Novel adaptive method reduces Hessian computation for large-scale ERM.
problem Efficiently solving large-scale empirical risk minimization problems.
method Truncated adaptive Newton method reducing Hessian computation costs.
result Single iteration of truncated Newton method achieves statistical accuracy.
Efficient algorithm for SSVM reduces computation time and scales to large datasets.
problem Efficiently training SSVM for large patient data sets.
method Directly optimizes primal objective function with truncated Newton optimisation and order statistic trees.
result Significantly reduces computational costs and scales to larger datasets.
A new optimisation method efficiently scales Hessian-vector products for neural networks.
problem Challenges in applying second-order quasi-Newton methods due to large Hessian and non-convexity.
method Proposes an optimisation algorithm that asymptotically uses the exact inverse Hessian with modified eigenvalues.
result Demonstrates scalability and comparable performance to other optimisation methods in neural networks.
The paper studies minimal resistance dynamics in radial fields, finding unique solutions for incompressible flows.
problem Nonlinear dynamics of minimal resistance in radial fields.
method Analysis of two non-equilibrium scenarios: scale-invariant free expansion and incompressible source flow.
result Incompressible flow acts as a structural regularizer, admitting unique, smooth, and strictly concave solutions.
This paper introduces the Metric-Free Natural Gradient (MFNG) algorithm for training Boltzmann Machines. Similar in spirit to the Hessian-Free method of Martens [8], our algorithm belongs to the family of truncated Newton methods and exploits an efficient matrix-vector product to avoid explicitely storing the natural g…
Stochastic Newton and quasi-Newton methods solve large linear least-squares problems efficiently.
problem Efficiently solve large linear least-squares problems with limited computational resources.
method Introduce stochasticity in Newton and quasi-Newton approaches to handle large datasets.
result Stochastic Newton iterates may not converge to the least-squares solution.
A new method for optimization in probability space using Newton's flows.
problem Optimization in probability space with information metrics.
method Information Newton's flows, including Fisher-Rao and Wasserstein-2 metrics, with Newton's Langevin dynamics and variational methods.
result Effective numerical implementation and convergence results for the proposed method.
Newton's method solves variational problems on manifolds.
problem Solving variational equations on manifolds.
method Newton's method with affine covariant damping strategy.
result Numerical results for variational problems demonstrated.
This research compares gradient and Newton boosting methods in classification and regression.
problem The distinction between gradient descent and Newton updates in boosting algorithms is not well understood.
method Presented a unified framework for gradient and Newton boosting, and compared them with tree base learners.
result Newton boosting outperforms gradient and hybrid boosting in predictive accuracy on most datasets.
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 ) . Muon with Newton-Schulz converges to the same stationary point as SVD-polar, up to a constant factor.
problem Improving the convergence rate of Muon optimizer.
method Using Newton-Schulz steps for momentum orthogonalization, proving convergence rate and constant factor.
result Muon with Newton-Schulz converges to the same stationary point as SVD-polar, up to a constant factor.
Newton methods improve CNN optimization, showing competitive accuracy.
problem Optimizing CNNs for deep learning applications.
method Detailed investigation and implementation of Newton methods for CNNs.
result Newton methods for CNNs achieve competitive test accuracy.
Simple stochastic Newton and cubic Newton methods with fast convergence.
problem Minimizing large numbers of smooth and strongly convex functions.
method Stochastic Newton and cubic Newton methods with simple local linear-quadratic rates.
result Local linear-quadratic convergence results with fast adaptation to problem's curvature.
Paper tackles zeroth-order optimization for nonconvex problems with constraints, high-dimensions, and saddle-points.
problem Optimization of nonconvex functions with constraints and high-dimensionality, avoiding saddle-points.
method Proposes zeroth-order stochastic approximation algorithms, including conditional gradient and truncated gradient methods, and a zeroth-order cubic regularization Newton's method.
result Demonstrates algorithms achieving rates similar to standard stochastic gradient methods, with rates dependent on poly-logarithmic dimensionality.
Generalizes Newton's Second Law for field theory.
problem Applying Newton's Second Law to higher-dimensional parameterized submanifolds.
method Introducing geodesic k-vector field and deriving Hamilton's equations.
result Different forces can lead to the same Hamilton's equations.
Modified Newton step for online learning reduces matrix size for large datasets.
problem Handling large multi-class datasets efficiently in online learning.
method Element-wise multiplication to reduce matrix size of second order matrices.
result Proposed method achieves similar mistake rates to popular methods but with faster computations.
RNN operators solve Newton's equations with large timesteps for molecular dynamics.
problem Solving Newton's equations of motion with large timesteps for molecular dynamics simulations.
method Recurrent Neural Networks (RNN) operators to solve Newton's equations using past trajectory data.
result Significant speedup in molecular dynamics simulations with timesteps up to 4000 times larger.
New algorithm improves convergence of gradient boosting trees.
problem Global convergence of Newton boosting in tabular machine learning.
method Introduces Gradient Regularized Newton Descent for GBDTs, proving linear convergence for smooth, strongly convex losses and O ( 1 k 2 ) \mathcal{O}(\frac{1}{k^2}) O ( k 2 1 ) rate for general convex losses. result Achieves globally convergent second-order GBDT algorithm with rate matching first-order boosting.
We generalize Newton-type methods for minimizing smooth functions to handle a sum of two convex functions: a smooth function and a nonsmooth function with a simple proximal mapping. We show that the resulting proximal Newton-type methods inherit the desirable convergence behavior of Newton-type methods for minimizing s…
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.
Study uses Newton polytopes to distinguish Lagrangian fillings of Legendrian submanifolds.
problem Distinguishing Lagrangian fillings of Legendrian submanifolds.
method Utilizes Newton polytopes associated with augmented values of Reeb chords.
result Newton polytopes can distinguish infinitely many distinct Lagrangian fillings.
Unified approach to Bayesian inference with guarantees on covariance matrices.
problem Approximate Bayesian inference with PSD guarantees.
method Bayes-Newton methods extending Newton's method for optimisation.
result Novel algorithms with PSD covariance matrices.
The problem of an arbitrary truncated Levy flight description using the method of cumulant approach has been solved. The set of cumulants of the truncated Levy distribution given the assumption of arbitrary truncation has been found. The influence of truncation shape on the truncated Levy flight properties in the Gauss…
Study on Newton-Sketch and Subsampled Newton methods for large-scale optimization.
problem Optimization of large-scale finite-sum problems with high-dimensional data.
method Hessian subsampling and randomized Hadamard transformations for dimensionality reduction in Newton's method.
result Advantages of conjugate gradient vs. stochastic gradient iterations revealed through experiments.
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.
New Q-Newton's method avoids saddle points and converges quadratically.
problem Optimizing functions with saddle points and ensuring convergence guarantees.
method Modified New Q-Newton's method with Backtracking line search.
result Theorem for Morse functions: quadratic convergence to local minima.
Truncated CauchyNMF robustly learns subspaces from noisy data.
problem Outliers in non-negative matrix factorization (NMF) cause failure.
method Proposes Truncated CauchyNMF loss to handle outliers.
result Theoretical analysis and experimental validation show Truncated CauchyNMF's robustness.
GIANT optimizes distributed computing by improving Newton method efficiency.
problem Efficiently solving empirical risk minimization problems in distributed environments.
method GIANT combines local ANT directions to form a GIANT direction, averaging communications and computations.
result GIANT achieves faster convergence compared to first-order and existing Newton-type methods.
Efficiently estimate Boolean product distribution parameters from truncated samples.
problem Estimating parameters of Boolean product distributions from truncated samples.
method Introducing fatness of truncation set, using membership queries, and adapting Stochastic Gradient Descent.
result Efficiently learn Boolean product distributions from truncated samples with small sample complexity.
A new optimization method improves deep learning accuracy without hyper-parameter tuning.
problem Computational demands and convergence behavior in deep learning training.
method Stochastic quasi-Gauss-Newton (SQGN) optimization method combining stochastic quasi-Newton, Gauss-Newton, and variance reduction.
result SQGN provides excellent accuracy without hyper-parameter experimentation, improving convergence and computational performance.
Proposes a Quasi-Newton trust region method for policy optimization in reinforcement learning.
problem Lack of stepsize selection criterion and slow convergence in gradient descent for policy optimization.
method Uses a trust region method with Quasi-Newton approximation for the Hessian.
result Demonstrates improved performance and efficiency in continuous control tasks.
This thesis disentangles Gauss-Newton and variational approximations in Bayesian deep learning.
problem Understanding the interplay between the Gauss-Newton method and variational approximations in Bayesian deep learning.
method Analysis of the Gauss-Newton method and Laplace/Gaussian variational approximations for neural networks.
result The combination of the Gauss-Newton method with approximate inference can be cast as inference in a linear or Gaussian process model.
In the paper "On Truncated Variation of Brownian Motion with Drift" (Bull. Pol. Acad. Sci. Math. 56 (2008), no.4, 267 - 281) we defined truncated variation of Brownian motion with drift, W t = B t + μ t , t ≥ 0 , W_t = B_t + μt, t\geq 0, W t = B t + μ t , t ≥ 0 , where ( B t ) (B_t) ( B t ) is a standard Brownian motion. Truncated variation differs from regular variation by neglect…
Newton's method converges linearly for stable Hessians, even with approximations.
problem Finding global linear convergence for functions without strong convexity or Lipschitz gradients.
method Global linear convergence of Newton's method for stable Hessians, using approximate Hessians and subproblems.
result Global linear convergence rate for a broad class of functions, superior to first-order methods.
SVRN accelerates Newton methods by reducing variance and improving performance.
problem Improving the efficiency of Newton methods for large-scale optimization problems.
method Stochastic Variance-Reduced Newton (SVRN) algorithm that accelerates Subsampled Newton and Iterative Hessian Sketch algorithms.
result SVRN accelerates Newton methods by reducing the number of passes over the data, achieving a significant improvement in performance.
Optimal algorithm learns Gaussian under halfspace truncation with minimal samples.
problem Learning a Gaussian distribution truncated to an unknown halfspace.
method Efficient algorithm using n = i l d e O ( d 2 / ε 2 ) n = ilde{O}(d^2/\varepsilon^2) n = i l d e O ( d 2 / ε 2 ) samples and runtime dominated by empirical covariance matrix computation. result Optimal sample and time complexity bounds for learning a Gaussian under halfspace truncation.
Study of measured laminations on surfaces using Newton polytopes and Poisson brackets.
problem Understanding the space of measured laminations on surfaces from a valuative perspective.
method Introducing Newton polytopes for character variety functions, defining tangent spaces, and identifying symplectic structures.
result Trace functions have unit coefficients at the extremal points of their Newton polytopes.
New method for constructing truncated vine copulas.
problem High-dimensional parameter space in vine copulas.
method Propose a new score and algorithm for constructing truncated vines.
result New algorithms exploit conditional independences.
Approximate Newton methods are a standard optimization tool which aim to maintain the benefits of Newton's method, such as a fast rate of convergence, whilst alleviating its drawbacks, such as computationally expensive calculation or estimation of the inverse Hessian. In this work we investigate approximate Newton meth…
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.
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.
Stochastic quasi-Newton tackles noisy gradients in optimization.
problem Optimizing with noisy data in stochastic settings.
method Extends quasi-Newton methods to handle stochastic gradients through flexible Hessian modeling and line-search regularization.
result Demonstrates superior performance in maximum likelihood estimation for complex models.
Paper develops a robust PP distributed quasi-Newton estimation for Byzantine machines.
problem Byzantine machines in distributed computing under Privacy Protection constraints.
method Robust PP distributed quasi-Newton estimation method that transmits only five vectors.
result Reduces privacy budgeting and transmission cost compared to gradient descent and Newton iteration.
Efficient methods for training deep neural networks using subsampled Gauss-Newton and natural gradient.
problem Training deep neural networks with large datasets and variables.
method Subsampled Gauss-Newton and natural gradient methods with subsampled gradient estimates.
result Methods converge to a stationary point and are efficient to implement.
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.
Four decades after their invention, quasi-Newton methods are still state of the art in unconstrained numerical optimization. Although not usually interpreted thus, these are learning algorithms that fit a local quadratic approximation to the objective function. We show that many, including the most popular, quasi-Newto…
A distributed Newton method speeds up deep neural network training.
problem Training deep neural networks with distributed computing.
method Variable and feature-wise data partitions, diagonalization method, subsampled Gauss-Newton matrices, and early termination.
result The method reduces communication and synchronization costs, leading to faster training.
A new optimizer combines Newton and ADMM for faster classification.
problem Slower convergence of first-order methods in distributed learning.
method Integrates GPU-accelerated Newton solver with ADMM for multiclass classification.
result Significantly reduces the time to solution and generalization error.