Schedule-free SGD is optimal for nonconvex optimization problems.
problem Nonconvex optimization in neural networks.
method Developed a general framework for online-to-nonconvex conversion, which converts schedule-free SGD into an effective nonconvex optimization algorithm.
result Schedule-free SGD achieves optimal iteration complexity for nonsmooth, nonconvex optimization problems.
PPGD solves nonconvex nonsmooth optimization problems without KL property.
problem Nonconvex and nonsmooth optimization problems in statistics and machine learning.
method Projective Proximal Gradient Descent (PPGD) for solving a class of nonconvex and nonsmooth problems.
result PPGD achieves a fast convergence rate of O(1/k^2) for k ≥ k_0.
Develops shuffling gradient-based methods for nonconvex-concave minimax optimization.
problem Nonconvex-concave minimax optimization problems.
method Two shuffling gradient-based algorithms for nonconvex-linear and nonconvex-strongly concave settings.
result Achieves state-of-the-art oracle complexity in nonconvex optimization and best-known complexity bounds for nonconvex-strongly concave setting.
Unified parametric assumption improves convergence guarantees for nonconvex optimization.
problem Weak convergence guarantees for nonconvex optimization.
method Introducing a novel unified parametric assumption.
result Unified convergence theorem for gradient-based methods.
Improved SGD methods converge faster for nonconvex optimization.
problem Nonconvex optimization challenges in machine learning.
method Adaptive SGD with line-search and Polyak stepsizes.
result Unified convergence rates for various nonconvex functions.
PAGE optimizes nonconvex problems with optimal convergence rates.
problem Nonconvex optimization problems.
method PAGE algorithm for achieving optimal convergence rates.
result PAGE achieves optimal convergence rates for nonconvex optimization.
Safe reinforcement learning with nonconvex constraints using convex approximations.
problem Safe reinforcement learning with nonlinear function approximation.
method Constructing surrogate convex constrained optimization problems by replacing nonconvex functions with convex quadratic functions.
result Solutions to surrogate problems converge to a stationary point of the original nonconvex problem.
New algorithm for privacy-preserving nonconvex optimization.
problem Privacy-preserving nonconvex empirical risk minimization.
method Differentially private stochastic gradient descent algorithm.
result Achieves strong privacy guarantees efficiently with improved utility.
Two-stage nonconvex algorithm and convex relaxation both achieve optimal accuracy in noisy blind deconvolution.
problem Solving bilinear systems of equations with random noise under different designs.
method Two-stage nonconvex algorithm and convex relaxation.
result Both methods achieve minimax-optimal accuracy in the presence of random noise.
Paper analyzes nonconvex bandit problems with improved adaptive methods.
problem Continuous armed bandit problems for nonconvex cost functions.
method Simple and adaptive bin splitting methods.
result Adaptive method achieves locally minimax optimal expected cumulative regret.
Study uncovers statistical optimality of nonconvex tensor completion methods.
problem Estimating a low-rank tensor from incomplete and corrupted observations.
method Two-stage estimation algorithm for nonconvex optimization.
result Nonconvex tensor completion achieves optimal ℓ 2 \ell_{2} ℓ 2 accuracy. Innovative method solves nonconvex optimization on manifolds.
problem Nonconvex optimization problems on Riemannian manifolds.
method Intrinsic Riemannian proximal gradient method.
result Converges for nonconvex or nonembedded problems.
Paper proposes a new method for training nonconvex models.
problem Training nonconvex models like neural networks.
method Successive functional gradient optimization using mirror descent in a function space.
result The method leads to better performance than standard training techniques.
Apollo improves nonconvex stochastic optimization efficiency.
problem Nonconvex stochastic optimization challenges.
method Adaptive parameter-wise diagonal quasi-Newton method approximating Hessian.
result Significant improvements in convergence speed and generalization over SGD and Adam.
Simple DP algorithms find approximate solutions for nonconvex ERM.
problem Finding approximate solutions to nonconvex ERM problems with privacy.
method Differential privacy, descent directions, line search, mini-batching, two-phase strategy.
result Effective algorithms for nonconvex ERM with privacy guarantees.
TiAda adapts adaptive gradient methods for nonconvex minimax optimization.
problem Nonconvex minimax optimization challenges in achieving convergence.
method TiAda is a time-scale adaptive GDA algorithm for nonconvex minimax optimization.
result TiAda achieves near-optimal complexities in deterministic and stochastic settings.
New algorithm tackles nonconvex machine learning problems with adaptive normalization and independent sampling.
problem Nonconvex machine learning problems with generalized-smoothness.
method Adaptive gradient normalization, independent sampling, and gradient clipping.
result Achieves an O(ε^(-4)) sample complexity for fast convergence.
In the paper, we study the stochastic alternating direction method of multipliers (ADMM) for the nonconvex optimizations, and propose three classes of the nonconvex stochastic ADMM with variance reduction, based on different reduced variance stochastic gradients. Specifically, the first class called the nonconvex stoch…
We analyze stochastic algorithms for optimizing nonconvex, nonsmooth finite-sum problems, where the nonconvex part is smooth and the nonsmooth part is convex. Surprisingly, unlike the smooth case, our knowledge of this fundamental problem is very limited. For example, it is not known whether the proximal stochastic gra…
PAGE is a simple gradient estimator for nonconvex optimization problems.
problem Nonconvex optimization problems in machine learning.
method PAGE is a probabilistic gradient estimator that uses vanilla SGD with probability and a small adjustment with probability 1-p.
result PAGE achieves optimal convergence rates for nonconvex finite-sum and online problems.
Smooth finite-sum optimization has been widely studied in both convex and nonconvex settings. However, existing lower bounds for finite-sum optimization are mostly limited to the setting where each component function is (strongly) convex, while the lower bounds for nonconvex finite-sum optimization remain largely unsol…
We study Frank-Wolfe methods for nonconvex stochastic and finite-sum optimization problems. Frank-Wolfe methods (in the convex case) have gained tremendous recent interest in machine learning and optimization communities due to their projection-free property and their ability to exploit structured constraints. However,…
The stochastic gradient descent has been widely used for solving composite optimization problems in big data analyses. Many algorithms and convergence properties have been developed. The composite functions were convex primarily and gradually nonconvex composite functions have been adopted to obtain more desirable prop…
The use of convex regularizers allows for easy optimization, though they often produce biased estimation and inferior prediction performance. Recently, nonconvex regularizers have attracted a lot of attention and outperformed convex ones. However, the resultant optimization problem is much harder. In this paper, for a …
This work studies low-rank approximation of a positive semidefinite matrix from partial entries via nonconvex optimization. We characterized how well local-minimum based low-rank factorization approximates a fixed positive semidefinite matrix without any assumptions on the rank-matching, the condition number or eigensp…
Unified Lagrangian-based methods for nonsmooth nonconvex optimization.
problem Minimizing nonsmooth nonconvex functions with constraints.
method Developed a unified framework for Lagrangian-based methods using subgradient updates.
result Global convergence guarantees for the proposed framework under mild conditions.
Paper proposes a faster SPIDER-EM variant for large-scale nonconvex optimization.
problem High computational cost of EM algorithm in large-scale learning.
method Extension of SPIDER-EM for nonconvex finite-sum optimization problems.
result Achieves state-of-the-art complexity bounds and linear convergence under certain conditions.
New SGD analysis for nonconvex optimization finds optimal rates.
problem Optimizing nonconvex functions in machine learning.
method Introduces a new expected smoothness assumption and analyzes SGD convergence rates.
result Optimal convergence rates for nonconvex smooth functions and global solutions under certain conditions.
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.
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-augmented Lagrangian method.
result Method achieves complexity bounds of O ~ ( ε − 11 / 2 ) \widetilde{\cal O}(ε^{-11/2}) O ( ε − 11/2 ) and O ~ ( ε − 11 / 2 min { n , ε − 5 / 4 } ) \widetilde{\cal O}(ε^{-11/2}\min\{n,ε^{-5/4}\}) O ( ε − 11/2 min { n , ε − 5/4 }) for finding an ( ε , ε ) (ε,\sqrtε) ( ε , ε ) -SOSP. Unified framework for nonconvex matrix completion with linearly parameterized factors.
problem Matrix completion with improved accuracy using linearly parameterized factors.
method Unified nonconvex optimization framework with Correlated Parametric Factorization condition.
result Uniform upper bounds for low-rank estimation at any local minimum.
New nonconvex methods improve SysID efficiency and accuracy.
problem Efficiently identify low-order linear systems from limited data.
method Proposes two nonconvex reformulations of Hankel-rank minimization for SysID.
result Nonconvex methods achieve lower statistical error rates and sample complexities.
New algorithm solves structured nonconvex-nonconcave min-max problems.
problem Min-max optimization challenges in deep learning.
method Generalized extragradient algorithm for structured nonconvex-nonconcave problems.
result Algorithm converges to stationary points in Euclidean and ℓ p \ell_p ℓ p spaces. NeAda solves nonconvex minimax optimization by balancing primal and dual variables adaptively.
problem Nonconvex minimax optimization challenges with parameter-agnostic adaptive algorithms.
method Nested Adaptive (NeAda) framework with inner and outer loops for primal and dual variables.
result Achieves near-optimal convergence rates for nonconvex-strongly-concave problems.
Paper develops methods for statistical inference with SGD in nonconvex optimization.
problem Statistical inference for nonconvex optimization problems.
method Proposes two online inferential procedures combining SGD and bootstrap techniques.
result Establishes error convergence rates and asymptotically valid bootstrap confidence intervals.
We study nonconvex finite-sum problems and analyze stochastic variance reduced gradient (SVRG) methods for them. SVRG and related methods have recently surged into prominence for convex optimization given their edge over stochastic gradient descent (SGD); but their theoretical analysis almost exclusively assumes convex…
Paper develops momentum schemes with variance reduction for non-convex composition optimization.
problem Lack of convergence guarantee and efficient momentum design in existing algorithms.
method Develops various momentum schemes with SPIDER-based variance reduction.
result Achieves near-optimal sample complexity and linear convergence rate.
The paper analyzes PPM for nonconvex-nonconcave problems, identifying three regions with varying convergence guarantees.
problem Challenges in nonconvex-nonconcave minimax optimization.
method Classic proximal point method with insights from the Moreau envelope.
result Identification of three regions with varying convergence guarantees for PPM.
Adam's convergence rate in nonsmooth nonconvex optimization is analyzed.
problem Understanding Adam's convergence behavior in nonsmooth nonconvex optimization.
method Finite-time analysis of Adam's classical form.
result Proven convergence rate of 1 / T 2 / 13 1/T^{2/13} 1/ T 2/13 for nonsmooth nonconvex optimization. In this paper, we study the stochastic gradient descent (SGD) method for the nonconvex nonsmooth optimization, and propose an accelerated SGD method by combining the variance reduction technique with Nesterov's extrapolation technique. Moreover, based on the local error bound condition, we establish the linear converge…
A new hybrid algorithm reduces stochastic gradient evaluations for nonconvex optimization.
problem Solving stochastic composite nonconvex optimization problems efficiently.
method Proposes a new hybrid variance-reduced proximal gradient method with a stochastic gradient estimator.
result Achieves optimal stochastic oracle complexity bound with one less gradient evaluation.
New lower bounds for bilevel optimization with first-order oracles.
problem Complexity of bilevel optimization with first-order oracles.
method Development of hard instances and proof of lower bounds.
result Nontrivial lower bounds for first-order zero-respecting algorithms.
Optimizes nonconvex optimization by converting it to static regret minimization.
problem Nonconvex optimization challenges in machine learning.
method Black-box online-to-nonconvex conversion with static regret minimization oracles.
result Achieves optimal convergence rates for nonconvex optimization.
Paper proposes a new method to find approximate SOSP for nonconvex constrained optimization problems.
problem Finding a second-order stationary point of nonconvex equality constrained optimization.
method Newton-CG based augmented Lagrangian method with a new Newton-CG subproblem solver.
result Achieves better complexity guarantees for finding approximate SOSP with high probability.
Paper characterizes optimization landscape of Tucker decomposition.
problem Finding exact Tucker decomposition is a nonconvex optimization problem.
method Characterized the optimization landscape and provided a local search algorithm.
result All local minima are globally optimal if tensor has an exact Tucker decomposition.
Freya PAGE optimizes nonconvex optimization with heterogeneous, asynchronous workers.
problem Optimizing nonconvex finite-sum problems with varying worker processing times.
method Freya PAGE, a parallel method robust to stragglers and adaptive to slow computations.
result Freya PAGE offers improved time complexity guarantees compared to previous methods.
New method reduces communication costs in distributed nonconvex optimization.
problem Large communication costs between central server and local workers in distributed learning.
method Communication-compressed AMSGrad for distributed nonconvex optimization.
result Converges to first-order stationary point with same iteration complexity as vanilla AMSGrad.
D-SPIDER-SFO solves nonconvex optimization problems faster on decentralized networks.
problem Finding a decentralized algorithm with similar convergence rate to SPIDER-SFO.
method Proposed D-SPIDER-SFO, a decentralized variant of SPIDER-SFO.
result Achieves a similar gradient computation cost to centralized SPIDER-SFO.