Features in machine learning problems are often time-varying and may be related to outputs in an algebraic or dynamical manner. The dynamic nature of these machine learning problems renders current higher order accelerated gradient descent methods unstable or weakens their convergence guarantees. Inspired by methods employed in adaptive control, this paper proposes new algorithms for the case when time-varying features are present, and demonstrates provable performance guarantees. In particular, we develop a unified variational perspective within a continuous time algorithm. This variational perspective includes higher order learning concepts and normalization, both of which stem from adaptive control, and allows stability to be established for dynamical machine learning problems where time-varying features are present. These higher order algorithms are also examined for provably correct learning in adaptive control and identification. Simulations are provided to verify the theoretical results.
This work continues in the tradition of @cite_26 and @cite_55 whereby insight is gained into accelerated gradient descent methods through a continuous lens. Future work will be to obtain discrete time implementations of our algorithms with matching rates @cite_48 @cite_27 , and to connect those back to classic discrete time adaptive control algorithms @cite_19 @cite_11 @cite_30 . Other fruitful directions forward would be to study these accelerated algorithms within the context of output feedback control, and to rigorously prove convergence rates when the regressor vectors are persistently exciting @cite_38 @cite_25 .
@cite_30 @cite_38 @cite_26 @cite_48 @cite_55 @cite_19 @cite_27 @cite_25 @cite_11
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Nesterov's accelerated gradient descent (AGD), an instance of the general family of momentum methods,'' provably achieves faster convergence rate than gradient descent (GD) in the convex setting. While these methods are widely used in modern nonconvex applications, including training of deep neural networks, whether they are provably superior to GD in the nonconvex setting remains open. This paper studies a simple variant of Nesterov's AGD, and shows that it escapes saddle points and finds a second-order stationary point in ( O (1 ^ 7 4 ) ) iterations, matching the best known convergence rate, which is faster than the ( O (1 ^ 2 ) ) iterations required by GD. To the best of our knowledge, this is the first direct acceleration (single-loop) algorithm that is provably faster than GD in general nonconvex setting---all previous nonconvex accelerated algorithms rely on more complex mechanisms such as nested loops and proximal terms. Our analysis is based on two key ideas: (1) the use of a simple Hamiltonian function, inspired by a continuous-time perspective, which AGD monotonically decreases on each step even for nonconvex functions, and (2) a novel framework called improve or localize, which is useful for tracking the long-term behavior of gradient-based optimization algorithms. We believe that these techniques may deepen our understanding of both acceleration algorithms and nonconvex optimization. Nesterov's accelerated gradient descent (AGD), an instance of the general family of "momentum methods", provably achieves faster convergence rate than gradient descent (GD) in the convex setting. However, whether these methods are superior to GD in the nonconvex setting remains open. This paper studies a simple variant of AGD, and shows that it escapes saddle points and finds a second-order stationary point in @math iterations, faster than the @math iterations required by GD. To the best of our knowledge, this is the first Hessian-free algorithm to find a second-order stationary point faster than GD, and also the first single-loop algorithm with a faster rate than GD even in the setting of finding a first-order stationary point. Our analysis is based on two key ideas: (1) the use of a simple Hamiltonian function, inspired by a continuous-time perspective, which AGD monotonically decreases per step even for nonconvex functions, and (2) a novel framework called improve or localize, which is useful for tracking the long-term behavior of gradient-based optimization algorithms. We believe that these techniques may deepen our understanding of both acceleration algorithms and nonconvex optimization. In this paper we employ a parallel version of a randomized (block) coordinate descent method for minimizing the sum of a partially separable smooth convex function and a fully separable nonsmooth convex function. Under the assumption of Lipschitz continuity of the gradient of the smooth function, this method has a sublinear convergence rate. Linear convergence rate of the method is obtained for the newly introduced class of generalized error bound functions. We prove that the new class of generalized error bound functions encompasses both global local error bound functions and smooth strongly convex functions. We also show that the theoretical estimates on the convergence rate depend on the number of blocks chosen randomly and a natural measure of separability of the smooth component of the objective function. In this paper we show how to accelerate randomized coordinate descent methods and achieve faster convergence rates without paying per-iteration costs in asymptotic running time. In particular, we show how to generalize and efficiently implement a method proposed by Nesterov, giving faster asymptotic running times for various algorithms that use standard coordinate descent as a black box. In addition to providing a proof of convergence for this new general method, we show that it is numerically stable, efficiently implementable, and in certain regimes, asymptotically optimal. To highlight the power of this algorithm, we show how it can used to create faster linear system solvers in several regimes: - We show how this method achieves a faster asymptotic runtime than conjugate gradient for solving a broad class of symmetric positive definite systems of equations. - We improve the convergence guarantees for Kaczmarz methods, a popular technique for image reconstruction and solving over determined systems of equations, by accelerating an algorithm of Strohmer and Vershynin. - We achieve the best known running time for solving Symmetric Diagonally Dominant (SDD) system of equations in the unit-cost RAM model, obtaining a running time of O(m log3 2n (log log n)1 2 log((log n) eps)) by accelerating a recent solver by Beyond the independent interest of these solvers, we believe they highlight the versatility of the approach of this paper and we hope that they will open the door for further algorithmic improvements in the future.