We propose a stochastic variance-reduced cubic regularized Newton method for non-convex optimization. At the core of our algorithm is a novel semi-stochastic gradient along with a semi-stochastic Hessian, which are specifically designed for cubic regularization method. We show that our algorithm is guaranteed to converge to an $(ε,\sqrtε)$-approximately local minimum within $\tilde{O}(n^{4/5}/ε^{3/2})$ second-order oracle calls, which outperforms the state-of-the-art cubic regularization algorithms including subsampled cubic regularization. Our work also sheds light on the application of variance reduction technique to high-order non-convex optimization methods. Thorough experiments on various non-convex optimization problems support our theory.
Matrix functions are utilized to rewrite smooth spectral constrained matrix optimization problems as smooth unconstrained problems over the set of symmetric matrices which are then solved via the cubic-regularized Newton method. A second-order chain rule identity for matrix functions is proven to compute the higher-order derivatives to implement cubic-regularized Newton, and a new convergence analysis is provided for cubic-regularized Newton for matrix vector spaces. We demonstrate the applicability of our approach by conducting numerical experiments on both synthetic and real datasets. In our…
Ohio State University（俄亥俄州立大学）；Georgia Institute of Technology（佐治亚理工学院）
This note considers the inexact cubic-regularized Newton's method (CR), which has been shown in \cite{Cartis2011a} to achieve the same order-level convergence rate to a secondary stationary point as the exact CR \citep{Nesterov2006}. However, the inexactness condition in \cite{Cartis2011a} is not implementable due to its dependence on future iterates variable. This note fixes such an issue by proving the same convergence rate for nonconvex optimization under an inexact adaptive condition that depends on only the current iterate. Our proof controls the sufficient decrease of the function value …
We propose a distributed, cubic-regularized Newton method for large-scale convex optimization over networks. The proposed method requires only local computations and communications and is suitable for federated learning applications over arbitrary network topologies. We show a $O(k^{{-}3})$ convergence rate when the cost function is convex with Lipschitz gradient and Hessian, with $k$ being the number of iterations. We further provide network-dependent bounds for the communication required in each step of the algorithm. We provide numerical experiments that validate our theoretical results.
King Abdullah University of Science and Technology（阿卜杜拉国王科技大学）；Catholic University of Louvain（鲁汶天主教大学）
In this paper, we propose a new randomized second-order optimization algorithm---Stochastic Subspace Cubic Newton (SSCN)---for minimizing a high dimensional convex function $f$. Our method can be seen both as a {\em stochastic} extension of the cubically-regularized Newton method of Nesterov and Polyak (2006), and a {\em second-order} enhancement of stochastic subspace descent of Kozak et al. (2019). We prove that as we vary the minibatch size, the global convergence rate of SSCN interpolates between the rate of stochastic coordinate descent (CD) and the rate of cubic regularized Newton, thus …
Krylov Cubic Regularized Newton: A Subspace Second-Order Method with Dimension-Free Convergence Rate
The University of Texas at Austin（德克萨斯大学奥斯汀分校）；Amazon Web Services（亚马逊云科技）；Technion – Israel Institute of Technology（以色列理工学院）；University of Minnesota（明尼苏达大学）；Ecole Polytechnique Fédérale de Lausanne（洛桑联邦理工学院）
Gradient Regularized Newton Boosting Trees with Global Convergence
Department of Mathematics, Imperial College London, UK（帝国理工学院伦敦数学系）；School of Computer Science, University of St. Gallen, Switzerland（圣加尔登大学计算机科学学院）；Department of Mathematics, Stockholm University, Sweden（斯德哥尔摩大学数学系）
This work is on constrained large-scale non-convex optimization where the constraint set implies a manifold structure. Solving such problems is important in a multitude of fundamental machine learning tasks. Recent advances on Riemannian optimization have enabled the convenient recovery of solutions by adapting unconstrained optimization algorithms over manifolds. However, it remains challenging to scale up and meanwhile maintain stable convergence rates and handle saddle points. We propose a new second-order Riemannian optimization algorithm, aiming at improving convergence rate and reducing …
The University of Texas at Austin（德克萨斯大学奥斯汀分校）；SAS Institute（SAS研究所）
In this work we describe an Adaptive Regularization using Cubics (ARC) method for large-scale nonconvex unconstrained optimization using Limited-memory Quasi-Newton (LQN) matrices. ARC methods are a relatively new family of optimization strategies that utilize a cubic-regularization (CR) term in place of trust-regions and line-searches. LQN methods offer a large-scale alternative to using explicit second-order information by taking identical inputs to those used by popular first-order methods such as stochastic gradient descent (SGD). Solving the CR subproblem exactly requires Newton's method,…
Alternating Iteratively Reweighted $\ell_1$ and Subspace Newton Algorithms for Nonconvex Sparse Optimization
School of Information Science and Technology, ShanghaiTech University（上海科技大学信息科学与技术学院）；School of Mathematics and Statistics, Henan University（河南大学数学与统计学院）；Center for Applied Mathematics of Henan Province, Henan University（河南大学河南省应用数学中心）
This paper proposes a stochastic variant of a classic algorithm---the cubic-regularized Newton method [Nesterov and Polyak 2006]. The proposed algorithm efficiently escapes saddle points and finds approximate local minima for general smooth, nonconvex functions in only $\mathcal{\tilde{O}}(ε^{-3.5})$ stochastic gradient and stochastic Hessian-vector product evaluations. The latter can be computed as efficiently as stochastic gradients. This improves upon the $\mathcal{\tilde{O}}(ε^{-4})$ rate of stochastic gradient descent. Our rate matches the best-known result for finding local minima withou…
This paper adapts a recently developed regularized stochastic version of the Broyden, Fletcher, Goldfarb, and Shanno (BFGS) quasi-Newton method for the solution of support vector machine classification problems. The proposed method is shown to converge almost surely to the optimal classifier at a rate that is linear in expectation. Numerical results show that the proposed method exhibits a convergence rate that degrades smoothly with the dimensionality of the feature vectors.
The problem of saddle-point avoidance for non-convex optimization is quite challenging in large scale distributed learning frameworks, such as Federated Learning, especially in the presence of Byzantine workers. The celebrated cubic-regularized Newton method of \cite{nest} is one of the most elegant ways to avoid saddle-points in the standard centralized (non-distributed) setup. In this paper, we extend the cubic-regularized Newton method to a distributed framework and simultaneously address several practical challenges like communication bottleneck and Byzantine attacks. Note that the issue o…
SOFIM: Stochastic Optimization Using Regularized Fisher Information Matrix
Swinburne University of Technology（斯威本科技大学）；Indian Institute of Technology Hyderabad（印度理工学院海得拉巴分校）；Mahindra University（马恒达大学）；Ritsumeikan University（立命馆大学）；Indian Institute of Technology Roorkee（印度理工学院鲁基分校）
We present a Newton-type method that converges fast from any initialization and for arbitrary convex objectives with Lipschitz Hessians. We achieve this by merging the ideas of cubic regularization with a certain adaptive Levenberg--Marquardt penalty. In particular, we show that the iterates given by $x^{k+1}=x^k - \bigl(\nabla^2 f(x^k) + \sqrt{H\|\nabla f(x^k)\|} \mathbf{I}\bigr)^{-1}\nabla f(x^k)$, where $H>0$ is a constant, converge globally with a $\mathcal{O}(\frac{1}{k^2})$ rate. Our method is the first variant of Newton's method that has both cheap iterations and provably fast global co…
University of Florida（佛罗里达大学）；Korea Institute of Industrial Technology (KITECH)（韩国产业技术研究院）
We focus on minimizing nonconvex finite-sum functions that typically arise in machine learning problems. In an attempt to solve this problem, the adaptive cubic regularized Newton method has shown its strong global convergence guarantees and ability to escape from strict saddle points. This method uses a trust region-like scheme to determine if an iteration is successful or not, and updates only when it is successful. In this paper, we suggest an algorithm combining negative curvature with the adaptive cubic regularized Newton method to update even at unsuccessful iterations. We call this new …
A Regularized Newton Method for Nonconvex Optimization with Global and Local Complexity Guarantees
Tsinghua University（清华大学）；The Hong Kong Polytechnic University（香港理工大学）；Beijing Institute of Mathematical Sciences and Applications（北京数学科学及应用研究所）；Chinese Academy of Sciences（中国科学院）
We analyze the performance of a variant of Newton method with quadratic regularization for solving composite convex minimization problems. At each step of our method, we choose regularization parameter proportional to a certain power of the gradient norm at the current point. We introduce a family of problem classes characterized by Hölder continuity of either the second or third derivative. Then we present the method with a simple adaptive search procedure allowing an automatic adjustment to the problem class with the best global complexity bounds, without knowing specific parameters of the p…
We present two new remarkably simple stochastic second-order methods for minimizing the average of a very large number of sufficiently smooth and strongly convex functions. The first is a stochastic variant of Newton's method (SN), and the second is a stochastic variant of cubically regularized Newton's method (SCN). We establish local linear-quadratic convergence results. Unlike existing stochastic variants of second order methods, which require the evaluation of a large number of gradients and/or Hessians in each iteration to guarantee convergence, our methods do not have this shortcoming. F…
Federated Cubic Regularized Newton Learning with Sparsification-amplified Differential Privacy
Hong Kong University of Science and Technology（香港科技大学）；KTH Royal Institute of Technology（瑞典皇家理工学院）；Southern University of Science and Technology（南方科技大学）
Department of Computer Science（计算机科学系）；ETH Zurich（苏黎世联邦理工学院）
We investigate the use of regularized Newton methods with adaptive norms for optimizing neural networks. This approach can be seen as a second-order counterpart of adaptive gradient methods, which we here show to be interpretable as first-order trust region methods with ellipsoidal constraints. In particular, we prove that the preconditioning matrix used in RMSProp and Adam satisfies the necessary conditions for provable convergence of second-order trust region methods with standard worst-case complexities on general non-convex objectives. Furthermore, we run experiments across different neura…
Optimization plays a key role in machine learning. Recently, stochastic second-order methods have attracted much attention due to their low computational cost in each iteration. However, these algorithms might perform poorly especially if it is hard to approximate the Hessian well and efficiently. As far as we know, there is no effective way to handle this problem. In this paper, we resort to Nesterov's acceleration technique to improve the convergence performance of a class of second-order methods called approximate Newton. We give a theoretical analysis that Nesterov's acceleration technique…
Bridging the Gap between Newton-Raphson Method and Regularized Policy Iteration
Laboratory for Information and Decision Systems, Massachusetts Institute of Technology（信息与决策系统实验室，麻省理工学院）；Department of Mechanical Engineering, Tsinghua University（机械工程系，清华大学）；College of Artificial Intelligence and the School of Vehicle and Mobility, Tsinghua University（人工智能学院和车辆与移动系统学院，清华大学）
National University of Singapore（新加坡国立大学）；Wisconsin Institute for Discovery（威斯康星发现研究所）；University of Wisconsin-Madison（威斯康星大学麦迪逊分校）
We propose a communication- and computation-efficient distributed optimization algorithm using second-order information for solving empirical risk minimization (ERM) problems with a nonsmooth regularization term. Our algorithm is applicable to both the primal and the dual ERM problem. Current second-order and quasi-Newton methods for this problem either do not work well in the distributed setting or work only for specific regularizers. Our algorithm uses successive quadratic approximations of the smooth part, and we describe how to maintain an approximation of the (generalized) Hessian and sol…
In this paper we study inexact dumped Newton method implemented in a distributed environment. We start with an original DiSCO algorithm [Communication-Efficient Distributed Optimization of Self-Concordant Empirical Loss, Yuchen Zhang and Lin Xiao, 2015]. We will show that this algorithm may not scale well and propose an algorithmic modifications which will lead to less communications, better load-balancing and more efficient computation. We perform numerical experiments with an regularized empirical loss minimization instance described by a 273GB dataset.
Optimization plays a key role in machine learning. Recently, stochastic second-order methods have attracted much attention due to their low computational cost in each iteration. However, these algorithms might perform poorly especially if it is hard to approximate the Hessian well and efficiently. As far as we know, there is no effective way to handle this problem. In this paper, we resort to Nestrov's acceleration technique to improve the convergence performance of a class of second-order methods called approximate Newton. We give a theoretical analysis that Nestrov's acceleration technique c…
