Proximal Oracles for Optimization and Sampling

The study on convex optimization with non-smooth objective functions and log-concave sampling presents innovative algorithms that utilize the proximal point framework and the alternating sampling framework (ASF). By implementing an efficient proximal map through the regularized cutting-plane method, the authors address the challenges posed by non-smoothness in optimization tasks. Their proposed adaptive proximal bundle method further refines this approach by dynamically adjusting stepsizes, which is crucial for improving convergence rates. The iteration-complexity results established in this work are new to the literature, providing valuable insights into the performance of these algorithms under H"older smoothness and hybrid settings. This research not only contributes to theoretical advancements but also has practical implications for various applications in machine learning and data analysis, where efficient optimization and sampling are essential.