Classical Langevin Monte Carlo algorithms, such as the unadjusted Langevin algorithm (ULA), are based on discretization schemes of the overdamped Langevin dynamics and are widely used for sampling in machine learning because they scale to high dimensions and large datasets. However, they require differentiable log-densities, and are less efficient for sampling heavy-tailed distributions. Although various approaches have been proposed in the literature for sampling non-differentiable and heavy-tailed targets, all of them are developed in the overdamped setting only. We propose a novel momentum-based underdamped anchored Langevin dynamics (UALD), and underdamped anchored Langevin Monte Carlo (UALMC) algorithm based on a discretization scheme of UALD. Our paper is the first to provide a kinetic (momentum-based) Langevin algorithm for non-differentiable and heavy-tailed targets that is free of asymptotic bias. The method replaces the original potential with a smooth reference potential and modifies the underdamped Langevin dynamics via multiplicative scaling in the drift term. We establish non-asymptotic guarantees in the 2-Wasserstein distance to the target distribution and show that the resulting iteration complexity has better dependence on dimension and accuracy level compared to overdamped-based anchored Langevin algorithms. We provide numerical experiments to illustrate the theory and practical performance of our proposed approach.
Modern image generation largely relies on specialized generative models that leave the process of image construction implicit in learned parameters. We explore an alternative: general-purpose coding agents that construct images through executable programs. Our study of zero-shot drawing on AgentDrawBench, a 200-prompt benchmark, reveals a distinctive capability profile. Agents excel at preserving structure and satisfying precise visual constraints, achieving competitive results in domains where compositional accuracy matters more than photorealistic appearance, while remaining limited in perceptual richness. Their visual expressiveness varies with the drawing interface, and a qualitative study on reconstruction shows how procedural skills extracted from human demonstrations support different artistic media. We then examine code as an external visual language of thought: a structured, executable representation that agents can construct and inspect. We test it in visual question answering task and find reconstructed visual code retains substantial information for question answering and, in some domains, supports higher accuracy than the original image. These abilities raise a further question: do agents draw on a shared internal representation of scene content beyond a particular drawing program? We explore an internal visual language-of-thought hypothesis, in which a shared, compositionally structured representation supports visual operations across drawing languages. Cross-harness drawings and images assembled from independent single-pixel queries exhibit related layouts and tones. Agents also maintain self-generated room arrangements across viewpoints after visible drawings are removed. These results demonstrate that the coding agent would maintain a consistent and coherent world inside its imagine even before the harness gets involved and the generation starts.