Universality and Convergence of Generative Flows

Leo Brunswic

cs.LGOct 4, 2026

Universality and Convergence of Generative Flows

Authors: Leo Brunswic

Organizations: LLM Suite Team, JP Morgan Chase & Co.

Abstract

Generative flows sample from an unnormalized target by training a flow to be balanced, and the training loss is the signal a practitioner watches. We ask what that signal is worth: whether a small loss certifies an accurate sampler, whether the loss can be driven to zero, and how fast gradient descent does so. The loss decides the first. Losses that compare the two sides of the balance by their difference bound, in total variation, the error of the sampler the flow implies, with explicit constants that do not involve the policy; flow-matching losses that compare them through a ratio admit no such bound, already on a single cycle, whenever their generator is continuous at balance. On graphs, the backward policy decides the other two. Once it is frozen, balance becomes invariance under the backward chain, so that existence is free on finite graphs, and one constant --- the norm of that chain's Green operator, which plays the role of an inverse spectral gap --- fixes the order of the curvature of the loss around the balanced flow, from above and below, and sets a floor under the rate at which training converges near it. The mechanism is that gradient descent diffuses the flow along the backward policy. For the squared-logarithm generator of detailed and trajectory balance, training the balance loss on states converges globally on every finite path-connected graph, from every positive initialization. The constant can be infinite while backward trajectories are short on average, and exact flow matching can then fail. The bounds and rates are tested by exact computation on enumerable state spaces, and every theorem carries a certification status computed from a Lean~4 development.

Figures & tables

Panel of Figure 1 — Figure 1: The results walked through in Section 1.3 . Each badge is computed from the dependency closure of the linked Lean declarations, never typed. An arrow is solid when the Lean proof of its target runs through a module proving its source, dashed when the link is in the argument only; the full proof structure is drawn in the appendix.

Figure 2: On the eight-cycle

C_{8}

the base flow samples

x_{2}

, a state the target

\kappa

barely charges: it is balanced toward the wrong target. Adding a circulation to it drives the ratio losses, taken against

\kappa

, to zero, while the sampling error and the stable loss stay put.

Figure 3: The local rate of the exact gradient flow on each of the 20 chains of the rate study, named as in Appendix C and ordered by their number of states, given after each name, against the rate

g^{\prime\prime}(1)w_{\min}/\widehat{B}^{2}

predicted from the Green operator, which with

w\equiv 1

is the smallest transverse curvature of the loss, and against

\varrho_{\sigma}/2

, the rate the local theorem guarantees at the hitting-time constant

\widehat{B}_{\sigma}

Panel of Figure 4 — Figure 4: The doubling graph: a source $s_{0}$ , a ladder of rungs and a sink $s_{f}$ , with forward edges from the source to the first rung, from each rung to the next, from each even rung to its half, and from every rung to the sink. Its backward chain therefore steps down one rung or jumps to twice its rung, doubling with probability $\varepsilon(j)$ at rung $j$ ; the family studied is $\varepsilon_{c,s}(j)=c/(j+1)^{s}$ , and geometric damping $\varepsilon(j)=a\rho^{j}$ is the contrast.

Appendix figures & tables6 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 5: The proof structure of the development at the level of its headlines, read off the compiled environment: one box per result stated in this appendix, grouped by chapter, and an arrow from a result to another when the proof term of the first invokes the second. Heavier arrows leave their chapter — the places where one part of the development stands on another — and the grey boxes are the results they pass through. The graph with every named result the proofs invoke is generated alongside this one and is not printed.

result	declaration	named results	declarations
The sampler, the certificates and the constant
14	SamplingGeneral.MFlow.sampling_theorem	6	55
15	SamplingGeneral.MFlow.negative_control_general	8	78
16	SamplingGeneral.stable_bound_general	39	128
17	SamplingGeneral.il_first_bullet_general	37	137
18	ILBoundFull.il_tv_le_sqrt	0	0

Appendix

Table 1: The results of this appendix as declarations of the Lean development, in the namespaces GFNBounds.Core , GFNBounds.Graph , GFNBounds.Balance , GFNBounds.Silva and GFNBounds.Doubling . The last two columns count what each proof rests on, unfolded to the bottom of the development: how many of the declarations it reaches are results the development names rather than steps internal to one, and how many it reaches in all. A zero means that the proof rests on Mathlib alone.

Figure 6: open: D5 Left: the exact total variation of the sampler against the certificate

C\mathcal{L}_{q}

, one point per perturbed flow on 4 of the hypergrids and permutation graphs, under a uniform

\nu_{T}

; every point lies below the diagonal. Right: at

q=1

, the median of their ratio against the size of the state space, one line per family with its fitted log-log slope: flat on permutation graphs, growing slowly on hypergrids, and on product-replacement graphs under the uniform policy nearly flat at first and growing on the largest groups.

Figure 7: The inverse spectral gap

1/(1-\lambda_{2})

of the uniform product-replacement walk, which is its

\widehat{B}

since the walk is symmetric, against the group order: lines join the ten groups carried at every

k

, open markers are larger groups. The ratio of the largest value to the smallest over the ten groups narrows as

k

passes the threshold of property (T).

Figure 8: The mean backward length against the doubling constant

c

s=1

, one curve per truncation

K

. For

c<1

the length is finite and the curves converge to it as

K

grows, slowly as

c

nears one: at

c=0.9

it is 25.4 at the smallest truncation and 38.6 at the largest, against the limit

\bar{\jmath}/(1-c)

of 45. Beyond, the curves fan out, the truncation hiding a divergence. The chain is positive recurrent for

c<1

, null recurrent for

1\leq c<1/\ln 2

and transient for

c>1/\ln 2

; at

c=1/\ln 2

it is only known not to be positive recurrent.

Figure 9: Left:

\widehat{B}_{K}

against the truncation, one curve per doubling constant

c

, with the proved lower bound; the dashed curves, at

c\geq 1

, lie outside the hypothesis

c<1

of the theorems. Right: the outflow bill of the universality certificate at

c=1/2

from a Dirac initial flow: flat for a target at the foot of the ladder, growing polynomially once the target reaches into it.

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