We analyze stochastic gradient algorithms for optimizing nonconvex problems. In particular, our goal is to find local minima (second-order stationary points) instead of just finding first-order stationary points which may be some bad unstable saddle points. We show that a simple perturbed version of stochastic recursiv…
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48 results for “Second order stationary points”
Variance reduction techniques like SVRG provide simple and fast algorithms for optimizing a convex finite-sum objective. For nonconvex objectives, these techniques can also find a first-order stationary point (with small gradient). However, in nonconvex optimization it is often crucial to find a second-order stationary…
New method finds stationary points in bilevel optimization problems.
problem Solving nonconvex-strongly-convex bilevel optimization problems.
method Restarted Accelerated HyperGradient Descent (RAHGD) method.
result Achieves best-known theoretical guarantees for finding stationary points in bilevel optimization.
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 stochastic gradient and Hessian-vector products.
result Optimal rate of for finding -approximate stationary points, matching lower bounds.
Improved method finds second-order stationary points privately with better efficiency.
problem Finding second-order stationary points privately under differential privacy constraints.
method Adaptive batch sizes and binary tree mechanism.
result Improved bound for privately finding SOSP, matching state-of-the-art for FOSP.
Escaping Saddle Points for Zeroth-order Nonconvex Optimization using Estimated Gradient Descentmath.OC
Gradient descent and its variants are widely used in machine learning. However, oracle access of gradient may not be available in many applications, limiting the direct use of gradient descent. This paper proposes a method of estimating gradient to perform gradient descent, that converges to a stationary point for gene…
Improved algorithm finds second-order stationary points in non-convex optimization.
problem Minimizing non-convex objectives while preserving training data privacy.
method SpiderBoost framework with two gradient oracles: precise and less precise.
result Improved rates for finding second-order stationary points.
Paper shows no spurious local minima in a specific matrix factorization problem.
problem Optimization of -norm rank-one symmetric matrix factorization.
method Second-order variational analysis to study the landscape of the problem.
result Any second-order stationary point is globally optimal.
Paper improves compressed SGD to reach second-order stationary points.
problem Efficiently reaching second-order stationary points in distributed machine learning.
method Gradient compression and RandomK compressor to improve convergence to second-order stationary points.
result Compressed SGD reaches second-order stationary points with improved communication efficiency.
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.
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.
This paper shows that a perturbed form of gradient descent converges to a second-order stationary point in a number iterations which depends only poly-logarithmically on dimension (i.e., it is almost "dimension-free"). The convergence rate of this procedure matches the well-known convergence rate of gradient descent to…
Bandit algorithms have been predominantly analyzed in the convex setting with function-value based stationary regret as the performance measure. In this paper, motivated by online reinforcement learning problems, we propose and analyze bandit algorithms for both general and structured nonconvex problems with nonstation…
Two new algorithms solve nonconvex-strongly concave problems efficiently.
problem Solving nonconvex-strongly concave minimax problems.
method Proposed MINIMAX-TR and MINIMAX-TRACE algorithms.
result Find -second order stationary points within iterations.
We consider the case of derivative-free algorithms for non-convex optimization, also known as zero order algorithms, that use only function evaluations rather than gradients. For a wide variety of gradient approximators based on finite differences, we establish asymptotic convergence to second order stationary points u…
Simple gradient descent algorithm escapes saddle points efficiently.
problem Escaping saddle points in nonconvex optimization.
method Gradient-based algorithm with polynomial iterations.
result Outputs ε-approximate second-order stationary points efficiently.
New algorithms solve complex minimax problems efficiently.
problem Nonconvex-strongly concave minimax problems in machine learning.
method Gradient norm regularized trust-region (GRTR) and Levenberg-Marquardt (LMNegCur) algorithms.
result Proved iteration complexities matching best known results.
The Hessian-vector product has been utilized to find a second-order stationary solution with strong complexity guarantee (e.g., almost linear time complexity in the problem's dimensionality). In this paper, we propose to further reduce the number of Hessian-vector products for faster non-convex optimization. Previous a…
PWGF escapes saddle points in nonconvex optimization.
problem Escaping saddle points in nonconvex optimization.
method PWGF uses noisy perturbations via Gaussian process to escape saddle points.
result PWGF achieves second-order optimality for nonconvex objectives.
Novel methods for accelerating optimization in complex bilevel and minimax problems.
problem Optimization challenges in bilevel and minimax problems, especially when strong convexity assumptions are not met.
method Accelerated fully first-order methods for Bilevel Optimization (BLO) and Minimax Optimization (NCSC).
result State-of-the-art complexity for finding approximate second-order stationary points in BLO and NCSC.
Stochastic gradient Langevin dynamics (SGLD) is a fundamental algorithm in stochastic optimization. Recent work by Zhang et al. [2017] presents an analysis for the hitting time of SGLD for the first and second order stationary points. The proof in Zhang et al. [2017] is a two-stage procedure through bounding the Cheege…
New algorithm provably converges to second-order stationary points in NMF.
problem Understanding convergence to local minima in NMF.
method Multiplicative weight update dynamics, concurrent updates, and simplex reduction.
result Provable convergence to second-order stationary points.
Paper develops a TR-SSQP method for noisy optimization with heavy-tailed noise.
problem Optimization problems with stochastic objectives and heavy-tailed noise.
method Trust-Region Stochastic Sequential Quadratic Programming (TR-SSQP) method.
result Achieves high-probability first-order and second-order stationarity bounds for heavy-tailed noise.
Second-order guarantees for federated learning algorithms.
problem Non-convex optimization in federated learning with saddle-points as bottlenecks.
method Drawing on recent results on second-order optimality in centralized and decentralized settings, establish second-order guarantees for federated learning algorithms.
result Established second-order guarantees for federated learning algorithms.
Adaptive methods such as Adam and RMSProp are widely used in deep learning but are not well understood. In this paper, we seek a crisp, clean and precise characterization of their behavior in nonconvex settings. To this end, we first provide a novel view of adaptive methods as preconditioned SGD, where the precondition…
Under appropriate cooperation protocols and parameter choices, fully decentralized solutions for stochastic optimization have been shown to match the performance of centralized solutions and result in linear speedup (in the number of agents) relative to non-cooperative approaches in the strongly-convex setting. More re…
Nesterov's accelerated gradient descent (AGD), an instance of the general family of "momentum methods", provably achieves faster convergence rate than gradient descent (GD) in the convex setting. However, whether these methods are superior to GD in the nonconvex setting remains open. This paper studies a simple variant…
The alternating gradient descent (AGD) is a simple but popular algorithm which has been applied to problems in optimization, machine learning, data ming, and signal processing, etc. The algorithm updates two blocks of variables in an alternating manner, in which a gradient step is taken on one block, while keeping the …
Two classes of methods have been proposed for escaping from saddle points with one using the second-order information carried by the Hessian and the other adding the noise into the first-order information. The existing analysis for algorithms using noise in the first-order information is quite involved and hides the es…
Structural Classification of Locally Stationary Time Series Based on Second-order Characteristicsstat.ME
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.
Gradient descent (GD) and stochastic gradient descent (SGD) are the workhorses of large-scale machine learning. While classical theory focused on analyzing the performance of these methods in convex optimization problems, the most notable successes in machine learning have involved nonconvex optimization, and a gap has…
In this paper, we study stochastic non-convex optimization with non-convex random functions. Recent studies on non-convex optimization revolve around establishing second-order convergence, i.e., converging to a nearly second-order optimal stationary points. However, existing results on stochastic non-convex optimizatio…
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.
New algorithms optimize non-smooth, non-convex objectives with improved complexity.
problem Optimizing non-smooth, non-convex stochastic objectives.
method Reduction to online learning, applying optimistic online learning techniques.
result Improved complexity for finding -stationary points.
In this paper, we study the problem of escaping from saddle points in smooth nonconvex optimization problems subject to a convex set . We propose a generic framework that yields convergence to a second-order stationary point of the problem, if the convex set is simple for a quadratic objectiv…
We introduce TrustVI, a fast second-order algorithm for black-box variational inference based on trust-region optimization and the reparameterization trick. At each iteration, TrustVI proposes and assesses a step based on minibatches of draws from the variational distribution. The algorithm provably converges to a stat…
Recent years have seen increased interest in performance guarantees of gradient descent algorithms for non-convex optimization. A number of works have uncovered that gradient noise plays a critical role in the ability of gradient descent recursions to efficiently escape saddle-points and reach second-order stationary p…
The paper analyzes second-order guarantees for optimization in various architectures.
problem Nonconvex optimization problems are difficult to solve, especially verifying local minima.
method Second-order guarantees for stochastic first-order optimization algorithms in different architectures.
result Gradient perturbations help algorithms distinguish and avoid undesirable stationary points.
We consider surfaces of class in the -dimensional sub-Riemannian Heisenberg group . Assuming the surface is area-stationary, i.e., a critical point of the sub-Riemannian perimeter under compactly supported variations, we show that its regular part is foliated by horizontal straight lines. In cas…
A new method helps escape saddle points in non-convex optimization.
problem Escaping saddle points in non-convex optimization problems.
method CNC-SCSG method using a separate SGD step to help escape from strict saddle points.
result The method converges to a second-order stationary point with a rate of .
Two new algorithms improve reinforcement learning by avoiding saddle points.
problem Control in reinforcement learning without model information.
method Cubic-regularized Policy Newton algorithms using likelihood ratio method.
result Convergence to a second-order stationary point with improved sample complexity.
A Newton-CG based barrier-augmented Lagrangian method for general nonconvex conic optimizationmath.OC
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 and for finding an -SOSP.
Computing Nash equilibrium (NE) of multi-player games has witnessed renewed interest due to recent advances in generative adversarial networks. However, computing equilibrium efficiently is challenging. To this end, we introduce the Gradient-based Nikaido-Isoda (GNI) function which serves: (i) as a merit function, vani…
This paper tackles the computational complexity of finding approximate stationary points in non-convex optimization.
problem Finding approximate stationary points in non-convex optimization problems.
method PLS-completeness, zero-order algorithms, and gradient queries.
result The query complexity of finding approximate stationary points is Θ(1/ε) for d=2.
SONIA optimizes machine learning problems with a novel algorithm.
problem Empirical risk minimization in machine learning.
method Symmetric Blockwise Truncated Optimization (SONIA) algorithm combining second-order and steepest descent steps.
result SONIA converges to stationary points in both convex and nonconvex cases.
This paper proposes low-complexity algorithms for finding approximate second-order stationary points (SOSPs) of problems with smooth non-convex objective and linear constraints. While finding (approximate) SOSPs is computationally intractable, we first show that generic instances of the problem can be solved efficientl…
In this paper we study general Schatten- quasi-norm (SPQN) regularized matrix minimization problems. In particular, we first introduce a class of first-order stationary points for them, and show that the first-order stationary points introduced in [11] for an SPQN regularized minimization problem are equiva…
New federated learning methods improve model performance on non-IID data.
problem Improving model performance on non-IID decentralized data.
method Proposed Federated AGMs using adaptive gradient methods with first-order and second-order momenta.
result The proposed Federated AGMs converge to a first-order stationary point under non-IID and unbalanced data settings for nonconvex optimization.