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Particle number and training time limits interchange qualitatively

Statement

Assume AC and the feature-energy hypotheses. Let $μ_{t}$ be a Wasserstein gradient flow from $μ_{0}$ concentrated on a common cutoff $Q_{r_{0}}$ , with $F (μ_{t}) \to F^{*} : = \inf_{λ \geq 0} F (λ)$ . For each $m \geq 1$ , let $μ_{m, t}$ be a finite-particle gradient flow for the same objective, with $μ_{m, 0}$ concentrated on $Q_{r_{0}}$ and $W_{2} (μ_{m, 0}, μ_{0}) \to 0$ . Assume that for every fixed $t$ , $W_{2} (μ_{m, t}, μ_{t}) \to 0$ as $m \to \infty$ , with both laws concentrated on a common cutoff for that finite time. Then both iterated limits exist and

$\lim_{t \to \infty} \lim_{m \to \infty} F (μ_{m, t}) = \lim_{m \to \infty} \lim_{t \to \infty} F (μ_{m, t}) = F^{*} .$

This is qualitative and gives no rate uniform in $m$ or $t$ .

Statement references

Feature-measure energy and its regularity assumptions

Finite-particle flows converge to the measure-valued flow in W2

Feature energy is W2-continuous on a common cutoff

Chizat and Bach, Lemma C.15

Chizat and Bach, Proposition 2.3, §2.2

Ambrosio, Gigli, and Savaré, Chapter 11

Facts & Assumptions

Given: AC, the limiting flow and its energy convergence, and the particle-flow assumptions in the Statement.

[A1]

AC is inherited from the finite-particle convergence theorem and the coupling construction. The Axiom of Choice

[F1]

For every fixed $t$ , $W_{2} (μ_{m, t}, μ_{t}) \to 0$ . Finite-particle flows converge to the measure-valued flow in W2

[F2]

Energy is continuous for W2-convergent measures on their common cutoff. Feature energy is W2-continuous on a common cutoff

[F3]

Every particle law is a finite nonnegative competitor, so $F (μ_{m, t}) \geq F^{*}$ . Feature-measure energy and its regularity assumptions

[F4]

Each classical finite-particle subgradient flow satisfies the finite-dimensional energy identity $F_{m} (x^{(m)} (t)) - F_{m} (x^{(m)} (s)) = - \int_{s}^{t} m^{- 1} \sum_{i} ∣ {\dot{x}}_{i}^{(m)} (q) ∣^{2} d q \leq 0$ ; this is proved by the semiconvex supporting-inequality argument in proof step 3.1 of Finite-particle flows converge to the measure-valued flow in W2, with the same weighted particle metric. Since $F_{m} (x^{(m)} (t)) = F (μ_{m, t})$ , the particle-law energy is nonincreasing.

Proof

Given: AC and the assumptions in the Statement.

1.1A1F1F2given

Fix $t$ . By [F1] and [F2], $F (μ_{m, t}) \to F (μ_{t})$ as $m \to \infty$ . Taking $t \to \infty$ after this inner limit gives $F^{*}$ by the assumed mean-field energy convergence.

2.1A1F3F4step 1.1choose∎

For each $m$ , [F3] and [F4] show that $e_{m} : = \lim_{t \to \infty} F (μ_{m, t})$ exists and satisfies $e_{m} \geq F^{*}$ . Let $ϵ > 0$ . Choose $t_{0}$ with $F (μ_{t_{0}}) < F^{*} + ϵ / 2$ . By step 1.1, there is $m_{0}$ such that for every $m \geq m_{0}$ , $F (μ_{m, t_{0}}) < F (μ_{t_{0}}) + ϵ / 2 < F^{*} + ϵ$ . Monotonicity [F4] gives $e_{m} \leq F (μ_{m, t_{0}}) < F^{*} + ϵ$ for every $m \geq m_{0}$ . Together with the lower bound, this proves $e_{m} \to F^{*}$ .

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Sources

Lénaïc Chizat and Francis Bach, On the Global Convergence of Gradient Descent for Over-parameterized Models using Optimal Transport (standard reference, not scraped)
Luigi Ambrosio, Nicola Gigli, and Giuseppe Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures (standard reference, not scraped)