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1. Krylov Cubic Regularized Newton: A Subspace Second-Order Method with Dimension-Free Convergence Rate

   Jiang, Ruichen; Raman, Parameswaran; Sabach, Shoham; Mokhtari, Aryan; Hong, Mingyi; Cevher, Volkan (May 2024, PMLR)

   Second-order optimization methods, such as cubic regularized Newton methods, are known for their rapid convergence rates; nevertheless, they become impractical in high-dimensional problems due to their substantial memory requirements and computational costs. One promising approach is to execute second order updates within a lower-dimensional subspace, giving rise to \textit{subspace second-order} methods. However, the majority of existing subspace second-order methods randomly select subspaces, consequently resulting in slower convergence rates depending on the problem's dimension $$d$$. In this paper, we introduce a novel subspace cubic regularized Newton method that achieves a dimension-independent global convergence rate of $$\bigO\left(\frac{1}{mk}+\frac{1}{k^2}\right)$$ for solving convex optimization problems. Here, $$m$$ represents the subspace dimension, which can be significantly smaller than $$d$$. Instead of adopting a random subspace, our primary innovation involves performing the cubic regularized Newton update within the \emph{Krylov subspace} associated with the Hessian and the gradient of the objective function. This result marks the first instance of a dimension-independent convergence rate for a subspace second-order method. Furthermore, when specific spectral conditions of the Hessian are met, our method recovers the convergence rate of a full-dimensional cubic regularized Newton method. Numerical experiments show our method converges faster than existing random subspace methods, especially for high-dimensional problems. 

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2. Achieving O(\epsilon^{-1.5}) Complexity in Hessian/Jacobian-free Stochastic Bilevel Optimization

   Yang, Y; Xiao, P; Ji, K (February 2024, Advances in Neural Information Processing Systems)

   In this paper, we revisit the bilevel optimization problem, in which the upper-level objective function is generally nonconvex and the lower-level objective function is strongly convex. Although this type of problem has been studied extensively, it still remains an open question how to achieve an $$\mathcal{O}(\epsilon^{-1.5})$$ sample complexity in Hessian/Jacobian-free stochastic bilevel optimization without any second-order derivative computation. To fill this gap, we propose a novel Hessian/Jacobian-free bilevel optimizer named FdeHBO, which features a simple fully single-loop structure, a projection-aided finite-difference Hessian/Jacobian-vector approximation, and momentum-based updates. Theoretically, we show that FdeHBO requires $$\mathcal{O}(\epsilon^{-1.5})$$ iterations (each using $$\mathcal{O}(1)$$ samples and only first-order gradient information) to find an $$\epsilon$$-accurate stationary point. As far as we know, this is the first Hessian/Jacobian-free method with an $$\mathcal{O}(\epsilon^{-1.5})$$ sample complexity for nonconvex-strongly-convex stochastic bilevel optimization. 

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3. Reduced Order Model Hessian Approximations in Newton Methods for Optimal Control

   https://doi.org/10.1007/978-3-030-95157-3_18  

   Heinkenschloss, Matthias; Magruder, Caleb (July 2022, Realization and Model Reduction of Dynamical Systems - A Festschrift in Honor of the 70th Birthday of Thanos Antoulas)

   Beattie, C.A.; Benner, P.; Embree, M.; Gugercin, S.; Lefteriu, S. (Ed.)

   This paper introduces reduced order model (ROM) based Hessian approximations for use in inexact Newton methods for the solution of optimization problems implicitly constrained by a large-scale system, typically a discretization of a partial differential equation (PDE). The direct application of an inexact Newton method to this problem requires the solution of many PDEs per optimization iteration. To reduce the computational complexity, a ROM Hessian approximation is proposed. Since only the Hessian is approximated, but the original objective function and its gradient is used, the resulting inexact Newton method maintains the first-order global convergence property, under suitable assumptions. Thus even computationally inexpensive lower fidelity ROMs can be used, which is different from ROM approaches that replace the original optimization problem by a sequence of ROM optimization problem and typically need to accurately approximate function and gradient information of the original problem. In the proposed approach, the quality of the ROM Hessian approximation determines the rate of convergence, but not whether the method converges. The projection based ROM is constructed from state and adjoint snapshots, and is relatively inexpensive to compute. Numerical examples on semilinear parabolic optimal control problems demonstrate that the proposed approach can lead to substantial savings in terms of overall PDE solves required. 

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4. mL-BFGS: A Momentum-based L-BFGS for Distributed Large-Scale Neural Network Optimization

   Niu, Yue; Fabian, Zalan; Lee, Sunwoo; Soltanolkotabi, Mahdi; Avestimehr, Salman (July 2023, Transactions on Machine Learning Research)

   Quasi-Newton methods still face significant challenges in training large-scale neural networks due to additional compute costs in the Hessian related computations and instability issues in stochastic training. A well-known method, L-BFGS that efficiently approximates the Hessian using history parameter and gradient changes, suffers convergence instability in stochastic training. So far, attempts that adapt L-BFGS to large-scale stochastic training incur considerable extra overhead, which offsets its convergence benefits in wall-clock time. In this paper, we propose mL-BFGS, a lightweight momentum-based L-BFGS algorithm that paves the way for quasi-Newton (QN) methods in large-scale distributed deep neural network (DNN) optimization. mL-BFGS introduces a nearly cost-free momentum scheme into L-BFGS update and greatly reduces stochastic noise in the Hessian, therefore stabilizing convergence during stochastic optimization. For model training at a large scale, mL-BFGS approximates a block-wise Hessian, thus enabling distributing compute and memory costs across all computing nodes. We provide a supporting convergence analysis for mL-BFGS in stochastic settings. To investigate mL-BFGS’s potential in large-scale DNN training, we train benchmark neural models using mL-BFGS and compare performance with baselines (SGD, Adam, and other quasi-Newton methods). Results show that mL-BFGS achieves both noticeable iteration-wise and wall-clock speedup. 

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5. Anderson acceleration for Seismic Inversion

   https://doi.org/10.1190/segam2020-3425521.1  

   Yang, Yunan (September 2020, SEG Technical Program Expanded Abstracts 2020)

   null (Ed.)

   Full waveform inversion (FWI) and least-squares reverse time migration (LSRTM) are popular imaging techniques that can be solved as PDE-constrained optimization problems. Due to the large-scale nature, gradient- and Hessian-based optimization algorithms are preferred in practice to find the optimizer iteratively. However, a balance between the evaluation cost and the rate of convergence needs to be considered. We propose the use of Anderson acceleration (AA), a popular strategy to speed up the convergence of fixed-point iterations, to accelerate a gradient descent method. We show that AA can achieve fast convergence that provides competitive results with some quasi-Newton methods. Independent of the dimensionality of the unknown parameters, the computational cost of implementing the method can be reduced to an extremely lowdimensional least-squares problem, which makes AA an attractive method for seismic inversion. 

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