This paper develops methods to solve saddle-point problems on Riemannian manifolds with exponential stability.
problem Solving saddle-point problems on Riemannian manifolds with exponential stability.
method Developed a projected dynamical system on a Riemannian manifold to solve saddle-point problems, leveraging the strong monotonicity of the gradient of the Lagrangian function.
result Established exponential stability and convergence of the projected dynamical system to the unique saddle-point.
Gradient-based methods struggle with saddle points; curvature exploitation helps.
problem Gradient-based methods struggle with saddle points, leading to undesired stable stationary points.
method Exploits curvature information to escape undesired stationary points.
result Different optimization methods, including gradient and Adagrad, can escape non-optimal stationary points when curvature exploitation is used.
DLNs dynamics change with variance, leading to saddle-to-saddle training phases.
problem Understanding the dynamics of DLNs with varying initialization variance.
method Analyzing the phase transition of DLNs' dynamics as variance changes.
result Gradient descent visits a sequence of saddles, reaching a sparse global minimum.
First-order methods avoid saddle points for most initializations.
problem Avoiding saddle points in optimization problems.
method First-order methods, including gradient descent and variants, analyzed using dynamical systems and the Stable Manifold Theorem.
result First-order methods avoid saddle points for almost all initializations.
The paper studies neural networks' convergence near origin and saddle points.
problem Directional convergence of neural networks near small initializations and saddle points.
method Gradient flow dynamics analysis of two-homogeneous neural networks.
result Neural networks' weights approximately converge in direction to KKT points for small initializations.
New ASGLD algorithm escapes saddle points and converges to local minima.
problem Non-convex optimization problems with saddle points.
method Adaptive stochastic gradient Langevin dynamics (ASGLD) and its specialized versions ASG and AGLD.
result ASGLD and ASG escape saddle points in nearly dimension-free iterations and converge to local minima.
Deep ReLU networks escape from the origin via saddle points with a low-rank bias.
problem Understanding the dynamics of gradient descent in deep ReLU networks.
method Analysis of escape directions and singular values of weight matrices.
result The first singular value of the ℓ-th layer weight matrix is at least ℓ41 larger than any other singular value. Paper defines saddle points in asymmetric Dynkin games using martingale theory.
problem Tackles saddle point conditions in asymmetric Dynkin games with partial information.
method Uses martingale theory to identify super and submartingales related to equilibrium payoffs.
result Characterizes saddle point strategies in terms of equilibrium payoffs' dynamics and Doob-Meyer decompositions.
Active learning reduces SP calculations by 90%.
problem Efficiently calculating saddle points in energy functions.
method Active learning framework with GPR and GAD.
result Significant reduction in the number of expensive evaluations.
The paper analyzes neural network dynamics after weights escape the origin.
problem Understanding gradient flow dynamics of neural networks after the origin.
method Analyzes gradient flow of homogeneous neural networks with locally Lipschitz gradients.
result Characterizes the first saddle point encountered after escaping the origin.
New ODE models show saddle-point optimization methods converge differently, with last-iterate convergence for OGDA.
problem Analyzing convergence properties of saddle-point optimization methods.
method High-Resolution Differential Equations (HRDEs) to design differential equation models for saddle-point optimization methods.
result HRDEs reveal last-iterate convergence for Optimistic Gradient Descent Ascent (OGDA) in bilinear games.
Paper analyzes Transformer learning dynamics, proving benign landscape for in-context learning.
problem Understanding how Transformers learn in context with nonlinear features.
method Mean-field and two-timescale analysis of Transformer dynamics, proving nonconvex but benign landscape.
result Proves mean-field dynamics avoid saddle points, leading to improved optimization.
GenFlow optimizes faster, avoiding saddle points in fixed time.
problem Designing efficient optimization algorithms for convex and non-convex functions.
method Introduces GenFlow and momentum variants with fixed-time convergence guarantees.
result GenFlow and momentum variants converge to optimal solutions in fixed time for PL functions and evade saddle points uniformly.
WSFN overcomes saddle points for non-convex functionals in Wasserstein space.
problem Minimizing non-convex functionals over the Wasserstein space with saddle point avoidance.
method WSFN is a second-order method that preconditions the Wasserstein gradient to avoid saddle points.
result WSFN escapes saddle regions and reaches a global minimizer in polynomial time.
DEO uses gradient information to escape saddle points in neural networks.
problem Training deep neural networks struggles with flat regions and saddle points.
method Dimer-Enhanced Optimization (DEO) uses gradient information to estimate curvature and escape saddle points.
result DEO improves training efficiency and performance compared to standard first-order methods.
SGD in DLNs reveals feature learning dynamics.
problem Understanding SGD dynamics in DLNs during saddle-to-saddle training.
method Stochastic Langevin dynamics with anisotropic, state-dependent noise; one-dimensional per-mode SDEs; Boltzmann distribution approximation.
result SGD noise encodes feature learning progression but does not alter saddle-to-saddle dynamics.
New method for LVEBMs using saddle-point optimization and Langevin updates.
problem Expressive generative modeling of latent variables with hidden structure.
method Reformulate LVEBM training as a saddle problem, using Langevin updates and gradient flows.
result Proves existence and convergence of the algorithm under standard assumptions, with improved ELBO bounds.
Study dynamics and topology of flows near non-saddle sets or W-sets.
problem Understanding the dynamics and topology of flows near specific invariant sets.
method Cohomological relations and global properties analysis.
result Dynamical classification of surfaces and robustness of non-saddle-sets.
Algorithm minimizes SP-Regret for online saddle point problem and related knapsack optimization.
problem Online saddle point problem and related online convex optimization with knapsacks.
method Proposed algorithms achieving sublinear SP-Regret in various settings.
result Achieved sublinear SP-Regret bounds for different problem settings.
Paper analyzes algorithms for nonstationary saddle-point optimization problems.
problem Nonstationary saddle-point optimization problems in game theory, reinforcement learning, and machine learning.
method Proposes extragradient and Frank-Wolfe algorithms for online and bandit settings.
result Establishes sub-linear regret bounds for the proposed algorithms.
Efficient algorithm converges to Nash equilibrium in bilinear problems with bandit feedback.
problem Learning dynamics in bilinear saddle-point problems with bandit feedback.
method Uncoupled learning algorithm combining experimental design and FTRL with a tailored regularizer.
result Last-iterate convergence rate of ildeO(T−1/4) in high probability. Riemannian gradient descent escapes some spurious critical points on low-rank matrix manifold.
problem Spurious critical points on the boundary of low-rank matrix manifold.
method Riemannian gradient descent with dynamical low-rank approximation and rescaled gradient flow.
result Riemannian gradient descent escapes some spurious critical points on the boundary of the manifold.
We consider discriminative dictionary learning in a distributed online setting, where a network of agents aims to learn a common set of dictionary elements of a feature space and model parameters while sequentially receiving observations. We formulate this problem as a distributed stochastic program with a non-convex o…
Heavy-ball algorithms can always avoid saddle points with random initialization.
problem Optimizing nonconvex functions with saddle points.
method Developed a new mapping to interpret heavy-ball algorithms as iterations, proving they can escape saddle points.
result Heavy-ball algorithms can escape saddle points with random initialization.
Develops algorithm to escape saddle points in Byzantine settings.
problem Byzantine workers create fake local minima near saddle points.
method ByzantinePGD, a robust first-order algorithm.
result Converges to approximate true local minimizer with low iteration complexity.
FeDualEx tackles saddle point optimization in federated learning with composite objectives.
problem Saddle point optimization with constraints and non-smooth regularization in federated learning.
method Federated Dual Extrapolation (FeDualEx) algorithm for saddle point optimization and composite objectives.
result FeDualEx effectively solves saddle point optimization problems with composite objectives in federated learning.
The paper identifies a problem with reinforcement learning causing performance plateaus.
problem Performance plateaus in reinforcement learning due to 'ray interference'.
method Investigation of learning dynamics and conditions for 'ray interference'.
result Conditions and properties of 'ray interference' are established.
Unified analysis of EG and OGDA for saddle point problems using proximal point method.
problem Solving saddle point problems in bilinear and strongly convex-strongly concave settings.
method Unified analysis as approximations of the proximal point method.
result Unified analysis of EG and OGDA for saddle point problems.
OKRidge solves sparse ridge regression problems for nonlinear systems.
problem Identifying sparse governing equations for nonlinear dynamical systems.
method OKRidge algorithm using saddle point formulation and ADMM-based approach with efficient proximal operators.
result OKRidge achieves provable optimality with significantly faster run times than Gurobi.
This paper explains why Adam generalizes worse than SGD by analyzing its components.
problem Understanding why Adam generalizes worse than Stochastic Gradient Descent (SGD).
method Diffusion theoretical framework to disentangle the effects of Adaptive Learning Rate and Momentum.
result Adaptive Learning Rate helps escape saddle points but not select flat minima, while Momentum provides a drift effect to help pass through saddle points.
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 O(ε−2log(1/ε)). Gradient descent can take exponentially long to escape saddle points in 2D.
problem Worst-case inefficiency of gradient descent in non-convex optimization.
method Analysis of gradient descent's performance on 2D functions.
result Gradient descent can take exponentially long to escape saddle points.
A new method avoids saddle points in training machine learning models.
problem Training machine learning models efficiently in the presence of saddle points.
method Modified Laplacian smoothing gradient descent (mLSGD).
result The attraction region for mLSGD is significantly smaller than for gradient descent, avoiding saddle points.
Develops path integral for spiked tensor model dynamics.
problem Dynamics of spiked tensor model with random initial conditions.
method Path integral approach applied to partial differential equations.
result Large-N saddle point equations dominated by melonic diagrams. New methods help escape strict saddle points in nonsmooth optimization.
problem Escaping strict saddle points in nonsmooth optimization.
method An inexact stochastically perturbed gradient method applied to the Moreau envelope.
result A variety of algorithms for nonsmooth optimization can efficiently escape strict saddle points of the Moreau envelope.
New insights into matrix factorization show strict saddles have bounded eigenvalues.
problem Understanding the nature of critical points in matrix factorization.
method Analyzing orbits of critical points under the general linear group and identifying canonical points.
result Minimum eigenvalue of strict saddles is not uniformly bounded below zero.
New algorithm speeds up solving saddle-point problems with large condition numbers.
problem Solving saddle-point problems with large condition numbers.
method Proposes a stochastic proximal point algorithm that accelerates variance reduction methods.
result Reduces logarithmic term of condition number for iteration complexity.
A new algorithm trains deep neural networks by adding neurons greedily.
problem Training deep neural networks efficiently and effectively.
method Neuron Pursuit (NP) algorithm, which alternates between neuron addition and loss minimization.
result The algorithm can train deep neural networks efficiently and effectively.
New method solves saddle-point problems faster than existing methods.
problem Large-scale saddle-point problems in optimization.
method Sequential subspace optimization with proximal regularization.
result Significantly better convergence compared to first-order methods.
Researchers compute gap distributions for saddle connection directions on specific translation surfaces.
problem Computing gap distributions for saddle connection directions on translation surfaces.
method Translation to dynamical question of return times to a transversal under the horocycle flow.
result Gap distributions have support at 0 and quadratic tail decay.
This paper extends Newton's method to distributed learning, avoiding saddle points and handling Byzantine workers.
problem Avoiding saddle points in distributed non-convex optimization, especially in the presence of Byzantine workers.
method Extends cubic-regularized Newton method to distributed framework, addressing communication bottlenecks and Byzantine attacks.
result The method achieves improved iteration complexity compared to first-order methods, with a 25% improvement in experiments.
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.
Last iterate of Extragradient algorithm converges slower than averaged iterates in saddle point problems.
problem Smooth convex-concave saddle point problems
method Analysis of Extragradient (EG) algorithm convergence rates
result The last iterate of EG converges at a rate of O(1/√T), compared to O(1/T) for averaged iterates
We optimize saddle-point problems for large-scale Markov decision processes.
problem Optimizing policies in large-scale Markov decision processes.
method Characterized conditions for convergence and designed an optimization algorithm.
result Our algorithm converges faster and is state-space independent.
Study on neural networks in overparameterized cases, focusing on flat minima and saddle points.
problem Understanding the landscape of training error in neural networks with overparameterization.
method Three methods of embedding a network into a wider one with more hidden units, analyzing the embedded point's properties.
result Smooth and ReLU activation networks have different partially flat landscapes around the embedded point.
Houdini finds high-dimensional saddle points under few constraints.
problem Escaping from saddle points in high-dimensional spaces with constraints.
method Gradient descent methods under logarithmic inequality constraints.
result Polynomial time algorithms for escaping saddle points under constraints.
Extends saddle-point method for large-time volatility smiles.
problem Analyzing large-time volatility smiles in financial models.
method Saddle-point approach to derive large-time model-implied volatility smiles.
result Provides theoretical foundation and wide class of arbitrage-free parametrizations.
Gradient descent can use larger step sizes to avoid strict saddle points.
problem Avoiding strict saddle points in non-convex optimization.
method Proving that gradient descent with step-size up to 2/L avoids strict saddle points with high probability.
result Gradient descent with step-size up to 2/L almost surely avoids strict saddle points.