Completing the Bolley–Guillin–Villani quantitative concentration program: dependent data, time discretization, phase-space truncation, and unbounded interaction curvature

Bolley, Guillin and Villani (2007) proved non-asymptotic Sanov-type deviation bounds in Wasserstein distance for the empirical measure of independent samples (under transport-entropy inequalities and square-exponential moments) and of a McKean–Vlasov mean-field particle system with bounded interaction Hessian (their Theorem 2.9); their Section 2.7 lists five further developments. This problem concerns four of these five directions; the fifth, path-space concentration, was settled by Bolley (2010). Deviation inequalities for Markov-chain empirical measures exist for uniformly or exponentially contractive chains (Boissard 2011; Riekert 2022) but are not known for general geometrically ergodic chains; uniform-in-time concentration for explicit-Euler-type discrete-time particle systems is known (Budhiraja–Pal Majumder 2015) but not for the implicit-Euler granular-media setting; the phase-space-truncation direction has no dedicated quantitative treatment; and the unbounded-$D^2W$ (cubic interaction) direction is open, adjacent progress on singular kernels and uniform functional inequalities concerning other settings. The open core is to extend BGV-type quantitative deviation bounds to these four settings.

Background

For $N\ge 1$ the empirical measure of $\mathbb{R}^d$-valued random variables $X_1,\dots,X_N$ is $\hat\mu^N=(1/N)\sum_{i=1}^N\delta_{X_i}$, a random probability measure. Natural metrics quantifying its deviation from the common law $\mu$ are the Wasserstein (optimal-transport) distances: for $p\ge 1$, $W_p(\mu,\nu)=(\inf\mathbb{E}|X-Y|^p)^{1/p}$ over couplings of $\mu$ and $\nu$; by Kantorovich duality $W_1(\mu,\nu)=\sup\{\int f\,d(\mu-\nu): f \text{ 1-Lipschitz}\}$, so a deviation bound in $W_p$ simultaneously controls the Monte Carlo error of all Lipschitz observables.

Sanov's theorem states that for i.i.d. samples the empirical measure satisfies a large deviation principle with speed $N$ and rate function given by the relative entropy $H(\nu|\mu)$ (the Kullback–Leibler divergence of $\nu$ from $\mu$); it is asymptotic and qualitative. Bolley, Guillin and Villani (PTRF 137:541–593, 2007; cited paper below) established quantitative (finite-$N$) versions. Their Theorem 2.1: if $\mu$ satisfies a transport-entropy (Talagrand) inequality $T_p(\lambda)$, i.e. $W_p(\nu,\mu)\le\sqrt{2H(\nu|\mu)/\lambda}$ for all $\nu$, and has a finite square-exponential moment $\int e^{\alpha_0|x|^2}\,d\mu(x)<\infty$ (a condition which in fact follows from $T_p(\lambda)$, since $T_p$ implies $T_1$ and $T_1$ is equivalent to the existence of a square-exponential moment), then for $p\in[1,2]$, any $d'>d$, $\lambda'<\lambda$ there is $N_0$ (depending only on $\lambda'$, $d'$ and a square-exponential moment of $\mu$) such that for all $\varepsilon>0$ and $N\ge N_0\max(\varepsilon^{-(d'+2)},1)$,

$$P\bigl(W_p(\mu,\hat\mu^N)>\varepsilon\bigr)\le e^{-\gamma_p\frac{\lambda'}{2}N\varepsilon^2},$$

with $\gamma_p$ an explicit constant.

The second half of the cited paper concerns the mean-field (McKean–Vlasov) particle system on $\mathbb{R}^d$: $N$ coupled diffusions $dX_t^i=\sqrt{2}\,dB_t^i-\nabla V(X_t^i)\,dt-\frac1N\sum_{j=1}^N\nabla W(X_t^i-X_t^j)\,dt$ with confining potential $V$ and interaction potential $W$ (here $W$ is a function on $\mathbb{R}^d$ describing pair interaction, not the Wasserstein distance). Their empirical measures $\hat\mu_t^N$ converge, as $N\to\infty$, to the solution $\mu_t$ of the granular media (McKean–Vlasov) equation — an example of propagation of chaos, the phenomenon that any fixed $k$ of the particles become asymptotically i.i.d. with law $\mu_t$. BGV Theorem 2.9: if $V$ and $W$ have locally bounded Hessians with $D^2V(x)\ge\beta I$ and $\gamma I\le D^2W(x)\le\gamma'I$ for all $x$ (so the interaction force $\nabla W$ is globally Lipschitz; the confining force $\nabla V$ need not be — it is only assumed to grow at most like $e^{a|x|^2}$ for every $a>0$), $W$ is symmetric in the sense $W(-z)=W(z)$, and $\mu_0$ has a finite square-exponential moment, then for each $T\ge 0$ and any $d'>d$ there are constants $C,K,N_0$ such that $N\ge N_0\max(\varepsilon^{-(d'+2)},1)$ implies

$$P\Bigl(\sup_{0\le t\le T}W_1(\hat\mu_t^N,\mu_t)>\varepsilon\Bigr)\le C(1+T\varepsilon^{-2})e^{-KN\varepsilon^2},$$

with a uniform-in-time variant (Theorem 2.12) when $\beta>0$ and $\beta+2\gamma>0$. Malrieu (2003) studied the same system, including concentration for single observables of its Euler time discretization; BGV treat the full empirical measure. Fournier–Guillin (2015) later settled the matching sharp expectation-level rates $\mathbb{E}W_p(\mu,\hat\mu^N)$.

Section 2.7 of the cited paper ('Remarks and further developments') lists five open extensions: (1) dependent sequences / Markov chains; (2) time-discretization of the particle system (e.g. implicit Euler); (3) errors from truncating the non-compact phase space; (4) interaction potentials with unbounded $D^2W$, the prototypical case being the cubic potential $W(z)=|z|^3/3$ of one-dimensional granular media (whose force $\nabla W(z)=|z|z$ and Hessian $D^2W(z)=2|z|$ are unbounded); (5) concentration on path space. This problem asks for the completion of this program in directions (1)–(4). Terminology for the directions: the occupation measure of a Markov chain is its time-averaged empirical distribution along a single trajectory, $(1/n)\sum_{k=1}^n\delta_{X_k}$ — the dependent-sequence analogue of the empirical measure; a chain is contractive (Dobrushin coefficient $r<1$) if $W_1(P(x,\cdot),P(y,\cdot))\le r|x-y|$ uniformly; it is geometrically ergodic if a total-variation mixing bound holds at geometric speed (no uniform $r<1$ required); implicit Euler is the time-discretization scheme solving at each step a nonlinear equation involving the current state, as opposed to the explicit scheme.

Problem Statement

Complete the quantitative-Sanov program of Bolley–Guillin–Villani (2007, Section 2.7) by removing, one restriction at a time, the structural assumptions under which its non-asymptotic concentration bounds for empirical measures are proved, while keeping the same style of conclusion: an explicit non-asymptotic inequality bounding the probability that an empirical measure deviates from its limit by more than $\varepsilon$ in a Wasserstein-type distance, with explicit dependence on $N$, $\varepsilon$, the dimension and the integrability/curvature constants, in the manner of BGV Theorems 2.1 and 2.9. Four of the five directions listed in the cited paper remain open (the fifth, path-space concentration, is resolved by Bolley 2010):

(a) Dependent data. Establish BGV-type non-asymptotic deviation inequalities in $W_1$ for the empirical (occupation) measure of a dependent sequence — in the first instance a Markov chain on a non-compact state space converging to its invariant measure — under appropriate ergodicity and integrability conditions, covering at least the geometrically ergodic case where no uniform Dobrushin contraction constant $r<1$ is available; clarify how the $N$-dependence degrades relative to the i.i.d. case as a function of the mixing rate.

(b) Time discretization. For the time-discretized mean-field particle system — in particular the implicit Euler scheme for the granular-media system of BGV Theorem 2.9 — establish concentration estimates for the full empirical measure on the non-compact space under BGV-type integrability (square-exponential moment of $\mu_0$), with the additional error due to the time step made explicit, ideally uniformly in time in the regime $\beta>0$, $\beta+2\gamma>0$.

(c) Phase-space truncation. Quantify the concentration and error introduced by replacing the non-compact phase space by a bounded region in numerical implementations: deviation bounds for the truncated empirical measure in terms of the truncation radius and the tail of the limit measure.

(d) Unbounded interaction curvature. Relax the assumption $\gamma I\le D^2W\le\gamma'I$ of BGV Theorem 2.9 to allow interaction potentials with Hessian unbounded above — the prototypical case being convex $W$ of cubic growth, $W(z)=|z|^3/3$ (force $\nabla W(z)=|z|z$, locally but not globally Lipschitz), keeping the assumptions on $V$ from Theorem 2.9 ($D^2V\ge\beta I$ and the growth condition on $\nabla V$) and the square-exponential moment of $\mu_0$ — and establish the analogous finite-$N$ concentration bounds for $W_1(\hat\mu_t^N,\mu_t)$ (uniformly in time where $\beta>0$, $\beta+2\gamma>0$).

These four directions are instances of a single objective (the completion of the program). Each of (a)–(d) is an open direction of that program, whose resolution is the corresponding BGV-type non-asymptotic deviation bound under the hypotheses of that direction, or a proof that no such bound can hold under those hypotheses. A result for one direction resolves that direction only.

References

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