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1. Understanding the acceleration phenomenon via high-resolution differential equations

   https://doi.org/10.1007/s10107-021-01681-8  

   Shi, Bin ; Du, Simon S. ; Jordan, Michael I. ; Su, Weijie J. ( July 2021 , Mathematical Programming)

   Gradient-based optimization algorithms can be studied from the perspective of limiting ordinary differential equations (ODEs). Motivated by the fact that existing ODEs do not distinguish between two fundamentally different algorithms—Nesterov’s accelerated gradient method for strongly convex functions (NAG-) and Polyak’s heavy-ball method—we study an alternative limiting process that yieldshigh-resolution ODEs. We show that these ODEs permit a general Lyapunov function framework for the analysis of convergence in both continuous and discrete time. We also show that these ODEs are more accurate surrogates for the underlying algorithms; in particular, they not only distinguish between NAG- and Polyak’s heavy-ball method, but they allow the identification of a term that we refer to as “gradient correction” that is present in NAG- but not in the heavy-ball method and is responsible for the qualitative difference in convergence of the two methods. We also use the high-resolution ODE framework to study Nesterov’s accelerated gradient method for (non-strongly) convex functions, uncovering a hitherto unknown result—that NAG- minimizes the squared gradient norm at an inverse cubic rate. Finally, by modifying the high-resolution ODE of NAG-, we obtain a family of new optimization methods that are shown to maintain the accelerated convergence rates of NAG- for smooth convex functions.

    

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2. 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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3. Accelerated Quasi-Newton Proximal Extragradient: Faster Rate for Smooth Convex Optimization

   Jiang, Ruichen ; Mokhtari, Aryan ( December 2023 , NeurIPS proceedings)

   In this paper, we present an accelerated quasi-Newton proximal extragradient method for solving unconstrained smooth convex optimization problems. With access only to the gradients of the objective function, we prove that our method can achieve a convergence rate of ${\bigO}\bigl(\min\{\frac{1}{k^2}, \frac{\sqrt{d\log k}}{k^{2.5}}\}\bigr)$, where $d$ is the problem dimension and $k$ is the number of iterations. In particular, in the regime where $k = \bigO(d)$, our method matches the \emph{optimal rate} of $\mathcal{O}(\frac{1}{k^2})$ by Nesterov's accelerated gradient (NAG). Moreover, in the the regime where $k = \Omega(d \log d)$, it outperforms NAG and converges at a \emph{faster rate} of $\mathcal{O}\bigl(\frac{\sqrt{d\log k}}{k^{2.5}}\bigr)$. To the best of our knowledge, this result is the first to demonstrate a provable gain for a quasi-Newton-type method over NAG in the convex setting. To achieve such results, we build our method on a recent variant of the Monteiro-Svaiter acceleration framework and adopt an online learning perspective to update the Hessian approximation matrices, in which we relate the convergence rate of our method to the dynamic regret of a specific online convex optimization problem in the space of matrices. 

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4. Nesterov Accelerated Shuffling Gradient Method for Convex Optimization

   Trang H. Tran, Katya Scheinberg ( July 2022 , Proceedings of Machine Learning Research)

   In this paper, we propose Nesterov Accelerated Shuffling Gradient (NASG), a new algorithm for the convex finite-sum minimization problems. Our method integrates the traditional Nesterov’s acceleration momentum with different shuffling sampling schemes. We show that our algorithm has an improved rate of O(1/T ) using unified shuffling schemes, where T is the number of epochs. This rate is better than that of any other shuffling gradient methods in convex regime. Our convergence analysis does not require an assumption on bounded domain or a bounded gradient condition. For randomized shuffling schemes, we improve the convergence bound further. When employing some initial condition, we show that our method converges faster near the small neighborhood of the solution. Numerical simulations demonstrate the efficiency of our algorithm. 

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5. 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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