关于约束优化中用于更新拉格朗日乘子的PI控制器
On PI Controllers for Updating Lagrange Multipliers in Constrained Optimization
AI总结：
本文提出νPI算法，利用PI控制器更新约束优化中的拉格朗日乘子，解决了梯度下降-上升法的振荡不稳定问题，并证明其推广了单目标动量方法且超参数行为稳健可预测。
摘要：
约束优化为在神经网络模型中规定期望行为提供了一个强大的框架。通常，约束问题通过其最小-最大拉格朗日公式来求解，当使用梯度下降-上升法优化时，这些公式会表现出不稳定的振荡动态。目前，机器学习社区对约束优化技术的采用受到缺乏可靠、通用的拉格朗日乘子更新方案的限制。本文提出了νPI算法，并基于PI控制器对拉格朗日乘子更新提供了一个优化视角，扩展了Stooke、Achiam和Abbeel（2020）的工作。我们提供了理论和实证见解，解释了动量方法无法解决梯度下降-上升法的缺陷，并将其与我们提出的νPI控制器的实证成功进行了对比。此外，我们证明了νPI推广了用于单目标最小化的流行动量方法。我们的实验表明，νPI能够可靠地稳定乘子动态，并且其超参数具有鲁棒且可预测的行为。
英文摘要：
Constrained optimization offers a powerful framework to prescribe desired behaviors in neural network models. Typically, constrained problems are solved via their min-max Lagrangian formulations, which exhibit unstable oscillatory dynamics when optimized using gradient descent-ascent. The adoption of constrained optimization techniques in the machine learning community is currently limited by the lack of reliable, general-purpose update schemes for the Lagrange multipliers. This paper proposes the $ν$PI algorithm and contributes an optimization perspective on Lagrange multiplier updates based on PI controllers, extending the work of Stooke, Achiam and Abbeel (2020). We provide theoretical and empirical insights explaining the inability of momentum methods to address the shortcomings of gradient descent-ascent, and contrast this with the empirical success of our proposed $ν$PI controller. Moreover, we prove that $ν$PI generalizes popular momentum methods for single-objective minimization. Our experiments demonstrate that $ν$PI reliably stabilizes the multiplier dynamics and its hyperparameters enjoy robust and predictable behavior.
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