Compact, non-convex curve flows are created.
problem Creating compact, non-convex ancient solutions for curve shortening flow.
method Constructed an ancient solution asymptotic to Yin-Yang curve.
result Compact, non-convex ancient solutions for curve shortening flow are demonstrated.
Paper uses integer programming for non-convex boosting in classification.
problem Improving classification performance using non-convex optimization.
method Non-convex boosting via integer programming.
result Results comparable to or better than state-of-the-art.
First order methods can take extremely long to find global minima of non-convex functions.
problem Finding global minimizers of non-convex functions.
method Designing a family of non-convex functions and using statistical lower bounds for parameter estimation.
result First order methods can take exponential time to converge to a global minimizer.
New algorithm improves convergence for non-convex problems with boundaries.
problem Optimizing non-convex problems with constraints.
method Reflected Gradient Langevin Dynamics with probabilistic representation.
result Promising convergence rates, faster than existing methods.
We introduce a new local regret framework for non-convex models in dynamic environments.
problem Challenges in online forecasting for non-convex models with frequent updates and concept drift.
method We propose a novel local regret framework and a time-smoothed gradient update rule.
result Our approach yields more stable, robust, and computationally efficient forecasting compared to state-of-the-art methods.
This work shows neural networks can solve non-convex constraints problems.
problem Training neural networks under non-convex constraints.
method Project stochastic gradient descent with no-regret analysis of online learning.
result Overparameterized neural networks achieve near-optimal and near-feasible solutions.
In this paper, we consider the problem of learning high-dimensional tensor regression problems with low-rank structure. One of the core challenges associated with learning high-dimensional models is computation since the underlying optimization problems are often non-convex. While convex relaxations could lead to polyn…
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 (δ,ε)-stationary points. SGD's uncertainty quantified in non-convex learning problems.
problem Uncertainty quantification in non-convex learning problems.
method Asymptotic normality of SGD iterates and bias characterization.
result SGD iterates are asymptotically normally distributed around the expected value of the invariant distribution.
Optimizers find approximate global minima in non-convex problems.
problem Understanding why local methods solve non-convex optimization problems.
method Formalizing the hypothesis that many local minima are approximately global minima.
result Most local minima of practical non-convex objectives are approximately global minima.
New insights into using momentum for non-convex optimization.
problem Improving training of non-convex models like deep neural networks.
method Developed a Lyapunov analysis of SGD with momentum using stochastic primal averaging.
result Precise conditions under which SGD+M outperforms SGD and optimal hyper-parameter schedules.
Here we study non-convex composite optimization: first, a finite-sum of smooth but non-convex functions, and second, a general function that admits a simple proximal mapping. Most research on stochastic methods for composite optimization assumes convexity or strong convexity of each function. In this paper, we extend t…
Generalizes smoothness conditions for optimization methods.
problem Optimization under non-uniform smoothness conditions.
method Develops a new analysis technique for bounding gradients.
result Obtains convergence rates for gradient descent and Nesterov's method.
This work explores the non-convex optimization in compressive learning and the performance of heuristics.
problem The challenge of learning from compressed representations in compressive learning.
method Numerical simulations of the non-convex optimization landscape and heuristic performance.
result Properties of the non-convex optimization landscape and heuristic performance are explored.
New method finds near-optimal solutions for non-convex optimization problems.
problem Finding near-optimal solutions for non-convex optimization problems.
method Riemannian stochastic recursive momentum method
result Achieves a near-optimal complexity of ildeO(ε−3). This paper improves convergence guarantees for SGD algorithms in non-convex smooth functions.
problem Theoretical convergence properties of SGD algorithms for non-convex smooth functions.
method Analysis of SGD algorithms with arbitrary data ordering for non-convex smooth functions.
result Enhanced convergence guarantees for incremental gradient and single shuffle SGD, improving the optimization term of convergence guarantee.
New approach for distributed online optimization of non-convex losses with sublinear regret.
problem Regret evaluation and consensus in distributed, multi-agent systems with non-convex losses.
method Composite regret metric and consensus-based online normalized gradient (CONGD) approach for pseudo-convex losses; offline optimization oracle for general non-convex losses.
result First sublinear regret bound for general distributed online non-convex learning.
Paper proposes a working set algorithm for non-convex sparse regression with provable convergence.
problem Estimating sparse linear models from high-dimensional data using non-convex regularizers.
method FireWorks algorithm based on non-convex reformulation and leveraging residual geometry.
result Convergence to a stationary point of the full problem with provable guarantees.
Improved convergence analysis for decentralized non-convex optimization.
problem Minimizing a sum of smooth non-convex functions over a network.
method Gradient tracking in decentralized stochastic gradient descent (GT-DSGD).
result GT-DSGD achieves network-independent performances matching centralized SGD under certain conditions.
Improved optimization guarantees for deep learning models with Nesterov acceleration.
problem Optimization in non-convex deep learning landscapes.
method Analysis of Nesterov acceleration in benignly non-convex landscapes.
result Identical guarantees can be obtained in optimization problems with weak geometric assumptions, especially in overparametrized deep learning.
Adaptive momentum method solves non-convex min-max problems.
problem Non-convex min-max optimization problems in training generative adversarial networks.
method Proposes an adaptive momentum algorithm for non-convex min-max optimization.
result Establishes non-asymptotic convergence rates for the proposed algorithm.
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…
Non-convex extremal length found in surface metrics.
problem Extremal length functions on surfaces are not always convex.
method Used harmonic maps to R-trees and minimal surfaces in Rn. result Found measured foliations with non-convex extremal length functions.
Study on equilibrium with non-convex preferences.
problem Existence of equilibrium in non-convex preference settings.
method Provided a necessary and sufficient condition for equilibrium existence.
result Standard equilibrium theory cannot be applied to non-convex preferences.
Paper solves curvature equations in Minkowski space for non-convex domains.
problem Solving curvature equations in non-convex domains of Minkowski space.
method Existence theorem proved via \emph{a priori} estimates and Serrin-type condition.
result Existence of solutions for curvature equations in non-convex domains.
Paper estimates differences in multi-attribute Gaussian graphical models using non-convex penalties.
problem Estimating differences in multi-attribute Gaussian graphical models with similar structure.
method Penalized D-trace loss function with non-convex (log-sum and SCAD) penalties, proximal gradient descent methods.
result Theoretical analysis and numerical examples support consistency in support recovery and estimation.
Online SGD from random init solves non-smooth, non-convex phase retrieval.
problem Solving phase retrieval with non-smooth, non-convex loss functions.
method Online stochastic gradient descent (SGD) with constant step size, starting from arbitrary initialization.
result SGD converges from arbitrary initializations for the amplitude squared loss objective.
Several recently proposed architectures of neural networks such as ResNeXt, Inception, Xception, SqueezeNet and Wide ResNet are based on the designing idea of having multiple branches and have demonstrated improved performance in many applications. We show that one cause for such success is due to the fact that the mul…
A fast method for decentralized non-convex optimization over networks.
problem Decentralized non-convex optimization problems over a network of nodes.
method GT-SAGA, a randomized incremental gradient method that evaluates one component gradient per node per iteration.
result GT-SAGA achieves almost sure and mean-squared convergence to a first-order stationary point for general smooth non-convex problems.
Non-convex sparsity-inducing penalties have recently received considerable attentions in sparse learning. Recent theoretical investigations have demonstrated their superiority over the convex counterparts in several sparse learning settings. However, solving the non-convex optimization problems associated with non-conv…
AGGLIO optimizes non-convex functions with local convexity guarantees.
problem Optimizing non-convex functions with local convexity.
method Stage-wise, graduated optimization technique for locally convex functions.
result Global convergence to the global optimum for non-convex and locally convex objectives.
AEGD optimizes non-convex functions with dynamic energy updates.
problem Optimizing non-convex functions efficiently and robustly.
method Adaptive Gradient Descent (AEGD) with a dynamically updated energy variable.
result AEGD achieves energy-dependent convergence rates for both convex and non-convex objectives.
Improved variance reduction for Riemannian non-convex optimization with adaptive batch size.
problem Optimizing non-convex functions on Riemannian manifolds.
method Batch size adaptation in R-SVRG, R-SRG, and R-SPIDER.
result Achieves lower total complexities for various non-convex functions.
Heavy Ball method speeds up finding global optima in non-convex problems.
problem Finding global optima in non-convex optimization problems.
method Heavy Ball momentum in non-convex optimization.
result Heavy Ball helps iterates enter a benign region faster, containing a global optimal point.
Note on the computational complexity of Gromov-Wasserstein distance.
problem Computational difficulty of Gromov-Wasserstein distance.
method Analysis of the optimization problem structure and providing explicit examples.
result Gromov-Wasserstein distance optimization problem is non-convex quadratic.
Preconditioned non-convex gradient descent improves noisy matrix estimation.
problem Estimating low-rank matrices from noisy measurements.
method Preconditioned non-convex gradient descent for noisy measurements.
result Preconditioned method converges to minimax optimal estimate at a linear rate.
Matrix completion is a well-studied problem with many machine learning applications. In practice, the problem is often solved by non-convex optimization algorithms. However, the current theoretical analysis for non-convex algorithms relies heavily on the assumption that every entry is observed with exactly the same pro…
We rigorously prove statistical physics predictions for non-convex GLMs in high dimensions.
problem Analyzing high-dimensional optimization problems in non-convex Generalized Linear Models.
method Developed a systematic framework using the Gaussian Min-Max Theorem and AMP to rigorously prove replica-symmetric formulas.
result Validated statistical physics predictions for non-convex GLMs, aligning with physicist's conjectures.
Study accelerates optimization methods in non-convex problems, but doesn't improve the algorithm's performance.
problem Understanding the behavior of momentum-based acceleration methods in non-convex, high-dimensional landscapes.
method Used dynamical mean field theory to describe the average dynamics of heavy-ball momentum and Nesterov acceleration in a non-convex model.
result Accelerated dynamics but did not improve the algorithm's performance with respect to gradient descent.
We consider the fundamental problem in non-convex optimization of efficiently reaching a stationary point. In contrast to the convex case, in the long history of this basic problem, the only known theoretical results on first-order non-convex optimization remain to be full gradient descent that converges in $O(1/\varep…
We show that any open subset of a contact manifold of dimension greater than three contains a certain non-convex hypersurface violating the Thurston-Bennequin inequality.
Proves convergence of PSGLA for sampling non-convex potentials.
problem Sampling from non-convex potentials with stability.
method Combines ULA and proximal optimization with stability analysis.
result First proof of convergence for PSGLA on non-convex potentials.
An Euler discretization of the Langevin diffusion is known to converge to the global minimizers of certain convex and non-convex optimization problems. We show that this property holds for any suitably smooth diffusion and that different diffusions are suitable for optimizing different classes of convex and non-convex …
Unified framework improves robust causal inference, overcoming Gaussian barriers and optimization issues.
problem Improving robust causal inference in non-Gaussian settings.
method Combines gamma-Divergence, GNC, and Gatekeeper mechanism.
result Enhanced robustness and global optimization in causal effect estimation.
Stochastic gradient descent~(SGD) and its variants have attracted much attention in machine learning due to their efficiency and effectiveness for optimization. To handle large-scale problems, researchers have recently proposed several lock-free strategy based parallel SGD~(LF-PSGD) methods for multi-core systems. Howe…
SGD converges to global minimum for certain non-convex functions.
problem Theoretical challenges in optimizing non-convex functions in machine learning.
method Perturbed SGD on a broad class of non-convex functions.
result SGD converges to global minimum for certain non-convex functions.
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 improves efficiency of non-convex matrix reconstruction.
problem Reconstructing a low-rank matrix from few linear measurements.
method Factorized gradient descent with spectral initialization.
result Non-convex approaches can match the sample complexity of convex methods.