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Efficient Methods for Structured Nonconvex-Nonconcave Min-Max Optimization
Jelena Diakonikolas, Constantinos Daskalakis, Michael I. Jordan
TL;DR
Smooth nonconvex-nonconcave min-max optimization can be intractable, motivating structural assumptions that permit efficient computation. The paper introduces a generalized extragradient method for such structured problems, with convergence results extending to ℓp norms and stochastic settings. It also identifies scope limits, including weaker results in some ℓp regimes and leaving constrained generalizations for future work.
Problem
Nonconvex-nonconcave min-max optimization can remain computationally intractable even for smooth objectives and approximate first-order solutions.
Method
The paper studies min-max problems as variational inequalities and proposes a generalized extragradient method under a structural condition weaker than existence of an MVI solution.
Results
The method’s accumulation points are SVI solutions, with convergence results also established for ℓp-normed settings and stochastic oracle access.
Takeaways & Limitations
Structured nonconvex-nonconcave problems can admit provably convergent algorithms despite the intractability of general smooth objectives.
Takeaways & Limitations
The analysis leaves constrained generalizations for future work and omits a weak local-convergence result for p > 2 whose usefulness is unclear.
Abstract
from arXiv · showhide
The use of min-max optimization in adversarial training of deep neural network classifiers and training of generative adversarial networks has motivated the study of nonconvex-nonconcave optimization objectives, which frequently arise in these applications. Unfortunately, recent results have established that even approximate first-order stationary points of such objectives are intractable, even under smoothness conditions, motivating the study of min-max objectives with additional structure. We introduce a new class of structured nonconvex-nonconcave min-max optimization problems, proposing a generalization of the extragradient algorithm which provably converges to a stationary point. The algorithm applies not only to Euclidean spaces, but also to general $\ell_p$-normed finite-dimensional real vector spaces. We also discuss its stability under stochastic oracles and provide bounds on its sample complexity. Our iteration complexity and sample complexity bounds either match or improve the best known bounds for the same or less general nonconvex-nonconcave settings, such as those that satisfy variational coherence or in which a weak solution to the associated variational inequality problem is assumed to exist.
1 Introduction
Nonconvex-nonconcave min-max optimization is difficult even for smooth objectives, so the paper identifies structural conditions enabling efficient solution methods. It proposes a generalized extragradient approach for structured problems and extends the analysis to broader norms and stochastic settings.
- Motivation: Smooth nonconvex-nonconcave min-max problems can be computationally intractable, even when approximate first-order solutions are sought.The paper relates this difficulty to intractability results for constrained problems and unconstrained variational inequalities.
- Contributions: The paper seeks structural properties that make smooth min-max optimization solvable and formulates the problem through unconstrained variational inequalities.The unconstrained min-max problem is treated as a special case of the Stampacchia variational inequality problem.
- Contributions: The generalized extragradient method converges to a variational inequality solution, or a stationary point in the min-max special case, under a weaker condition than existence of an MVI solution.The main result uses an aggressive interpolation step and achieves a 1/√k convergence rate under the paper’s weaker assumption.
- Extensions and scope: The results extend to general ℓp-normed settings and stochastic oracle access, while related constrained setups remain outside the paper’s scope.The paper compares deterministic iteration complexities and defers constrained generalizations to future work.
- Contributions: The weaker structural condition includes negatively comonotone and positively cohypomonotone operators, for which the paper provides rates matching those under the stronger MVI assumption.The paper states that these rates are asymptotically identical to the stronger-assumption rates and avoid the cited intractability results for general smooth objectives.
2 Notation and Preliminaries
The paper formulates smooth nonconvex-nonconcave min-max optimization through variational inequalities and introduces weak MVI structure that is weaker than standard MVI assumptions. It also develops the notation and convex-analytic tools used for Euclidean and ℓp-normed analyses.
- Min-max formulation: Stationary points are points where the gradient of the smooth min-max objective is zero, with ε-approximate stationarity defined by ∥∇f(x, y)∥p* ≤ ε.The objective may be nonconvex-nonconcave, and d1 + d2 = d.
- Variational inequalities: Without monotonicity, MVI solutions form a subset of SVI solutions and may not exist even when SVI solutions do.The paper therefore assumes at least one SVI solution and introduces additional structure beyond Lipschitzness.
- Weak MVI structure: Weak MVI structure is weaker than requiring an MVI solution and includes operators that are negatively comonotone or positively cohypomonotone.When ρ = 0, the framework recovers MVI-based settings including variationally coherent and several structured min-max objective families.
- Implications and examples: Assumption 1 does not imply convex-concavity or monotonicity, even near an SVI solution, and permits local nonconvex behavior controlled by the gradient.A concrete ratio-game example has SVI solutions satisfying weak MVI while lacking MVI solutions and cohypomonotonicity.
- Convex-analytic tools: The analysis uses uniform convexity, convex conjugates, and Bregman divergence, with uniform convexity generalizing strong convexity beyond the Euclidean case.These tools support the later ℓp-norm analysis.
3 Generalized Extragradient for Problems with Weak MVI Solutions
The paper proposes Extragradient+ for Lipschitz operators satisfying weak MVI structure and proves convergence of its iterates toward SVI solutions. The result includes bounded iterates, vanishing operator residuals, and a locally sufficient structural assumption.
- Algorithm: Extragradient+ generalizes standard extragradient through an interpolation parameter β, with β = 1 recovering standard EG.The method is analyzed using a merit or gap function and tunable step sizes.
- Convergence analysis: Theorem 3.2 applies to arbitrary L-Lipschitz operators satisfying Assumption 1 and uses parameter choices that ensure the required descent conditions.The proof bounds the merit-function sequence and controls the operator residual.
- Convergence guarantee: All accumulation points of the extrapolated iterates are SVI solutions.The proof shows that the operator residual converges to zero and uses boundedness to extract convergent subsequences.
- Rates: The theorem also provides an iteration-dependent bound on the residual and a corresponding bound for a uniformly selected iterate.These guarantees follow by telescoping the one-step inequality.
- Locality: The iterates remain within a ball of radius at most 2∥u0 − u∗∥ around a solution, so Assumption 1 need only hold locally on that ball.This is a scope relaxation from requiring the weak MVI condition globally.
- Existence: The analysis further shows that Assumption 1 itself guarantees existence of an SVI solution under the stated Lipschitz conditions.The argument extends beyond choosing u∗ from the SVI solution set.
4 Extensions: ℓp Norms and Stochastic Setups
The paper extends EG+ to arbitrary ℓp norms and stochastic evaluations, with convergence guarantees under Assumption 1. The generalized method differs from standard Extragradient and Mirror-Prox and includes deterministic and stochastic complexity bounds.
- Scope: The extension covers non-Euclidean ℓp-normed setups for p ∈ (1, ∞) and stochastic evaluations of F.The analysis allows different uniform-convexity regimes depending on p.
- Algorithm: EGp+ generalizes EG+ to arbitrary ℓp norms and is equivalent to EG+ when p = 2.Unlike standard Extragradient or Mirror-Prox, its interpolation and update steps need not correspond to a Bregman divergence except in special cases.
- Deterministic convergence: Under exact oracle access and Assumption 1 with ρ = 0, suitable parameters ensure all accumulation points of the iterates lie in U∗.The theorem applies to L-Lipschitz operators with p ∈ (1, ∞).
- Deterministic convergence: For any ε > 0, EGp+ can output u with ∥F(u)∥p∗ ≤ ε within a stated iteration bound whose hidden constants depend only on p.The guarantee is obtained by selecting δ proportional to ε^2 and using the theorem’s parameter choices.
- Scope boundary: The analysis faces a limitation for p > 2: with ρ > 0, only a local-convergence result with ρ scaling polynomially in ε is known and is omitted as weak.For p > 2, the difficulty involves the relationship between step sizes and error terms.
- Stochastic oracle: The stochastic extension provides guarantees for expected residuals, allowing EGp+ to output u with E[∥˜F(u)∥p∗] ≤ ε under specified parameter choices.The stochastic analysis separately bounds errors from the variance of the oracle estimates.
5 Discussion
The paper introduces a structured class of nonconvex-nonconcave min-max problems and a generalized extragradient method that converges to stationary points. It extends these results to stochastic setups and ℓp-normed settings under an (MVI) solution assumption, while identifying further assumption relaxation as future work.
- The proposed structured class generalizes nonconvex-nonconcave problems in which an (MVI) solution exists.
- The generalized extragradient method provably converges to a stationary point in Euclidean setups.
- The algorithmic results guarantee that problems in the introduced class contain at least one stationary point, corresponding to an (SVI) solution.
- The results extend to stochastic setups and ℓp-normed setups when an (MVI) solution exists.
- Further relaxing structural assumptions while retaining computational feasibility remains an open direction.