New quasi-Newton method guarantees global superlinear convergence.
problem Global convergence and superlinear convergence of quasi-Newton methods.
method Hybrid proximal extragradient method with online learning for Hessian approximation.
result First globally convergent quasi-Newton method with explicit superlinear convergence rate.
Paper proposes an efficient online Newton method with Nesterov's acceleration for streaming data.
problem Efficient inference of online Newton methods with robustness to noise and ill-conditioning.
method Online Newton method with Hessian averaging and Nesterov's accelerated sketch-and-project solver.
result Global almost-sure convergence and asymptotic normality of the last iterate with non-asymptotic convergence guarantees.
Paper proposes a quasi-Newton method for nonlinear equations with global convergence guarantees.
problem Solving smooth and monotone nonlinear equations efficiently and globally.
method Hybrid proximal extragradient framework combined with online learning for Jacobian approximation.
result First global convergence results showing quasi-Newton method's advantage over extragradient method.
This paper introduces online algorithms to estimate robust geometric median in large data streams.
problem Detecting outliers in large data sets using robust statistical measures.
method Online stochastic Newton methods for estimating the geometric median.
result Rates of convergence for online estimation of the geometric median.
Improved complexity for smooth nonconvex optimization using quasi-Newton methods.
problem Finding ε-first-order stationary points of smooth functions with gradient information only.
method Two-level online learning approach involving quasi-Newton methods.
result Gradient complexity improved to O(d^(1/4)ε^(-13/8)) for d = O(ε^(-1/2)).
Paper proposes an online covariance estimator for sketched Newton methods.
problem Estimating the limiting covariance matrix of sketched Newton methods.
method Proposes a fully online covariance matrix estimator from Newton iterates.
result Establishes the consistency and convergence rate of the proposed estimator.
We describe a framework for deriving and analyzing online optimization algorithms that incorporate adaptive, data-dependent regularization, also termed preconditioning. Such algorithms have been proven useful in stochastic optimization by reshaping the gradients according to the geometry of the data. Our framework capt…
New method solves convex optimization faster than NAG.
problem Unconstrained smooth convex optimization problems.
method Accelerated quasi-Newton proximal extragradient (A-QPNE) method.
result Achieves a faster convergence rate of O ( min { 1 k 2 , d log k k 2.5 } ) {O}\bigl(\min\{\frac{1}{k^2}, \frac{\sqrt{d\log k}}{k^{2.5}}\}\bigr) O ( min { k 2 1 , k 2.5 d l o g k } ) . A new method for machine learning updates reduces complexity and improves robustness.
problem Stochastic gradient updates are inefficient and sensitive to feature scaling.
method Incremental Gauss-Newton Descent (IGND) reduces the need for matrix operations and improves robustness.
result IGND improves robustness to sensitivity scaling and can be competitive with common stochastic optimizers.
The second order method as Newton Step is a suitable technique in Online Learning to guarantee regret bound. The large data is a challenge in Newton method to store second order matrices as hessian. In this paper, we have proposed an modified online Newton step that store first and second order matrices of dimension m …
A new algorithm reduces the time and space complexity for multinomial logistic bandits.
problem High-dimensional feedback in multinomial logistic bandits makes existing algorithms inefficient.
method Integrates frequent directions matrix sketching into OFUL-MLogB to reduce time and space complexity.
result Achieves a regret bound of i l d e O ( Δ T ( K d ln Δ T + m ) T ) ilde{\mathcal{O}}(Δ_T(Kd\lnΔ_T+m)\sqrt{T}) i l d e O ( Δ T ( K d ln Δ T + m ) T ) . New adaptive first-order methods improve on quasi-Newton variants.
problem Designing efficient gradient methods for practical applications.
method Online scaled gradient methods (OSGM) with new adaptive methods OSGM-Best.
result OSGM-Best matches quasi-Newton variants but requires less memory and cheaper iterations.
A new Bayesian filtering method speeds up stochastic Newton optimization.
problem Minimizing log-convex functions using stochastic methods.
method Contextualizes the problem as Bayesian inference, applying Bayesian filtering to update estimates.
result Establishes conditions for diminishing effect of older observations, akin to momentum.
Improved neural network ensembles using Stein Variational Newton updates.
problem Lack of efficient second-order information in current ensemble methods.
method Proposes a novel approximate Bayesian inference method integrating Stein Variational Newton updates with scalable Hessian approximations.
result Significantly faster convergence and more accurate posterior distribution approximations.
Online learning with limited information feedback (bandit) tries to solve the problem where an online learner receives partial feedback information from the environment in the course of learning. Under this setting, Flaxman et al.[8] extended Zinkevich's classical Online Gradient Descent (OGD) algorithm [29] by proposi…
Newton's method tackles nonlinear mappings into vector bundles with connections and retractions.
problem Finding zeros of mappings from a manifold into a vector bundle.
method Local convergence using differentiability concepts, Banach space Riemannian distance, and affine covariant damping strategy.
result Illustrated application to generalized non-symmetric eigenvalue problems.
A new quasi-Newton method uses cubic regularization to avoid saddle points in deep learning.
problem Avoiding saddle points and poor local minima in deep learning models.
method Limited-memory symmetric rank-one quasi-Newton approach with adaptive regularized cubics.
result The method effectively avoids saddle points and converges to better local minima.
RONM method reduces regret in stochastic convex bandits with decreasing noise.
problem Stochastic convex bandit problem with decreasing noise.
method Regularized Online Newton Method (RONM) based on Online Newton Method (ONM).
result RONM achieves polylogarithmic regret in time horizon n.
AuON is a linear-time optimizer that improves upon Muon's performance without approximate orthogonal matrices.
problem High memory and computational costs of orthogonal momentum updates.
method AuON uses normalized nonlinear scaling and a 'emergency brake' to handle exploding attention logits.
result AuON achieves strong performance without approximate orthogonal matrices, preserving structural alignment and reconditioning.
Boosting algorithms are frequently used in applied data science and in research. To date, the distinction between boosting with either gradient descent or second-order Newton updates is often not made in both applied and methodological research, and it is thus implicitly assumed that the difference is irrelevant. The g…
New method solves constrained stochastic optimization problems efficiently.
problem Online statistical inference of constrained stochastic nonlinear optimization problems.
method Stochastic Sequential Quadratic Programming (StoSQP) with iterative sketching solver.
result The rescaled primal-dual sequence converges to a mean-zero Gaussian distribution.
The question of how to incorporate curvature information in stochastic approximation methods is challenging. The direct application of classical quasi- Newton updating techniques for deterministic optimization leads to noisy curvature estimates that have harmful effects on the robustness of the iteration. In this paper…
An online learning framework for survival analysis with real-time adaptation.
problem Real-time adaptation to dynamic environments and censored data in survival analysis.
method Online Newton Step (ONS) for optimal second order online convex optimization.
result Logarithmic stochastic regret for ONS with adaptive aggregation method.
Develops a new screening method called Newton screening for faster and more accurate sparse learning.
problem Sparse learning problems with large-scale optimization.
method Newton screening method with built-in working set and dual variable updates.
result Newton screening achieves one-step local convergence and sharp estimation error bound.
Global convergence of an online (stochastic) limited memory version of the Broyden-Fletcher- Goldfarb-Shanno (BFGS) quasi-Newton method for solving optimization problems with stochastic objectives that arise in large scale machine learning is established. Lower and upper bounds on the Hessian eigenvalues of the sample …
Newton's method converges faster than gradient descent in overparameterized neural networks.
problem Training neural networks efficiently in the overparameterized limit.
method Developed a convergence analysis for the regularized Newton method in this context.
result The NN training dynamics converge to the solution of a deterministic limit equation involving a Newton neural tangent kernel (NNTK).
Paper analyzes regret bounds for unconstrained online optimization.
problem Minimizing regret in dynamic online learning for strongly convex and smooth functions.
method Preconditioned OGD, Online Optimistic Newton (OON), multiple gradient queries.
result Achieves O ( C 2 , T ∗ ) O(C^*_{2,T}) O ( C 2 , T ∗ ) regret bound with one gradient query per round. A new method for faster optimization in high dimensions.
problem Slow convergence in high-dimensional optimization problems.
method Subspace cubic regularized Newton method within Krylov subspace.
result Achieves a dimension-independent convergence rate of O(1/mk + 1/k^2).
Efficient algorithm for zeroth-order bandit convex optimization with bounds on regret.
problem Optimizing in unknown, noisy environments with limited information.
method Online Newton Method for bandit convex optimization, proving regret bounds.
result Regret bounds for both adversarial and stochastic settings.
Paper develops efficient methods for estimating Hessian inverses in stochastic optimization.
problem Estimating the inverse Hessian for convex function minimization.
method Robbins-Monro procedure for recursive estimation of the inverse Hessian.
result Develops universal stochastic Newton methods with improved efficiency.
New method solves optimization problems faster than existing methods.
problem Large-scale nonconvex optimization problems.
method ARC method using LQN matrices with exact CR subproblem solution.
result Exact solutions to CR subproblem found in matrix-free manner.
FedNew improves federated learning efficiency and privacy.
problem Low communication efficiency and privacy issues in Newton-type methods for federated learning.
method Introduces a two-level framework using ADMM for inverse Hessian-gradient approximation and Newton's method for global model updates, reducing communication overhead.
result FedNew achieves superior communication efficiency and privacy compared to existing methods.
A new method for faster optimization on statistical manifolds.
problem Slow convergence of first-order methods in manifold optimization.
method Dual Riemannian Newton method on manifolds with dual connections.
result Local quadratic convergence of the dual Riemannian Newton method.
Develops a new SPP algorithm with variance reduction for weakly convex optimization.
problem Weakly convex, composite optimization problems.
method Inexact semismooth Newton framework with variance reduction for stochastic proximal point updates.
result Establishes convergence results for the proposed algorithm.
Newton-LESS sparsifies Gaussian sketching for faster optimization.
problem Computing the Hessian matrix in optimization is computationally expensive.
method Uses a sparsified version of a dense Gaussian sketching matrix.
result Achieves nearly the same convergence rate as dense Gaussian embeddings without the computational cost.
EGN optimizes deep neural networks with exact Gauss-Newton for large-scale problems.
problem Training deep neural networks efficiently and accurately.
method Stochastic second-order optimization using low-rank linear algebra and matrix factorization.
result Converges to stationary points of the objective function under mild assumptions.
A new EM gradient algorithm for mixture models with skewed components.
problem Fitting mixture models with skewed components derived from the Manly transformation.
method Proposes an alternative EM gradient algorithm using Newton's method for better parameter updates.
result Shows improved convergence and parameter estimation compared to the Nelder-Mead optimization.
We propose an algorithm, semismooth Newton coordinate descent (SNCD), for the elastic-net penalized Huber loss regression and quantile regression in high dimensional settings. Unlike existing coordinate descent type algorithms, the SNCD updates each regression coefficient and its corresponding subgradient simultaneousl…
Kernel interpolation speeds up online Gaussian process updates.
problem Efficiently updating Gaussian process posteriors with new data.
method Structured kernel interpolation for constant-time updates.
result Exact inference maintained with constant-time updates.
Bayesian approach improves AdaLoRA's performance and efficiency.
problem Improving the efficiency and performance of adaptive low-rank adaptation.
method Utilized Bayesian metrics and the Improved Variational Online Newton (IVON) optimizer for adaptive parameter budget allocation.
result Bayesian counterpart outperforms sensitivity-based importance metric and is faster than AdaLoRA.
A new sequential method estimates Poisson means in streaming data, achieving optimality and efficiency.
problem Estimating Poisson means in a streaming, or online, framework.
method A quasi-Bayesian approach based on Newton's algorithm for a sequential estimate.
result Established frequentist guarantees including consistency and asymptotic optimality.
In stochastic gradient descent, especially for neural network training, there are currently dominating first order methods: not modeling local distance to minimum. This information required for optimal step size is provided by second order methods, however, they have many difficulties, starting with full Hessian having…
A new method avoids saddle points in Newton's method.
problem Avoiding saddle points in optimization problems.
method New Q-Newton's method with specific update rule.
result The method guarantees convergence to a critical point that is not a saddle point.
In this paper, we address the problem of adaptive learning for autoregressive moving average (ARMA) model in the quaternion domain. By transforming the original learning problem into a full information optimization task without explicit noise terms, and then solving the optimization problem using the gradient descent a…
Apollo improves nonconvex stochastic optimization efficiency.
problem Nonconvex stochastic optimization challenges.
method Adaptive parameter-wise diagonal quasi-Newton method approximating Hessian.
result Significant improvements in convergence speed and generalization over SGD and Adam.
An online decision-making algorithm using stochastic gradient descent for big data.
problem Efficiently updating decision rules in online decision making with big data.
method Stochastic gradient descent for online updates, asymptotic normality of estimators.
result Asymptotic normality of parameter and value estimators, enabling statistical inference.
A framework for certified unlearning in decentralized federated learning.
problem Privacy-preserving machine learning in decentralized federated learning.
method Newton-style updates to quantify and correct data influence, using Fisher information matrices for scalability.
result The proposed framework ensures that the unlearned model is difficult to distinguish from a retrained model without the deleted data.
Predictive coding networks are shown to be stable, robust, and converge faster than backpropagation.
problem Stability, robustness, and convergence of predictive coding networks.
method Dynamical systems theory and Lyapunov stability analysis.
result Predictive coding networks are Lyapunov stable and converge faster than backpropagation.