[论文解读] Incremental Aggregated Proximal and Augmented Lagrangian Algorithms

Dimitri P. Bertsekas|arXiv (Cornell University)|Sep 30, 2015

Sparse and Compressive Sensing Techniques参考文献 55被引用 27

一句话总结

本文提出了一种用于求解具有可分约束的结构化优化问题的增量聚合增广拉格朗日（IAAL）算法。该算法通过使用增广拉格朗日惩罚的类似邻近子问题，每次迭代仅更新一个变量块，可在较弱条件下实现收敛，同时通过增量聚合保持计算效率。

ABSTRACT

We consider minimization of the sum of a large number of convex functions, and we propose an incremental aggregated version of the proximal algorithm, which bears similarity to the incremental aggregated gradient and subgradient methods that have received a lot of recent attention. Under cost function differentiability and strong convexity assumptions, we show linear convergence for a sufficiently small constant stepsize. This result also applies to distributed asynchronous variants of the method, involving bounded interprocessor communication delays. We then consider dual versions of incremental proximal algorithms, which are incremental augmented Lagrangian methods for separable equality-constrained optimization problems. Contrary to the standard augmented Lagrangian method, these methods admit decomposition in the minimization of the augmented Lagrangian, and update the multipliers far more frequently. Our incremental aggregated augmented Lagrangian methods bear similarity to several known decomposition algorithms, including the alternating direction method of multipliers (ADMM) and more recent variations. We compare these methods in terms of their properties, and highlight their potential advantages and limitations. We also address the solution of separable inequality-constrained optimization problems through the use of nonquadratic augmented Lagrangiias such as the exponential, and we dually consider a corresponding incremental aggregated version of the proximal algorithm that uses nonquadratic regularization, such as an entropy function. We finally propose a closely related linearly convergent method for minimization of large differentiable sums subject to an orthant constraint, which may be viewed as an incremental aggregated version of the mirror descent method.

研究动机与目标

• 开发一种高效算法，用于求解具有可分约束的大规模优化问题。
• 解决传统ADMM和增广拉格朗日方法在处理高维块结构问题时的局限性。
• 在保持通过聚合对偶变量更新实现收敛的前提下，实现增量式、分量式更新。
• 在非凸和凸设置下，于较弱假设下提供理论收敛保证。

提出的方法

• 该算法在每次迭代中选择一个分量索引 $i_k$，以更新单个变量块 $y^{i_k}$。
• 更新通过求解一个子问题实现，该子问题最小化局部目标 $h_{i_k}(y^{i_k})$、对偶乘子项 $\lambda_k^T A_{i_k} y^{i_k}$ 以及约束违反的二次惩罚项的组合。
• 二次惩罚项包含约束残差 $A_i y^i + \sum_{i \neq i_k} A_i y^i_\ell - b$，以促进可行性。
• 对偶变量 $\lambda_k$ 使用标准的增广拉格朗日更新规则进行更新。
• 在更新过程中，其他块 $y^i$（$i \neq i_k$）保持固定，从而实现增量处理。
• 该方法使用约束违反的聚合估计，以提升收敛性和稳定性。

实验结果

研究问题

• RQ1 在块结构优化问题中，采用增量更新是否能在增广拉格朗日框架下保持收敛性？
• RQ2 与标准增量方法相比，约束违反的聚合如何改善收敛性？
• RQ3 在非凸和凸问题中，哪些条件可确保IAAL算法的收敛性？
• RQ4 该算法是否能在每次迭代仅更新一个块的情况下实现全局收敛？

主要发现

• 在较弱假设下（包括目标函数和约束函数的凸性），IAAL算法可收敛至KKT点。
• 增量更新策略相比全数据集方法显著降低了每次迭代的计算成本，使其适用于大规模问题。
• 在惩罚项中使用聚合的约束残差可提高收敛稳定性，并减少对偶变量的振荡。
• 即使每次迭代仅更新一个块，该算法仍能保持收敛性，表明其对部分更新具有鲁棒性。