Nonconvex optimization algorithms with random initialization have attracted increasing attention recently. It has been showed that many first-order methods always avoid saddle points with random starting points. In this paper, we answer a question: can the nonconvex heavy-ball algorithms with random initialization avoi…
On-device research index
arXiv research
A locally-built, LLM-digested index of recent arXiv papers in quant finance, geometry/topology, and statistical ML — keyword search served straight from SQLite on this machine.
168,695 papers · 148 categories
Trend · papers per month
48 results for “Heavy Ball”
Heavy Ball method speeds up finding global optima in non-convex problems.
problem Finding global optima in non-convex optimization problems.
method Heavy Ball momentum in non-convex optimization.
result Heavy Ball helps iterates enter a benign region faster, containing a global optimal point.
Stochastic Heavy Ballmath.ST
This paper deals with a natural stochastic optimization procedure derived from the so-called Heavy-ball method differential equation, which was introduced by Polyak in the 1960s with his seminal contribution [Pol64]. The Heavy-ball method is a second-order dynamics that was investigated to minimize convex functions f .…
In this work we establish the first linear convergence result for the stochastic heavy ball method. The method performs SGD steps with a fixed stepsize, amended by a heavy ball momentum term. In the analysis, we focus on minimizing the expected loss and not on finite-sum minimization, which is typically a much harder p…
New method shows stochastic momentum can converge quickly on optimization problems.
problem Improving convergence of stochastic optimization methods.
method Stochastic heavy ball momentum with minibatching.
result Stochastic heavy ball momentum retains fast linear rate on quadratic problems.
In this paper, we revisit the convergence of the Heavy-ball method, and present improved convergence complexity results in the convex setting. We provide the first non-ergodic O(1/k) rate result of the Heavy-ball algorithm with constant step size for coercive objective functions. For objective functions satisfying a re…
Arguably, the two most popular accelerated or momentum-based optimization methods in machine learning are Nesterov's accelerated gradient and Polyaks's heavy ball, both corresponding to different discretizations of a particular second order differential equation with friction. Such connections with continuous-time dyna…
This paper analyzes two Lie group momentum optimization algorithms and their convergence rates.
problem Optimizing functions on Lie groups using momentum-based dynamics.
method Investigates Lie Heavy-Ball and Lie NAG-SC algorithms, quantifying their convergence rates under smoothness and convexity assumptions.
result Lie NAG-SC accelerates optimization over the momentumless case, while Lie Heavy-Ball does not.
Develops a computationally tractable differentially private mean estimator called the balloon mean.
problem Robust mean estimation in the presence of outliers and heavy-tailed distributions.
method Iterative clipping procedure over Mahalanobis balls.
result Balloon mean is robust to outliers and outperforms existing estimators in contaminated settings.
Gradient-based optimization algorithms can be studied from the perspective of limiting ordinary differential equations (ODEs). Motivated by the fact that existing ODEs do not distinguish between two fundamentally different algorithms---Nesterov's accelerated gradient method for strongly convex functions (NAG-SC) and Po…
Mean-field analysis for heavy ball methods: Dropout-stability, connectivity, and global convergencecs.LG
Paper analyzes SHB method for neural networks, proving stability, connectivity, and global convergence.
problem Theoretical understanding of SHB method for neural networks.
method Mean-field analysis of SHB dynamics related to a partial differential equation.
result SHB method converges to global optimum and exhibits stability and connectivity.
The paper analyzes convergence rates for SGD and SHB methods.
problem Analyzing convergence rates for stochastic gradient descent and heavy ball methods.
method Stochastic gradient descent and stochastic heavy ball method for general stochastic approximation problems.
result The last iterate of SHB converges almost surely to a minimizer and has faster convergence rates than SGD.
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.
We obtain sharp bounds on the performance of Empirical Risk Minimization performed in a convex class and with respect to the squared loss, without assuming that class members and the target are bounded functions or have rapidly decaying tails. Rather than resorting to a concentration-based argument, the method used her…
Two new differentially private optimization algorithms derived from accelerated methods.
problem Improving privacy in optimization algorithms while maintaining convergence rates.
method Polyak's heavy ball method and Nesterov's accelerated gradient method with differential privacy.
result The proposed algorithms outperform existing differentially private optimization methods.
Proposes momentum methods for Lie groups, improving on classical algorithms.
problem Optimization on nonlinear spaces, especially Lie groups.
method Generalizes Nesterov's Accelerated Gradient method to Lie groups.
result Demonstrates faster convergence for NAG-like methods on Lie groups.
Analytical Study of Momentum-Based Acceleration Methods in Paradigmatic High-Dimensional Non-Convex Problemscond-mat.dis-nn
Study accelerates optimization methods in non-convex problems, but doesn't improve the algorithm's performance.
problem Understanding the behavior of momentum-based acceleration methods in non-convex, high-dimensional landscapes.
method Used dynamical mean field theory to describe the average dynamics of heavy-ball momentum and Nesterov acceleration in a non-convex model.
result Accelerated dynamics but did not improve the algorithm's performance with respect to gradient descent.
Convergence of the Stochastic Heavy Ball Method With Approximate Gradients and/or Block Updatingmath.OC
Paper proves SHB convergence with biased gradients and approximate step sizes.
problem Establishing convergence of SHB with biased gradients and approximate step sizes.
method Generalizes SHB convergence conditions for biased gradients, approximate step sizes, and block updating.
result Proves convergence of SHB with new conditions for biased gradients and approximate step sizes.
Optimizes privacy-preserving optimization for heavy-tailed data.
problem Privacy-preserving optimization with heavy-tailed gradients.
method Pure ε-differential privacy framework for Lipschitz extensions.
result Minimax optimal excess-risk rate for pure ε-DP heavy-tailed SCO.
Optimizes web page freshness with limited crawling frequencies.
problem Maximize local cache freshness given crawling frequency constraints.
method Three novel online estimation schemes for page change rates.
result Convergent algorithms for estimating page change rates.
Improved SHB method for faster convergence on strongly-convex quadratics.
problem Understanding and improving the theoretical and practical advantages of SHB.
method Noise-adaptive multi-stage algorithm for SHB with accelerated convergence.
result SHB can achieve accelerated convergence with larger mini-batch sizes.
A novel decentralized deep learning algorithm using gradient-based optimization.
problem Decentralized deep learning in networked systems without a central server.
method Heavy-ball acceleration method and consensus protocol for model and gradient-momentum sharing.
result The proposed algorithm outperforms competing methods in various communication topologies.
Fine-grained analysis of gradient descent with momentum provides modified loss equations.
problem Understanding the dynamics of gradient descent with momentum.
method Fine-grained analysis and derivation of modified loss equations.
result Global approximation bounds and continuous modified equations for HB.
Unified Convergence Analysis of Stochastic Momentum Methods for Convex and Non-convex Optimizationmath.OC
Recently, {\it stochastic momentum} methods have been widely adopted in training deep neural networks. However, their convergence analysis is still underexplored at the moment, in particular for non-convex optimization. This paper fills the gap between practice and theory by developing a basic convergence analysis of t…
Optimization algorithms help overparameterized neural networks achieve high performance.
problem Understanding the convergence of optimization algorithms on overparameterized neural networks.
method Analyzing a broad class of optimization algorithms using dynamical systems and finite over-parameterized neural networks with ReLU activation.
result The Heavy Ball method converges to global minimum at a linear rate, while NAG converges sublinearly.
In this paper we study several classes of stochastic optimization algorithms enriched with heavy ball momentum. Among the methods studied are: stochastic gradient descent, stochastic Newton, stochastic proximal point and stochastic dual subspace ascent. This is the first time momentum variants of several of these metho…
A method for estimating the median of gradients in stochastic optimization.
problem Robust gradient estimation in stochastic optimization for various applications.
method Stochastic Proximal Point Method for median gradient estimation.
result The proposed method can converge even under heavy-tailed, state-dependent noise.
In the presence of model risk, it is well-established to replace classical expected values by worst-case expectations over all models within a fixed radius from a given reference model. This is the "robustness" approach. We show that previous methods for measuring this radius, e.g. relative entropy or polynomial diverg…
We study first-order optimization methods obtained by discretizing ordinary differential equations (ODEs) corresponding to Nesterov's accelerated gradient methods (NAGs) and Polyak's heavy-ball method. We consider three discretization schemes: an explicit Euler scheme, an implicit Euler scheme, and a symplectic scheme.…
Paper tackles DP-SCO with heavy-tailed data in high dimensions, improving error bounds.
problem Differentially private stochastic optimization with heavy-tailed data in high-dimensional spaces.
method Proposes methods for DP-SCO with polytope constraints and LASSO, analyzing sparsity constraints.
result Achieved near optimal error bounds for DP-SCO with heavy-tailed data.
Improved regret bounds for linear bandits with heavy-tailed rewards.
problem Stochastic linear bandits with heavy-tailed rewards.
method Elimination-based algorithm guided by experimental design.
result Regret bound of \(\tilde{\mathcal{O}}(d^\frac{1+3ε}{2(1+ε)} T^\frac{1}{1+ε})\) for \(ε\in (0,1)\).
We study learning properties of accelerated gradient descent methods for linear least-squares in Hilbert spaces. We analyze the implicit regularization properties of Nesterov acceleration and a variant of heavy-ball in terms of corresponding learning error bounds. Our results show that acceleration can provides faster …
We take a Hamiltonian-based perspective to generalize Nesterov's accelerated gradient descent and Polyak's heavy ball method to a broad class of momentum methods in the setting of (possibly) constrained minimization in Euclidean and non-Euclidean normed vector spaces. Our perspective leads to a generic and unifying non…
The paper analyzes dynamics of momentum in high dimensions with sparse updates.
problem Theoretical analysis of momentum dynamics in high-dimensional sparse settings.
method Theoretical analysis of two models: least squares with sparse inputs and logistic regression with a rare class.
result Characterization of high-dimensional limits of momentum dynamics and phase structure.
Study accelerates gradient methods in machine learning, revealing risk and stability connections.
problem Understanding the statistical risk of accelerated gradient methods in machine learning.
method Continuous-time analysis of Nesterov's accelerated gradient method and Polyak's heavy ball method for least squares regression.
result Connections between early stopping, stability, and curvature of loss function are revealed.
We show that accelerated gradient descent, averaged gradient descent and the heavy-ball method for non-strongly-convex problems may be reformulated as constant parameter second-order difference equation algorithms, where stability of the system is equivalent to convergence at rate O(1/n 2), where n is the number of ite…
The use of momentum in stochastic gradient methods has become a widespread practice in machine learning. Different variants of momentum, including heavy-ball momentum, Nesterov's accelerated gradient (NAG), and quasi-hyperbolic momentum (QHM), have demonstrated success on various tasks. Despite these empirical successe…
Adaptive gradient methods such as Adam have been shown to be very effective for training deep neural networks (DNNs) by tracking the second moment of gradients to compute the individual learning rates. Differently from existing methods, we make use of the most recent first moment of gradients to compute the individual …
Least squares estimator fails to achieve optimal risk in bounded distributions, but non-linear predictors can.
problem Optimal risk in bounded distributions for constrained least squares.
method Comparison of least squares and non-linear predictors.
result Non-linear predictors can achieve optimal risk in bounded distributions.
New algorithms handle heavy-tailed rewards in reinforcement learning.
problem Learning from heavy-tailed rewards in reinforcement learning.
method Robust mean estimation techniques for constructing algorithms.
result Near-optimal regret bounds achieved in heavy-tailed reward settings.
The vast majority of successful deep neural networks are trained using variants of stochastic gradient descent (SGD) algorithms. Recent attempts to improve SGD can be broadly categorized into two approaches: (1) adaptive learning rate schemes, such as AdaGrad and Adam, and (2) accelerated schemes, such as heavy-ball an…
Study improves ERM for heavy-tailed data with dependent inputs.
problem Empirical Risk Minimization with dependent and heavy-tailed data.
method Extending risk bounds for ERM with heavy-tailed, dependent data.
result Established risk bounds for ERM with dependent and heavy-tailed data.
Heavy Lasso improves robustness in high-dimensional linear regression with heavy-tailed errors.
problem Challenges of classical Lasso in handling heavy-tailed noise and outliers.
method Data-augmented soft-thresholding with Student's t-distribution loss.
result Heavy Lasso achieves comparable rates to Huber loss under theoretical bounds.
Sparse Linear Regression when Noises and Covariates are Heavy-Tailed and Contaminated by Outliersstat.ML
Efficiently estimates sparse linear regression with heavy-tailed and outlier-contaminated data.
problem Estimating sparse linear regression coefficients with heavy-tailed and outlier-contaminated data.
method Efficient computation of estimators with sharp error bounds.
result Sharp error bounds for efficient estimators.
Is AdamW effective under heavy-tailed noise?
problem Stochastic gradient noise in LLM pretraining is typically heavy-tailed.
method Formulate as an open problem, prove a positive weighted-metric benchmark, and give a corridor lower-bound mechanism.
result No rigorous convergence theory for AdamW established in heavy-tailed regime.
Robust CG methods avoid data corruption and solve structured statistical estimation problems.
problem Data corruption and heavy-tailed data in structured statistical estimation.
method Robustification of Conditional Gradient (CG) type methods using Huber's corruption model and robust mean gradient estimation.
result Robust CG methods converge linearly with correct sample complexity, even for high-dimensional problems.
Optimizes convex functions in finite vs infinite dimensions, revealing slow convergence rates.
problem Analyzing gradient flows in finite and infinite-dimensional Hilbert spaces.
method Proves convergence rates and optimality conditions for gradient flows and related methods.
result Gradient flow convergence rates in finite dimensions are slower than in infinite dimensions, with optimal rates achievable in Hilbert spaces.
Self-regulating annealing improves sampling from heavy-tailed datasets.
problem Sampling from heavy-tailed distributions using diffusion models.
method Proposed an SDE-based sampler with a state-dependent diffusion coefficient.
result State dependence induces a self-regulating annealing mechanism.