On the Role of Constraints in the Complexity of Min-Max Optimization

TL;DR

It is shown that with general constraints, even convex-concave min-max optimization becomes PPAD-hard, and PPAD-membership is provided of a general problem related to quasi-variational inequalities, which has applications beyond the problem.

Abstract

We investigate the role of constraints in the computational complexity of min-max optimization. The work of Daskalakis, Skoulakis, and Zampetakis [2021] was the first to study min-max optimization through the lens of computational complexity, showing that min-max problems with nonconvex-nonconcave objectives are PPAD-hard. However, their proof hinges on the presence of joint constraints between the maximizing and minimizing players. The main goal of this paper is to understand the role of these constraints in min-max optimization. The first contribution of this paper is a fundamentally new proof of their main result, which improves it in multiple directions: it holds for degree 2 polynomials, it is essentially tight in the parameters, and it is much simpler than previous approaches, clearly highlighting the role of constraints in the hardness of the problem. Second, we show that with general constraints (i.e., the min player and max player have different constraints), even convex-concave min-max optimization becomes PPAD-hard. Along the way, we also provide PPAD-membership of a general problem related to quasi-variational inequalities, which has applications beyond our problem.

Figures (9)

• Figure 1: Summary of known results regarding the complexity of min-max optimization for a different combination of structures of utilities and constraints. The arrows point in the direction of increased generality and thus hardness.
• Figure 2: Our results nearly settle the complexity of finding local equilibria of nonconvex-nonconcave objective functions under jointly convex constraints as a function of the parameter $\delta$.
• Figure 3: The two embeddings of the field $F:\mathbb{R}^d\to\mathbb{R}^d$ into the pseudo-gradient of $f:\mathbb{R}^{2d}\to\mathbb{R}$. The box on the right is the field $x\to\nabla_x f$, while the one on top represents the field $y\to -\nabla_y f$. The embedded field is $F(y)=\left(y-\frac{1}{3}\right)\cdot\left(y-\frac{2}{3}\right)$; the blue dots are in correspondence with its zeros and are inserted only for reference. In both figures we have that the $x$-projection of the pseudo-gradient field on the $x=y$ subspace corresponds to the field $F$, while only in \ref{['fig:embedding2']}, both the $x$ and $y$ projections are aligned with the field $F$.
• Figure 4: (a) Two possible deviations of $z$ to $\tilde{z}$ and $\tilde{z}'$. A deviation towards $\tilde{z}$ is possible without violating the constraint $\lVert z-w\rVert_{\infty}\le \Delta$, while a deviation toward $\tilde{z}'$ is not feasible. (b) The deviation toward $\tilde{z}$ satisfying the constraint is achievable by either deviating from $z$ or by deviating from $w$. The brown arrow represents the gradient $\nabla_z f(z,w)$ while the orange arrow represents the gradient $-\nabla_w f(z,w)$.
• Figure 5: Neither $w$ nor $z$ can be moved towards $\tilde{z}$. However, we can decompose $\tilde{z}-z$ into $v_w$ and $v_z$ such that $v_w+v_z$ is parallel to $\tilde{z}-z$.

Theorems & Definitions (70)

• Theorem 1.1: Informal version of \ref{['th:nonconvexconcavehardness']}
• Theorem 1.2: Informal version of \ref{['th:qvihardness']}
• Corollary 1.3: Informal version of \ref{['cor:qvihard']}
• Theorem 1.4: Informal version of \ref{['th:qvippad']}
• Proposition 2.1: rubinstein2015inapproximability
• Theorem 2.2: daskalakis2021complexity
• Definition 2.3: Correspondence
• Theorem 3.1: daskalakis2021complexity
• Proposition 3.1
• Theorem 3.2: Two-way reduction between and