Slowly Annealed Langevin Dynamics: Theory and Applications to Training-Free Guided Generation

Abstract: We study Slowly Annealed Langevin Dynamics (SALD), a sampler for tracking a path of moving target distributions and approximating the terminal target through time slowdown. We establish non-asymptotic convergence guarantees via a KL differential inequality, showing that slowdown improves tracking through contraction of intermediate targets and the complexity of the path. Motivated by training-free guided generation with pretrained score-based generative models, we further introduce Velocity-Aware SALD (VA-SALD), which explicitly incorporates the underlying marginal distributions of the pretrained model and uses slowdown to correct the additional deviation induced by guidance. This yields a principled framework for training-free guided generation for diffusion-based and related generative model families, together with convergence guarantees that clarify the roles of intermediate functional inequalities and guidance bias. Code is available at https://github.com/anitan0925/sald.

• The paper introduces SALD, a method that decelerates annealing to improve target tracking through a non-asymptotic KL convergence analysis.
• It presents VA-SALD, which leverages pretrained marginal velocities to enable robust, training-free guided generation with controlled complexity.
• Empirical evaluations show VA-SALD reduces terminal KL divergence and avoids artifacts, outperforming traditional guided generation baselines.

Slowly Annealed Langevin Dynamics for Training-Free Guided Generation

Introduction

This work introduces a rigorous non-asymptotic theory of Slowly Annealed Langevin Dynamics (SALD) and expands its applicability to training-free guided generation with pretrained score-based generative models. SALD addresses high-dimensional sampling from complex, evolving distributions, a core problem in generative modeling, statistics, and machine learning. The paper analyzes how SALD improves distributional tracking via time slowdown, leveraging functional inequalities of intermediate targets and an explicit path complexity term. A novel extension, Velocity-Aware SALD (VA-SALD), is proposed to perform inference-time guided generation, exploiting marginal dynamics of pretrained models and yielding favorable theoretical and empirical results.

Slowly Annealed Langevin Dynamics: Methodology and Theory

Traditional annealed Langevin approaches incrementally transform a reference measure into a target via a sequence of intermediates, guided by score functions, as in diffusion-based generative models. SALD generalizes this by introducing a slowdown factor $r\geq 1$, rescaling time such that the intermediate distributions evolve more slowly, thus allowing the sampler to track moving targets more closely. The sampler evolves according to

$$dX_s = \nabla \log f_s(X_s)\, ds + \sqrt{2}\, dW_s,\quad X_0 \sim p_0$$

where $f_s$ follows a rescaled target path and $W_s$ is standard Brownian motion.

The central theoretical contribution is a non-asymptotic convergence analysis via a forward Kullback-Leibler (KL) differential inequality. The analysis makes explicit two sources of contraction: (1) functional inequalities (notably the log-Sobolev inequality, LSI) of the intermediate targets that provide exponential decrease in the KL divergence with the slowdown, and (2) a path complexity term, i.e., the “energy” accumulated along the trajectory. Importantly, this analysis captures effects missed by earlier path-space arguments, notably contraction towards a moving target even when the initial distribution is mismatched.

For discrete-time implementations, an Euler-Maruyama discretized SALD is shown to achieve iteration complexity $\mathcal{O}(\varepsilon^{-6})$ for $\varepsilon$-accurate solutions in forward KL divergence, comparable to prior bounds but under weaker initialization assumptions.

Velocity-Aware SALD: Guided Generation without Training

A significant limitation of existing guided generation methods—based on Doob's $h$-transform or direct path tracking—is reliance on learning or estimating time-dependent guide functions, or high-variance inference-time Monte Carlo corrections. By contrast, VA-SALD leverages pretrained marginal velocities from diffusion, flow-matching, or Schrödinger bridge models while incorporating a smooth guide function. Specifically, VA-SALD adds only a guidance correction to the dynamics, rather than requiring the full drift for the guided path. The VA-SALD SDE is

$$dX_s = \left[t(s) u_{t(s)}(X_s) - t(s) \nabla f_{t(s)}(X_s)\right]\, ds + \sigma_{t(s)} dW_s$$

where $u_t$ is the transport velocity of the (pretrained) marginal $p_t$, and $$dX_s = \nabla \log f_s(X_s)\, ds + \sqrt{2}\, dW_s,\quad X_0 \sim p_0$$0 is a possibly time-varying guidance function targeting the desired conditional.

The key theoretical result is that VA-SALD's convergence is governed by the complexity of the guide-induced correction vector field $$dX_s = \nabla \log f_s(X_s)\, ds + \sqrt{2}\, dW_s,\quad X_0 \sim p_0$$1, rather than the full guided dynamics. Most notably, in the unguided case $$dX_s = \nabla \log f_s(X_s)\, ds + \sqrt{2}\, dW_s,\quad X_0 \sim p_0$$2, the correction vanishes and the convergence rate is determined solely by the initial mismatch, with no unnecessary complexity penalty.

Theoretical Results: Non-Asymptotic KL Bounds

For both SALD and VA-SALD, the main theorems establish that under mild regularity conditions (LSI and finite path energy), the KL divergence to the terminal target at the end of the annealing path decays as

$$dX_s = \nabla \log f_s(X_s)\, ds + \sqrt{2}\, dW_s,\quad X_0 \sim p_0$$3

The contraction term decays exponentially in the slowdown parameter $$dX_s = \nabla \log f_s(X_s)\, ds + \sqrt{2}\, dW_s,\quad X_0 \sim p_0$$4, directly reflecting the accumulated functional inequalities along the intermediate targets. The complexity term depends on the total “action” (energy) of the transport velocity field along the path ($$dX_s = \nabla \log f_s(X_s)\, ds + \sqrt{2}\, dW_s,\quad X_0 \sim p_0$$5 complexity), and, for VA-SALD, only on the guide correction vector field, which is typically much smaller than the velocity needed to track the full guided path. Importantly, these results hold for both continuous-time and discrete-time Euler-Maruyama implementations.

Empirical Evaluation

The empirical section validates the theoretical predictions on both synthetic and realistic high-dimensional generative tasks:

• Synthetic Data: On controlled 2D guided diffusion tasks (e.g., moving between Gaussian mixtures with nonlinear guides), monotonically increasing the slowdown $$dX_s = \nabla \log f_s(X_s)\, ds + \sqrt{2}\, dW_s,\quad X_0 \sim p_0$$6 systematically reduces the terminal KL divergence for both SALD and VA-SALD. VA-SALD consistently outperforms both vanilla SALD and the DOIT method (a simulation-based Doob's $$dX_s = \nabla \log f_s(X_s)\, ds + \sqrt{2}\, dW_s,\quad X_0 \sim p_0$$7-transform estimator) at matched computational budgets, attaining lower terminal errors and more stable objective values.
• Image Generation: Using Stable Diffusion 3.5-Medium as the pretrained backbone and applying VA-SALD for black-box reward guidance (e.g., aesthetic score, PickScore, CLIPScore), VA-SALD produces higher reward scores and more stable outputs compared to FM-ZG and FM-Evolv, which are leading classifier-free training-free baselines. Notably, VA-SALD avoids the semantic collapse or artifact generation that plagues these baselines as guidance strength increases, attributable to its control over the deviation from the pretrained marginal path.

Implications and Future Directions

The marginal-KL differential analysis clarifies when and how slowdown in annealed Langevin dynamics improves sampling efficiency, especially for non-log-concave or rapidly evolving targets. For practitioners, SALD and VA-SALD provide a principled framework for training-free, test-time guidance in diffusion or flow-based generative models, accommodating arbitrary reward or constraint functions and controlling the complexity induced by guidance.

The practical efficacy of VA-SALD, particularly its robustness at large guidance strengths and black-box rewards, suggests broad applicability for preference alignment, conditional generation, and safety-critical sampling.

Several open questions are highlighted:

• Acceleration and Efficiency: Large slowdown comes at increased computational cost. Combining SALD with acceleration techniques or distillation (e.g., time-adaptive schedules, estimator-based step compression) could further improve efficiency.
• Guide Schedule Design: The impact of the guidance schedule $$dX_s = \nabla \log f_s(X_s)\, ds + \sqrt{2}\, dW_s,\quad X_0 \sim p_0$$8 on the complexity of $$dX_s = \nabla \log f_s(X_s)\, ds + \sqrt{2}\, dW_s,\quad X_0 \sim p_0$$9 is not fully understood. Systematic design of such schedules may yield further improvements in guided generation.
• Generalization and Extensions: The machinery developed here could theoretically extend to other non-reversible or non-diffusive transport processes, broader classes of guides (e.g., non-differentiable, black-box), and high-dimensional state spaces.

Conclusion

This work provides a comprehensive non-asymptotic theory and practical algorithmic methods for annealed Langevin dynamics with explicit time slowdown, both for unbiased sampling of evolving targets and for efficient, training-free guided generation with pretrained score-based generative models. The introduction of VA-SALD establishes that principled control over the complexity introduced by guidance enables robust, stable, and theoretically well-characterized generation, as substantiated by both rigorous analysis and strong empirical results.

Reference: "Slowly Annealed Langevin Dynamics: Theory and Applications to Training-Free Guided Generation" (2605.07950)