Linear Convergence
Linear convergence, the phenomenon where an iterative algorithm's error decreases at a geometric rate, is a central objective in optimization and machine learning. Current research focuses on establishing linear convergence guarantees for various algorithms, including adaptive gradient methods (like AdaGrad and Adam), policy gradient methods (often with log-linear policies), and distributed optimization algorithms (especially in federated learning settings), often under specific conditions like strong convexity or entropy regularization. These advancements provide crucial theoretical underpinnings for the practical success of these algorithms and improve our understanding of their performance in diverse applications, ranging from training neural networks to solving control problems. The resulting theoretical frameworks offer valuable insights into algorithm design and efficiency, leading to improved convergence rates and reduced computational costs.