Saddle-point optimization problems are an important class of optimization problems with applications to game theory, multi-agent reinforcement learning and machine learning. A majority of the rich literature available for saddle-point optimization has focused on the offline setting. In this paper, we study nonstationar…
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48 results for “saddle point optimization”
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.
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.
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.
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…
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 Convergence of Saddle-Point Optimizers via High-Resolution Differential Equationsmath.OC
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.
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.
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 .
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 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.
Nonconvex optimization problems such as the ones in training deep neural networks suffer from a phenomenon called saddle point proliferation. This means that there are a vast number of high error saddle points present in the loss function. Second order methods have been tremendously successful and widely adopted in the…
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.
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.
This paper analyzes saddle points and minimax points in non-convex smooth games.
problem Understanding local optimal points in non-convex smooth games.
method Comprehensive analysis of local minimax points, including their optimality conditions and stability.
result Local saddle points are uniformly local minimax points under mild continuity assumptions.
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 algorithm helps escape saddle points in optimization problems.
problem Optimizing smooth non-convex functions to avoid saddle points.
method Perturbed Saddle-escape Descent (PSD) algorithm with explicit constants.
result PSD finds approximate second-order stationary points efficiently.
Equivalence of convex optimization, saddle-point problems, and variational inequalities is a well-established concept. The variational inequality (VI) is a static problem which is studied under dynamical settings using a framework called the projected dynamical system, whose stationary points coincide with the static s…
We consider the problem of computing optimal policies in average-reward Markov decision processes. This classical problem can be formulated as a linear program directly amenable to saddle-point optimization methods, albeit with a number of variables that is linear in the number of states. To address this issue, recent …
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.
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…
Nonconvex optimization algorithms with random initialization have attracted increasing attention recently. It has been showed that many first-order methods always avoid saddle points with random starting points. In this paper, we answer a question: can the nonconvex heavy-ball algorithms with random initialization avoi…
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.
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.
We analyze stochastic gradient descent for optimizing non-convex functions. In many cases for non-convex functions the goal is to find a reasonable local minimum, and the main concern is that gradient updates are trapped in saddle points. In this paper we identify strict saddle property for non-convex problem that allo…
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.
We extend the Frank-Wolfe (FW) optimization algorithm to solve constrained smooth convex-concave saddle point (SP) problems. Remarkably, the method only requires access to linear minimization oracles. Leveraging recent advances in FW optimization, we provide the first proof of convergence of a FW-type saddle point solv…
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 -th layer weight matrix is at least larger than any other singular value.
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…
Optimal privacy-preserving algorithm for solving saddle point problems.
problem Solving convex-concave stochastic saddle point problems under differential privacy constraints.
method Recursive regularization technique repurposed for saddle point problems, achieving strong gap rate of O(1/√n + √d/nε).
result Achieves nearly optimal strong gap rate of O(1/√n + √d/nε) with gradient complexity O(min{n^2ε^(1.5)/√d, n^(3/2)}).
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.
A central challenge to many fields of science and engineering involves minimizing non-convex error functions over continuous, high dimensional spaces. Gradient descent or quasi-Newton methods are almost ubiquitously used to perform such minimizations, and it is often thought that a main source of difficulty for these l…
Oracle-efficient algorithm for offline RL with partial data coverage.
problem Offline reinforcement learning with partial data coverage and constraints.
method PDOCRL, a primal-dual algorithm with decomposed linear-programming formulation.
result Near-optimal, near-feasible policy with \(\widetilde{\mathcal O}(ε^{-2})\) sample guarantee.
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…
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.
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…
New saddle network architectures preserve convex-concave geometry in optimization problems.
problem Optimization models with convex x and concave y components.
method Structured separable decomposition and saddle network architectures.
result Proven one-dimensional approximation theorem and high accuracy on various test functions.
We solve a complex optimization problem for Wasserstein barycenters using stochastic methods.
problem Optimizing the average of multiple probability distributions in a streaming data setting.
method We reformulate the problem as a convex-concave saddle-point problem and propose a stochastic optimization algorithm.
result Our algorithm has better complexity than existing methods for arbitrary distributions.
Perturbed Proximal Descent to Escape Saddle Points for Non-convex and Non-smooth Objective Functionscs.LG
We consider the problem of finding local minimizers in non-convex and non-smooth optimization. Under the assumption of strict saddle points, positive results have been derived for first-order methods. We present the first known results for the non-smooth case, which requires different analysis and a different algorithm…
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…
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…
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.
In this paper, we propose and analyze zeroth-order stochastic approximation algorithms for nonconvex and convex optimization, with a focus on addressing constrained optimization, high-dimensional setting and saddle-point avoiding. To handle constrained optimization, we first propose generalizations of the conditional g…
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…
New method for causal effect estimation with hidden confounders.
problem Estimating causal effects in the presence of hidden confounders.
method Singular value decomposition of a conditional expectation operator followed by saddle-point optimization.
result Our method outperforms existing methods on common benchmarks.
We consider convex-concave saddle point problems with a separable structure and non-strongly convex functions. We propose an efficient stochastic block coordinate descent method using adaptive primal-dual updates, which enables flexible parallel optimization for large-scale problems. Our method shares the efficiency an…
A commonly used heuristic in non-convex optimization is Normalized Gradient Descent (NGD) - a variant of gradient descent in which only the direction of the gradient is taken into account and its magnitude ignored. We analyze this heuristic and show that with carefully chosen parameters and noise injection, this method…
Auto-conditioned primal-dual hybrid gradient method and alternating direction method of multipliersmath.OC
New methods solve saddle point problems without line search.
problem Solving saddle point problems efficiently and adaptively.
method Auto-conditioned primal-dual hybrid gradient (AC-PDHG) and auto-conditioned ADMM (AC-ADMM) methods.
result Methods achieve optimal complexity and convergence guarantees.