Dimension-Free Multimodal Sampling via Preconditioned Annealed Langevin Dynamics

Conference: ICML 2026
arXiv: 2605.30396
Code: TBD
Area: Optimization / Sampling Algorithms / Diffusion Model Theory
Keywords: Annealed Langevin Dynamics, Multimodal Distribution, Dimension-Free Convergence, Hessian Preconditioning

TL;DR

The first dimension-free non-asymptotic convergence analysis for Preconditioned Annealed Langevin Dynamics (PALD) is provided—reducing the sampling complexity for multimodal distributions from \(\tilde{O}(d/\epsilon^2)\) to \(\tilde{O}(1/\epsilon^2)\), effectively liberating diffusion-based sampling algorithms from the "curse of dimensionality" in high-dimensional settings.

Background & Motivation

Background: Sampling from multimodal distributions is a central challenge in machine learning and statistics. Standard Langevin Dynamics (LD) requires exponential time to cross "energy barriers" between modes. Annealed LD (ALD) gradually reduces the energy landscape through temperature annealing and has proven practical in NCSN and diffusion models.

Limitations of Prior Work: Existing convergence analyses for ALD, while providing guarantees, show a complexity linear in dimension \(d\) or worse. This leads to an explosion in required sample steps for high-dimensional data (e.g., ImageNet where \(d \approx 10^6\)).

Key Challenge: ALD efficiently samples in millions of dimensions in practice, but theoretical analysis fails to explain this phenomenon, creating a "dimensionality gap" between theory and practice.

Goal: Establish a dimension-free convergence guarantee for ALD on high-dimensional multimodal distributions to bridge the gap between theory and practice.

Key Insight: It is observed that the dimension dependence in existing analyses stems from the assumption of isotropic step sizes. By utilizing preconditioning (local Hessian adaptation), effective step sizes can be maintained across high-dimensional directions, enabling dimension-free convergence.

Core Idea: The update rule of Langevin Dynamics is replaced with a local Hessian-based preconditioned version—\(\theta_{t+1} = \theta_t - \eta H(\theta_t)^{-1} \nabla U(\theta_t) + \sqrt{2\eta H(\theta_t)^{-1}} \xi_t\)—to achieve dimension-free convergence while preserving the annealing framework.

Method

Overall Architecture

(1) Target distribution \(\pi(\theta) \propto \exp(-U(\theta))\); (2) Construction of a temperature sequence \(\beta_1 < \beta_2 < ... < \beta_K = 1\); (3) Execution of preconditioned Langevin updates at each temperature; (4) Obtaining the preconditioner \(H(\theta_t)\) via Hessian adaptation or low-rank approximation; (5) Obtaining target samples at the final temperature.

Key Designs

1. Preconditioned Hessian Adaptation:

   • Function: Compensates for curvature differences in different directions of the potential energy, ensuring identical effective step sizes for all directions.
   • Mechanism: \(H(\theta) = \nabla^2 U(\theta)\) (or a regularized version \(H + \lambda I\)) is used as the preconditioner. It reduces step sizes in sharp directions (large Hessian eigenvalues) to maintain stability and increases step sizes in flat directions (small eigenvalues) to accelerate exploration.
   • Design Motivation: Standard LD uses uniform step sizes restricted by the sharpest direction. After preconditioning, the relative step size \(\eta / \lambda_i\) for each direction reaches the stability threshold, making the effective "number of steps" dimension-free.

2. Annealing Schedule + Dimension-Free Barrier Crossing:

   • Function: Bridges global exploration and local refinement via temperature annealing.
   • Mechanism: High temperatures (small \(\beta_k\)) flatten the potential function to facilitate mode jumping, while low temperatures refine the sampling. A geometric annealing schedule \(\beta_k = \beta_0 \cdot r^k\) (\(r > 1\)) is designed. Analysis shows that the barrier height \(\Delta\) is no longer linearly related to \(d\) because the "effort" required for crossing is determined by the effective curvature after preconditioning.
   • Design Motivation: Traditional annealing complexity proofs rely on maximum barrier height (roughly \(O(d)\)); preconditioning decouples this height from the dimension.

3. Theoretical Analysis Framework:

   • Function: Establishes the dimension-free complexity \(\tilde{O}(\log(1/\epsilon) / \epsilon^2)\).
   • Mechanism: The KL divergence \(\text{KL}(p_k \| \pi_{\beta_k})\) is shown to decrease monotonically along the temperature sequence. Complexity bounds are derived using log-Sobolev inequalities and Talagrand transport inequalities. A preconditioner-assisted synchronous coupling is explicitly constructed to avoid dimension explosion.
   • Design Motivation: While log-Sobolev constants are typically \(O(d^{-1})\), preconditioning is equivalent to analyzing in a transformed isometric space.

Key Experimental Results

Convergence Complexity

Method	Sampling Complexity	Dimension Dependence
Standard LD	\(\tilde{O}(d \beta^* / \epsilon^2)\)	Linear \(d\)
Standard ALD	\(\tilde{O}(d \log K / \epsilon^2)\)	Linear \(d\)
PALD (Ours)	\(\tilde{O}(\log K / \epsilon^2)\)	None
MCMC (HMC)	\(\tilde{O}(d^{1/4} / \epsilon^{1/2})\)	\(d^{1/4}\)

Synthetic Multimodal Exp

Distribution	Dim	Modes	LD Jump Rate	ALD Jump Rate	PALD Jump Rate
Mix of 2 Gaussians	100	2	12%	89%	97%
Mix of 2 Gaussians	10000	2	0%	23%	94%
4-Mix (Rotated)	100	4	8%	73%	96%
4-Mix (Rotated)	10000	4	0%	12%	91%

PALD maintains high jump rates in high dimensions while ALD/LD degrade significantly.

High-dim Specific Benchmarks

Task	Algorithm	Dim	Convergence Time (vs ALD)
NN Posterior Sampling	PALD vs ALD	50000	0.07× Time
High-dim GMM	PALD vs ALD	100000	0.02× Time

Key Findings

• Experimental Verification of Dimension-Freedom: PALD convergence time remains relatively stable from 100 to 10,000 dimensions, whereas ALD degrades sharply.
• Multimodal Retention: In 4-mode distributions, PALD accurately captures the relative weights of all modes, while ALD biases towards the initial mode in high dimensions.
• Preconditioner Update Frequency: Updating every 100 steps is optimal; excessive updates increase computational overhead.

Highlights & Insights

• First Dimension-Free Convergence Proof: Breakthrough in the "curse of dimensionality" for multimodal sampling, providing theoretical support for high-dimensional diffusion models.
• Elegant Combination of Preconditioning + Annealing: The synergy between these two independent techniques exceeds their individual use—preconditioning ensures step size efficiency, while annealing ensures global exploration.
• Rigorous Experimental Validation: Systematically demonstrates dimension-freedom from low (100) to high (\(10^5\)) dimensions, highly consistent with theoretical predictions.

Limitations & Future Work

• Hessian Computation Cost: Requires \(O(d^2)\) storage or \(O(d^3)\) factorization per step; remains difficult for ultra-high dimensions (\(d > 10^7\)).
• Precision Loss in Low-rank Approximations: Theoretical analysis assumes an exact Hessian preconditioner; practical low-rank or diagonal approximations might violate dimension-free conditions.
• Non-smooth Potentials: Current analysis requires \(U\) to be twice differentiable; non-smooth potentials or distributions on Stiefel manifolds are not directly applicable.
• Improvements: Explore fast approximations based on efficient preconditioners like K-FAC or Shampoo; extend analysis to non-smooth or geometrically constrained distributions.

Related Work & Insights

• vs Standard ALD (Song-Ermon 2019): The primary innovation lies in the preconditioning mechanism and theoretical analysis, providing the dimension-free proof.
• vs Hamiltonian Monte Carlo (HMC): HMC accelerates mixing via momentum, but its theoretical analysis remains dimension-dependent; PALD addresses dimensionality directly through preconditioning.
• vs Second-order Preconditioning in Adam/SGD: This work represents the first application of preconditioning to sampling scenarios rather than optimization.

Rating

• Novelty: ⭐⭐⭐⭐⭐ First dimension-free guarantee for multimodal sampling; a major theoretical breakthrough.
• Experimental Thoroughness: ⭐⭐⭐⭐ Comprehensive synthetic multimodal experiments; validation on real-world high-dimensional tasks is limited.
• Writing Quality: ⭐⭐⭐⭐ Mathematically rigorous with clear proof steps; theory and experiments corroborate each other.
• Value: ⭐⭐⭐⭐⭐ Lays a theoretical foundation for diffusion models and high-dimensional Bayesian inference.

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