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1. High-Resolution Modeling of the Fastest First-Order Optimization Method for Strongly Convex Functions

   https://doi.org/10.1109/CDC42340.2020.9304444  

   Sun, Boya ; George, Jemin ; Kia, Solmaz ( December 2020 , 2020 59th IEEE Conference on Decision and Control (CDC))

   null (Ed.)

   Motivated by the fact that the gradient-based optimization algorithms can be studied from the perspective of limiting ordinary differential equations (ODEs), here we derive an ODE representation of the accelerated triple momentum (TM) algorithm. For unconstrained optimization problems with strongly convex cost, the TM algorithm has a proven faster convergence rate than the Nesterov's accelerated gradient (NAG) method but with the same computational complexity. We show that similar to the NAG method, in order to accurately capture the characteristics of the TM method, we need to use a high-resolution modeling to obtain the ODE representation of the TM algorithm. We propose a Lyapunov analysis to investigate the stability and convergence behavior of the proposed high-resolution ODE representation of the TM algorithm. We compare the rate of the ODE representation of the TM method with that of the NAG method to confirm its faster convergence. Our study also leads to a tighter bound on the worst rate of convergence for the ODE model of the NAG method. In this paper, we also discuss the use of the integral quadratic constraint (IQC) method to establish an estimate on the rate of convergence of the TM algorithm. A numerical example verifies our results. 

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2. Continuous Time Analysis of Momentum Methods

   Kovachki, Nikola ; Stuart, Andrew M. ( January 2021 , Journal of machine learning research)

   null (Ed.)

   Gradient descent-based optimization methods underpin the parameter training of neural networks, and hence comprise a significant component in the impressive test results found in a number of applications. Introducing stochasticity is key to their success in practical problems, and there is some understanding of the role of stochastic gradient descent in this context. Momentum modifications of gradient descent such as Polyak’s Heavy Ball method (HB) and Nesterov’s method of accelerated gradients (NAG), are also widely adopted. In this work our focus is on understanding the role of momentum in the training of neural networks, concentrating on the common situation in which the momentum contribution is fixed at each step of the algorithm. To expose the ideas simply we work in the deterministic setting. Our approach is to derive continuous time approximations of the discrete algorithms; these continuous time approximations provide insights into the mechanisms at play within the discrete algorithms. We prove three such approximations. Firstly we show that standard implementations of fixed momentum methods approximate a time-rescaled gradient descent flow, asymptotically as the learning rate shrinks to zero; this result does not distinguish momentum methods from pure gradient descent, in the limit of vanishing learning rate. We then proceed to prove two results aimed at understanding the observed practical advantages of fixed momentum methods over gradient descent, when implemented in the non-asymptotic regime with fixed small, but non-zero, learning rate. We achieve this by proving approximations to continuous time limits in which the small but fixed learning rate appears as a parameter; this is known as the method of modified equations in the numerical analysis literature, recently rediscovered as the high resolution ODE approximation in the machine learning context. In our second result we show that the momentum method is approximated by a continuous time gradient flow, with an additional momentum-dependent second order time-derivative correction, proportional to the learning rate; this may be used to explain the stabilizing effect of momentum algorithms in their transient phase. Furthermore in a third result we show that the momentum methods admit an exponentially attractive invariant manifold on which the dynamics reduces, approximately, to a gradient flow with respect to a modified loss function, equal to the original loss function plus a small perturbation proportional to the learning rate; this small correction provides convexification of the loss function and encodes additional robustness present in momentum methods, beyond the transient phase. 

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3. From differential equation solvers to accelerated first-order methods for convex optimization

   https://doi.org/10.1007/s10107-021-01713-3  

   Luo, Hao ; Chen, Long ( October 2021 , Mathematical Programming)

   Convergence analysis of accelerated first-order methods for convex optimization problems are developed from the point of view of ordinary differential equation solvers. A new dynamical system, called Nesterov accelerated gradient (NAG) flow, is derived from the connection between acceleration mechanism andA-stability of ODE solvers, and the exponential decay of a tailored Lyapunov function along with the solution trajectory is proved. Numerical discretizations of NAG flow are then considered and convergence rates are established via a discrete Lyapunov function. The proposed differential equation solver approach can not only cover existing accelerated methods, such as FISTA, Güler’s proximal algorithm and Nesterov’s accelerated gradient method, but also produce new algorithms for composite convex optimization that possess accelerated convergence rates. Both the convex and the strongly convex cases are handled in a unified way in our approach.

    

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4. Accelerated Linear Convergence of Stochastic Momentum Methods in Wasserstein Distances

   Can, Bugra ; Gurbuzbalaban, Mert ; Zhu, Lingjiong ( July 2019 , Proceedings of Machine Learning Research)

   Momentum methods such as Polyak's heavy ball (HB) method, Nesterov's accelerated gradient (AG) as well as accelerated projected gradient (APG) method have been commonly used in machine learning practice, but their performance is quite sensitive to noise in the gradients. We study these methods under a first-order stochastic oracle model where noisy estimates of the gradients are available. For strongly convex problems, we show that the distribution of the iterates of AG converges with the accelerated linear rate to a ball of radius " centered at a unique invariant distribution in the 1-Wasserstein metric where  is the condition number as long as the noise variance is smaller than an explicit upper bound we can provide. Our analysis also certifies linear convergence rates as a function of the stepsize, momentum parameter and the noise variance; recovering the accelerated rates in the noiseless case and quantifying the level of noise that can be tolerated to achieve a given performance. To the best of our knowledge, these are the first linear convergence results for stochastic momentum methods under the stochastic oracle model. We also develop finer results for the special case of quadratic objectives, extend our results to the APG method and weakly convex functions showing accelerated rates when the noise magnitude is sufficiently small. 

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5. Conformal symplectic and relativistic optimization

   https://doi.org/10.1088/1742-5468/abcaee  

   França, Guilherme ; Sulam, Jeremias ; Robinson, Daniel P ; Vidal, René ( December 2020 , Journal of Statistical Mechanics: Theory and Experiment)

   Abstract Arguably, the two most popular accelerated or momentum-based optimization methods in machine learning are Nesterov’s accelerated gradient and Polyaks’s heavy ball, both corresponding to different discretizations of a particular second order differential equation with friction. Such connections with continuous-time dynamical systems have been instrumental in demystifying acceleration phenomena in optimization. Here we study structure-preserving discretizations for a certain class of dissipative (conformal) Hamiltonian systems, allowing us to analyse the symplectic structure of both Nesterov and heavy ball, besides providing several new insights into these methods. Moreover, we propose a new algorithm based on a dissipative relativistic system that normalizes the momentum and may result in more stable/faster optimization. Importantly, such a method generalizes both Nesterov and heavy ball, each being recovered as distinct limiting cases, and has potential advantages at no additional cost. 

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