[论文解读] Learning for Convex Optimization

Sidhant Misra, Line Roald+1 more|arXiv (Cornell University)|Feb 26, 2018

Machine Learning and Algorithms被引用 8

一句话总结

本文提出一种基于学习的方法，通过直接预测最优解的活跃约束集，加速凸优化，利用理论支持的流式算法。在最优功率分配基准测试中，通常仅有少数活跃约束集是相关的，使其成为实时决策系统中可扩展且可解释的抽象方法。

ABSTRACT

In many engineered systems, optimization is used for decision making at time-scales ranging from real-time operation to long-term planning. This process often involves solving similar optimization problems over and over again with slightly modified input parameters, often under stringent time requirements. We consider the problem of using the information available through this solution process to directly learn the optimal solution as a function of the input parameters, thus reducing the need of solving computationally expensive large-scale parametric programs in real time. Our proposed method is based on learning relevant sets of active constraints, from which the optimal solution can be obtained efficiently. Using active sets as features preserves information about the physics of the system, enables more interpretable learning policies, and inherently accounts for relevant safety constraints. Further, the number of relevant active sets is finite, which make them simpler objects to learn. To learn the relevant active sets, we propose a streaming algorithm backed up by theoretical results. Through extensive experiments on benchmarks of the Optimal Power Flow problem, we observe that often only a few active sets are relevant in practice, suggesting that this is the appropriate level of abstraction for a learning algorithm to target.

研究动机与目标

• 通过直接从重复的问题实例中学习最优解，减少实时优化中的计算开销。
• 识别一种紧凑且可解释的抽象——活跃约束集，以保留系统物理特性和安全约束。
• 开发一种流式学习算法，以高效识别并泛化相关活跃约束集。
• 验证在实践中通常仅需少量活跃约束集，从而实现高效学习。

提出的方法

• 该方法通过识别相关活跃约束集，将最优解表示为输入参数的函数。
• 它使用流式算法，从连续的优化解中逐步学习活跃约束集。
• 活跃约束集被用作特征，保留了问题结构中固有的物理和安全约束。
• 理论保证支持流式学习过程的收敛性和正确性。
• 该方法通过直接将输入映射到活跃约束集，避免了反复求解大规模参数化规划问题。
• 该方法在最优功率分配基准问题上进行评估，以检验其可扩展性和准确性。

实验结果

研究问题

• RQ1 活跃约束集能否作为学习凸优化中最优解的有效且可解释的抽象？
• RQ2 在实际应用中，对于现实世界的优化问题，通常有多少个活跃约束集是相关的？
• RQ3 流式学习算法能否有效识别并泛化来自连续解的相关活跃约束集？
• RQ4 在时间受限的实时应用中，基于活跃约束集的学习方法是否优于传统优化方法？

主要发现

• 即使在像最优功率分配这样复杂的问题中，实践中通常仅有少量活跃约束集是相关的。
• 将活跃约束集用作特征可保留物理系统约束，并提高所学策略的可解释性。
• 所提出的流式算法成功识别了相关活跃约束集，并具有理论保证。
• 该方法显著减少了在实时环境中重新求解计算成本高昂的参数化规划问题的需求。