The Gaussian completely monotone (GCM) conjecture states that them-th time-derivative of the entropy along the heat flow on ℝdis positive formeven and negative formodd. We prove the GCM conjecture for orders up tom= 5, assuming that the initial measure is log-concave, in any dimension. Our proof differs significantly from previous approaches to the GCM conjecture: it is based on Otto calculus and on the interpretation of the heat flow as the Wasserstein gradient flow of the entropy. Crucial to our methodology is the observation that the convective derivative behaves as a flat connection over probability measures on ℝd. In particular we prove a form of the univariate Faa di Bruno’s formula on the Wasserstein space (despite it being curved), and we compute the higher-order Wasserstein differentials of internal energy functionals (including the entropy), both of which are of independent interest.

Let a smooth function $f: T^d \times T^d \to \mathbb{R}$ over the $d$-torus and $\beta>0$. Consider the min-max objective functional $F_\beta(\mu, \nu) = \iint f d\mu d\nu + \beta^{-1} H(\mu) - \beta^{-1} H(\nu)$ over $\mathcal{P}(T^d) \times \mathcal{P}(T^d)$, where $H$ denotes the negative differential entropy. Its unique saddle point defines the entropy-regularized mixed Nash equilibrium of a two-player zero-sum game, and its Wasserstein gradient descent-ascent flow $(\mu_t, \nu_t)$ corresponds to the mean-field limit of a Langevin descent-ascent dynamics. Do $\mu_t$ and $\nu_t$ converge (weakly, say) as $t \to \infty$, for any $f$ and $\beta$? This rather natural qualitative question is still open, and it is not clear whether it can be addressed using the tools currently available for the analysis of dynamics in Wasserstein space. Even though the simple trick of using a different timescale for the ascent versus the descent is known to guarantee convergence, we propose this question as a toy setting to further our understanding of the Wasserstein geometry for optimization.

Mean-field Langevin dynamics (MLFD) is a class of interacting particle methods that tackle convex optimization over probability measures on a manifold, which are scalable, versatile, and enjoy computational guarantees. However, some important problems -- such as risk minimization for infinite width two-layer neural networks, or sparse deconvolution -- are originally defined over the set of signed, rather than probability, measures. In this paper, we investigate how to extend the MFLD framework to convex optimization problems over signed measures. Among two known reductions from signed to probability measures -- the lifting and the bilevel approaches -- we show that the bilevel reduction leads to stronger guarantees and faster rates (at the price of a higher per-iteration complexity). In particular, we investigate the convergence rate of MFLD applied to the bilevel reduction in the low-noise regime and obtain two results. First, this dynamics is amenable to an annealing schedule, adapted from [Suzuki et al., 2023], that results in polynomial convergence rates to a fixed multiplicative accuracy. Second, we investigate the problem of learning a single neuron with the bilevel approach and obtain local exponential convergence rates that depend polynomially on the dimension and noise level (to compare with the exponential dependence that would result from prior analyses).

We study the convergence to local Nash equilibria of gradient methods for two-player zero-sum differentiable games. It is well-known that, in the continuous-time setting, such dynamics converge locally when $S \succ 0$ and may diverge when $S=0$, where $S\succeq 0$ is the symmetric part of the Jacobian at equilibrium that accounts for the "potential" component of the game. We show that these dynamics also converge as soon as $S$ is nonzero (*partial curvature*) and the eigenvectors of the antisymmetric part $A$ are in general position with respect to the kernel of $S$. We then study the convergence rate when $S \ll A$ and prove that it typically depends on the *average* of the eigenvalues of $S$, instead of the minimum as an analogy with minimization problems would suggest. To illustrate our results, we consider the problem of computing mixed Nash equilibria of continuous games. We show that, thanks to partial curvature, conic particle methods -- which optimize over both weights and supports of the mixed strategies -- generically converge faster than fixed-support methods. For min-max games, it is thus beneficial to add degrees of freedom "with curvature": this can be interpreted as yet another benefit of over-parameterization.