No Trick, No Treat: Pursuits and Challenges Towards Simulation-free Training of Neural Samplers

We consider the sampling problem, where the aim is to draw samples from a distribution whose density is known only up to a normalization constant. Recent breakthroughs in generative modeling to approximate a high-dimensional data distribution have sparked significant interest in developing neural network-based methods for this challenging problem. However, neural samplers typically incur heavy computational overhead due to simulating trajectories during training. This motivates the pursuit of simulation-free training procedures of neural samplers. In this work, we propose an elegant modification to previous methods, which allows simulation-free training with the help of a time-dependent normalizing flow. However, it ultimately suffers from severe mode collapse. On closer inspection, we find that nearly all successful neural samplers rely on Langevin preconditioning to avoid mode collapsing. We systematically analyze several popular methods with various objective functions and demonstrate that, in the absence of Langevin preconditioning, most of them fail to adequately cover even a simple target. Finally, we draw attention to a strong baseline by combining the state-of-the-art MCMC method, Parallel Tempering (PT), with an additional generative model to shed light on future explorations of neural samplers.

Debiasing Guidance for Discrete Diffusion with Sequential Monte Carlo

Discrete diffusion models are a class of generative models that produce samples from an approximated data distribution within a discrete state space. Often, there is a need to target specific regions of the data distribution. Current guidance methods aim to sample from a distribution with mass proportional to $p_0(x_0) p(\zeta|x_0)^\alpha$ but fail to achieve this in practice. We introduce a Sequential Monte Carlo algorithm that generates unbiasedly from this target distribution, utilising the learnt unconditional and guided process. We validate our approach on low-dimensional distributions, controlled images and text generations. For text generation, our method provides strong control while maintaining low perplexity compared to guidance-based approaches.

Iterative Importance Fine-tuning of Diffusion Models

Diffusion models are an important tool for generative modelling, serving as effective priors in applications such as imaging and protein design. A key challenge in applying diffusion models for downstream tasks is efficiently sampling from resulting posterior distributions, which can be addressed using the $h$-transform. This work introduces a self-supervised algorithm for fine-tuning diffusion models by estimating the $h$-transform, enabling amortised conditional sampling. Our method iteratively refines the $h$-transform using a synthetic dataset resampled with path-based importance weights. We demonstrate the effectiveness of this framework on class-conditional sampling and reward fine-tuning for text-to-image diffusion models.

Beyond ELBOs: A Large-Scale Evaluation of Variational Methods for Sampling

Monte Carlo methods, Variational Inference, and their combinations play a pivotal role in sampling from intractable probability distributions. However, current studies lack a unified evaluation framework, relying on disparate performance measures and limited method comparisons across diverse tasks, complicating the assessment of progress and hindering the decision-making of practitioners. In response to these challenges, our work introduces a benchmark that evaluates sampling methods using a standardized task suite and a broad range of performance criteria. Moreover, we study existing metrics for quantifying mode collapse and introduce novel metrics for this purpose. Our findings provide insights into strengths and weaknesses of existing sampling methods, serving as a valuable reference for future developments. The code is publicly available here.

Improving Antibody Design with Force-Guided Sampling in Diffusion Models

Antibodies, crucial for immune defense, primarily rely on complementarity-determining regions (CDRs) to bind and neutralize antigens, such as viruses. The design of these CDRs determines the antibody's affinity and specificity towards its target. Generative models, particularly denoising diffusion probabilistic models (DDPMs), have shown potential to advance the structure-based design of CDR regions. However, only a limited dataset of bound antibody-antigen structures is available, and generalization to out-of-distribution interfaces remains a challenge. Physics based force-fields, which approximate atomic interactions, offer a coarse but universal source of information to better mold designs to target interfaces. Integrating this foundational information into diffusion models is, therefore, highly desirable. Here, we propose a novel approach to enhance the sampling process of diffusion models by integrating force field energy-based feedback. Our model, DiffForce, employs forces to guide the diffusion sampling process, effectively blending the two distributions. Through extensive experiments, we demonstrate that our method guides the model to sample CDRs with lower energy, enhancing both the structure and sequence of the generated antibodies.

DEFT: Efficient Finetuning of Conditional Diffusion Models by Learning the Generalised $h$-transform

Generative modelling paradigms based on denoising diffusion processes have emerged as a leading candidate for conditional sampling in inverse problems. In many real-world applications, we often have access to large, expensively trained unconditional diffusion models, which we aim to exploit for improving conditional sampling. Most recent approaches are motivated heuristically and lack a unifying framework, obscuring connections between them. Further, they often suffer from issues such as being very sensitive to hyperparameters, being expensive to train or needing access to weights hidden behind a closed API. In this work, we unify conditional training and sampling using the mathematically well-understood Doob's h-transform. This new perspective allows us to unify many existing methods under a common umbrella. Under this framework, we propose DEFT (Doob's h-transform Efficient FineTuning), a new approach for conditional generation that simply fine-tunes a very small network to quickly learn the conditional $h$-transform, while keeping the larger unconditional network unchanged. DEFT is much faster than existing baselines while achieving state-of-the-art performance across a variety of linear and non-linear benchmarks. On image reconstruction tasks, we achieve speedups of up to 1.6$\times$, while having the best perceptual quality on natural images and reconstruction performance on medical images.

A framework for conditional diffusion modelling with applications in motif scaffolding for protein design

Many protein design applications, such as binder or enzyme design, require scaffolding a structural motif with high precision. Generative modelling paradigms based on denoising diffusion processes emerged as a leading candidate to address this motif scaffolding problem and have shown early experimental success in some cases. In the diffusion paradigm, motif scaffolding is treated as a conditional generation task, and several conditional generation protocols were proposed or imported from the Computer Vision literature. However, most of these protocols are motivated heuristically, e.g. via analogies to Langevin dynamics, and lack a unifying framework, obscuring connections between the different approaches. In this work, we unify conditional training and conditional sampling procedures under one common framework based on the mathematically well-understood Doob's h-transform. This new perspective allows us to draw connections between existing methods and propose a new variation on existing conditional training protocols. We illustrate the effectiveness of this new protocol in both, image outpainting and motif scaffolding and find that it outperforms standard methods.

Transport, Variational Inference and Diffusions: with Applications to Annealed Flows and Schrödinger Bridges

This paper explores the connections between optimal transport and variational inference, with a focus on forward and reverse time stochastic differential equations and Girsanov transformations.We present a principled and systematic framework for sampling and generative modelling centred around divergences on path space. Our work culminates in the development of a novel score-based annealed flow technique (with connections to Jarzynski and Crooks identities from statistical physics) and a regularised iterative proportional fitting (IPF)-type objective, departing from the sequential nature of standard IPF. Through a series of generative modelling examples and a double-well-based rare event task, we showcase the potential of the proposed methods.

Expressiveness Remarks for Denoising Diffusion Models and Samplers

Denoising diffusion models are a class of generative models which have recently achieved state-of-the-art results across many domains. Gradual noise is added to the data using a diffusion process, which transforms the data distribution into a Gaussian. Samples from the generative model are then obtained by simulating an approximation of the time reversal of this diffusion initialized by Gaussian samples. Recent research has explored adapting diffusion models for sampling and inference tasks. In this paper, we leverage known connections to stochastic control akin to the F\"ollmer drift to extend established neural network approximation results for the F\"ollmer drift to denoising diffusion models and samplers.

Dimensionality Reduction as Probabilistic Inference

Dimensionality reduction (DR) algorithms compress high-dimensional data into a lower dimensional representation while preserving important features of the data. DR is a critical step in many analysis pipelines as it enables visualisation, noise reduction and efficient downstream processing of the data. In this work, we introduce the ProbDR variational framework, which interprets a wide range of classical DR algorithms as probabilistic inference algorithms in this framework. ProbDR encompasses PCA, CMDS, LLE, LE, MVU, diffusion maps, kPCA, Isomap, (t-)SNE, and UMAP. In our framework, a low-dimensional latent variable is used to construct a covariance, precision, or a graph Laplacian matrix, which can be used as part of a generative model for the data. Inference is done by optimizing an evidence lower bound. We demonstrate the internal consistency of our framework and show that it enables the use of probabilistic programming languages (PPLs) for DR. Additionally, we illustrate that the framework facilitates reasoning about unseen data and argue that our generative models approximate Gaussian processes (GPs) on manifolds. By providing a unified view of DR, our framework facilitates communication, reasoning about uncertainties, model composition, and extensions, particularly when domain knowledge is present.