正则风险最小化的小批量割平面法

卢孟龙; 乔林波; 冯大为; 李东升; 卢锡城

doi:10.1631/FITEE.1800596

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正则风险最小化的小批量割平面法

常规文章 | Updated：2022-05-19

- 正则风险最小化的小批量割平面法
- Mini-batch cutting plane method for regularized risk minimization
- 信息与电子工程前沿（英文） 2019年20卷第11期页码：1551-1563
- Affiliations：
  
  Science and Technology on Parallel and Distributed Laboratory, National University of Defense Technology, Changsha 410073, China
- Author bio：
  
  [ "Meng-long LU, E-mail: lumenglong2018@163.com" ]
  Da-wei FENG, E-mail: davyfeng.c@gmail.com
- Funds：
  
  Project supported by the National Key R&D Program of China (No.2018YFB0204300) and the National Natural Science Foundation of China (Nos.61872376 and 61806216
- DOI：10.1631/FITEE.1800596
  中图分类号： TP391
- 收稿：2018-09-25，
  
  修回：2019-10-10，
  
  纸质出版：2019-11
- Accepted：
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卢孟龙, 乔林波, 冯大为, 等. 正则风险最小化的小批量割平面法[J]. 信息与电子工程前沿（英文）, 2019,20(11):1551-1563.

Meng-long LU, Lin-bo QIAO, Da-wei FENG, et al. Mini-batch cutting plane method for regularized risk minimization[J]. Frontiers of Information Technology & Electronic Engineering, 2019, 20(11): 1551-1563.
卢孟龙, 乔林波, 冯大为, 等. 正则风险最小化的小批量割平面法[J]. 信息与电子工程前沿（英文）, 2019,20(11):1551-1563. DOI： 10.1631/FITEE.1800596.

Meng-long LU, Lin-bo QIAO, Da-wei FENG, et al. Mini-batch cutting plane method for regularized risk minimization[J]. Frontiers of Information Technology & Electronic Engineering, 2019, 20(11): 1551-1563. DOI： 10.1631/FITEE.1800596.

摘要

虽然最近求解非凸问题的研究十分热门，尤其在机器学习需要可解释性模型情况下，凸优化仍然重要。求解凸问题可得到全局最优解，故而最终模型可用数学方法解释。通常为防止过度拟合，凸问题被重新描述为一个正则风险最小化问题。无论目标函数是否可微，割平面法是求解凸问题最佳方法之一。然而，割平面法及其变体无法充分应对大规模密集型数据，因为这些算法每次迭代都需访问整个数据集，大大增加了计算负担和内存成本。为解决这一问题，提出一种新的小批量割平面法。该算法通过对小批量采样数据进行计算，得到估计切割平面用于迭代，使其能处理大规模数据。此外，小批量割平面法使用“sink”算子检测和调整噪声估计以保证收敛性。在大量实际数据集上的数值实验证明了小批量割平面法有效性，收敛速度优于正则风险最小化的bundle法和普遍使用的随机梯度下降法。

Abstract

Although concern has been recently expressed with regard to the solution to the non-convex problem

convex optimization is still important in machine learning

especially when the situation requires an interpretable model. Solution to the convex problem is a global minimum

and the final model can be explained mathematically. Typically

the convex problem is re-casted as a regularized risk minimization problem to prevent overfitting. The cutting plane method (CPM) is one of the best solvers for the convex problem

irrespective of whether the objective function is differentiable or not. However

CPM and its variants fail to adequately address large-scale data-intensive cases because these algorithms access the entire dataset in each iteration

which substantially increases the computational burden and memory cost. To alleviate this problem

we propose a novel algorithm named the mini-batch cutting plane method (MBCPM)

which iterates with estimated cutting planes calculated on a small batch of sampled data and is capable of handling large-scale problems. Furthermore

the proposed MBCPM adopts a "sink" operation that detects and adjusts noisy estimations to guarantee convergence. Numerical experiments on extensive real-world datasets demonstrate the effectiveness of MBCPM

which is superior to the bundle methods for regularized risk minimization as well as popular stochastic gradient descent methods in terms of convergence speed.

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