VA-SALD: Velocity-Aware Guided Diffusion

• The paper introduces VA-SALD, a method that augments SALD by incorporating pretrained transport velocity and a time slowdown schedule to control guidance-induced deviation.
• It decouples the base generative flow from the guidance correction via a correction field, reducing bias by focusing on the corrective energy rather than the full guided transport.
• Empirical results demonstrate that VA-SALD achieves lower terminal KL and improved stability in both synthetic scenarios and large-scale guided image generation compared to baseline methods.

Velocity-Aware SALD (VA-SALD) is a guided sampling framework for pretrained diffusion-based and related generative models that extends Slowly Annealed Langevin Dynamics (SALD) by explicitly incorporating the transport velocity of the pretrained marginal path and using time slowdown to control guidance-induced deviation. In the formulation developed in "Slowly Annealed Langevin Dynamics: Theory and Applications to Training-Free Guided Generation" (Nitanda et al., 8 May 2026), VA-SALD is designed for training-free guided generation: it uses a fixed pretrained score model, a guide or reward $f_t$, and a slowdown schedule $t=t(s)$ to approximate a terminal guided target $\pi_T$, while providing non-asymptotic convergence guarantees in KL divergence.

1. SALD, moving targets, and path velocity

SALD begins from a nonstationary sampling problem. One is given a path of target distributions $(\pi_t)_{t\in[0,T]}$ on $\mathbb{R}^d$, typically with oracle access to the time-dependent score $\nabla \log \pi_t(x)$, and seeks to approximately sample from $\pi_T$. Standard annealed Langevin dynamics along the path is

$dX_t=\nabla\log \pi_t(X_t)\,dt+\sqrt{2}\,dW_t.$

SALD slows this traversal by introducing algorithmic time $s\in[0,S]$ and a monotone schedule $t=t(s)$, with slowed path $t=t(s)$0. The continuous-time SALD dynamics are

$t=t(s)$1

and the Euler–Maruyama discretization is

$t=t(s)$2

A standard choice is linear slowdown $t=t(s)$3, so the same physical path is traversed in $t=t(s)$4 times more algorithmic time (Nitanda et al., 8 May 2026).

The paper’s notion of “velocity” is not a particle kinematic observable but a transport velocity field for a curve of measures. For a path $t=t(s)$5, a vector field $t=t(s)$6 is a transport velocity if

$t=t(s)$7

This continuity-equation viewpoint quantifies how fast probability mass moves along the path. The paper defines the exponential-moment functional

$t=t(s)$8

and the corresponding path complexity

$t=t(s)$9

SALD’s convergence depends on this complexity and on functional inequalities, especially log-Sobolev and Poincaré inequalities, along the intermediate targets. A common misconception is that slowdown is merely a heuristic increase in step count; in the paper’s formulation, it is a continuous-time reparametrization whose benefit is expressed analytically through contraction and path-complexity terms.

2. Construction of VA-SALD for guided diffusion models

VA-SALD specializes SALD to the setting of pretrained Itô diffusions and guided generation. The forward diffusion is

$\pi_T$0

with reverse marginal family

$\pi_T$1

The associated forward and reverse transport velocities are

$\pi_T$2

Guidance is introduced through a potential or reward $\pi_T$3, or more generally a time-dependent guide $\pi_T$4 with $\pi_T$5, defining

$\pi_T$6

The framework assumes oracle access to $\pi_T$7 and to $\pi_T$8, or to a stochastic zeroth-order estimate in black-box guidance settings (Nitanda et al., 8 May 2026).

VA-SALD combines slowdown with explicit use of the pretrained velocity $\pi_T$9. With schedule $(\pi_t)_{t\in[0,T]}$0, $(\pi_t)_{t\in[0,T]}$1, and $(\pi_t)_{t\in[0,T]}$2, the SDE is

$(\pi_t)_{t\in[0,T]}$3

Equivalently, using the explicit form of $(\pi_t)_{t\in[0,T]}$4,

$(\pi_t)_{t\in[0,T]}$5

The defining distinction from SALD is structural. SALD uses only $(\pi_t)_{t\in[0,T]}$6, whereas VA-SALD uses both the pretrained path velocity $(\pi_t)_{t\in[0,T]}$7 and a Langevin drift based on $(\pi_t)_{t\in[0,T]}$8. In the unguided case $(\pi_t)_{t\in[0,T]}$9, so $\mathbb{R}^d$0, the paper shows that if $\mathbb{R}^d$1, then $\mathbb{R}^d$2 exactly for any schedule $\mathbb{R}^d$3. Slowdown therefore does not itself introduce bias in the unguided setting; VA-SALD exactly tracks the pretrained marginal path (Nitanda et al., 8 May 2026). This property is the basis of the “velocity-aware” terminology: the base transport is handled explicitly, and only the guidance-induced deviation must be corrected.

3. Guided-path deviation and the correction field

A central contribution of the theory is the decomposition of guided transport into a pretrained component and a correction component. If

$\mathbb{R}^d$4

then the pretrained velocity $\mathbb{R}^d$5, which generates $\mathbb{R}^d$6, need not generate $\mathbb{R}^d$7. The paper defines

$\mathbb{R}^d$8

and proves the residual identity

$\mathbb{R}^d$9

This equation gives an exact characterization of the mismatch between the guided marginal evolution and the pretrained transport field (Nitanda et al., 8 May 2026).

To restore a valid transport description of the guided path, the paper introduces a correction field $\nabla \log \pi_t(x)$0 solving the Poisson-type equation

$\nabla \log \pi_t(x)$1

Then $\nabla \log \pi_t(x)$2 is a transport velocity for $\nabla \log \pi_t(x)$3: $\nabla \log \pi_t(x)$4 In this decomposition, $\nabla \log \pi_t(x)$5 is the pretrained marginal transport velocity and $\nabla \log \pi_t(x)$6 is the additional velocity required to bend the pretrained path $\nabla \log \pi_t(x)$7 into the guided path $\nabla \log \pi_t(x)$8.

This distinction is what makes VA-SALD different from simply applying SALD to $\nabla \log \pi_t(x)$9. Standard SALD on the guided path must track the full guided transport, whose complexity is governed by $\pi_T$0. VA-SALD instead incorporates $\pi_T$1 directly in the sampler, so the convergence bounds depend only on $\pi_T$2, the complexity of the correction field. When guidance is relatively mild, the paper states that $\pi_T$3 and typically $\pi_T$4. This suggests that slowdown can be concentrated on compensating for guidance-induced deviation rather than re-tracking the entire pretrained generative flow.

The unguided limit makes the decomposition transparent. If $\pi_T$5, then $\pi_T$6, the residual vanishes, and $\pi_T$7. In that regime, VA-SALD reduces to exact tracking of the pretrained marginal path, and the guidance-complexity term disappears from the theory.

4. KL theory, slowdown, and functional inequalities

The paper establishes non-asymptotic convergence guarantees through KL differential inequalities. For SALD, if each $\pi_T$8 satisfies a log-Sobolev inequality with constant $\pi_T$9, and $dX_t=\nabla\log \pi_t(X_t)\,dt+\sqrt{2}\,dW_t.$0 is a transport velocity for $dX_t=\nabla\log \pi_t(X_t)\,dt+\sqrt{2}\,dW_t.$1, the law $dX_t=\nabla\log \pi_t(X_t)\,dt+\sqrt{2}\,dW_t.$2 at terminal algorithmic time satisfies a bound of the form

$dX_t=\nabla\log \pi_t(X_t)\,dt+\sqrt{2}\,dW_t.$3

For linear slowdown $dX_t=\nabla\log \pi_t(X_t)\,dt+\sqrt{2}\,dW_t.$4, the bound simplifies to

$dX_t=\nabla\log \pi_t(X_t)\,dt+\sqrt{2}\,dW_t.$5

As $dX_t=\nabla\log \pi_t(X_t)\,dt+\sqrt{2}\,dW_t.$6, the paper writes

$dX_t=\nabla\log \pi_t(X_t)\,dt+\sqrt{2}\,dW_t.$7

Two effects are explicit: slowdown improves contraction of initialization mismatch through accumulated intermediate log-Sobolev constants, and it reduces tracking error through the $dX_t=\nabla\log \pi_t(X_t)\,dt+\sqrt{2}\,dW_t.$8 scaling of path energy. The discrete-time SALD theory yields analogous rates and an iteration complexity $dX_t=\nabla\log \pi_t(X_t)\,dt+\sqrt{2}\,dW_t.$9 to reach $s\in[0,S]$0 KL accuracy (Nitanda et al., 8 May 2026).

For VA-SALD, the general theory introduces a comparison field $s\in[0,S]$1 and mismatch $s\in[0,S]$2, where $s\in[0,S]$3 is the true transport velocity of $s\in[0,S]$4. The continuous-time KL bound has the form

$s\in[0,S]$5

In the actual diffusion-model specialization, $s\in[0,S]$6, so $s\in[0,S]$7. Under linear slowdown and with $s\in[0,S]$8, Corollary 2 gives

$s\in[0,S]$9

For small $t=t(s)$0, the asymptotic form is

$t=t(s)$1

The interpretation is direct: guidance bias is controlled by the action of the correction field $t=t(s)$2, not by the action of the full guided velocity $t=t(s)$3. In the unguided case $t=t(s)$4, the path-complexity term vanishes and only contraction of initialization mismatch remains.

The appendix further connects correction complexity to a Poincaré inequality. If $t=t(s)$5 satisfies PI with constant $t=t(s)$6, then the solution $t=t(s)$7 of the Poisson equation satisfies

$t=t(s)$8

and therefore

$t=t(s)$9

This identifies guide-induced complexity with the variance of the directional time derivative $t=t(s)$00, giving a functional-analytic interpretation of when training-free guidance is easy or hard.

5. Algorithms, implementations, and empirical results

The general Euler–Maruyama discretization of VA-SALD is

$t=t(s)$01

with $t=t(s)$02 in the diffusion-model setting. For linear slowdown, $t=t(s)$03, and the conceptual Itô-diffusion implementation uses

$t=t(s)$04

together with the guide gradient $t=t(s)$05. The algorithm samples $t=t(s)$06, iterates the VA-SALD-EM update, and outputs $t=t(s)$07 as an approximate sample from $t=t(s)$08. The main hyperparameters are the slowdown factor $t=t(s)$09, step size $t=t(s)$10, guidance scale $t=t(s)$11, and model noise schedule $t=t(s)$12. The paper states that larger $t=t(s)$13 improves tracking and bias suppression at linear computational cost, and suggests choosing $t=t(s)$14 with $t=t(s)$15 for discretization control (Nitanda et al., 8 May 2026).

A concrete large-scale implementation is given for flow-matching models, including experiments with Stable Diffusion 3.5 Medium. For flow matching, the forward ODE is

$t=t(s)$16

and an SDE with the same marginals is

$t=t(s)$17

with score

$t=t(s)$18

After reversing time and applying slowdown, the implementation uses a specialized VA-SALD SDE and its Euler discretization. For black-box guidance, $t=t(s)$19 is approximated by zeroth-order gradient estimation,

$t=t(s)$20

with $t=t(s)$21, together with group reward normalization.

The empirical study has two parts. In synthetic VP diffusion experiments, the paper considers a two-Gaussian data distribution with a two-moons guide, and an eight-Gaussian ring with a left-half mode penalty. SALD’s terminal KL to the guided target decreases monotonically with the slowdown factor $t=t(s)$22, matching the theory. VA-SALD achieves lower terminal KL for smaller budgets, with flatter and better KL curves than SALD under matched budgets. DOIT, under matched proposal-particle budgets, remains significantly worse in terminal KL.

In guided image generation, the backbone is Stable Diffusion 3.5 Medium in flow-matching form, and the rewards include Aesthetic score, PickScore, and CLIPScore. Under matched numbers of steps and batch query size, and with guidance scale $t=t(s)$23, VA-SALD consistently achieves higher reward and visually better images than the baselines FM-ZG and FM-Evolv. Those baselines often exhibit instability and artifacts at high guidance strength, whereas VA-SALD produces samples that evolve stably as $t=t(s)$24 increases and remain high quality. The paper also reports qualitative examples for prompts such as “bear” and “wolf” across varying $t=t(s)$25 and guidance strengths $t=t(s)$26, showing gradual and stable guidance under VA-SALD (Nitanda et al., 8 May 2026).

6. Methodological position, misconceptions, and open problems

VA-SALD occupies a specific position within guided generative modeling. Relative to standard Langevin or ancestral sampling, it does not merely add a heuristic guide term to the pretrained sampler; it incorporates the pretrained transport velocity $t=t(s)$27 and applies slowdown to a principled mismatch term. Relative to standard SALD on the guided path, it does not pay for the action of the full guided transport; its KL bounds depend only on the correction energy $t=t(s)$28. Relative to classifier or classifier-free guidance, it is framed through marginal KL control, intermediate functional inequalities, and path complexity rather than through heuristic score modification. Relative to Doob $t=t(s)$29-transform approaches such as DOIT and Doob’s matching, it does not promise exact terminal tilting, but it avoids training a time-dependent guide network or performing expensive on-the-fly simulations (Nitanda et al., 8 May 2026).

The relation to Doob $t=t(s)$30-transforms is especially instructive. The exact terminal tilt $t=t(s)$31 can be realized through

$t=t(s)$32

with a reverse-process drift involving $t=t(s)$33. VA-SALD instead works with a prescribed guide schedule $t=t(s)$34, uses only $t=t(s)$35, and controls the resulting bias through slowdown and the correction complexity $t=t(s)$36. The paper explicitly characterizes this as a trade: no Doob optimality guarantee, but a training-free inference procedure with quantitative KL bounds.

Several limitations and open questions are highlighted. First, slowdown is computationally costly: increasing $t=t(s)$37 improves tracking but linearly increases the number of iterations. Second, the choice of guide schedule $t=t(s)$38 is unresolved; the correction complexity depends on $t=t(s)$39 and $t=t(s)$40 through $t=t(s)$41, so designing schedules that minimize $t=t(s)$42 remains open. Third, the theory assumes log-Sobolev or Poincaré inequalities, smoothness, and dissipativity conditions that are natural for many annealing paths and dissipative diffusions but are not trivial to verify for large-scale high-dimensional generative models. Fourth, extremely strong or adversarial guides may produce large $t=t(s)$43, requiring very large slowdown factors. Fifth, the empirical evaluation is illustrative rather than exhaustive: it covers two-dimensional synthetic experiments and a limited set of SD3.5 prompts rather than large prompt suites or human preference studies.

Taken together, these features define VA-SALD as a training-free guided sampling method whose novelty lies in its explicit separation of base transport from guidance correction. Its theory formalizes the intuition that pretrained generative models already encode a useful path of marginals $t=t(s)$44, and that guidance should be treated as a controlled perturbation of that path rather than as an entirely new trajectory to be tracked from scratch (Nitanda et al., 8 May 2026).