Non Asymptotic Analysis

Non-asymptotic analysis focuses on deriving finite-sample guarantees for the performance of machine learning algorithms, moving beyond traditional asymptotic analyses that only hold for infinitely large datasets. Current research emphasizes developing such guarantees for a range of algorithms, including stochastic gradient descent (SGD) variants, Hamiltonian Monte Carlo, actor-critic methods, and diffusion models, often focusing on characterizing convergence rates and error bounds in various metrics (e.g., Wasserstein distance, total variation distance). This rigorous approach provides crucial insights into algorithm behavior for practical sample sizes, improving model reliability and enabling more informed hyperparameter tuning. The resulting theoretical advancements directly impact the design and application of machine learning across diverse fields, from generative modeling and reinforcement learning to control systems and federated learning.