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Semi-discrete quadratic Wasserstein energy and state-dependent Langevin exploration
Ran Gu, Gaoyue Guo, Kelvin Shuangjian Zhang
TL;DR
The paper addresses nonsmoothness at site collisions and asks both which state-dependent randomized temperatures solve an entropy-regularized control problem and which discrete rules ensure global optimization through best-so-far records. It develops collision-valid regularity and relaxed-control machinery, then proves well-posed collision-free dynamics and geometric ergodicity for fixed positive-temperature Euler schemes.
Problem
Site collisions obstruct differentiability of the semi-discrete Wasserstein energy, while temperature control and asymptotic guarantees require separate continuous- and discrete-time analyses.
Method
The paper combines a whole-space splitting formulation with differentiable Laguerre-cell analysis off collisions, entropy-regularized relaxed control, and Euler-chain ergodicity analysis for fixed Borel temperature rules.
Results
The value function is finite and continuous, solves the exploratory HJB equation in the viscosity sense, and the controlled dynamics has a unique global strong solution avoiding collisions; fixed positive-temperature Euler chains are irreducible, aperiodic, and admit unique full-support invariant laws.
Takeaways & Limitations
With a strictly positive temperature floor, raw iterates are not expected to converge to a minimizer, so the relevant optimization object is the running record.
Takeaways & Limitations
On the full configuration space, the available estimate is linear growth rather than a globally bounded gradient condition.
Abstract
from arXiv · showhide
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
The paper develops a regularity theory for semi-discrete quadratic Wasserstein energy and applies it to entropy-regularized Langevin temperature control and robust discrete exploration. It establishes classical interior HJB regularity and proves that fixed positive-temperature Euler schemes converge through their running records, despite nonconvergent raw iterates.
- Scope and limitations: The positive temperature floor supports the asymptotic record guarantee but means the optimal-control result does not establish fastest finite-time exploration or convergence of raw iterates.The exact HJB feedback and any fixed clipped Borel approximation retain the record-convergence guarantee; the theorem does not cover time- or history-dependent temperature rules without enlarging the state space.
- Energy regularity and geometry: The energy is globally locally Lipschitz, semiconcave, coercive, and dissipative, while global minimizers are interior and collision free.On the collision-free configuration space, the energy is C^2; its gradient uses balanced Laguerre-cell barycenters and its Hessian has an explicit facet representation with a global one-sided bound.
- Explicit cases and Lloyd behavior: In one dimension, each ordering chamber has a unique minimizer explicitly characterized by the quantiles of the target measure.The paper also classifies two-site critical configurations for the uniform measure on the unit square, including non-minimizing Lloyd fixed points.
- Exploratory temperature control: The value function is finite and continuous, solves the exploratory HJB equation classically in the collision-free interior, and has locally C^1 Laplacian, producing a locally Lipschitz optimal feedback.The HJB equation uses the scalar Hamiltonian Hλ(∆vλ), and the optimal feedback is determined through the Laplacian of the value function.
- Exploratory temperature control: For d ≥2, entropy-regularized relaxed temperature control yields a unique global strong, nonexplosive, collision-free diffusion under every relaxed control.The punctured configuration space requires collision avoidance because the drift is only locally Lipschitz there and no boundary condition is imposed on collision diagonals.
- Discrete exploration and records: For every fixed Borel temperature rule in [a, 1] and sufficiently small step size, the Gaussian Euler chain is irreducible, aperiodic, geometrically ergodic, and has a full-support invariant law.Its running record converges almost surely to the global minimum and approaches the global minimizer set, whereas the raw iterates do not converge.
2 Regularity of the semi-discrete energy
The paper establishes global regularity and coercive structure for the semi-discrete Wasserstein energy, then derives smooth Laguerre-cell formulas on collision-free configurations. It characterizes minimizers, solves the one-dimensional problem chamberwise, and identifies saddle configurations on the unit square.
- Global estimates: E is continuous and locally Lipschitz on the full configuration space, with global semiconcavity, coercive growth, and dissipativity estimates.The full-space estimates remain valid at colliding configurations, where differentiability can fail.
- Minimizers: Every global minimizer lies in the interior and is collision free, because coincident sites can be replaced by distinct barycenters with strictly lower competitor cost.The minimizer set is therefore a nonempty compact subset of the interior collision-free configuration space.
- Differentiability: On the collision-free configuration space, E is C^2, with gradient given by balanced Laguerre-cell barycenters and Hessian represented explicitly through cell facets.Continuity of balancing weights and facet derivatives yields continuous second derivatives on compact subsets.
- Explicit cases: In one dimension, each ordering chamber has a strictly convex quadratic restriction with a unique chamber minimizer, and global minimizers are obtained by comparing finitely many orderings.For equal weights, the labeled minimizer is unique only up to label permutations.
- Explicit cases: For two equally weighted sites on the unit square, the critical configurations include global minimizers and diagonal saddle points, so Lloyd fixed points need not minimize the energy.The diagonal configurations and their label exchanges are saddle points, while the two off-diagonal configurations and exchanges are global minimizers.
3 Exploratory temperature control
The paper formulates entropy-regularized relaxed control of the Langevin temperature on the collision-free state space and proves well-posedness, value-function regularity, and verification. It also establishes continuity and optimality properties for the value function while identifying scope limits for numerical and control interpretations.
- Control formulation: The scalar Hamiltonian is smooth, strictly increasing, and strictly concave, and its pointwise entropy-regularized minimizer is the Gibbs law λq.The Gibbs identity identifies the unique minimizer in the relaxed temperature optimization.
- Controlled dynamics: For d ≥ 2, the controlled Langevin dynamics has a unique global strong solution that is nonexplosive and never reaches the collision set.The construction works for every relaxed control, with estimates uniform over admissible controls.
- HJB analysis: The value function is finite and continuous on the collision-free state space, has at most quadratic growth, and solves the exploratory HJB equation in the interior.Dynamic programming first yields viscosity-solution status, while the regularity argument supplies the stronger interior solution properties.
- HJB analysis: The feedback constructed from the value function is continuous, locally Lipschitz after the Laplacian regularity bootstrap, and satisfies the verification identity for the discounted criterion.The proof avoids requiring a full C^3 estimate for the value function or an unproved Hölder modulus for moving-facet derivatives.
4 Langevin discretization and global record convergence
The section separates optimal HJB feedback from broader long-time guarantees for fixed Borel temperature rules. Small-step Gaussian Euler chains are geometrically ergodic, while their best-so-far energies converge globally even though raw iterates do not.
- Global record convergence: The long-time convergence theorem applies independently of HJB optimality to every fixed Borel temperature rule bounded away from zero.The rule must remain the same at every iteration; changing rules produce a nonhomogeneous chain outside the stated theorem.
- Scope and limitations: The guarantee is robust to fixed clipped Borel temperature approximations but does not extend to errors in the semi-discrete Wasserstein gradient.The convergence proof uses the exact drift −∇E and global dissipativity; clipping temperature alone cannot preserve the result under perturbed gradients.
- Algorithm: The deterministic drift is an under-relaxed weighted Lloyd step followed by Gaussian exploration when 0 < η ≤1.The algorithm constructs balanced Laguerre cells, computes their moments, applies the approximate temperature rule, and retains the earliest best-so-far record.
- Collision avoidance: The chain remains collision free almost surely in every dimension because each update has a nondegenerate Gaussian law and the collision set has Lebesgue measure zero.In one dimension, the chain may cross a collision hyperplane without landing on it.
- Geometric ergodicity: For every sufficiently small step size, the Gaussian Euler chain is geometrically ergodic and has a unique full-support invariant law.Aperiodicity and small-set minorization yield convergence in total variation from every initial state.
- Global record convergence: The raw iterates do not converge, but the best-so-far energy converges almost surely to the global minimum and the running record approaches the global minimizers.Full support ensures relevant energy sublevel regions are visited infinitely often, enabling the record-convergence argument.
A Joint C2 regularity of the semi-dual
The appendix establishes joint C^2 regularity of the semi-dual on the open set where sites are distinct and all Laguerre cells have positive volume. It then differentiates moving cell integrals to obtain continuous derivatives of mass and moment maps.
- Positive-cell set: The positive-cell parameter set is relatively open, so distinct sites and positive Laguerre-cell volumes persist throughout a neighborhood.Continuity of cell volumes gives openness, while local nondegeneracy preserves the defining conditions.
- Applications: The mass map is jointly C^1, and choosing q(y)=y supplies the moment derivative used later for the energy gradient.Uniform continuity conclusions on compact subsets support the implicit-function argument, without asserting a Hölder exponent for joint second derivatives.
- Joint regularity: The semi-dual K is jointly C^2 in sites and weights on the positive-cell set, with continuous Hessian.The proof verifies the interface, multiple-intersection, fixed-boundary, and transversality hypotheses for the quadratic Euclidean cost.
- Moving-domain differentiation: Differentiating moving Laguerre-cell integrals produces facet formulas whose derivatives remain continuous with respect to sites and weights.The outer boundary contributes no velocity, and lower-dimensional multiple intersections contribute no surface term.