José M. Moreu

José M. Moreu, Anuradha M. Annaswamy

Iterative gradient-based algorithms have been increasingly applied for the training of a broad variety of machine learning models including large neural-nets. In particular, momentum-based methods, with accelerated learning guarantees, have received a lot of attention due to their provable guarantees of fast learning in certain classes of problems and multiple algorithms have been derived. However, properties for these methods hold only for constant regressors. When time-varying regressors occur, which is commonplace in dynamic systems, many of these momentum-based methods cannot guarantee stability. Recently, a new High-order Tuner (HT) was developed for linear regression problems and shown to have 1) stability and asymptotic convergence for time-varying regressors and 2) non-asymptotic accelerated learning guarantees for constant regressors. In this paper, we extend and discuss the results of this same HT for general convex loss functions. Through the exploitation of convexity and smoothness definitions, we establish similar stability and asymptotic convergence guarantees. Finally, we provide numerical simulations supporting the satisfactory behavior of the HT algorithm as well as an accelerated learning property.

Joseph E. Gaudio, Anuradha M. Annaswamy, José M. Moreu et al.

High order momentum-based parameter update algorithms have seen widespread applications in training machine learning models. Recently, connections with variational approaches have led to the derivation of new learning algorithms with accelerated learning guarantees. Such methods however, have only considered the case of static regressors. There is a significant need for parameter update algorithms which can be proven stable in the presence of adversarial time-varying regressors, as is commonplace in control theory. In this paper, we propose a new discrete time algorithm which 1) provides stability and asymptotic convergence guarantees in the presence of adversarial regressors by leveraging insights from adaptive control theory and 2) provides non-asymptotic accelerated learning guarantees leveraging insights from convex optimization. In particular, our algorithm reaches an $ε$ sub-optimal point in at most $\tilde{\mathcal{O}}(1/\sqrtε)$ iterations when regressors are constant - matching lower bounds due to Nesterov of $Ω(1/\sqrtε)$, up to a $\log(1/ε)$ factor and provides guaranteed bounds for stability when regressors are time-varying. We provide numerical experiments for a variant of Nesterov's provably hard convex optimization problem with time-varying regressors, as well as the problem of recovering an image with a time-varying blur and noise using streaming data.