1,447 research outputs found
Adaptive Quasi-Newton and Anderson Acceleration Framework with Explicit Global (Accelerated) Convergence Rates
Despite the impressive numerical performance of quasi-Newton and
Anderson/nonlinear acceleration methods, their global convergence rates have
remained elusive for over 50 years. This paper addresses this long-standing
question by introducing a framework that derives novel and adaptive
quasi-Newton or nonlinear/Anderson acceleration schemes. Under mild
assumptions, the proposed iterative methods exhibit explicit, non-asymptotic
convergence rates that blend those of gradient descent and Cubic Regularized
Newton's method. Notably, these rates are achieved adaptively, as the method
autonomously determines the optimal step size using a simple backtracking
strategy. The proposed approach also includes an accelerated version that
improves the convergence rate on convex functions. Numerical experiments
demonstrate the efficiency of the proposed framework, even compared to a
fine-tuned BFGS algorithm with line search
Stochastic Trust Region Methods with Trust Region Radius Depending on Probabilistic Models
We present a stochastic trust-region model-based framework in which its
radius is related to the probabilistic models. Especially, we propose a
specific algorithm, termed STRME, in which the trust-region radius depends
linearly on the latest model gradient. The complexity of STRME method in
non-convex, convex and strongly convex settings has all been analyzed, which
matches the existing algorithms based on probabilistic properties. In addition,
several numerical experiments are carried out to reveal the benefits of the
proposed methods compared to the existing stochastic trust-region methods and
other relevant stochastic gradient methods
Newton-Type Methods for Non-Convex Optimization Under Inexact Hessian Information
We consider variants of trust-region and cubic regularization methods for
non-convex optimization, in which the Hessian matrix is approximated. Under
mild conditions on the inexact Hessian, and using approximate solution of the
corresponding sub-problems, we provide iteration complexity to achieve -approximate second-order optimality which have shown to be tight.
Our Hessian approximation conditions constitute a major relaxation over the
existing ones in the literature. Consequently, we are able to show that such
mild conditions allow for the construction of the approximate Hessian through
various random sampling methods. In this light, we consider the canonical
problem of finite-sum minimization, provide appropriate uniform and non-uniform
sub-sampling strategies to construct such Hessian approximations, and obtain
optimal iteration complexity for the corresponding sub-sampled trust-region and
cubic regularization methods.Comment: 32 page
Cubic Regularization is the Key! The First Accelerated Quasi-Newton Method with a Global Convergence Rate of for Convex Functions
In this paper, we propose the first Quasi-Newton method with a global
convergence rate of for general convex functions. Quasi-Newton
methods, such as BFGS, SR-1, are well-known for their impressive practical
performance. However, they may be slower than gradient descent for general
convex functions, with the best theoretical rate of . This gap
between impressive practical performance and poor theoretical guarantees was an
open question for a long period of time. In this paper, we make a significant
step to close this gap. We improve upon the existing rate and propose the Cubic
Regularized Quasi-Newton Method with a convergence rate of . The key
to achieving this improvement is to use the Cubic Regularized Newton Method
over the Damped Newton Method as an outer method, where the Quasi-Newton update
is an inexact Hessian approximation. Using this approach, we propose the first
Accelerated Quasi-Newton method with a global convergence rate of
for general convex functions. In special cases where we can improve the
precision of the approximation, we achieve a global convergence rate of
, which is faster than any first-order method. To make these methods
practical, we introduce the Adaptive Inexact Cubic Regularized Newton Method
and its accelerated version, which provide real-time control of the
approximation error. We show that the proposed methods have impressive
practical performance and outperform both first and second-order methods
Newton-MR: Inexact Newton Method With Minimum Residual Sub-problem Solver
We consider a variant of inexact Newton Method, called Newton-MR, in which
the least-squares sub-problems are solved approximately using Minimum Residual
method. By construction, Newton-MR can be readily applied for unconstrained
optimization of a class of non-convex problems known as invex, which subsumes
convexity as a sub-class. For invex optimization, instead of the classical
Lipschitz continuity assumptions on gradient and Hessian, Newton-MR's global
convergence can be guaranteed under a weaker notion of joint regularity of
Hessian and gradient. We also obtain Newton-MR's problem-independent local
convergence to the set of minima. We show that fast local/global convergence
can be guaranteed under a novel inexactness condition, which, to our knowledge,
is much weaker than the prior related works. Numerical results demonstrate the
performance of Newton-MR as compared with several other Newton-type
alternatives on a few machine learning problems.Comment: 35 page
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