Accelerates coordinate descent methods for machine learning problems.
problem Slowness of coordinate descent methods in machine learning.
method Extrapolation-based accelerated coordinate descent.
result Significant speed-up in practice compared to existing methods.
New analysis shows GMD can converge linearly under PL-like conditions.
problem Establishing linear convergence for generalized mirror descent.
method PL-based analysis for time-dependent mirrors, Taylor-series approach for stochastic GMD.
result Linear convergence of stochastic GMD under PL-like conditions.
New adaptive step-size method for convex optimization without tuning.
problem Optimizing convex functions efficiently with stochastic gradients.
method Adapted Adaptive Gradient Descent Without Descent to stochastic setting.
result Stochastic gradient descent converges under various assumptions.
A new method improves stochastic gradient descent for faster and more efficient estimation.
problem Efficient and fast parametric estimation methods.
method Projected stochastic gradient descent corrected by Fisher scoring.
result The method is faster and more efficient than traditional methods.
Gravilon improves gradient descent for neural networks.
problem Improving efficiency and accuracy of gradient descent methods.
method Uses geometric modification of gradient step lengths.
result Promising experimental results on MNIST classification.
Blind Descent avoids gradient issues, using a different learning approach.
problem Gradient issues like exploding and vanishing gradients.
method Does not use gradients to guide learning; instead, it is a more fundamental learning process.
result Gradient descent is a specific case of Blind Descent.
Information geometry applies concepts in differential geometry to probability and statistics and is especially useful for parameter estimation in exponential families where parameters are known to lie on a Riemannian manifold. Connections between the geometric properties of the induced manifold and statistical properti…
Gradient descent methods for deep ReLU networks achieve optimal generalization rates.
problem Generalization of gradient descent methods for deep neural networks
method Establishing minimax-optimal rates for GD and SGD with deep ReLU networks
result Gradient descent methods for deep ReLU networks achieve optimal generalization rates
Paper proposes a pre-conditioning technique to speed up gradient-descent convergence in distributed linear least-squares problems.
problem Expediting convergence of gradient-descent method for ill-conditioned distributed linear least-squares problems.
method Iterative pre-conditioning technique to improve convergence rate of gradient-descent method.
result Pre-conditioned gradient-descent achieves superlinear convergence for unique solutions and improved linear convergence otherwise.
Differentially private random block coordinate descent improves utility in machine learning.
problem Lack of privacy in classical CD methods when handling sensitive information.
method Proposes a differentially private random block coordinate descent method using sketch matrices and importance sampling.
result Demonstrates improved convergence rates and utility guarantees compared to non-private methods.
Any gradient descent optimization requires to choose a learning rate. With deeper and deeper models, tuning that learning rate can easily become tedious and does not necessarily lead to an ideal convergence. We propose a variation of the gradient descent algorithm in the which the learning rate is not fixed. Instead, w…
This paper explores a new framework for reinforcement learning based on online convex optimization, in particular mirror descent and related algorithms. Mirror descent can be viewed as an enhanced gradient method, particularly suited to minimization of convex functions in highdimensional spaces. Unlike traditional grad…
Derives Mirror Descent from gradient flow on a Riemannian manifold.
problem No specific problem stated; focuses on derivation.
method Derives Mirror Descent from gradient flow on a Riemannian manifold with a natural discretization.
result Generalizes Mirror Descent to non-Hessian metrics.
Gradient descent stagnates in low-precision, but unbiased rounding schemes improve convergence.
problem Stagnation of gradient descent in low-precision computation.
method Proposed unbiased stochastic rounding schemes that trade zero bias for larger probability of preserving small gradients.
result Unbiased rounding methods typically improve convergence rate of gradient descent for convex problems.
In this paper, we provide an overview of first-order and second-order variants of the gradient descent method that are commonly used in machine learning. We propose a general framework in which 6 of these variants can be interpreted as different instances of the same approach. They are the vanilla gradient descent, the…
Efficiently performs robust and sparse kernel regression.
problem Robust and sparse kernel regression.
method Sign gradient descent and early stopping.
result Sign gradient descent achieves robust and sparse kernel regression efficiently.
Explains gradient descent methods and their convergence, focusing on simple analysis.
problem Understanding and analyzing gradient descent methods and their variants.
method Elementary mathematical analysis focusing on structures and assumptions of objective functions.
result Unified convergence analysis of various gradient descent methods and variants.
Gradient descent outperforms ridge regression under certain covariance matrix decay conditions.
problem Comparing the performance of gradient descent and ridge regression in linear models.
method Investigated gradient descent and ridge regression for linear regression with random isotropic ground truth.
result Gradient descent outperforms ridge regression under specific covariance matrix decay conditions.
We propose a new randomized coordinate descent method for a convex optimization template with broad applications. Our analysis relies on a novel combination of four ideas applied to the primal-dual gap function: smoothing, acceleration, homotopy, and coordinate descent with non-uniform sampling. As a result, our method…
Unified signSGD and gradient descent analysis for neural networks.
problem Performance of sign-based optimization methods in neural networks.
method Unified analysis of separable smoothness and ℓ∞-smoothness, isolating geometric properties affecting performance. result Sign-based methods are preferable over gradient descent under specific Hessian properties in deep networks.
Gradient descent benefits from tangent kernel advantages under specific conditions.
problem Comparing gradient descent with tangent kernel methods in learning.
method Analysis of gradient descent and tangent kernel methods under different conditions.
result Gradient descent can achieve small error only if tangent kernel methods have a non-trivial advantage, but this advantage can be very small.
LES optimizes designs by sampling descent sequences, achieving strong sample efficiency.
problem Optimizing large, complex design spaces is infeasible and unnecessary.
method LES uses Bayesian optimization to target solutions reachable by iterative optimizers.
result LES achieves strong sample efficiency compared to existing methods.
This paper extends Mirror Descent to Riemannian manifolds for optimization.
problem Optimization on Riemannian manifolds.
method Developed a Riemannian Mirror Descent (RMD) framework and a stochastic variant.
result Established non-asymptotic convergence guarantees for RMD and stochastic RMD.
This paper investigates different vector step-size adaptation approaches for non-stationary online, continual prediction problems. Vanilla stochastic gradient descent can be considerably improved by scaling the update with a vector of appropriately chosen step-sizes. Many methods, including AdaGrad, RMSProp, and AMSGra…
Developing a singular dimension descent method for positive scalar curvature obstructions
problem Positive scalar curvature obstructions in arbitrary dimensions
method Schoen--Yau type singular dimension descent method
result Proving obstructions to positive scalar curvature on enlargeable manifolds
In this paper we generalize the framework of the feasible descent method (FDM) to a randomized (R-FDM) and a coordinate-wise random feasible descent method (RC-FDM) framework. We show that the famous SDCA algorithm for optimizing the SVM dual problem, or the stochastic coordinate descent method for the LASSO problem, f…
New method improves accuracy in computing implied volatility.
problem Computing implied volatility from the Black-Scholes model.
method Adaptive gradient descent optimizers for numerical computation.
result More accurate results compared to close form approximation and Newton-Raphson method.
Study shows how steepest descent algorithms' geometric margin increases during training.
problem Understanding implicit bias in steepest descent algorithms for neural networks.
method Analysis of steepest descent algorithms with infinitesimal learning rates in homogeneous neural networks.
result Limit points of training trajectories correspond to KKT points of margin-maximization problems.
Stochastic gradient descent outperforms traditional force-directed methods.
problem Improving graph layout quality and efficiency.
method Applying stochastic gradient descent for stress minimization.
result Stochastic gradient descent is simpler and more robust than traditional methods.
New method improves scalability of SGD for large datasets.
problem High variance in stochastic gradient descent.
method Adaptive measure reduction with Carathéodory's theorem.
result Improved scalability to high-dimensional spaces.
This work proposes a new method for variational inference using Wasserstein gradient descent.
problem Optimizing variational parameters to match a true posterior distribution.
method Reinterpreting VI as an optimization problem over a variational parameter space, using Wasserstein gradient descent.
result The proposed Wasserstein gradient descent can be seen as a generalization of existing optimization techniques in VI.
First-order methods play a central role in large-scale machine learning. Even though many variations exist, each suited to a particular problem, almost all such methods fundamentally rely on two types of algorithmic steps: gradient descent, which yields primal progress, and mirror descent, which yields dual progress. W…
Coordinate descent methods usually minimize a cost function by updating a random decision variable (corresponding to one coordinate) at a time. Ideally, we would update the decision variable that yields the largest decrease in the cost function. However, finding this coordinate would require checking all of them, which…
AutoGD automatically adjusts learning rates for gradient descent.
problem Optimizing learning rates for gradient descent methods.
method AutoGD automatically adjusts learning rates based on iteration.
result AutoGD can recover the optimal rate of GD for a broad class of functions.
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.
Stochastic gradient descent improves Gaussian process regression.
problem Efficiently solving large linear systems in Gaussian process regression.
method Developed a stochastic dual descent algorithm using insights from optimisation and kernel communities.
result Stochastic gradient descent is highly effective when done right.
Frank-Wolfe optimization applied to a small deep network shows slower convergence compared to gradient descent.
problem Training deep neural networks is challenging due to many parameters and optimization difficulties.
method Frank-Wolfe optimization method applied to a deep network.
result Frank-Wolfe optimization converges slowly and is unstable in a stochastic setting.
Paper uses Mirror Descent for efficient risk budgeting portfolios.
problem Computing optimal risk budgeting weights for various risk measures.
method Employed Mirror Descent algorithms in deterministic and stochastic settings.
result Established convergence and quantitative rate for averaged Mirror Descent algorithm.
Paper proposes a pre-conditioning method to speed up gradient descent in multi-agent optimization.
problem Speed up convergence of gradient descent in multi-agent optimization problems.
method Iterative pre-conditioning approach to mitigate the effect of problem conditioning.
result Significant improvement in convergence speed of gradient descent method.
We establish that first-order methods avoid saddle points for almost all initializations. Our results apply to a wide variety of first-order methods, including gradient descent, block coordinate descent, mirror descent and variants thereof. The connecting thread is that such algorithms can be studied from a dynamical s…
New method escapes local optima in neural architecture optimization.
problem Escaping local optima in neural architecture optimization.
method Signed neural splitting in steepest descent framework.
result Escapes local optima, leading to better performance.
Recently several methods were proposed for sparse optimization which make careful use of second-order information [10, 28, 16, 3] to improve local convergence rates. These methods construct a composite quadratic approximation using Hessian information, optimize this approximation using a first-order method, such as coo…
We demonstrate that distributed block coordinate descent can quickly solve kernel regression and classification problems with millions of data points. Armed with this capability, we conduct a thorough comparison between the full kernel, the Nyström method, and random features on three large classification tasks from va…
Develops DP-SCD for stochastic coordinate descent, making it differentially private.
problem Privacy leak in auxiliary information during stochastic coordinate descent training.
method Develops DP-SCD, leveraging independent noise addition and decoupling/parallelizing coordinate updates.
result Demonstrates competitive performance against DP-SGD with less tuning.
CBO interprets as SGD, leading to global convergence for nonconvex functions.
problem Understanding and improving gradient-based learning algorithms.
method Interpreting CBO as a stochastic relaxation of SGD.
result CBO provably converges globally to minimizers for nonsmooth nonconvex functions.
Quantum models show improved performance in overparameterized regimes.
problem Overfitting in quantum machine learning models.
method Analytical demonstration and numerical experiments on quantum kernel methods.
result Quantum models can operate in the modern, overparameterized regime without overfitting.
This paper proves AdaGrad and Adam converge linearly under PL inequality.
problem Understanding the convergence of adaptive gradient methods.
method Unified approach proving AdaGrad and Adam converge linearly under PL inequality.
result AdaGrad and Adam converge linearly when the cost function is smooth and satisfies PL inequality.
Stochastic gradient descent (\textsc{Sgd}) methods are the most powerful optimization tools in training machine learning and deep learning models. Moreover, acceleration (a.k.a. momentum) methods and diagonal scaling (a.k.a. adaptive gradient) methods are the two main techniques to improve the slow convergence of \text…