As machine learning models grow in complexity and scale, the limitations of traditional sampling methods become increasingly apparent. In large-scale models, especially deep neural networks with millions or billions of parameters, classic approaches like Markov chain Monte Carlo (MCMC) and importance sampling often struggle to remain effective. The challenges stem from high-dimensional parameter spaces, intricate energy landscapes, and computational constraints. For instance, MCMC methods may require an impractical number of steps to adequately explore the posterior, leading to poor mixing and biased estimates. Importance sampling, meanwhile, can suffer from high variance and degeneracy when the proposal distribution does not sufficiently cover the target. These issues are amplified in deep generative models, where both the expressiveness of the model and the size of the data make efficient sampling a formidable challenge.

Open problems in sampling for modern machine learning revolve around three main themes: scalability, mixing, and evaluation. Scalability refers to the ability of sampling algorithms to handle models and datasets of ever-increasing size without prohibitive computational cost. Many existing methods do not scale gracefully, leading to slow convergence and high memory usage. Mixing, the process by which a sampler explores the target distribution, remains a persistent difficulty in high dimensions or with multi-modal distributions. Poor mixing can result in samples that are highly correlated or fail to represent the full diversity of the distribution. Evaluation of generative models introduces another layer of complexity: how can you reliably assess the quality and diversity of samples from models like GANs or VAEs? Metrics such as likelihood estimates or sample-based visual inspection often provide incomplete or misleading information, making it hard to compare models or diagnose failures.

To address these limitations, researchers are exploring a range of new ideas that build on or depart from classical sampling. One promising direction involves score-based generative models and diffusion models. Score-based models, for example, learn to estimate the gradient of the data distribution’s log-density (the score function) and use this information to construct samplers that can generate new data. Diffusion models, on the other hand, progressively transform simple noise into structured data by reversing a diffusion process, effectively turning sampling into a sequence of denoising steps. Both approaches offer advantages in terms of sample quality and diversity, and they connect to classical methods through concepts like Langevin dynamics and stochastic differential equations. These models demonstrate how insights from traditional sampling can inspire new architectures and algorithms that are better suited to the demands of modern machine learning.

In summary, the field of sampling in machine learning is evolving rapidly, driven by the challenges posed by large-scale and deep models. Key concepts such as scalability, mixing, and evaluation remain central, but new approaches like score-based and diffusion models are expanding the toolkit available to practitioners. Understanding the limitations and open problems of current methods, as well as the intuition behind emerging techniques, is essential for pushing the boundaries of what machine learning models can achieve. As research continues, advances in sampling will play a crucial role in enabling more powerful, reliable, and interpretable models for a wide range of applications.