Adjoint Sampling: Teaching AI to Create Better Samples, Faster

Adjoint Sampling is a novel and efficient algorithm for training generative models, particularly those based on diffusion processes, to sample from complex distributions defined by unnormalized densities or energy functions. It’s particularly useful in scenarios where explicit training data is limited or unavailable, and the goal is to generate high-quality samples based on a scalar reward signal, such as an energy function.

Here’s a summary of its key features and advantages:

• Scalability and Efficiency: Adjoint Sampling offers improved scalability and allows for more gradient updates per energy evaluation and model sample, making it suitable for larger problems.
• On-Policy Training: The method utilizes an on-policy approach, enabling it to use the energy function’s gradient to refine samples.
• Reduced Energy Evaluations: It requires fewer energy evaluations compared to traditional methods by addressing bottlenecks in SDE simulation and energy evaluation.
• Theoretical Grounding: Based on stochastic optimal control and Adjoint Matching, it can train models without corrective measures like importance sampling.
• Handling of Symmetries and Boundary Conditions: It can incorporate geometric symmetries and periodic boundary conditions, important for applications like molecular modeling.
• Applications: Adjoint Sampling has shown effectiveness in applications like molecular modeling and conformer generation, demonstrating strong performance on tasks like generating conformers from energy models.
• Novel Benchmarks: It introduces amortized molecule sampling benchmarks to encourage the development of more scalable sampling methods.

In essence, Adjoint Sampling is an efficient approach for training generative models using scalar reward signals and stochastic optimal control principles, which is particularly beneficial for fields like computational chemistry where generating high-quality samples from complex distributions is essential.

The document presents a novel algorithm called Adjoint Sampling for efficiently learning diffusion processes to sample from unnormalized densities, with applications in computational chemistry and molecular modeling.

Adjoint Sampling: A Scalable Diffusion Algorithm ​

Adjoint Sampling is a novel algorithm designed for efficiently learning diffusion processes that sample from unnormalized densities. ​ It significantly increases the number of gradient updates possible per energy evaluation, enabling scalability to larger problem settings.

• Introduces Adjoint Sampling for learning diffusion processes. ​
• Allows more gradient updates than energy evaluations, enhancing scalability.
• Theoretically grounded in stochastic optimal control.
• Incorporates symmetries and periodic boundary conditions for molecular modeling. ​
• Aims to open-source benchmarks to advance computational chemistry. ​

Challenges in Sampling High-Dimensional Distributions ​

Sampling from complex, high-dimensional distributions is crucial in computational science but remains challenging due to intricate energy landscapes and computational costs. ​ Traditional methods like MCMC and SMC often struggle with slow mixing and scalability. ​

• High-dimensional distributions are essential in molecular modeling and Bayesian inference. ​
• Traditional methods (MCMC, SMC) face slow mixing and scalability issues. ​
• Energy functions can be computationally expensive, complicating sampling. ​
• Recent approaches use learned proposal distributions to improve sampling efficiency. ​

Theoretical Foundations of Adjoint Sampling ​

Adjoint Sampling is based on stochastic optimal control principles, specifically addressing the optimization of diffusion processes for sampling. ​ The method minimizes the control energy required to transport distributions towards the target density. ​

• Built on stochastic optimal control and Schrödinger-Bridge problems.
• Aims to minimize control energy for efficient sampling. ​
• The objective is to transport the Dirac distribution to the target density. ​
• Theoretical guarantees ensure performance without corrective measures.

Implementation and Efficiency of Adjoint Sampling ​

The algorithm employs a unique approach to optimize the sampling process, significantly reducing the need for computationally expensive simulations. ​ It utilizes a replay buffer to enhance efficiency and allows multiple gradient updates without frequent energy evaluations. ​

• Utilizes a replay buffer to store samples for efficiency. ​
• Reduces computational costs by avoiding frequent simulations.
• Allows multiple gradient updates per energy evaluation. ​
• The algorithm is designed to be highly scalable.

Geometric Extensions and Symmetries in Sampling ​

Adjoint Sampling incorporates geometric symmetries to improve sampling efficiency, particularly in molecular systems. It employs equivariant neural networks to respect symmetry constraints during the sampling process.

• Incorporates symmetries to enhance sampling efficiency. ​
• Uses Equivariant Graph Neural Networks (EGNNs) for modeling. ​
• Ensures models are G-invariant for all time steps. ​
• Supports periodic boundary conditions for complex state spaces. ​

Experimental Results and Performance Metrics

The performance of Adjoint Sampling is evaluated against various methods using synthetic energy functions, demonstrating its effectiveness in achieving lower energy evaluations per gradient update. The results indicate significant improvements in sampling efficiency.

• Adjoint Sampling shows improved performance metrics across various experiments. ​
• Achieves lower energy evaluations per gradient update compared to other methods. ​
• Results indicate a geometric Wasserstein distance improvement in sampling quality.
• Demonstrates effectiveness in both synthetic and real-world molecular systems.

Learning Augmented MCMC and SMC Methods ​

Recent advancements in Markov Chain Monte Carlo (MCMC) and Sequential Monte Carlo (SMC) methods have integrated deep learning techniques to enhance sampling efficiency from complex distributions. ​

• MCMC and SMC methods face challenges like long mixing times and poor scaling in high dimensions. ​
• Techniques such as variational inference and normalizing flows have been employed to improve MCMC proposal distributions. ​
• Learning proposal distributions for SMC has also been explored, with notable contributions from Albergo et al. ​ (2024) and Holderrieth et al. (2025).

MCMC-reliant Diffusion Samplers

MCMC-reliant diffusion samplers utilize auxiliary sampling mechanisms to train diffusion processes effectively, often leveraging score-based diffusion models for improved sample efficiency. ​

• Recent works focus on score matching and iterated Denoising Energy Matching (iDEM) to enhance sampling tasks. ​
• These methods typically require ground truth samples or rely on importance-weighted estimation, which can be computationally intensive. ​
• Phillips et al. ​ (2024) and De Bortoli et al. (2024) highlight the effectiveness of regression objectives in these contexts. ​

SOC-based Diffusion Samplers

Stochastic optimal control (SOC) based samplers reformulate sampling tasks as optimization problems, but they often face high computational costs. ​

• SOC methods have been generalized to optimize controlled processes for matching target distributions. ​
• Adjoint Sampling offers a solution by enabling more gradient updates per generated sample, reducing computational burdens. ​
• Challenges include expensive differentiation through sampling procedures and the need for higher-order derivatives. ​

Off-policy Methods in Sampling

Off-policy methods do not require samples from the current model, which can lead to inefficiencies in gradient utilization. ​

• These methods often parameterize models using energy function gradients, but this approach is not feasible for computationally expensive energy functions. ​
• Adjoint Sampling, as an on-policy method, effectively uses energy gradients while requiring fewer samples for training. ​

Molecular Conformer Generative Models ​

Adjoint Sampling is positioned as a novel approach for generating molecular conformers directly from energy functions without relying on existing data.

• Traditional methods like Boltzmann generators require offline molecular dynamics data, limiting their scalability. ​
• Adjoint Sampling demonstrates state-of-the-art performance in generating conformers for complex molecular datasets, such as SPICE and GEOM-DRUGS.

Evaluation of Adjoint Sampling

Adjoint Sampling is evaluated against synthetic energy functions and a challenging molecular conformer generation benchmark, showcasing its efficiency and effectiveness.

• The method is compared to state-of-the-art samplers like iDEM and PIS, with results indicating superior performance in terms of energy evaluations per gradient update.
• Metrics such as recall and precision are reported, with Adjoint Sampling achieving high recall rates across various datasets. ​

Results of Synthetic Energy Functions ​

The performance of Adjoint Sampling is benchmarked against synthetic energy functions, demonstrating its scalability and efficiency. ​

• Three energy functions are evaluated: 2D 4-particle Double-Well Potential, 3D 13-particle Lennard-Jones potential, and 55-particle Lennard-Jones energy. ​
• Adjoint Sampling outperforms iDEM in terms of energy evaluations, making it more suitable for complex energy functions. ​

Sampling Conformers from Energy Functions ​

Adjoint Sampling is applied to generate conformers from a molecular energy model, emphasizing its ability to explore configuration spaces effectively. ​

• The eSEN energy model is utilized, which predicts DFT energy with high accuracy.
• Two approaches are designed: Cartesian Adjoint Sampling and Torsional Adjoint Sampling, with the former showing superior performance after pretraining. ​

Datasets and Baselines for Evaluation

The evaluation of Adjoint Sampling is conducted using the SPICE and GEOM-DRUGS datasets, with comparisons to traditional methods like RDKit ETKDG. ​

• SPICE contains over 23,000 diverse drug-like molecules, while GEOM-DRUGS serves as a benchmark for generalization capabilities. ​
• Adjoint Sampling outperforms RDKit in recall metrics, particularly benefiting from relaxation techniques post-generation. ​

Double-Well and Lennard-Jones Potentials ​

The text discusses the use of double-well (DW-4) and Lennard-Jones (LJ-13, LJ-55) potentials in modeling energy functions for particle systems. These potentials are defined mathematically and are used to evaluate energy based on particle interactions and distances.

• DW-4 potential is defined for 4 particles in 2D, with specific parameters: a = 0, b = -4, c = 0.9, and temperature τ = 1. ​
• Lennard-Jones potential is based on pairwise distances in 3D, with constants rm = 1, τ = 1, ϵ = 1, and an additional harmonic potential. ​
• The total energy for the Lennard-Jones potential includes contributions from both the Lennard-Jones and harmonic potentials. ​

Architectures and Hyperparameters for Adjoint Sampling

This section outlines the architectures and hyperparameters used for training Adjoint Sampling on synthetic energy functions. ​ Different configurations are specified for each energy model.

• DW-4 uses an EGNN with 3 layers and 128 hidden features, trained for 1000 iterations with a learning rate of 3 × 10^-4. ​
• LJ-13 employs an EGNN with 5 layers and 128 hidden features, also trained for 1000 iterations but generates 1024 samples per iteration. ​
• LJ-55 uses a similar architecture to LJ-13 but generates only 128 samples per iteration.

Metrics for Evaluating Generated Samples

The text describes various metrics used to evaluate the quality of generated samples from the energy models. ​ These metrics account for symmetries and energy distributions.

• Geometric W2 metric measures the distance between generated and ground truth point clouds, considering rotational and permutation symmetries. ​
• Energy W2 metric assesses the energy distribution of generated samples against ground truth distributions from MCMC simulations. ​
• Path effective sample size (path-ESS) is calculated using importance weights over paths, normalized by the number of samples. ​

Sampling Molecular Conformers and Torsion Angles ​

This section explains the process of sampling molecular conformers, focusing on torsion angles as critical degrees of freedom. ​ The methodology for defining and adjusting torsion angles is detailed. ​

• Torsion angles are defined between four atoms, with adjustments made by rotating atoms around a bond axis. ​
• A regularizer is added to the energy function to maintain desired molecular structures during sampling. ​
• The process involves using RDKit to identify torsional degrees of freedom and applying stochastic differential equations for sampling. ​

Coverage Recall and Precision Metrics

The text presents metrics for evaluating the coverage and precision of generated conformers against reference conformers. ​ These metrics help assess the effectiveness of the sampling methods.

• Average Minimum RMSD (AMR) and Coverage (COV) metrics are used to evaluate the quality of generated conformers. ​
• Coverage Recall is calculated based on the number of generated conformers that match reference conformers within a specified RMSD threshold.
• The average number of conformers generated is analyzed in relation to the number of rotatable bonds in the molecules. ​

Hyperparameters and Architecture for SPICE and GEOM-DRUGS ​

This section details the hyperparameters and architectures used for the SPICE and GEOM-DRUGS datasets in conformer generation tasks. ​ Different models and configurations are specified for each dataset.

• Cartesian Adjoint Sampling uses an EGNN with 12 layers and 128 hidden features, trained for 5000 iterations with a batch size of 64. ​
• Torsional Adjoint Sampling employs a different architecture with 6 convolutional layers and specialized layers for predicting torque quantities. ​
• Both models utilize a geometric noise schedule and weight gradient clipping to enhance training stability. ​

Energy Histograms and Runtime Analysis

The text provides qualitative comparisons of energy distributions from synthetic benchmarks and analyzes the runtime efficiency of the Adjoint Sampling technique.

• Energy histograms illustrate the effectiveness of Adjoint Sampling in avoiding high-energy regions, particularly in the LJ55 system. ​
• A runtime analysis shows that Adjoint Sampling significantly reduces computational bottlenecks, allowing for more gradient updates within the same execution time compared to previous methods. ​