[Paper Review] A Generic Descent Aggregation Framework for Gradient-based Bi-level Optimization.

Risheng Liu, Pan Mu+3 more|arXiv (Cornell University)|Feb 16, 2021

Sparse and Compressive Sensing Techniques50 references4 citations

TL;DR

This paper proposes a generic Bi-level Descent Aggregation (BDA) framework that replaces the restrictive Lower-Level Singleton (LLS) assumption in gradient-based bi-level optimization. By aggregating hierarchical objectives and enabling task-specific iteration dynamics, BDA ensures convergence without LLS and accelerates back-propagation via a one-stage technique, outperforming prior methods in hyper-parameter tuning and meta-learning.

ABSTRACT

In recent years, gradient-based methods for solving bi-level optimization tasks have drawn a great deal of interest from the machine learning community. However, to calculate the gradient of the best response, existing research always relies on the singleton of the lower-level solution set (a.k.a., Lower-Level Singleton, LLS). In this work, by formulating bi-level models from an optimistic bi-level viewpoint, we first establish a novel Bi-level Descent Aggregation (BDA) framework, which aggregates hierarchical objectives of both upper level and lower level. The flexibility of our framework benefits from the embedded replaceable task-tailored iteration dynamics modules, thereby capturing a wide range of bi-level learning tasks. Theoretically, we derive a new methodology to prove the convergence of BDA framework without the LLS restriction. Besides, the new proof recipe we propose is also engaged to improve the convergence results of conventional gradient-based bi-level methods under the LLS simplification. Furthermore, we employ a one-stage technique to accelerate the back-propagation calculation in a numerical manner. Extensive experiments justify our theoretical results and demonstrate the superiority of the proposed algorithm for hyper-parameter optimization and meta-learning tasks.

Motivation & Objective

• Address the limitation of existing gradient-based bi-level optimization methods that rely on the restrictive Lower-Level Singleton (LLS) assumption.
• Develop a flexible framework capable of capturing diverse bi-level learning tasks through replaceable, task-tailored iteration dynamics.
• Establish a new theoretical proof recipe to demonstrate convergence of the BDA framework without requiring the LLS condition.
• Improve convergence guarantees for conventional gradient-based bi-level methods under the LLS simplification using the proposed proof technique.
• Accelerate back-propagation in bi-level optimization through a novel one-stage numerical computation technique.

Proposed method

• Formulate bi-level optimization from an optimistic bi-level viewpoint to enable hierarchical objective aggregation across both upper and lower levels.
• Design the Bi-level Descent Aggregation (BDA) framework with modular, replaceable iteration dynamics for task-specific adaptation.
• Derive a new convergence proof methodology that does not require the LLS assumption, enabling broader theoretical validity.
• Apply the new proof recipe to refine convergence results for existing gradient-based bi-level methods under the LLS simplification.
• Implement a one-stage technique to accelerate back-propagation by reducing computational overhead in gradient computation.
• Integrate the BDA framework into standard bi-level learning pipelines for hyper-parameter optimization and meta-learning.

Experimental results

Research questions

• RQ1 Can a generic bi-level optimization framework be designed to operate without the restrictive Lower-Level Singleton (LLS) assumption?
• RQ2 How can hierarchical objectives from both upper and lower levels be effectively aggregated in a way that supports diverse learning tasks?
• RQ3 Can a new theoretical proof framework be developed to establish convergence of bi-level methods without relying on LLS?
• RQ4 To what extent can the proposed proof recipe improve convergence guarantees for existing gradient-based bi-level methods under the LLS condition?
• RQ5 How effective is the one-stage back-propagation acceleration technique in reducing computational cost in practice?

Key findings

• The BDA framework achieves convergence without requiring the Lower-Level Singleton (LLS) assumption, expanding the theoretical scope of bi-level optimization.
• The proposed proof recipe improves convergence results for conventional gradient-based bi-level methods under the LLS simplification.
• The one-stage back-propagation technique significantly accelerates gradient computation in the BDA framework.
• Extensive experiments confirm the theoretical claims, showing superior performance in hyper-parameter optimization tasks.
• The BDA framework demonstrates strong empirical advantages in meta-learning benchmarks compared to existing methods.
• The modular design of BDA enables effective adaptation to various bi-level learning tasks through task-tailored iteration dynamics.