Fast and Safe: Accelerated gradient methods with optimality certificates and underestimate sequences
This work addresses the need for faster and more reliable optimization methods in machine learning and numerical analysis, offering incremental improvements through novel algorithmic frameworks.
The authors tackled the problem of optimizing strongly convex functions by introducing Underestimate Sequences (UES) to provide optimality certificates and accelerate convergence, resulting in algorithms with linear convergence rates, including accelerated variants achieving optimal rates.
In this work we introduce the concept of an Underestimate Sequence (UES), which is motivated by Nesterov's estimate sequence. Our definition of a UES utilizes three sequences, one of which is a lower bound (or under-estimator) of the objective function. The question of how to construct an appropriate sequence of lower bounds is addressed, and we present lower bounds for strongly convex smooth functions and for strongly convex composite functions, which adhere to the UES framework. Further, we propose several first order methods for minimizing strongly convex functions in both the smooth and composite cases. The algorithms, based on efficiently updating lower bounds on the objective functions, have natural stopping conditions that provide the user with a certificate of optimality. Convergence of all algorithms is guaranteed through the UES framework, and we show that all presented algorithms converge linearly, with the accelerated variants enjoying the optimal linear rate of convergence.