  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 $\nu$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 $\nu$PI controller. Moreover,
we prove that $\nu$PI generalizes popular momentum methods for single-objective
minimization. Our experiments demonstrate that $\nu$PI reliably stabilizes the
multiplier dynamics and its hyperparameters enjoy robust and predictable
behavior.
