Negative step sizes improve second-order methods for neural networks.
problem Second-order methods discard negative curvature, limiting their effectiveness.
method Introduce negative step sizes in second-order methods combined with Wolfe line search.
result Negative step sizes lead to global convergence and improved performance.
Exact second-order optimization for deep learning reduces computational cost and improves performance.
problem Inadequate use of second-order optimization methods in deep learning due to high computational cost and non-convexity.
method Developed an exact stochastic second-order Newton method that addresses the non-convexity issue and provides an expression for the stochastic Hessian.
result Exact second-order Newton direction formula and its application in deep learning datasets.
SOLBP extends efficient inference to uncertain Bayesian networks.
problem Inference in uncertain Bayesian networks with second-order probabilities.
method Extends Loopy Belief Propagation to second-order Bayesian networks.
result Generates inferences consistent with sum-product networks, more efficient and scalable.
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.
New methods using natural gradient for structured optimization.
problem Structured optimization problems.
method Structured second-order methods via natural gradient descent.
result Efficiency demonstrated on non-convex and deep learning problems.
Second-order methods improve differential privacy in convex optimization.
problem Improving differential privacy in convex optimization.
method Developed a private variant of the regularized cubic Newton method for strongly convex loss functions.
result Achieves quadratic convergence and optimal excess loss for strongly convex loss functions.
AdamQLR optimizes Adam with K-FAC heuristics, achieving comparable performance to tuned benchmarks.
problem Improving the performance of Adam optimizers with stabilizing heuristics.
method Combining Adam's update directions with K-FAC's heuristics (damping and learning rate selection).
result Untuned AdamQLR can achieve comparable performance to tuned benchmarks.
New method explains predictive uncertainty by focusing on second-order effects.
problem Explaining predictive uncertainty in machine learning models.
method CovLRP, CovGI, etc., based on second-order effects.
result Predictive uncertainty is dominated by second-order effects.
A new method for faster optimization of noisy functions.
problem Optimizing noisy functions efficiently.
method A universal and adaptive second-order method for convex functions.
result Achieves O ( σ / T ) O(σ/ \sqrt{T}) O ( σ / T ) convergence for stochastic oracles and O ( 1 / T 3 ) O( 1 / T^3) O ( 1/ T 3 ) for deterministic oracles. 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).
Develops unbiased averaging methods for second order optimization in distributed systems.
problem Computing the Hessian is challenging and communication is a bottleneck in distributed optimization.
method Unbiased parameter averaging methods using sampling and sketching of the Hessian.
result Provably minimizes bias for sketched Newton directions.
First-order stochastic methods are the state-of-the-art in large-scale machine learning optimization owing to efficient per-iteration complexity. Second-order methods, while able to provide faster convergence, have been much less explored due to the high cost of computing the second-order information. In this paper we …
Second-order optimizers retain residual information after data deletion, affecting machine unlearning.
problem Residual information in second-order optimizers after data deletion.
method Comparison of first-order and second-order learners, eigendecomposition analysis.
result Second-order optimizers retain residual information, not detectable by first-order analysis.
Paper proposes a new method for efficient second-order neural network training.
problem Infeasibility of Hessian calculation and noisy second-order information in deep learning.
method Adopting complex-step directional derivative (CSFD) for accurate Hessian computation and designing an effective Newton Krylov procedure.
result Our method outperforms existing methods and often converges one-order faster.
AdaSub optimizes with second-order info in low-dims subspace.
problem Efficiently use second-order optimization methods with low computational cost.
method Adaptive subspace selection for second-order optimization.
result AdaSub outperforms other stochastic optimizers in time and iterations.
Efficient method classifies locally stationary time series based on second-order characteristics.
problem Classifying locally stationary time series for various applications.
method Autoregressive approximation, ensemble aggregation, distance-based threshold.
result Zero misclassification error rate asymptotically for mildly differing second-order characteristics.
A first-order model for a stock market assigns to each stock a return parameter and a variance parameter that depend only on the rank of the stock. A second-order model assigns these parameters based on both the rank and the name of the stock. First- and second-order models exhibit stability properties that make them a…
New algorithm finds approximate stationary points in non-convex optimization.
problem Finding approximate stationary points in non-convex stochastic optimization.
method Design of an algorithm using O ( ε − 3 ) O(ε^{-3}) O ( ε − 3 ) stochastic gradient and Hessian-vector products. result Optimal rate of O ( ε − 3 ) O(ε^{-3}) O ( ε − 3 ) for finding ε ε ε -approximate stationary points, matching lower bounds. Optimistic method adapted for faster convex-concave min-max problems.
problem Solving convex-concave min-max optimization problems efficiently.
method Adaptive, line search-free second-order methods combining optimistic updates and second-order information.
result Achieves optimal convergence rate without line search or backtracking.
Paper proposes a second-order method for faster SVI convergence.
problem Poor convergence rate of first-order SVI algorithms.
method Derives Hessian matrix and implements two numerical schemes for efficient second-order SVI.
result Proposed approach achieves faster convergence compared to first-order SVI.
New method speeds up deep learning optimization.
problem Scalable second-order optimization for deep learning.
method Second-order optimization with algorithmic and numerical improvements.
result Significant convergence and wall-clock time improvements.
Proposes a new stock prediction method that accounts for market dynamics.
problem The dynamic nature of the stock market invalidates traditional machine learning assumptions.
method Develops a second-order learning paradigm with multi-scale patterns.
result Demonstrates effectiveness in stock prediction on real-world data.
We apply the Cartan equivalence method to the study of real analytic second order ODEs under the local real analytic diffeomorphism of $\C^2$ which are area-preserving. This enables us to give a characterization of the second order ODEs which are equivalent to y ′ ′ = 0 y^{\prime\prime}=0 y ′′ = 0 under such transformations. Moreover w…
Clarifies method of phase synchronization for decoupling linear differential equations.
problem Velocity-dependent transformations in linear second-order differential equations.
method Linear transformation of coordinates and velocities.
result Velocity-dependent transformations do not preserve second-order character and define their own system.
Finite-sum optimization problems are ubiquitous in machine learning, and are commonly solved using first-order methods which rely on gradient computations. Recently, there has been growing interest in \emph{second-order} methods, which rely on both gradients and Hessians. In principle, second-order methods can require …
New matrix approximation method speeds up optimization for deep learning.
problem Efficient computation of matrix inverse and square root for high-dimensional optimization.
method Divide matrix into blocks and represent each block by one or two numbers.
result Improved performance of AdaGrad in training deep neural networks compared to diagonal approximation.
Paper introduces STSL, a second-order Tweedie sampler for efficient posterior sampling in inverse problems.
problem Computational challenges in sampling from posterior distributions using latent diffusion models.
method Introduces STSL, a novel second-order Tweedie sampler with tractable reverse process.
result STSL achieves 4X and 8X reduction in neural function evaluations compared to state-of-the-art solvers.
A new GGN method speeds up training of deep neural networks for regression tasks.
problem Training deep neural networks efficiently for regression problems.
method Proposes a Gram-Gauss-Newton (GGN) algorithm for overparameterized neural networks.
result For sufficiently wide neural networks, GGN achieves quadratic convergence rate.
Adaptive norms improve neural network optimization with second-order methods.
problem Optimizing neural networks with second-order methods.
method Regularized Newton methods with adaptive norms, interpreting adaptive gradient methods as trust region methods.
result Ellipsoidal constraints outperform spherical constraints in terms of backpropagations and asymptotic loss value.
Solves a differential equation problem using Cartan's method.
problem Equivalence of second order ordinary differential equations under point transformations.
method E. Cartan's method of equivalence
result Solves the point equivalence problem for second order ODEs.
Paper proposes a method to find approximate SOSP for nonconvex conic optimization problems.
problem Finding approximate second-order stationary points in nonconvex conic optimization.
method Newton-CG based barrier method with complexity guarantees.
result Achieves iteration complexity of O(ε^(-3/2)) for finding (ε,√ε)-SOSP.
Method provides bounds for sparse PCA and nuclear norm problems.
problem Semidefinite optimization problems (SDOs).
method Cutting-plane method with focus on initial outer approximation as a second-order cone approximation.
result Method provides bound gaps of 0.5-6.5% for sparse PCA problems with 1000 covariates and solves nuclear norm problems over 500x500 matrices.
New findings show second-order scoring rules can't accurately represent epistemic uncertainty.
problem Lack of epistemic uncertainty representation in second-order learners.
method Generalised second-order scoring rules introduced to prove theoretical limitations.
result No loss function incentivizes second-order learners to accurately represent epistemic uncertainty.
Method studies equivalence of second order ODEs under specific transformations.
problem Classifying second order ODEs modulo fibre-preserving transformations.
method Using Moser's method of normal forms and Lie algebra computations.
result Normal forms can be used to prove fibre-preserving equivalence.
Paper introduces FoMoH for optimization without backpropagation.
problem Optimizing machine learning models without backpropagation.
method Second-order hyperplane search, forward-mode stochastic gradient method, hyper-dual numbers, FoMoH.
result Developed a novel optimization algorithm that avoids backpropagation.
Enhances SMC² with Hessian info for more efficient posterior approximation.
problem Improving accuracy and efficiency in Bayesian inference.
method Integrates second-order information (Hessian) into SMC²'s proposal distribution.
result Second-order proposals lead to more accurate posterior approximations and better step-size selection.
ADAHESSIAN optimizes machine learning models with adaptive second-order methods.
problem Efficiently optimizing machine learning models with second-order methods.
method Dynamic Hessian estimation via adaptive estimates, incorporating fast approximations and moving averages.
result ADAHESSIAN achieves state-of-the-art performance across various tasks.
Two new algorithms improve federated optimization under second-order similarity.
problem Federated learning under communication constraints and second-order similarity.
method SVRP and Catalyzed SVRP algorithms combining proximal point evaluations, client sampling, and variance reduction.
result Achieves superior performance and uniformly improves upon existing algorithms for federated optimization under second-order similarity and strong convexity.
Stochastic second-order methods converge fast under interpolation conditions.
problem Minimizing smooth and strongly-convex functions efficiently.
method Regularized subsampled Newton method (R-SSN) and stochastic BFGS algorithms.
result R-SSN achieves global linear convergence and quadratic rate in a local neighbourhood.
NMC improves MCMC convergence by analyzing gradients to determine optimal proposal densities.
problem Improving MCMC convergence in structured relational models.
method Newtonian Monte Carlo (NMC) uses first and second order gradients to determine a suitable proposal density.
result NMC outperforms existing methods in various domains, including non-conjugate models.
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 superintegrable systems derived from Frobenius structures.
problem Constructing second-order superintegrable systems.
method Using conification and direct product construction, applying to semi-simple and nilpotent algebras.
result Explicitly constructed second-order superintegrable systems in three dimensions.
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.
A new method improves ICA performance by approximating MDI.
problem Improving F astICA's performance with nonlinear functions.
method Second-order approximation of MDI for joint maximization.
result Efficiency validated through experiments compared to other ICA algorithms.
Improved SVRG method using BB techniques for faster convergence.
problem Improving the convergence speed of stochastic variance reduction methods.
method Incorporates Barzilai-Borwein (BB) techniques as second-order information into SVRG.
result Proves linear convergence of the proposed method and its variants.
New method solves stochastic optimization problems with random models.
problem Optimizing stochastic objectives with deterministic constraints.
method Trust-Region Sequential Quadratic Programming with random model.
result Global convergence guarantees for first- and second-order stationary points.
New distributed algorithm for second-order optimization with reduced Hessian computation.
problem Efficiently using second-order methods in distributed optimization with limited communication.
method Distributed training of generalized linear models with adaptive trust-region approach.
result Achieves state-of-the-art results on multiple large datasets.
Stein variational Newton method accelerates SVGD for faster inference.
problem Efficient nonparametric variational inference.
method Accelerates SVGD by incorporating second-order information and optimal kernel selection.
result Significant computational gains over original SVGD in multiple test cases.