In this paper, we follow Rodomanov and Nesterov's work to study quasi-Newton methods. We focus on the common SR1 and BFGS quasi-Newton methods to establish better explicit (local) superlinear convergence rates. First, based on the greedy quasi-Newton update which greedily selects the direction to maximize a certain measure of progress, we improve the convergence rate to a condition-number-free superlinear convergence rate. Second, based on the random quasi-Newton update that selects the direction randomly from a spherically symmetric distribution, we show the same superlinear convergence rate established as above. Our analysis is closely related to the approximation of a given Hessian matrix, unconstrained quadratic objective, as well as the general strongly convex, smooth, and strongly self-concordant functions.

's work to study quasi-Newton methods.

's work to study quasi-Newton methods.

's work to study quasi-Newton methods.

's work to study quasi-Newton methods.

Normalizing flows are promising generative models with advantages such as theoretical rigor, analytical log-likelihood computation, and end-to-end training. However, the architectural constraints to ensure invertibility and tractable Jacobian computation limit their expressive power and practical usability. Recent advancements utilize autoregressive modeling, significantly enhancing expressive power and generation quality. However, such sequential modeling inherently restricts parallel computation during inference, leading to slow generation that impedes practical deployment. In this paper, we first identify that strict sequential dependency in inference is unnecessary to generate high-quality samples. We observe that patches in sequential modeling can also be approximated without strictly conditioning on all preceding patches. Moreover, the models tend to exhibit low dependency redundancy in the initial layer and higher redundancy in subsequent layers. Leveraging these observations, we propose a selective Jacobi decoding (SeJD) strategy that accelerates autoregressive inference through parallel iterative optimization. Theoretical analyses demonstrate the method's superlinear convergence rate and guarantee that the number of iterations required is no greater than the original sequential approach. Empirical evaluations across multiple datasets validate the generality and effectiveness of our acceleration technique. Experiments demonstrate substantial speed improvements up to 4.7 times faster inference while keeping the generation quality and fidelity.

In this paper, we follow Rodomanov and Nesterov's work to study quasi-Newton methods. We focus on the common SR1 and BFGS quasi-Newton methods to establish better explicit (local) superlinear convergence rates. First, based on the greedy quasi-Newton update which greedily selects the direction to maximize a certain measure of progress, we improve the convergence rate to a condition-number-free superlinear convergence rate. Second, based on the random quasi-Newton update that selects the direction randomly from a spherically symmetric distribution, we show the same superlinear convergence rate established as above. Our analysis is closely related to the approximation of a given Hessian matrix, unconstrained quadratic objective, as well as the general strongly convex, smooth, and strongly self-concordant functions.

This paper addresses the challenge of solving large-scale nonlinear equations with H\"older continuous Jacobians. We introduce a novel Incremental Gauss--Newton (IGN) method within explicit superlinear convergence rate, which outperforms existing methods that only achieve linear convergence rate. In particular, we formulate our problem by the nonlinear least squares with finite-sum structure, and our method incrementally iterates with the information of one component in each round. We also provide a mini-batch extension to our IGN method that obtains an even faster superlinear convergence rate. Furthermore, we conduct numerical experiments to show the advantages of the proposed methods.

Quasi-Newton algorithms are among the most popular iterative methods for solving unconstrained minimization problems, largely due to their favorable superlinear convergence property. However, existing results for these algorithms are limited as they provide either (i) a global convergence guarantee with an asymptotic superlinear convergence rate, or (ii) a local non-asymptotic superlinear rate for the case that the initial point and the initial Hessian approximation are chosen properly. In particular, no current analysis for quasi-Newton methods guarantees global convergence with an explicit superlinear convergence rate. In this paper, we close this gap and present the first globally convergent quasi-Newton method with an explicit non-asymptotic superlinear convergence rate. Unlike classical quasi-Newton methods, we build our algorithm upon the hybrid proximal extragradient method and propose a novel online learning framework for updating the Hessian approximation matrices. Specifically, guided by the convergence analysis, we formulate the Hessian approximation update as an online convex optimization problem in the space of matrices, and we relate the bounded regret of the online problem to the superlinear convergence of our method.