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Semi-discrete quadratic Wasserstein energy and state-dependent Langevin exploration

Ran Gu, Gaoyue Guo, Kelvin Shuangjian Zhang

arXiv:2609.03405v1math.NAmath.OCmath.PR

TL;DR

The paper addresses how to analyze a nonsmooth semi-discrete Wasserstein energy and use it for state-dependent Langevin control and exploration. It develops collision-aware regularity and control analyses, proving well-posed collision-free dynamics, an HJB characterization, and geometrically ergodic Euler chains with full-support invariant laws.

  • Problem

    The paper studies regularity and optimization for a Wasserstein energy that is generally nondifferentiable at site collisions, alongside optimal temperature control and global exploration.

    Source: §1 · p. 2

  • Method

    It combines a global splitting formulation with classical Laguerre-cell differentiation on collision-free configurations, then applies entropy-regularized control and Markov-chain stability analysis.

    Source: §1 · p. 2, §3 · p. 23, §4.3 · p. 47

  • Results

    The analysis establishes collision-free strong solutions, an HJB viscosity characterization, and geometrically ergodic Euler chains with unique full-support invariant laws.

    Source: §1.3 · pp. 7–8, §3.4 · p. 39

  • Takeaways & Limitations

    With a strictly positive temperature floor, raw iterates are not expected to converge to a minimizer, so the running record is the relevant exploration outcome.

    Source: §1.2 · p. 4

    The discrete exploration result assumes a temperature rule bounded away from zero, so it does not target raw-iterate convergence to a minimizer.

    Source: §1.2 · p. 4

Abstract

from arXiv · show

We study the semi-discrete quadratic Wasserstein energy. The energy is nonsmooth at collisions of sites. We prove local Lipschitz continuity on the full configuration space, together with global semiconcavity, coercivity, and dissipativity; show that every global minimizer is interior and collision free; and establish $C^2$ regularity on the collision-free configuration space. The gradient is expressed through the barycenters of the balanced Laguerre cells, while the Hessian is given by an explicit facet formula and satisfies a global one-sided bound. We also solve the one-dimensional problem explicitly in each ordering chamber and give a two-site example on the unit square with non-minimizing Lloyd fixed points. For $d\ge 2$, we then formulate an entropy-regularized relaxed control of the Langevin temperature. The controlled dynamics is strongly well posed, nonexplosive, and collision free. Its value function is a classical interior solution of the exploratory Hamilton-Jacobi-Bellman equation; the Laplacian of the value function is locally $C^1$, which yields a locally Lipschitz optimal temperature feedback. Independently of this optimal-control result, for every fixed Borel temperature rule bounded away from zero, and every sufficiently small step size, the associated Gaussian Euler chain is geometrically ergodic with a full-support invariant law. The raw iterates do not converge, whereas the best-so-far energy converges almost surely to the global minimum and the running record approaches the set of global minimizers.

1 Introduction

This section develops the regularity theory for semi-discrete quadratic Wasserstein energy and applies it to entropy-regularized Langevin temperature control and persistent Euler exploration. It outlines the scope and limitations of the optimal feedback, illustrates escape from non-global Lloyd configurations, and previews the paper’s analytical, control, and Euler-scheme results.

1.3 Main results

  • Weighted Lloyd iterations can remain trapped at non-minimizing critical points, whereas positive-temperature Euler exploration escapes local basins and its running record reaches the global minimum.For eight equally weighted sites on the unit square, Lloyd stagnates at a high-energy local minimizer while fixed-temperature Euler exploration achieves a significantly lower running-record energy; this does not establish finite-time superiority of HJB feedback.Source: §1 · p. 2, §1.3 · p. 9

1.1 Problem formulation

  • The paper formulates fixed-mass semi-discrete Wasserstein minimization over site configurations, with collisions handled by a splitting representation valid on the full configuration space.Away from collisions, optimal splittings are induced by balanced Laguerre cells and admit differential analysis.Source: §1.1 · p. 3

1.2 Relation to previous work and contributions

  • Its energy theory proves collision-valid semiconcavity, coercivity, dissipativity, explicit supergradients, collision-free global minimizers, and a symmetric facet-form Hessian with a global one-sided bound.In one dimension, the unique minimizer in each ordering chamber is given explicitly through target quantiles; the unit-square two-site problem is classified separately.Source: §1.2 · p. 4, §1.3 · pp. 5–6
  • For d ≥2, the entropy-regularized control problem yields a classical interior exploratory HJB solution with locally C^1 Laplacian, locally Lipschitz optimal temperature feedback, and collision-free strong dynamics.The analysis uses global dissipativity and polarity of collision diagonals instead of bounded gradients or artificial boundary conditions.Source: §1.2 · p. 4, §1.3 · p. 7
  • For any fixed Borel temperature rule bounded in [a, 1] and sufficiently small step size, the Gaussian Euler chain is geometrically ergodic with a full-support invariant law, while its running record converges almost surely to the global minimum.The raw iterates do not converge to a finite limit; the theorem applies to the exact feedback, clipped approximations, and constant positive temperatures.Source: §1.2 · pp. 4–5, §1.3 · p. 8

1.3 Main results

  • The optimal relaxed feedback τλ uses dissipative confinement and uniformly positive Gaussian exploration but does not establish hitting-time or finite-time-record optimality.It also does not compare convergence rates with constant temperatures or annealing schedules, and concerns only homogeneous state-feedback chains.Source: §1.3 · p. 9
  • The Langevin scheme escapes a non-global two-row Lloyd fixed point and reaches the numerically identified global minimum in the running-record energy.For N = 8 sites, the comparison uses η = 0.4 and τ = 5 × 10−4; Lloyd energy stabilizes at a non-global critical value.Source: Figure 1 · p. 10

1.4 Organization and notation

  • The paper organizes its contributions across global and Laguerre analysis, relaxed control and HJB verification, and Euler-scheme ergodicity and running-record convergence.Appendix A verifies the joint semi-dual differentiation input.Source: §1.4 · p. 11

2 Regularity of the semi-discrete energy

This section develops the regularity, coercivity, and differentiability theory of the semi-discrete energy, including its gradient and Hessian structure on collision-free configurations. It also treats explicit low-dimensional examples, including the one-dimensional problem and non-minimizing fixed points in two dimensions.

2.1 Local Lipschitz continuity, semiconcavity, supergradients, and dissipativity

  • E is continuous and locally Lipschitz, globally semiconcave through convexity of P^−2E, with supergradients having linear growth on the full configuration space.The linear-growth estimate, rather than globally bounded gradients, is the key condition for later nonexplosion and ergodicity.Source: §2.1 · pp. 11–13

2.2 Reduction of the minimization domain and proof of Theorem 1.1

  • Every global minimizer exists in the interior, is collision free, and the compact nonempty minimizer set lies within the bounded configuration domain.Projection reduces minimization to the compact domain, while barycenter arguments exclude boundary sites and collisions.Source: §2.2 · pp. 13–14
  • On the collision-free configuration space, normalized balancing weights are unique and continuous, and E is C^1 with gradient given by balanced Laguerre-cell barycenters.The superdifferential is therefore a singleton on this space.Source: §2.2 · pp. 14–15
  • For two equal-mass sites on [0,1], E is smooth within each ordering chamber but nondifferentiable on the collision diagonal x1 = x2.This example motivates restricting classical Hessian statements to collision-free configurations.Source: §2.2 · p. 15
  • The second derivative on collision-free configurations is represented explicitly by balanced-facet integrals, with a nonnegative bilinear form appearing in the Hessian formula.The balancing-weight derivative is characterized through the weighted Laguerre graph Laplacian.Source: §2.2 · pp. 16, 18
  • On the collision-free configuration space, E is C2, its gradient is given componentwise by cell centroids, and D2E is continuous and uniformly controlled on compact subsets.The proof derives these properties from differentiability of balancing weights and centroids, with a global one-sided Hessian bound.Source: §2.2 · p. 20

2.3 The explicit one-dimensional problem

  • In one dimension, each ordering chamber has a strictly convex quadratic restriction with a unique barycentric minimizer, and every global minimizer is collision free and equals one chamber minimizer.For equal weights, the unlabeled optimal atomic measure is unique, while labeled minimizers are unique only up to permutations.Source: §2.3 · p. 21

2.4 A two-dimensional saddle-point example

  • For two equally weighted sites on the unit square, axis-aligned configurations are exactly the global minimizers, whereas diagonal critical configurations and label exchanges are saddle points.The example shows that convergence of deterministic Lloyd iteration to a critical point does not imply global optimization.Source: §2.4 · p. 22

3 Exploratory temperature control

This section develops entropy-regularized relaxed temperature control on the collision-free configuration space, establishing well-posedness, collision avoidance, dynamic programming, and regularity of the value function. It proves that the value function satisfies the exploratory Hamilton–Jacobi–Bellman equation classically in the interior and admits an admissible feedback yielding a global, collision-free controlled process.

3 Exploratory temperature control

  • The section’s additional analytical challenges are the punctured state space, merely local Lipschitz drift, collision avoidance, and the bootstrap needed for locally Lipschitz feedback.The entropy-regularized relaxed-control setup follows the exploratory-control framework of and exploratory HJB work.Source: §3 · p. 23

3.1 Relaxed temperature controls and the canonical formulation

  • The relaxed formulation uses raw-progressive probability-valued temperature controls, discounted entropy-regularized costs, and a smooth strictly increasing, strictly concave scalar Hamiltonian.The Gibbs law is the unique minimizer of the entropy-penalized Hamiltonian representation, while the lower temperature bound supplies the exploration floor and uniform ellipticity.Source: §3.1 · pp. 24–25

3.2 Well-posedness, localization, and continuity

  • Controlled dynamics admit a unique global strong solution that remains collision free, with uniform second-moment bounds over all relaxed controls.The proof combines localization, nonexplosion estimates, and a Girsanov comparison reducing collision attainment to hitting a point by multidimensional Brownian motion.Source: §3.2 · pp. 26, 29
  • Uniform fourth-moment and near-collision estimates provide compact localization over bounded initial sets, supporting continuity of costs and value functions.The localization is uniform over initial states in compact subsets and over all relaxed controls, and is also used for later Euler approximation.Source: §3.2 · pp. 29–30
  • The value function vλ is finite and continuous on the collision-free configuration space, with compact-set equicontinuity for controls having finite cost at one point.The proof combines growth bounds, localization, and coupling of trajectories driven by the same Brownian motion.Source: §3.2 · pp. 31, 33

3.3 Canonical shifts, concatenation, and dynamic programming

  • Canonical shifts, concatenation, and countable near-optimal continuation selectors yield dynamic programming identities at bounded stopping times and control-dependent localized exit times.The identities apply when the relevant displayed expectation is finite; controls with infinite expectation may be omitted from the infimum.Source: §3.3 · p. 35

3.4 The exploratory Hamilton–Jacobi–Bellman equation

  • The value function satisfies the exploratory HJB equation pointwise in the interior, and its Laplacian-related expression is locally C1 after interior viscosity regularity is established.The additional derivative follows from the HJB identity and the smooth diffeomorphism Hλ, rather than from C2,β regularity alone.Source: §3.4 · p. 37
  • The value function is a viscosity solution, and the HJB equation holds classically on the collision-free configuration space.Interior regularity also provides the differentiability of its Laplacian needed for the subsequent feedback construction.Source: §3.4 · p. 39
  • The minimizing temperature feedback is locally Lipschitz, admissible, and generates a global strong solution that remains collision free.The proof combines local Lipschitz coefficients, Lyapunov estimates, and a Girsanov-based pairwise collision argument.Source: §3.4 · p. 39
  • The feedback attains the value function: Jλ(X; π∗) = vλ(X), while every finite-cost control has cost at least vλ(X).The feedback cost is finite, and verification follows by applying the classical HJB equation with stopped-process estimates.Source: §3.4 · p. 40
  • The argument obtains the extra Laplacian derivative through algebraic inversion of the trace-form HJB equation, avoiding full C3 estimates for the value function.It also avoids assuming a Hölder modulus for moving-facet second derivatives of E.Source: §3.4 · p. 40
  • The theorem does not claim uniqueness among arbitrary classical solutions on the punctured state space without growth conditions or data on the collision set.The stated numerical implication is to approximate the discounted value-function solution through consistent finite-domain stochastic representations.Source: §3.4 · p. 40

4 Langevin discretization and global record convergence

The section separates finite-horizon Euler consistency for the exact HJB feedback from a dimension-independent long-time theorem for any fixed Borel temperature rule bounded away from zero. The chain is geometrically ergodic and collision free, while its running record converges to global minimizers although the raw iterates do not.

4 Langevin discretization and global record convergence

  • For every fixed Borel temperature rule in [a,1], the homogeneous Euler chain has long-time record convergence independently of HJB optimality, in every dimension d ≥1.The guarantee applies to exact feedback and fixed clipped Borel approximations, but not to time-varying or history-dependent rules.Source: §4 · p. 40, §4.1 · p. 42

4.1 The Euler exploration algorithm

  • The Euler update combines an under-relaxed weighted Lloyd step with nondegenerate Gaussian exploration, and it remains collision free almost surely at every iteration.The Gaussian increment has full ambient support, while the collision set is Lebesgue-null; this also permits jumps across collision hyperplanes when d = 1.Source: §4.1 · pp. 41–42

4.2 Finite-horizon consistency of the exact-feedback scheme

  • For d ≥2, the exact-feedback Euler interpolation converges to the feedback diffusion on every finite time horizon, uniformly in probability for initial conditions in compact collision-free sets.The proof localizes to collision-free domains, applies a standard Euler estimate to globally Lipschitz truncated coefficients, and then removes the localization.Source: §4.2 · pp. 42–44

4.3 Geometric ergodicity and convergence of the record

  • For sufficiently small step size, every fixed Borel temperature rule bounded below by a > 0 yields a unique full-support invariant law and geometric ergodicity in total variation.Global dissipativity supplies a Foster–Lyapunov drift, while the strictly positive Gaussian density gives irreducibility, aperiodicity, and small sets.Source: §4.3 · pp. 46–47
  • The running minimum energy converges almost surely to the global minimum and the running record approaches the global minimizer set, whereas persistent Gaussian exploration prevents raw-iterate convergence.The positive temperature floor is essential for irreducibility and full support; the theorem does not claim finite-time optimality or invariant-measure concentration as η ↓0.Source: §4.3 · pp. 47–48

A Joint C2 regularity of the semi-dual

On the positive-cell parameter set, the semi-dual K is jointly C2 in sites and weights, with a jointly C1 mass map. The appendix verifies the required Laguerre-cell hypotheses and derives continuous moving-facet differentiation formulas, including moment derivatives and compact-uniform consequences.

A Joint C2 regularity of the semi-dual

  • Proposition A.1 establishes that the positive-cell set O+ is relatively open and that the semi-dual K belongs to C2(O+).The result is obtained by verifying the hypotheses of the semi-discrete regularity theorem [4].Source: A Joint C2 regularity of the semi-dual · pp. 48, 50
  • Cell-volume continuity makes O+ relatively open, while a neighborhood argument preserves pairwise distinct sites and positive Laguerre-cell volumes.These properties hold throughout a full parameter neighborhood rather than only at the base point.Source: A Joint C2 regularity of the semi-dual · p. 49
  • Throughout neighborhood N, sites are pairwise distinct and all Laguerre cells have positive volume by (A.7), satisfying the additional hypotheses imposed in [4, Proposition 1.1] for quadratic Euclidean cost.The proposition verifies the pairwise-interface, multiple-intersection, and fixed-boundary conditions required by the general second-order theorem.Source: A Joint C2 regularity of the semi-dual · p. 49
  • Differentiating K with respect to weights shows that the mass map is jointly C1 on O+, as required for the implicit-function argument.Taking q = 1 gives mass derivatives, while q(y) = y gives the moment derivative used in (2.51).Source: A Joint C2 regularity of the semi-dual · p. 50
  • Moving-domain differentiation applies componentwise to qρ, and internal-facet velocities yield continuous derivatives of cell integrals without extra terms from the fixed boundary or multiple intersections.The assumptions q ∈ C1 and ρ ∈ W 1,1 ∩ L∞ support the formula, with multiple intersections invisible to the surface integrals.Source: A Joint C2 regularity of the semi-dual · pp. 48, 50
  • On compact subsets of the balanced graph, these continuity and invertibility properties are uniform, but no Hölder exponent for joint second derivatives is asserted.Balanced cells have positive mass and volume, placing the graph compactly inside O+.Source: A Joint C2 regularity of the semi-dual · p. 50

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