Optimizes bond portfolios to avoid worst-case losses.
problem Finding the worst-case value of a bond portfolio over a range of yield curves and spreads.
method Solves a convex-concave saddle point optimization problem to find the worst-case value and construct a robust portfolio.
result Constructs a bond portfolio that includes the worst-case value, ensuring robustness against market uncertainties.
When a boudnary-parabolic representation of a link group to PSL(2,C) is given, Inoue and Kabaya suggested a combinatorial method to obtain the developing map of the representation using the octahedral triangulation and the shadow-coloring of certain quandle. Quandle is an algebraic system closely related wit…
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.
Investment and consumption strategies optimized with uncertain parameters.
problem Investment and consumption preferences in an incomplete financial market with uncertain parameters.
method PDE characterization and semi-explicit saddle-point construction of forward preferences and optimal strategies.
result A specific relationship between initial investment preference and forward consumption preference is necessary.
Study on scalar curvature minimizability loss and saddle point solutions.
problem Loss of minimizability in prescribing scalar curvature.
method Variational approach, focusing on saddle point solutions.
result Existence of saddle point solutions despite minimizability loss.
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.
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.
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.
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.
Optimizes portfolio construction using Bayesian methods and variational techniques.
problem Balancing reward and risk in portfolio construction.
method Bayesian decision-theoretic formulation, saddle-point problem, variational Bayes relaxation, efficient algorithm, provable convergence.
result Proves statistical consistency of proposed decision with optimal Bayesian decision.
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.
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.
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.
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.
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.
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)). 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.
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.
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
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.
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.
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.
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.
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.
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.
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.
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 optimization methods are the most popular choice for finding local optima for classical minimization and saddle point problems. Here, we highlight a systemic issue of gradient dynamics that arise for saddle point problems, namely the presence of undesired stable stationary points that are no local optima…
Study on convergence of OMD for saddle point problems, correcting previous claims.
problem Convergence of OMD for saddle point problems with exact gradients.
method Analysis of Mirror Descent and Optimistic Mirror Descent for saddle point problems.
result Monotone convergence only occurs after a large number of iterations for coherent saddle point problems.
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.
New algorithm solves saddle point problems in Banach spaces.
problem Solving saddle point problems in real reflexive Banach spaces.
method Stochastic Bregman Primal-Dual Splitting Algorithm with relative smoothness and strong convexity assumptions.
result Almost sure convergence to saddle points under various conditions.
We study robust distributed learning that involves minimizing a non-convex loss function with saddle points. We consider the Byzantine setting where some worker machines have abnormal or even arbitrary and adversarial behavior. In this setting, the Byzantine machines may create fake local minima near a saddle point tha…
A new method avoids saddle points in Newton's method.
problem Avoiding saddle points in optimization problems.
method New Q-Newton's method with specific update rule.
result The method guarantees convergence to a critical point that is not a saddle point.
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.
Although gradient descent (GD) almost always escapes saddle points asymptotically [Lee et al., 2016], this paper shows that even with fairly natural random initialization schemes and non-pathological functions, GD can be significantly slowed down by saddle points, taking exponential time to escape. On the other hand, g…
Local search heuristics for non-convex optimizations are popular in applied machine learning. However, in general it is hard to guarantee that such algorithms even converge to a local minimum, due to the existence of complicated saddle point structures in high dimensions. Many functions have degenerate saddle points su…
Adaptive methods like Adam help escape saddle points in deep learning.
problem Escaping saddle points in nonconvex optimization problems.
method Viewed as preconditioned SGD, where the preconditioner estimates noise isotropy.
result Adaptive methods can efficiently estimate a preconditioner that helps escape saddle points.
New algorithm finds local minima in non-convex, non-smooth problems.
problem Finding local minimizers in non-convex and non-smooth optimization.
method Perturbed Proximal Descent, tailored for non-smooth cases.
result First known results for non-smooth optimization.
Study proves existence and properties of shrinkers in area-preserving curve-shortening flow.
problem Existence and properties of shrinkers in area-preserving curve-shortening flow.
method Using known results on λ-curves, we prove existence of non-circular shrinkers and deduce a saddle-point property.
result Existence and properties of shrinkers in area-preserving curve-shortening flow, including a saddle-point property.
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.
We establish that first-order methods avoid saddle points for almost all initializations. Our results apply to a wide variety of first-order methods, including gradient descent, block coordinate descent, mirror descent and variants thereof. The connecting thread is that such algorithms can be studied from a dynamical s…
New method simplifies optimization landscapes by transforming saddle points.
problem Saddle points hinder non-convex optimization in machine learning.
method Variable elimination algorithms, like VarPro, are compared to reveal geometric insights.
result Variable elimination reshapes critical point structure, creating local maxima from saddle points.
A distributed optimization method solves saddle point problems with strong concavity and convexity.
problem Solving saddle point problems with distributed and heterogeneous data.
method GT-GDA, a distributed first-order method using gradient tracking and consensus over coupling matrices.
result GT-GDA converges linearly to the unique saddle point solution under specific conditions.
New method stabilizes saddle-point optimization with unbounded gradients.
problem Stochastic saddle-point optimization faces instability due to large gradients.
method Proposes a regularization technique to stabilize iterates.
result Yields meaningful performance guarantees even with unbounded gradients.
New analysis shows GD and SGD avoid saddle points efficiently in high dimensions.
problem Gradient descent and SGD struggle with saddle points in high-dimensional nonconvex optimization problems.
method Perturbed versions of GD and SGD analyzed for efficiency in high dimensions.
result Perturbed GD and SGD converge to second-order stationary points efficiently, avoiding saddle points.
Riemannian gradient descent helps escape saddle points on curved spaces.
problem Minimizing nonconvex functions on curved spaces (Riemannian manifolds).
method Perturbed Riemannian gradient descent algorithm.
result Converges to second-order stationary points, matching unconstrained smooth minimization rates.
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.