[Paper Review] A Value-Function-based Interior-point Method for Non-convex Bi-level Optimization

Risheng Liu, Xuan Liu+3 more|arXiv (Cornell University)|Jun 15, 2021

Sparse and Compressive Sensing Techniques35 references4 citations

TL;DR

This paper proposes BVFIM, a novel gradient-based method for non-convex bi-level optimization that uses a value-function-based log-barrier interior-point approach to transform the bi-level problem into a sequence of smooth, unconstrained subproblems. By avoiding expensive Hessian- or Jacobian-vector products and proving convergence without convexity assumptions, BVFIM achieves superior efficiency and robustness on non-convex tasks such as hyper-parameter optimization and data hyper-cleaning.

ABSTRACT

Bi-level optimization model is able to capture a wide range of complex learning tasks with practical interest. Due to the witnessed efficiency in solving bi-level programs, gradient-based methods have gained popularity in the machine learning community. In this work, we propose a new gradient-based solution scheme, namely, the Bi-level Value-Function-based Interior-point Method (BVFIM). Following the main idea of the log-barrier interior-point scheme, we penalize the regularized value function of the lower level problem into the upper level objective. By further solving a sequence of differentiable unconstrained approximation problems, we consequently derive a sequential programming scheme. The numerical advantage of our scheme relies on the fact that, when gradient methods are applied to solve the approximation problem, we successfully avoid computing any expensive Hessian-vector or Jacobian-vector product. We prove the convergence without requiring any convexity assumption on either the upper level or the lower level objective. Experiments demonstrate the efficiency of the proposed BVFIM on non-convex bi-level problems.

Motivation & Objective

• Address the challenge of solving non-convex bi-level optimization problems efficiently in machine learning applications.
• Overcome the limitations of existing gradient-based methods that require convexity or suffer from high computational cost due to Hessian/Jacobian products.
• Develop a scalable and convergent algorithm for bi-level problems where neither the upper-level nor lower-level objectives are convex.
• Enable effective optimization in hierarchical learning tasks such as hyper-parameter tuning, neural architecture search, and meta-learning.
• Provide theoretical convergence guarantees under minimal assumptions, including no requirement for a unique lower-level solution.

Proposed method

• Reformulate the bi-level problem using the value function of the lower-level problem to embed the lower-level optimality condition into the upper-level objective.
• Apply a log-barrier penalty to the regularized value function to enforce feasibility, transforming the constrained bi-level problem into a sequence of smooth, unconstrained subproblems.
• Use a regularization technique on the lower-level value function to ensure differentiability and numerical stability.
• Apply gradient-based methods to solve the resulting smooth subproblems without computing Hessian-vector or Jacobian-vector products.
• Employ an adaptive penalty parameter strategy that decreases over iterations to balance convergence and accuracy.
• Leverage the implicit function theorem and smoothing techniques to maintain differentiability and enable efficient optimization.

Experimental results

Research questions

• RQ1 Can a gradient-based method be designed for non-convex bi-level problems without requiring convexity in either the upper or lower-level objectives?
• RQ2 Can the computational cost of Hessian- or Jacobian-vector products be avoided in bi-level optimization while maintaining convergence?
• RQ3 How does the proposed value-function-based interior-point method compare to existing gradient-based methods in terms of convergence speed and solution quality?
• RQ4 What is the impact of regularization and penalty parameter adaptation on the convergence behavior and final solution quality?
• RQ5 Can the method effectively handle real-world non-convex bi-level tasks such as data hyper-cleaning and hyper-parameter optimization?

Key findings

• BVFIM achieves convergence to a stationary point of the bi-level problem without requiring convexity in the upper-level or lower-level objectives.
• The method avoids computing expensive Hessian-vector or Jacobian-vector products, significantly reducing computational overhead compared to existing gradient-based approaches.
• On the FashionMNIST dataset with corrupted labels, BVFIM achieved a test F1 score of 91.19%, outperforming EGBM (88.24%) and IGBM (84.31%) in both accuracy and training time.
• In hyper-parameter optimization tasks, BVFIM demonstrated faster convergence and better generalization, particularly on larger datasets like CIFAR10 with high-dimensional lower-level parameters.
• Experiments show that using a decreasing regularization parameter leads to optimal convergence, while fixed or zero regularization degrades performance on the upper-level variable.
• The convergence behavior is sensitive to the number of inner iterations L; smaller L values (e.g., L=1) accelerate convergence on the upper-level variable x, making it suitable when x* is the primary target.