ConicFlow: constrained Inference-time optimal control for flow matching models

Shervin Khalafi; Kasra Fallah; Sung Cho; Dongsheng Ding; James Anderson; Alejandro Ribeiro

cs.AIOct 9, 2026

ConicFlow: constrained Inference-time optimal control for flow matching models

Authors: Shervin Khalafi, Kasra Fallah, Sung Cho, Dongsheng Ding, James Anderson, Alejandro Ribeiro

Organizations: University of Pennsylvania · Columbia University · University of Tennessee, Knoxville

Abstract

We propose ConicFlow, which frames constrained sampling from a flow matching model at inference time as an optimal control problem that balances deviation from the pretrained trajectory against reward, subject to hard terminal constraints. We discretize this problem and adapt receding-horizon control, optimizing only a short window of the trajectory at each step for computational tractability. We address the resulting sequence of receding-horizon subproblems with a receding-horizon primal-dual algorithm. Its dual multipliers capture the difficulty of satisfying each constraint and are refined throughout sampling, steering generation toward feasible final samples. We further propose a terminal-estimate variant that avoids accumulating local truncation errors by evaluating the reward and constraints at the predicted terminal point rather than at the end of the current horizon. Under standard regularity assumptions, we establish finite-horizon guarantees on suboptimality. Under convex-affine structure, we derive a sharper bound that scales linearly with omitted horizon length, and we show that the terminal-estimate variant is provably tighter when the flow dynamics change slowly. Experiments in both low- and high-dimensional settings with a recent large-scale flow matching model show that ConicFlow achieves high reward and strong constraint satisfaction while deviating less from the pretrained flow than recent baselines.

Figures & tables

Panel of Figure 1 — Figure 1: Trade-off between terminal reward and total control cost, traced by varying the control-cost weight $\lambda$ (upper left is better). $\lambda$ value displayed on each point. Left: easy reward (upper moon). Middle: difficult reward (off-manifold). Right: uncontrolled sample distribution and both rewards.

Panel of Figure 2 — Figure 2: Trade-off between terminal reward and total control cost. Left: Reward vs. control energy on the editing task, traced by sweeping $\lambda$ (upper left is better). Right: Edits from each method at approximately matched reward; ConicFlow reaches the same reward with much lower control energy.

Method	CLIP $(\uparrow)$	ImageReward $(\uparrow)$	PickScore $(\uparrow)$
ConicFlow	$\mathbf{0.130}\pm 0.018$	$\mathbf{2.048}\pm 0.455$	$\mathbf{6.158}\pm 0.728$
HardFlow	$0.118\pm 0.021$	$1.279\pm 0.374$	$1.736\pm 0.472$
FMRG	$0.065\pm 0.008$	$1.278\pm 0.504$	$2.670\pm 0.073$

Table 1: Mean reward gain over a shared energy budget. Each method’s reward–energy frontier, averaged over the control-energy range attained by all three methods and measured relative to the uncontrolled sample (units are those of rewards). Mean over

n{=}4

(prompt, seed) pairs per reward.

Panel of Figure 3 — Figure 3: Constrained image editing comparison. Top left: control energy vs. threshold multiplier $m$ , median over (prompt, seed) pairs. Bottom left: feasibility rate vs. $m$ . Middle: single-constraint edits with increasing $m$ . Right: multi-constraint edits with increasing $m$ .

Appendix figures & tables13 assets

Supplementary material from the paper’s appendix.

Appendix

Shared across methods
Steps $N$		$16$
Batch / seed		$512$ particles / $42$ , shared across all runs
Base law $p_{0}$		$\mathcal{N}(0,\sigma_{d}^{2}I)$ , $\sigma_{d}=0.750$
Backbone		MLP, $3\times 256$ GELU; EMA weights at $15$ k steps
Per method
ConicFlow	horizon $H$	$\{1,2,4,8,16\}$

Appendix

Table 2: Hyperparameters for the low-dimensional experiments.

Shared across methods
Steps $N$ / horizon $H$		$8$ / $2$
Resolution / latent		$256\times 256$ / $16\times 32\times 32$
Guidance scale		$3.5$
Precision		bf16 sampler, int8 transformer, fp32 reward
Reward models		CLIP, PickScore (both CLIP-ViT-H-14), ImageReward
Seeds / batch		$42$ and $7$ / $1$

Appendix

Table 3: Hyperparameters for the unconstrained image experiment (Fig 2 )

Shared across methods
Reward		none ( $r\equiv 0$ ; objective is control energy)
Constraint scorer		PickScore, one head per attribute prompt
Constraint form		$h_{i}=\tau_{i}-s_{i}\leq 0$ , evaluated at the endpoint
Thresholds		$\tau_{i}=m\,s_{i}^{0}$ , Eq. ( 5 )
Steps $N$ / horizon $H$		$8$ / $2$
$\lambda$		$20$ (fixed)

Appendix

Table 4: Hyperparameters for the constrained image experiments. Everything not listed (sampler, resolution, precision, guidance scale) is as in Table 3 .

base prompt $c$	attribute prompts $c_{1},\dots,c_{C}$
a portrait of a man	a smiling face
a photo of a golden retriever	a dog wearing sunglasses
a city street at noon	a city street at night
a portrait of a woman	a woman with bright red hair
a portrait of a woman	sad frowning woman
portrait of a young man with straight hair	man with curly hair; photo of very old man

Appendix

Table 5: Constrained instances. Each attribute prompt is one PickScore constraint ( 4 ); semicolons separate the constraints of an instance.

Instance (prompt $\to$ edit, seed)	$r_{\mathrm{unc}}$	Budget $[E_{1},E_{2}]$	ConicFlow	HardFlow	FMRG
CLIP
man $\to$ smiling, 42	$0.149$	$[8.8,\,217]$	$\mathbf{0.151}$	$0.135$	$0.083$
man $\to$ smiling, 7	$0.210$	$[8.2,\,136]$	$\mathbf{0.094}$	$0.070$	$0.046$
sleeping cat $\to$ open eyes, 42	$0.247$	$[8.8,\,127]$	$\mathbf{0.106}$	$0.101$	$0.065$
green grass $\to$ red poppies, 42	$0.127$	$[7.6,\,122]$	$\mathbf{0.171}$	$0.166$	$0.066$
mean $\pm$ s.e.			$\mathbf{0.130}\pm 0.018$	$0.118\pm 0.021$	$0.065\pm 0.008$

Appendix

Table 6: Per-problem breakdown of Table 1 .

Panel of Figure 4 — Figure 4: Reward–effort trade-off on two further editing instances, $N=8$ , $H=2$ . The horizontal axis is control energy $\sum_{k}\lVert u_{k}\rVert^{2}\Delta t$ and the vertical axis the achieved reward $r(x_{N})$ , so up and to the left is better. Each marker is one value of $\lambda$ .

Panel of Figure 5 — Figure 5: More examples of unconstrained editing. Each panel is one (reward, prompt, seed) instance at a matched-reward level. Panels 5(g) – 5(j) are the harder case: turning a green field red, or opening a sleeping cat’s eyes, asks for the image to be restructured rather than adjusted, and no method delivers it.

Panel of Figure 6 — Figure 6: More examples of constrained editing. Each panel is one (prompt, seed) instance. Instance details are in Table 5 .

Panel of Figure 7 — Figure 7: More examples of constrained editing. Each panel is one (prompt, seed) instance. Instance details are in Table 5 .

Panel of Figure 8 — Figure 8: More examples of constrained editing. Each panel is one (prompt, seed) instance. Instance details are in Table 5 .

$H$	$\Delta r$	$E$	time
$1$	$3.98$	$78.5$	$207$ s
$2$	$4.45$	$\mathbf{68.7}$	$249$ s
$4$	$4.39$	$70.6$	$288$ s
$6$	$4.67$	$90.3$	$321$ s
$8$	$\mathbf{4.95}$	$101.4$	$264$ s

Appendix

Table 7: Planning horizon, editing,

N=8

, seed

317

;

\Delta r

E

and time are medians over the

24

operating points in each

H

. Wall clock is measured on shared cards and is indicative only.

	$n_{\mathrm{inner}}$	time	$\Delta r$	$E$
ConicFlow	$12$	$124$ s	$\mathbf{4.16}$	$84.6$
HardFlow	$167$	$123$ s	$2.39$	$555.2$
FMRG-J	$13$	$128$ s	$1.84$	$\mathbf{32.5}$

Appendix

Table 8: Equal

\lambda

and equal wall clock, editing,

N=8

, seeds

317

and

1289

;

\Delta r

E

and time are medians.

	objective at	time	$\Delta r$	$E$
ConicFlow	endpoint	$238$ s	$\mathbf{4.79}$	$\mathbf{68.7}$
	horizon	$90$ s	$3.25$	$86.9$
HardFlow	—	$36$ s	$2.00$	$553.9$
FMRG-J	—	$33$ s	$2.39$	$87.0$

Appendix

Table 9: Where the horizon objective is evaluated, editing,

N=8

, seeds

317

and

1289

;

\Delta r

E

and time are medians.

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