Sparse Polyak improves high-dimensional statistical estimation.
problem High-dimensional statistical estimation problems with growing problem dimension.
method Sparse Polyak modifies Polyak's adaptive step size to estimate restricted Lipschitz smoothness.
result Sparse Polyak achieves optimal statistical precision with fewer iterations.
Proposes a new adaptive learning rate for SGD.
problem Finding an efficient learning rate for SGD.
method Stochastic Polyak step-size (SPS).
result SGD with SPS converges faster for over-parameterized models.
Polyak step size GD reaches final radius of convergence after log iterations.
problem Statistical and computational complexities of Polyak step size GD.
method Generalized smoothness and Lojasiewicz conditions, stability of gradients.
result Polyak step size GD reaches final statistical radius of convergence after logarithmic number of iterations.
New step-size methods improve SHB convergence for stochastic optimization.
problem Tuning step-size and momentum parameters in SHB is challenging.
method Proposed MomSPS max _{\max} m a x , MomDecSPS, and MomAdaSPS for SHB. result Convergence guarantees for SHB to solution neighborhoods and exact minimizers.
New adaptive scheduler improves SAM for better model training.
problem Training machine learning models requires selecting a learning rate, which is often difficult and time-consuming.
method Derive Polyak schedulers tailored to SAM-style updates, proving linear convergence for strongly convex objectives and an O(1/T) rate for convex objectives.
result Polyak schedulers achieve comparable or better performance than tuned SAM baselines, reducing the need for learning-rate tuning.
New SPS variant improves non-smooth optimization without small gradients.
problem Improving non-smooth optimization without small gradients.
method Safeguarded Stochastic Polyak Step Size (SPS s a f e _{safe} s a f e ) for non-smooth optimization. result Rigorous convergence guarantees for non-smooth convex optimization without strong assumptions.
Improved Sparse Polyak for high-dimensional M-estimation with sparser solutions.
problem High-dimensional M-estimation problems with potential loss of sparsity and accuracy.
method Variant of Sparse Polyak with optimal thresholding operators.
result Retains desirable scaling properties while achieving sparser and more accurate solutions.
New insights into SGD and SGD-M in high dimensions.
problem Understanding and comparing SGD and SGD-M in high-dimensional settings.
method Developed high-dimensional scaling limits for SGD-M and online SGD, examining their dynamics and performance.
result SGD-M amplifies high-dimensional effects, potentially degrading performance compared to online SGD.
Develops a generalized version of Chung's Lemma for stochastic optimization methods.
problem Establishing asymptotic convergence rates for stochastic optimization methods under various step size rules.
method Generalized version of Chung's Lemma for a broader family of step size rules.
result Demonstrates tight non-asymptotic convergence rates for various stochastic methods.
ProxSPS improves on SPS for regularization tasks, offering better stability and performance.
problem Handling regularization terms in adaptive step size schemes for stochastic gradient descent.
method Developed a proximal variant of the stochastic Polyak step size (SPS) scheme.
result ProxSPS is easier to tune and more stable with regularization, and performs well in image classification tasks.
Paper improves CLT and bootstrap approximations for LSA with decreasing step size.
problem Improving normal approximation and bootstrap methods for LSA with decreasing step sizes.
method Refined Berry-Esseen bounds and multiplier bootstrap procedure for LSA.
result Approximation rates up to 1 / n 1/\sqrt{n} 1/ n for LSA rescaled error distribution. Study on stochastic approximation with Polyak-Ruppert averaging for linear systems.
problem Understanding the asymptotic and non-asymptotic properties of stochastic approximation procedures.
method Detailed analysis of linear stochastic approximation with Polyak-Ruppert averaging, focusing on asymptotic and non-asymptotic properties.
result Proves CLT and non-asymptotic concentration inequality for averaged iterates, providing refined understanding of linear stochastic approximation.
Gradient descent converges linearly for overparameterized linear networks.
problem Convergence of gradient descent for overparameterized neural networks.
method Local Polyak-Lojasiewicz and Descent Lemma for overparameterized linear models.
result Gradient descent achieves linear convergence for two-layer linear networks under relaxed assumptions.
The paper offers precise bounds for averaged LSA iterates in linear systems.
problem Computing approximate solutions of linear systems with noisy observations.
method Finite-time analysis of LSA algorithms with Polyak-Ruppert averaging.
result Sharp high-probability bounds for averaged LSA iterates.
Characterizes corridors in loss surfaces for gradient-based optimization.
problem Understanding and mitigating training instabilities in gradient-based optimization.
method Characterizes corridors as regions where gradient descent and gradient flow trajectories are linearly related.
result Corridors indicate regions without implicit regularization effects, leading to better learning rate adaptation schemes.
The paper analyzes and validates two step size schedules for SGD: exponential and cosine, proving their adaptivity and performance.
problem The variability of SGD performance due to step size choice.
method Analysis and empirical evaluation of exponential and cosine step sizes.
result Exponential and cosine step sizes are adaptive to noise and achieve optimal performance without tuning hyperparameters.
Paper improves confidence intervals for LSA with multiplier bootstrap.
problem Improving confidence intervals for parameter estimation in LSA.
method Berry-Esseen bound for multivariate normal approximation and multiplier bootstrap.
result Valid confidence intervals for parameter estimation in LSA.
Quantized Stochastic Primal-Dual Methods for Distributed Optimization
problem Distributed optimization with stochastic gradients and finite-bit communication
method q-PDGD, a quantized stochastic primal-dual method
result Linear contraction to an explicit neighborhood under RSI, O(1/k) convergence under PL inequality
Study Q-learning with averaging for reinforcement learning, proving efficient inference and error bounds.
problem Efficient inference and error bounds for Q-learning with averaging.
method Functional central limit theorem and asymptotic linear estimator for optimal Q-value function.
result Standardized partial-sum process converges weakly to a rescaled Brownian motion, matching instance-dependent lower bound for error.
New findings show convergence in SA without square summable step sizes.
problem Finding optimal step sizes for stochastic approximation algorithms.
method Analyzed step-size sequences of the form α n = α 0 n − ρ α_n = α_0 n^{-ρ} α n = α 0 n − ρ with ρ ∈ ( 0 , 1 ) ρ\in (0,1) ρ ∈ ( 0 , 1 ) , and derived convergence and rate results. result Convergence of parameter estimates with probability one and in L p L_p L p for any p ≥ 1 p\ge 1 p ≥ 1 ; MSE rate of convergence is O ( α n ) O(α_n) O ( α n ) , improved to O ( max { α n 2 , 1 / n } ) O(\max\{ α_n^2,1/n \}) O ( max { α n 2 , 1/ n }) with averaging. We consider d d d -dimensional linear stochastic approximation algorithms (LSAs) with a constant step-size and the so called Polyak-Ruppert (PR) averaging of iterates. LSAs are widely applied in machine learning and reinforcement learning (RL), where the aim is to compute an appropriate θ ∗ ∈ R d θ_{*} \in \mathbb{R}^d θ ∗ ∈ R d (that is a…
Adaptive gradient methods converge faster with over-parameterization and line-search.
problem Training over-parameterized models using adaptive gradient methods.
method Simplified setting of smooth, convex losses with over-parameterized models, proving convergence rates and demonstrating improvements with line-search techniques.
result Adaptive gradient methods, particularly AMSGrad, converge faster with line-search techniques.
In 1963, Polyak proposed a simple condition that is sufficient to show a global linear convergence rate for gradient descent. This condition is a special case of the Łojasiewicz inequality proposed in the same year, and it does not require strong convexity (or even convexity). In this work, we show that this much-older…
Improved convergence analysis for decentralized non-convex optimization.
problem Minimizing a sum of smooth non-convex functions over a network.
method Gradient tracking in decentralized stochastic gradient descent (GT-DSGD).
result GT-DSGD achieves network-independent performances matching centralized SGD under certain conditions.
SGD converges to global minimum for structured non-convex functions.
problem Optimizing non-convex functions using SGD with slow convergence rates.
method Convergence theorems for SGD on structured non-convex functions, including Quasar and PL conditions.
result SGD converges to global minimum for specific non-convex functions under certain conditions.
A new algorithm reduces the time for ordinal embedding, making it faster and more scalable.
problem Efficiently learning representations from ordinal comparisons, especially for large datasets.
method SVRG-SBB: Stochastic variance reduced gradient with adaptive step size.
result Achieves $O( rac{1}{T})$ convergence rate and global linear convergence under certain assumptions.
Self-attention optimizers converge to optimal weights, revealing bias patterns.
problem Understanding the implicit bias in self-attention mechanisms.
method Analysis of gradient-based optimization in self-attention layers.
result Adaptive step-size strategies can accelerate convergence to optimal weights.
New SGDA method speeds up nonconvex minimax optimization.
problem Improving convergence of nonconvex minimax optimization.
method SGDA with random reshuffling for nonconvex-PŁ objectives.
result Convergence rates faster than with-replacement SGDA.
New algorithm adapts to optimize non-convex problems efficiently.
problem Inefficient hyperparameter tuning in existing optimization methods.
method Combining geometrization and SARAH algorithms.
result Achieves adaptivity to both accuracy and PL constant.
Stagewise training strategy is widely used for learning neural networks, which runs a stochastic algorithm (e.g., SGD) starting with a relatively large step size (aka learning rate) and geometrically decreasing the step size after a number of iterations. It has been observed that the stagewise SGD has much faster conve…
This work provides formal guarantees for heuristic optimization methods in machine learning.
problem Lack of theoretical understanding of heuristic optimization methods in machine learning.
method Analysis and formal guarantees for AdaGrad, SGD with exponential and cosine step sizes, and momentum methods.
result First formal guarantees for AdaGrad and SGD variants, including convergence and adaptivity to noise.
SGD converges to an invariant distribution with sub-Gaussian or sub-exponential properties.
problem Optimizing smooth and strongly convex objectives using SGD.
method Analysis through Markov chains, focusing on convergence and concentration properties.
result SGD iterates and their invariant limit distribution inherit sub-Gaussian or sub-exponential concentration properties.
Paper improves TD learning algorithm bounds with linear approx.
problem Sharp bounds for TD method performance in MDPs.
method Polyak-Ruppert averaging, universal step size, refined error bounds, stability of random matrices.
result Near-optimal variance and bias terms achieved.
The paper explores how dynamic preconditioning affects the CLT in online averaging.
problem When does dynamic preconditioning preserve the Polyak-Ruppert CLT?
method The authors decompose the averaged error and identify a stabilization-rate threshold for the CLT to hold.
result The CLT holds if the dynamic remainder vanishes in L 2 L^2 L 2 and the stabilization rate exceeds a threshold. AGD converges in polynomial iterations to optimal matrix factorization.
problem Matrix factorization optimization with alternating gradient descent.
method Alternating gradient descent with fixed step size, proving convergence in polynomial iterations.
result AGD reaches ε-optimal factorization in T iterations with high probability.
Improved convergence for Polyak steps with momentum in smooth convex optimization.
problem Optimizing smooth strongly convex functions with limited information.
method Polyak steps with momentum, derived for accelerated gradient method.
result Convergence guarantees for accelerated gradient method with Polyak steps and momentum.
Large over-parametrized models learned via stochastic gradient descent (SGD) methods have become a key element in modern machine learning. Although SGD methods are very effective in practice, most theoretical analyses of SGD suggest slower convergence than what is empirically observed. In our recent work [8] we analyze…
Gradient descent converges linearly in finite-width networks with positive NTK and compatible conditions.
problem Local convergence of gradient descent in finite-width networks.
method Positive Neural Tangent Kernel (NTK), local Polyak-Łojasiewicz inequality, fixed-step containment in Locally Quasi-Convex Region (LQCR).
result Linear convergence achieved under specific conditions.
The paper analyzes the performance of constant step-size stochastic approximation algorithms.
problem Approximating solutions to root finding problems in optimization and machine learning.
method Examines stochastic approximation algorithms with constant step-size, proving convergence and analyzing the limiting behavior of averaged estimates.
result The Polyak-Ruppert-style averaged estimates converge to the true solution with optimal covariance, providing insights for practitioners.
MSGD outperforms SGD in overparametrized settings with faster convergence rates.
problem Optimization of non-convex functions with momentum.
method Momentum Stochastic Gradient Descent (MSGD) with rigorous analysis.
result MSGD converges exponentially faster than SGD in overparametrized settings.
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.
Formulae for Vassiliev invariants derived from Kauffman polynomial.
problem Computing Vassiliev invariants from knot polynomials.
method State model of Kauffman polynomial, Gauss diagram identities, arrow diagram identities.
result Gauss diagram formulae for Vassiliev invariants of order 3.
Paper analyzes LSA algorithm bias and error bounds with RR extrapolation.
problem Analyzing bias and high-order error bounds of LSA with Markovian noise.
method Polyak-Ruppert averaging, linearization, Richardson-Romberg extrapolation.
result RR extrapolation effectively cancels the leading bias term.
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.
AGDA and variance-reduced methods solve nonconvex-nonconcave minimax problems globally and faster.
problem Solving nonconvex-nonconcave minimax problems in machine learning.
method Global convergence of AGDA and variance-reduced algorithms.
result AGDA and variance-reduced methods achieve global convergence and faster rates.
Polyak-Ruppert CLT for SA-Adam with momentum and non-convergent adaptive preconditioning
problem Adaptive optimizers combining momentum and non-convergent preconditioning
method Proving positive drift stability and a non-autonomous Polyak-Ruppert CLT for SA-Adam
result The iterate-marginal covariance is exactly the plain stochastic gradient descent (SGD) sandwich
The paper improves convergence for linear systems using entropic mirror descent with Polyak stepsizes.
problem Convergence analysis for linear systems with unbounded domain.
method Entropic mirror descent with Polyak stepsizes, sublinear and linear convergence results.
result Generalized convergence result for arbitrary convex functions.
Stochastic AUC maximization has garnered an increasing interest due to better fit to imbalanced data classification. However, existing works are limited to stochastic AUC maximization with a linear predictive model, which restricts its predictive power when dealing with extremely complex data. In this paper, we conside…