[Paper Review] Branch-and-Pruning Optimization Towards Global Optimality in Deep Learning.

Yuanwei Wu, Ziming Zhang+1 more|arXiv (Cornell University)|Apr 5, 2021

Stochastic Gradient Optimization Techniques64 references4 citations

TL;DR

This paper proposes BPGrad, a novel branch-and-pruning optimization algorithm that leverages Lipschitz continuity to adaptively set step sizes for achieving global optimality in deep learning. By iteratively refining search regions through branching and pruning, BPGrad guarantees convergence to the global optimum within finite iterations and empirically outperforms Adam, Adagrad, Adadelta, and RMSProp on object recognition, detection, and segmentation tasks.

ABSTRACT

It has been attracting more and more attention to understand the global optimality in deep learning (DL) recently. However, conventional DL solvers, have not been developed intentionally to seek for such global optimality. In this paper, we propose a novel approximation algorithm, {\em BPGrad}, towards optimizing deep models globally via branch and pruning. The proposed BPGrad algorithm is based on the assumption of Lipschitz continuity in DL, and as a result, it can adaptively determine the step size for the current gradient given the history of previous updates, wherein theoretically no smaller steps can achieve the global optimality. We prove that, by repeating such a branch-and-pruning procedure, we can locate the global optimality within finite iterations. Empirically an efficient adaptive solver based on BPGrad for DL is proposed as well, and it outperforms conventional DL solvers such as Adagrad, Adadelta, RMSProp, and Adam in the tasks of object recognition, detection, and segmentation. The code is available at \url{this https URL}.

Motivation & Objective

• To address the lack of global optimality guarantees in conventional deep learning solvers.
• To develop an optimization framework that systematically seeks the global minimum rather than settling for local minima.
• To theoretically and empirically demonstrate convergence to global optimality within finite iterations using a branch-and-pruning strategy.
• To outperform standard adaptive optimizers like Adam, Adagrad, Adadelta, and RMSProp in downstream vision tasks.

Proposed method

• The method is based on the assumption of Lipschitz continuity in deep neural networks to bound gradient changes.
• It adaptively determines step sizes using historical updates, ensuring no smaller steps could achieve global optimality.
• A branch-and-pruning mechanism is applied iteratively to narrow down the search space around the global optimum.
• The algorithm maintains theoretical guarantees of finite convergence to the global minimum under the Lipschitz assumption.
• An efficient adaptive solver is derived from BPGrad for practical deployment in deep learning.
• The approach dynamically refines candidate regions by eliminating suboptimal regions through pruning.

Experimental results

Research questions

• RQ1 Can a deep learning optimization algorithm guarantee convergence to the global optimum under reasonable assumptions?
• RQ2 How can adaptive step size selection be formalized to ensure global optimality without requiring excessively small steps?
• RQ3 What is the theoretical convergence behavior of a branch-and-pruning strategy in non-convex deep learning landscapes?
• RQ4 How does the proposed BPGrad method compare in practice to standard adaptive optimizers like Adam and RMSProp?
• RQ5 Can the branch-and-pruning framework be efficiently implemented for large-scale deep learning models?

Key findings

• BPGrad guarantees convergence to the global optimum within finite iterations under the Lipschitz continuity assumption.
• The adaptive step size mechanism ensures theoretical optimality by preventing suboptimal descent directions.
• The empirical solver based on BPGrad achieves superior performance on object recognition, detection, and segmentation benchmarks.
• BPGrad outperforms standard optimizers including Adam, Adagrad, Adadelta, and RMSProp across all evaluated vision tasks.
• The method maintains theoretical rigor while enabling practical deployment through an efficient adaptive implementation.
• The code for BPGrad is publicly available, supporting reproducibility and further research.