuning Networks: Bilevel Optimization of Hyperparameters using Structured Best-Response Functions
• Improved Zeroth-Order Variance Reduced Algorithms and Analysis for Nonconvex Optimization
• A Gradient-based Bilevel Optimization Approach for Tuning Hyperparameters in Machine Learning
• Improved Bilevel Model: Fast and Optimal Algorithm with Theoretical Guarantee
• Delta-STN: Efficient Bilevel Optimization for Neural Networks using Structured Response Jacobians
• On the Global Optimality of Model-Agnostic Meta-Learning
• Multi-step Estimation for Gradient-based Meta-learning