For distributed computing environment, we consider the empirical risk minimization problem and propose a distributed and communication-efficient Newton-type optimization method. At every iteration, each worker locally finds an Approximate NewTon (ANT) direction, which is sent to the main driver. The main driver, then, averages all the ANT directions received from workers to form a Globally Improved ANT (GIANT) direction. GIANT is highly communication efficient and naturally exploits the trade-offs between local computations and global communications in that more local computations result in fewer overall rounds of communications. Theoretically, we show that GIANT enjoys an improved convergence rate as compared with first-order methods and existing distributed Newton-type methods. Further, and in sharp contrast with many existing distributed Newton-type methods, as well as popular first-order methods, a highly advantageous practical feature of GIANT is that it only involves one tuning parameter. We conduct large-scale experiments on a computer cluster and, empirically, demonstrate the superior performance of GIANT.

Computational optimal transport (OT) has received massive interests in the machine learning community, and great advances have been gained in the direction of entropic-regularized OT. The Sinkhorn algorithm, as well as its many improved versions, has become the solution to large-scale OT problems. However, most of the existing methods behave like first-order methods, which typically require a large number of iterations to converge. More recently, Newton-type methods using sparsified Hessian matrices have demonstrated promising results on OT computation, but there still remain a lot of unresolved open questions. In this article, we make major new progresses towards this direction: first, we propose a novel Hessian sparsification scheme that promises a strict control of the approximation error; second, based on this sparsification scheme, we develop a Newton-type method that is guaranteed to avoid singularity in computing the search directions; third, the developed algorithm has a clear implementation for practical use, avoiding most hyperparameter tuning; and remarkably, we provide rigorous global and local convergence analysis of the proposed algorithm, which is lacking in the prior literature. Various numerical experiments are conducted to demonstrate the effectiveness of the proposed algorithm in solving large-scale OT problems.

For distributed computing environment, we consider the empirical risk minimization problem and propose a distributed and communication-efficient Newton-type optimization method. At every iteration, each worker locally finds an Approximate NewTon (ANT) direction, which is sent to the main driver. The main driver, then, averages all the ANT directions received from workers to form a Globally Improved ANT (GIANT) direction. GIANT is highly communication efficient and naturally exploits the trade-offs between local computations and global communications in that more local computations result in fewer overall rounds of communications. Theoretically, we show that GIANT enjoys an improved convergence rate as compared with first-order methods and existing distributed Newton-type methods. Further, and in sharp contrast with many existing distributed Newton-type methods, as well as popular first-order methods, a highly advantageous practical feature of GIANT is that it only involves one tuning parameter. We conduct large-scale experiments on a computer cluster and, empirically, demonstrate the superior performance of GIANT.

Computational optimal transport (OT) has received massive interests in the machine learning community, and great advances have been gained in the direction of entropic-regularized OT. The Sinkhorn algorithm, as well as its many improved versions, has become the de facto solution to large-scale OT problems. However, most of the existing methods behave like first-order methods, which typically require a large number of iterations to converge. More recently, Newton-type methods using sparsified Hessian matrices have demonstrated promising results on OT computation, but there still remain a lot of unresolved open questions. In this article, we make major new progresses towards this direction: first, we propose a novel Hessian sparsification scheme that promises a strict control of the approximation error; second, based on this sparsification scheme, we develop a safe Newton-type method that is guaranteed to avoid singularity in computing the search directions; third, the developed algorithm has a clear implementation for practical use, avoiding most hyperparameter tuning; and remarkably, we provide rigorous global and local convergence analysis of the proposed algorithm, which is lacking in the prior literature.

In this paper, the authors propose a distributed Newton method for gradient-norm optimization. The method does not impose any specific form on the underlying objective function. The authors present convergence analysis for the method and illustrate the performance of the method on a convex problem (in the main paper). Originality: The topic of the paper, in my opinion, is very interesting. The paper presents an efficient Newton method that is motivated via the optimization of the norm of the gradient.

R is a convex but not necessarily differentiable function whose proximal mapping can be evaluated efficiently. We derive a generalization of Newton-type methods to handle such convex but nonsmooth objective functions. We prove such methods are globally convergent and achieve superlinear rates of convergence in the vicinity of an optimal solution. We also demonstrate the performance of these methods using problems of relevance in machine learning and statistics.

We propose a novel and efficient lifting approach for the optimal control of rigid-body systems with contacts to improve the convergence properties of Newton-type methods. To relax the high nonlinearity, we consider the state, acceleration, contact forces, and control input torques, as optimization variables and the inverse dynamics and acceleration constraints on the contact frames as equality constraints. We eliminate the update of the acceleration, contact forces, and their dual variables from the linear equation to be solved in each Newton-type iteration in an efficient manner. As a result, the computational cost per Newton-type iteration is almost identical to that of the conventional non-lifted Newton-type iteration that embeds contact dynamics in the state equation. We conducted numerical experiments on the whole-body optimal control of various quadrupedal gaits subject to the friction cone constraints considered in interior-point methods and demonstrated that the proposed method can significantly increase the convergence speed to more than twice that of the conventional non-lifted approach.

To account for the sequential and nonconvex nature, new solution concepts and algorithms have been developed. In this work, we provide a detailed analysis of existing algorithms and relate them to two novel Newton-type algorithms. We argue that our Newton-type algorithms nicely complement existing ones in that (a) they converge faster to (strict) local minimax points; (b) they are much more effective when the problem is ill-conditioned; (c) their computational complexity remains similar. We verify our theoretical results by conducting experiments on training GANs.

For distributed computing environment, we consider the empirical risk minimization problem and propose a distributed and communication-efficient Newton-type optimization method. At every iteration, each worker locally finds an Approximate NewTon (ANT) direction, which is sent to the main driver. The main driver, then, averages all the ANT directions received from workers to form a Globally Improved ANT (GIANT) direction. GIANT is highly communication efficient and naturally exploits the trade-offs between local computations and global communications in that more local computations result in fewer overall rounds of communications. Theoretically, we show that GIANT enjoys an improved convergence rate as compared with first-order methods and existing distributed Newton-type methods.

First-order optimization methods, such as stochastic gradient descent (SGD) and its variants, are widely used in machine learning applications due to their simplicity and low per-iteration costs. However, they often require larger numbers of iterations, with associated communication costs in distributed environments. In contrast, Newton-type methods, while having higher per-iteration costs, typically require a significantly smaller number of iterations, which directly translates to reduced communication costs. In this paper, we present a novel distributed optimizer for classification problems, which integrates a GPU-accelerated Newton-type solver with the global consensus formulation of Alternating Direction of Method Multipliers (ADMM). By leveraging the communication efficiency of ADMM, GPU-accelerated inexact-Newton solver, and an effective spectral penalty parameter selection strategy, we show that our proposed method (i) yields better generalization performance on several classification problems; (ii) significantly outperforms state-of-the-art methods in distributed time to solution; and (iii) offers better scaling on large distributed platforms.