PaperlayerView exact revision on arXiv

Source-linked AI summary

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

Zahra Dehghani, Pablo Piantanida, Mohammadhadi Shateri

arXiv:2610.20636v1cs.LGcs.CV

TL;DR

Sampling budgets for flow-matching models are often inherited or selected through repeated quality-versus-NFE evaluation, while the numerical resolution needed for a fixed model and solver is the complementary question. FLOWBUDGET estimates that resolution from interpolation probes; experiments report budgets near the low-FID plateau and a 4.5× wall-clock reduction for the stated comparison.

  • Problem

    For a fixed pretrained model and solver, the sufficient numerical sampling budget is not directly specified, and budgets are often inherited or selected through trial and error.

    Source: §1 · p. 1

  • Method

    FLOWBUDGET estimates a scale-normalized leading local-error coefficient from interpolation probes and combines it with solver order to obtain a step budget without simulating sampling trajectories.

    Source: §1 · p. 1, §2 · p. 2, §3 · p. 2

  • Results

    FLOWBUDGET selects budgets near the low-FID plateau, and for 3-RF its 2K-probe estimate and evaluation took 24.3 minutes versus 110.3 minutes for a six-budget FID–NFE curve, a 4.5× reduction.

    Source: §4 · p. 4

  • Takeaways & Limitations

    The estimator provides a solver-relative numerical-resolution budget, with ODE coefficients also controlling terminal integration error under a bounded-velocity assumption.

    Source: §5 · p. 4

    Interpolation probes use training-distribution samples that only approximate the sampler trajectory law, and the tolerance is not theoretically calibrated to FID.

    Source: §3.2 · p. 3, §5 · p. 4

Abstract

from arXiv · show

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

1. INTRODUCTION

Flow-matching sampling cost depends on solver NFEs, yet budgets are often inherited from prior work or chosen through trial and error. FLOWBUDGET instead estimates the numerical resolution needed for a fixed pretrained model and solver, without an FID-versus-NFE sweep.

1. INTRODUCTION

  • Each integration step requires neural-network evaluations, so solver choice and NFE count directly determine sampling cost.Too few NFEs can produce substantial discretization error; increasing NFEs reduces it according to solver convergence order.Source: §1 · p. 1
  • Sampling budgets are often inherited from reference implementations, such as 250 NFEs, or selected by trial and error using generation quality.The paper tests its estimator on pretrained SiT, Rectified Flow, and FlowDCN models without constructing it through FID-versus-NFE sweeps.Source: §1 · p. 1
  • FLOWBUDGET reframes fixed-solver budget selection as estimating numerical error, rather than changing the solver or reallocating a prescribed budget.Its novelty is using local truncation error to estimate solver-relative complexity before a quality-versus-NFE sweep.Source: §1 · p. 1
  • FLOWBUDGET estimates a scale-normalized numerical-complexity coefficient from simulation-free interpolation probes, then combines it with solver order to map a tolerance to a step budget.The paper derives estimators for Euler and Heun ODE sampling and additive-noise Euler–Maruyama SDE sampling, and connects ODE coefficients to terminal error under standard stability assumptions.Source: §1 · p. 1

2. BACKGROUND AND PROBLEM FORMULATION

The paper specifies the flow-matching interpolant and the reverse-time sampling setups, then formulates budget estimation as finding the smallest uniform step count meeting a relative local-error tolerance. The estimate uses data-noise interpolation probes rather than simulated sampling trajectories.

2. BACKGROUND AND PROBLEM FORMULATION

  • The flow-matching interpolant is x_t = (1 − t)x_0 + tϵ, with conditional velocity ϵ − x_0 under the stated linear schedules.The schedules satisfy a_0 = b_1 = 1 and a_1 = b_0 = 0.Source: §2 · p. 2
  • Inference starts from Gaussian noise and integrates backward on a uniform grid, using signed step size h = −1/K.For ODE sampling, the velocity field is integrated with Euler or Heun.Source: §2 · p. 2
  • For SDE sampling, the reverse-time dynamics combine a drift with prescribed time-dependent noise, and the drift includes a score correction to the learned velocity.The setup uses a reverse-time Wiener process and estimates the score from the learned velocity for the linear interpolant.Source: §2 · p. 2
  • Euler–Maruyama is the stated SDE solver on the same uniform time grid.The supplied passage identifies its update but does not provide enough readable detail to characterize the full update.Source: §2 · p. 2
  • The budget target is the smallest uniform step count whose leading-order mean relative local error is at most a prescribed tolerance.Relative local error divides one-step discretization error by a stabilized displacement scale, estimated from data-noise interpolation probes without simulating trajectories.Source: §2 · p. 2

3. METHODOLOGY

FLOWBUDGET estimates a fixed solver’s sampling budget from interpolation-probe estimates of scale-normalized local error, mapping a tolerance to a step count through solver order. The methodology derives this scaling for Euler and Heun ODE sampling and additive-noise Euler–Maruyama, with stated assumptions on terminal-error interpretation and SDE probe times.

3. METHODOLOGY

  • FLOWBUDGET evaluates local error and displacement at interpolation probes, normalizing the former by the latter with a displacement-scaled stabilizer.For SDEs, the exact and numerical updates share a Brownian path, and the error and displacement use conditional RMS norms.Source: §3 · p. 2
  • The probe-averaged relative local error scales as κ_S C_S K^-γ_S to leading order, so solving for a prescribed tolerance yields the leading-order budget estimate.The empirical coefficient is a Monte Carlo estimate from independent interpolation probes; a stated boundedness condition permits a confidence interval for the unrounded budget.Source: §3 · p. 2

3.1. Euler and Heun

  • Euler’s normalized coefficient uses trajectory acceleration, while Heun’s uses a solver-specific third-order combination; their NFE multipliers are one and two, respectively.Under standard Lipschitz stability and uniform local expansions, these numerator coefficients also control terminal ODE integration error; probe averages approximate trajectory-time averages, and no numerical-error-to-FID implication is assumed.Source: §3.1 · p. 3

3.2. Euler–Maruyama

  • For additive-noise SDE sampling, Euler–Maruyama has relative-error scaling exponent γ_EM = 1 when diffusion dominates the displacement on the probe-time domain.This is a strong local-error statement, consistent with order-one strong convergence for sufficiently regular additive-noise SDEs, not the generic order-1/2 rate for multiplicative noise.Source: §3.2 · p. 3
  • SDE probes are restricted to times where G and its time derivative are bounded and G stays nonzero; endpoints where the normalization or expansion may fail are excluded.The resulting tolerance controls a strong numerical-resolution proxy, with no theoretical claim that it is calibrated to FID.Source: §3.2 · p. 3
  • In practice, FLOWBUDGET averages solver coefficients over interpolation probes, applies the budget scaling, and converts steps to NFEs using multipliers of one for Euler or EM and two for Heun.The EM coefficient is estimated using the stated additive-noise local-error construction.Source: §3.2 · p. 3

4. EXPERIMENTS

FLOWBUDGET selects budgets near low-FID plateaus while reducing sampling costs, and its estimates are stable across probe counts. For 3-RF, estimating one budget is faster than constructing a multi-budget FID–NFE curve.

Experimental Setup

  • Across six pretrained models, one tolerance per dynamics class is used to estimate budgets from 2K interpolation probes, without tuning tolerance per model.The study tests whether fixed numerical tolerances yield useful budgets across architectures, not whether they are universally calibrated to perceptual quality.Source: §4 · pp. 3–4

Main Results

  • At reference NFE 250, FLOWBUDGET reduces budgets 16.0–76.8% for SiT-XL/2 and 28.8–83.2% for FlowDCN-XL/2, selecting near low-FID plateaus.Under the same ODE tolerance, Heun requires 58 versus 146 NFEs for SiT-XL/2 and 42 versus 150 for FlowDCN-XL/2.Source: §4 · p. 4, Table 1 · p. 4

Stability Analysis

  • NFE estimates remain stable from 1K to 20K probes with narrow 95% confidence bands; FID varies modestly around selected tolerances.The reported results support using 2K probes.Source: §4 · p. 4

Computational Cost Analysis

  • 4.5× less wall-clock time is required to estimate and evaluate the 3-RF budget than to construct its FID–NFE curve: 24.3 versus 110.3 minutes.The 3-RF evaluation covers only up to NFE 32, and the advantage is expected to grow for models needing broader sweeps.Source: §4 · p. 4

5. CONCLUSION

FLOWBUDGET estimates a fixed solver’s numerical resolution from scale-normalized local discretization error, without an NFE–quality sweep. Its probes use training-distribution samples and only approximate the sampler trajectory law.

Conclusion

  • FLOWBUDGET estimates a fixed solver’s numerical resolution from scale-normalized local error; its Euler, Heun, and additive-noise Euler–Maruyama laws are numerical statements.For ODE solvers, normalized coefficients also control terminal integration error under a bounded-velocity assumption.Source: §5 · p. 4

Limitations and Future Work

  • Interpolation probes use training-distribution samples and only approximate the sampler trajectory law; calibration between τ and perceptual metrics such as FID remains future work.Source: §5 · p. 4

Related papers

Loading 2610.20636v1…