LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6-luna)

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Correct finite-time bound for partial-homogeneity zero crossing

Statement

Let $M \geq 0$ , $η > 0$ , $L \geq 0$ , $ϵ > 0$ , and let $w : [t_{0}, t_{0} + 2 M / η] \to [- M, 0]$ and $θ : [t_{0}, t_{0} + 2 M / η] \to R^{q}$ be absolutely continuous until the first time $w$ reaches zero. Assume $w (t_{0}) \in [- M, 0]$ , $θ (t_{0}) = θ_{0}$ , and, almost everywhere before the crossing,

$w^{'} (t) \geq η / 2, q^{'} (t) \leq ∣ w (t) ∣ (ϵ + L q (t)), q (t) = ∣ θ (t) - θ_{0} ∣ .$

Then $w$ reaches zero at some $τ \leq t_{0} + 2 M / η$ . At the first such time,

$q (τ) \leq {\begin{cases} \frac{ϵ}{L} (e^{L M^{2} / η} - 1), & L > 0, \\ \frac{ϵ M^{2}}{η}, & L = 0. \end{cases}$

Thus the crossing remains in $∣ θ - θ_{0} ∣ < r_{0}$ whenever the displayed bound is less than $r_{0}$ . The crossing-time bound depends on the initial negative-weight magnitude $M$ .

Statement references

Chizat and Bach, Lemma C.18 and its proof, Appendix C.4.2

Facts & Assumptions

Given: The differential inequalities and initial data in the Statement.

[A1]

The functions $w$ and $θ$ are absolutely continuous on each interval before the first zero of $w$ .

Proof

Given: $M, η, L, ϵ, w, θ$ as in the Statement.

1.1givenA1algebra

Suppose first that $M > 0$ and $w$ has not reached zero by $t_{0} + 2 M / η$ . Integrating $w^{'} \geq η / 2$ gives $w (t_{0} + 2 M / η) \geq w (t_{0}) + M \geq 0$ , contradicting the assumed strict negativity before the first crossing. Hence the first crossing time $τ$ exists and satisfies the stated bound. If $M = 0$ , then $w (t_{0}) = 0$ and $τ = t_{0}$ .

2.1A1step 1.1algebra∎

On $[t_{0}, τ]$ , $w$ is strictly increasing with $w^{'} \geq η / 2$ , so it is an admissible absolutely continuous time change. Its inverse satisfies $d t / d w \leq 2 / η$ . If $L > 0$ , set $p = ϵ + L q$ . The inequality for $q$ gives $d p / d w \leq (2 L / η) ∣ w ∣ p$ , and integration from $w (t_{0})$ to $0$ yields $p (τ) \leq ϵ e^{L M^{2} / η}$ . Rearranging gives $q (τ) \leq ϵ (e^{L M^{2} / η} - 1) / L$ . If $L = 0$ , direct integration gives $q (τ) \leq (2 ϵ / η) \int_{w (t_{0})}^{0} ∣ s ∣ d s = ϵ w (t_{0})^{2} / η \leq ϵ M^{2} / η$ . The same bounds hold with zero right-hand side when $M = 0$ .

Source note

Chizat and Bach's printed proof claims a common crossing time $T < 2 / η$ for every $w_{0} \in [- M, 0]$ . Integrating $w^{'} \geq η / 2$ gives the correct worst-case bound $2 M / η$ . The estimate above proves hitting at an actual time $τ$ and the corresponding $θ$ -displacement bound; it does not assert $w (T) = 0$ at one fixed time for all initial weights.

Used by

Nothing in the library uses this result yet.

Dependency tree · 0 levels

Nothing. This result depends on no other item in the library.

Sources

Lénaïc Chizat and Francis Bach, On the Global Convergence of Gradient Descent for Over-parameterized Models using Optimal Transport (standard reference, not scraped)