FlowBudget: Training-Free Sampling-Budget Estimation for Flow Matching Models

arXiv:2610.20636 · cs.LG, cs.CV · Submitted 2026-10-01 · Read on arXiv

Listen

Transcript

Introduction to the show: ident: AI Radio. Generated commentary on the latest Artificial Intelligence papers.

Tom: Today's paper: "FlowBudget: Training-Free Sampling-Budget Estimation for Flow Matching Models".

Jane: The gist The FLOWBUDGET method introduces a training-free estimator for solver-relative sampling complexity in flow matching models, allowing researchers to determine sufficient numerical resolution without performing expensive FID-versus-NFE sweeps.

Tom: First, who's behind it and why it matters.

Title and authors: Tom: Now we're moving into the specifics. The paper is titled "FlowBudget: Training-Free Sampling-Budget Estimation for Flow Matching Models."

Jane: That title tells you right away what it’s about, but I want to unpack that for someone who isn't deep in the weeds one <ref:2610.20636#pg1>.

Lu: It means they are proposing a method that estimates the sampling budget—that is, how many steps you need—without needing any training data or expensive sweeps one <ref:2610.20636#pg1>.

Meng: Without training data is good because we don't have to spend time collecting more samples just to find the right setting one <ref:2610.20636#pg1>.

Lalam: It’s about finding that numerical resolution needed for flow matching models based on the dynamics learned by the model, not by running trials one <ref:2610.20636#pg1>.

The paper's summary: Tom: So, what they’re actually doing is taking the dynamics learned by a flow matching model and using those local properties—the interpolation probes—to estimate how much error we expect for any given step size one <ref:2610.20636#pg1>.

Jane: They construct these probes xt = (one-t)xzero + tϵ and use them to define the one-step error ES(xt, t; h), which is the local numerical error between the exact and numerical updates starting from that state one <ref:2610.20636#pg1>.

Lu: And they show how this local error scales with step size h through Taylor expansion, yielding eS(xt, t; h) = κSh αS qS(xt, t) + O(h αS+δ)

one page three of that work reads: "Taylor expansion along the exact trajectory gives xt+h = xt+hvθ+(h two /two)aθ+(h three /six)(cθ+gθ)+O(h four)": <ref:2610.20636#pg3,Taylor expansion along the exact trajectory gives xt+h = xt+hvθ+(h>.

Meng: They then define a displacement scale DS(xt, t; h), which is proportional to h times the derivative of some function dS one <ref:2610.20636#pg1>.

Jane: And they then normalize these two quantities by taking the ratio, ES divided by DS, to get ES(xt, t; h) which has a stabilization factor one <ref:2610.20636#pg1>.

Tom: This normalized coefficient rSθ(xt, t) is what they average over all those probes to get CS one <ref:2610.20636#pg1>.

Lalam: And they show that this averages out nicely in terms of K, leading to the proposition where ES(K) scales with K to the power of negative gamma S

one page three of that work reads: "Proposition one states that for sufficiently small h, ES(K):= Eµ [ES(xt, t; h): = κSCSK−γS + O(h γS+δ)") <ref:2610.20636#pg1>.

Jane: So the budget is proportional to tau raised to the power of negative gamma S one <ref:2610.20636#pg1>.

The paper's improvements: Tom: The main improvement they propose is that solving the equation κSCSK−γS equals tau lets us define the population leading-order budget CbS one <ref:2610.20636#pg1>.

Lu: And from that, we get a Monte Carlo plug-in estimate for the budget, CbS, which is one over N times an average of rSθ evaluated at some probes

one page three of that work reads: "KbS = κSCbS/τ": <ref:2610.20636#pg1>.

Meng: This is important because it allows us to select the optimal number of function evaluations based on a tolerance tau, potentially reducing total inference cost if we're near the low-FID plateau one <ref:2610.20636#pg1>.

Jane: And they provide probabilistic bounds using Hoeffding’s inequality to show how sure we are about our estimate CbS

one page three of that work reads: "If zero ≤ rSθ(xt, t) ≤ RS µ-a.s., Hoeffding’s inequality gives, with probability at least one − η, CbS − CS ≤ RS p log(two/η)/2N": <ref:2610.20636#pg1>.

Tom: They also give specific estimates for ODE solvers like Euler and Heun errors eE and eH one <ref:2610.20636#pg1>.

Lu: For Euler, the error is eE = (h two/two)aθ + O(h three), which gives a cost CE = Eµ aθ/vθ + ε

one page three of that work reads: "CE = Eµ aθ/vθ + ε": <ref:2610.20636#pg1>.

Meng: And for Heun, the error is eH = (h three/twelve)(2gθ − cθ) + O(h four), which yields a cost CH = Eµ 2gθ − cθ/vθ + ε

one page three of that work reads: "CH = Eµ 2gθ - cθ/vtheta + ε": <ref:2610.20636#pg3,eH = (h^3/12)(2gθ − cθ) + O(h^4>.

Jane: So the improvement here is that for ODE solvers, they can determine if Euler or Heun requires fewer steps to meet a target numerical error tolerance one <ref:2610.20636#pg1>.

Conclusion: Tom: To wrap up, the paper "FlowBudget: Training-Free Sampling-Budget Estimation for Flow Matching Models" gives us a way to estimate the solver-relative sampling complexity without doing those expensive FID versus NFE sweeps one <ref:2610.20636#pg1,FlowBudget: Training-Free Sampling-Budget Estimation for Flow Matching Models>.

Jane: The core idea is using simulation-free interpolation probes of the learned dynamics to normalize the leading one-step discretization error by the motion one <ref:2610.20636#pg1,using simulation-free interpolation probes of the learned dynamics>.

Lu: This lets us determine sufficient numerical resolution based on local properties, giving us a budget CbS that we can plug into an estimate for K one <ref:2610.20636#pg1>.

Meng: And they even give explicit cost estimates for Euler and Heun solvers to show how many steps each requires for a target error tolerance one <ref:2610.20636#pg1>.

Lalam: Overall, this is a solid way to get solver-relative sampling complexity without the heavy empirical evaluation overhead that was previously required in this area one <ref:2610.20636#pg1>.

Tom: It’s about moving from guessing the budget based on visual quality to a more principled estimate using simulation-free interpolation probes for these flow matching workflows one <ref:2610.20636#pg1,using simulation-free interpolation probes>.

Montréal AI Institute · CNRS · CentraleSupelec - University of Paris-Saclay

cs.LG, cs.CV

Submitted: 2026-10-01

Updated: 2026-10-01

Code: https://github.com/Yasaman-dt/FlowBudget

License: http://creativecommons.org/licenses/by/4.0/

Importance score: 89/100

The gist: The gist The FLOWBUDGET method introduces a training-free estimator for solver-relative sampling complexity in flow matching models, allowing researchers to determine sufficient numerical resolution

Key concepts

Flow Matching Models: These are generative models that learn a velocity field $v_ heta(x_t, t)$ which guides the transformation from noise to data distribution. They are trained by minimizing a loss function that ensures the learned velocity field matches the true conditional velocity between noise and data points.
Simulation-Free Interpolation Probes: Instead of generating full samples, FLOWBUDGET uses interpolated states $x_t = (1-t)x_0 + t ilde{oldsymbol{ u}}$ to probe the learned dynamics. These probes are used to estimate how the numerical solver's error scales with the motion of the flow, providing local information about discretization errors.
Solver-Relative Sampling Complexity: This refers to determining the minimum number of steps ($K$) required for a specific numerical solver (like Euler or Heun) to achieve a desired accuracy ($ au$). FLOWBUDGET estimates this complexity by relating the local error of one step to the overall budget, allowing users to select $K$ based on model dynamics and solver order.

Terminology

Flow Matching — 2× in this paper · explained in 15 episodes
Flow Matching Model — 2× in this paper · explained in 3 episodes
Generator — 1× in this paper · explained in 2 episodes
Rectified Flow — 1× in this paper · explained in 3 episodes

Summary

The gist The FLOWBUDGET method introduces a training-free estimator for solver-relative sampling complexity in flow matching models, allowing researchers to determine sufficient numerical resolution without performing expensive FID-versus-NFE sweeps.

Background and Problem Formulation

Flow matching models learn a time-dependent velocity field whose integration transports samples between noise and data distributions Given data x0 ∼ pdata and Gaussian noise ϵ ∼ N (0, I), consider the interpolant xt = atx0 + btϵ, t ∈ [0, 1], where the differentiable schedules satisfy a0 = b1 = 1 and a1 = b0 = 0 The resulting interpolant is xt = (1 − t)x0 + tϵ, with conditional velocity vt = dxt/dt = ϵ − x0 A neural velocity field vθ(xt, t) is trained to predict this conditional velocity by minimizing LFM(θ) = Ex0,ϵ,t[∥vθ(xt, t) − vt∥2/2], where t ∼ U[0, 1] At inference, generation starts from Gaussian noise x1 ∼ N (0, I) and proceeds backward from t = 1 to t = 0 For K uniform steps, we define the time grid tn = 1 − n/K, n = 0..., K, and the signed step size h = tn+1 − tn = -1/K We write xtn for the numerical state at time tn, initialized with xt0 = x1 For ODE sampling we integrate dxt/dt = vθ(xt, t) using Euler or Heun [12]. Euler uses the update xEtn+1 = xtn + h vθ(xtn, tn), while Heun uses xHtn+1 = xtn + h/2 [vθ(xtn, tn) + vθ(xtn + h vθ(xtn, tn), tn+1]. For SDE sampling, we consider dxt = fθ(xt, t) dt + √wt dWt, where Wt is a reverse-time Wiener process, wt ≥ 0 is a prescribed time-dependent noise variance rate, and fθ(xt, t) = vθ(xt, t) - 1/2wtsθ(xt, t) is the reverse-time drift. On the same time grid, Euler–Maruyama gives xEMtn+1 = xtn + h/vθ(xtn, tn) - 1/2wtn sθ(xtn, tn) + p wtn h ξn, where ξn ∼ N (0, I) are independent [20]. Given a pretrained velocity field vθ, a fixed sampling dynamics and solver S, and a tolerance τ > 0, we estimate the smallest uniform step budget K whose leading-order mean relative local error is at most τ.

Methodology

FLOWBUDGET estimates the sampling budget from local properties of the dynamics evaluated at interpolation probes We construct xt = (1−t)x0+tϵ and denote µ as a simulationfree surrogate for the sampler-induced state–time law For a probe (xt, t), consider a step of magnitude h in the sampling direction Let LS(xt, t; h) be the one-step local error between exact and numerical updates starting from the same state, and let DS(xt, t; h) denote the one-step displacement scale. For SDEs, these quantities use the conditional root-mean-square norm, with the exact and numerical updates sharing the same Brownian path Their small-step expansions take the form LS(xt, t; h) = κSh αS qS(xt, t) + O(h αS+δ), (1) and DS(xt, t; h) = h βS dS(xt, t) + O(h βS+δ), (2). To measure local numerical error relative to this motion, define ES(xt, t; h) = LS(xt, t; h)/DS(xt, t; h) + εh βS, (3). The stabilization scales with the displacement, preserving the leading step-size dependence The corresponding normalized coefficient and its probe average are rSθ(xt, t) = qS(xt, t)/dS(xt, t) + ε, CS = Eµ rSθ(xt, t).

Budget Scaling

Proposition 1 states that for sufficiently small h, ES(K):= Eµ [ES(xt, t; h)] = κSCSK−γS + O(h γS+δ), where γS = αS − βS. Accordingly, solving κSCSK−γS = τ defines the population leading-order budget and its Monte Carlo plug-in estimate CbS = 1/N X N i=1 rSθ(x(i)t(i), t(i)), KbS(τ) = κSCbS/τ1/γS. If 0 ≤ rSθ(xt, t) ≤ RS µ-a.s., Hoeffding’s inequality gives, with probability at least 1 − η, CbS − CS ≤ RS p log(2/η)/2N.

Solver Specific Estimates

For ODE solvers, the Euler error is eE = (h 2/2)aθ + O(h 3), and the Heun error is eH = (h 3/12)(2gθ − cθ) + O(h 4), yielding CE = Eµ∥aθ/∥vθ + ε, KbE = lCbE/(2τ m, and CH = Eµ∥2gθ − cθ/∥vθ + ε, KbH = qCbH/(12τ. Euler uses one NFE per step and Heun two. The normalized coefficients also control terminal ODE discretization error under standard Lipschitz stability and uniform local expansions, with backward step magnitude h = 1/K, error propagation over a unit interval gives x0 − x(K)E ≤ eL 2K 2/K X−1 n=0∥aθ(xtn, tn) + O(K−2), and x0 − x(K)H ≤ eL12K3/K X−1 n=0∥2gθ − cθ (xtn, tn) + O(K−3).

SDE Sampling

For SDEs, the local error LEM(xt, t; ∆t) = E[Xt+∆t − XEM t+∆t] squared Xt = x] 1/2 is given by LEM(xt, t; ∆t) = (∆t 3/2) √3 A 2F + B 2F + F 1/2 + O(∆t 2), where A = Jxfθ(xt, t)G(t) and B = ∂tG(t), and the diffusion term dominates and DEM = ∆t 1/2 GF + O(∆t). Hence αEM = 3/2, βEM = 1/2, and γEM = 1. The EM displacement scale is DEM(xt, t; ∆t) = E[XEM t+∆t − x] squared Xt = x] 1/2. On a probe-time domain T = [tmin, tmax] ⊂ (0, 1) where G and ∂tG are bounded and inf t∈T G(t)F > 0, the diffusion term dominates and DEM = ∆t 1/2 GF + O(∆t).

Improvements for AI systems

Improve sampling efficiency by estimating solver-relative sampling complexity without performing expensive FID-versus-NFE sweeps, allowing for a direct mapping of a numerical tolerance to a sampling-step budget.
The improved system can select the optimal number of function evaluations (NFEs) required for inference based on fixed pretrained models and solvers, potentially reducing total inference cost by identifying budgets near the low-FID plateau of the empirical curves.
Implement solver-dependent quality–cost operating points, meaning the system can determine if a specific solver (e.g., Heun vs. Euler) requires fewer steps to achieve a target numerical error tolerance, leading to more efficient deployment across different model architectures like SiT and FlowDCN.

Abstract

Sampling from continuous flow-based generators such as flow matching models requires numerical ODE or SDE integration, making inference cost depend on the number of function evaluations (NFEs). In practice, the sampling budget is inherited from prior work or selected through expensive FID-versus-NFE sweeps, using FID to assess image-generation quality, requiring sample generation and evaluation at multiple budgets. This makes sampling-budget selection dependent on repeated empirical evaluation. We therefore ask a complementary question to solver design and timestep allocation: for a fixed pretrained model and solver, what numerical resolution is sufficient? We introduce FLOWBUDGET, a training-free estimator of solver-relative sampling complexity. Using simulation-free interpolation probes of the learned dynamics, FLOWBUDGET normalizes the leading one-step discretization error by the motion, estimates its coefficient from interpolation probes, and maps a numerical tolerance to a step budget using the solver order. We derive estimators for numerical solvers in flow matching sampling, namely Euler and Heun for ODE sampling and additive-noise Euler--Maruyama for SDE sampling, and connect the ODE coefficients to terminal integration error. Experiments on SiT, FlowDCN and Rectified Flow models, spanning flow-model architectures and formulations, test whether FLOWBUDGET can estimate the NFE budget without generating samples at multiple budgets or performing an FID-versus-NFE sweep. The code is available at https://github.com/Yasaman-dt/FlowBudget