arXiv research

Adaptive algorithm AMSGrad converges for weakly convex constrained optimization problems.

In this work we propose to fit a sparse logistic regression model by a weakly convex regularized nonconvex optimization problem. The idea is based on the finding that a weakly convex function as an approximation of the $\ell_0$ pseudo norm is able to better induce sparsity than the commonly used $\ell_1$ norm. For a cl…

New adaptive methods solve weakly convex stochastic optimization problems.

Neural network approximates weakly efficient frontier of convex vector optimization problems.

Expanding FCCO to non-smooth weakly-convex problems, improving deep learning performance.

Paper analyzes convergence of stochastic methods under heavy-tailed noise.

New single-loop algorithm tackles weakly convex constraints in stochastic optimization.

New algorithm solves complex non-convex problems efficiently.

Develops a new SPP algorithm with variance reduction for weakly convex optimization.

SGD avoids critical points on weakly convex functions.

We introduce a geometrically transparent strict saddle property for nonsmooth functions. This property guarantees that simple proximal algorithms on weakly convex problems converge only to local minimizers, when randomly initialized. We argue that the strict saddle property may be a realistic assumption in applications…

The paper proves optimal estimates and inequalities for spectral functions on certain manifolds.

Paper extends SMM to weakly convex and multi-convex surrogates for non-convex optimization.

Weakly convex polyhedra which are star-shaped with respect to one of their vertices are infinitesimally rigid. This is a partial answer to the question whether every decomposable weakly convex polyhedron is infinitesimally rigid. The proof uses a recent result of Izmestiev on the geometry of convex caps.

A distributed subgradient method tackles non-convex optimization problems in networks.

Unified approach tackles high-dimensional tensor bandits with convex optimization and weakly decomposable regularizers.

Smooth analog of Gromov's dihedral rigidity for 3D weakly convex domains.

In this paper, we show that if the optimization function is restricted-strongly-convex (RSC) and restricted-smooth (RSM) -- a rich subclass of weakly submodular functions -- then a streaming algorithm with constant factor approximation guarantee is possible. More generally, our results are applicable to any monotone we…

On R^n endowed with a riemannian metric of bounded nonpositive curvature, the weakly convex closed subsets are topologically trivial. The stability of such subsets under intersection characterizes the euclidean spaces.

This paper improves inverse problem solving with weakly convex regularisers and proves convergence.

Optimized method tackles convex optimization with heavy-tailed noise.

Paper tackles efficient learning of non-convex hypotheses in metric spaces.

This work establishes uniform convergence of subdifferentials in stochastic optimization.

We present $\textit{Free-MESSAGE}^{p}$, the first zeroth-order algorithm for (weakly-)convex mean-semideviation-based risk-aware learning, which is also the first three-level zeroth-order compositional stochastic optimization algorithm whatsoever. Using a non-trivial extension of Nesterov's classical results on Gaussia…

The paper proves a Schwarz lemma for weakly Kähler-Finsler manifolds.

MAGIC method optimally estimates model predictions changes.

In this paper, we consider first-order convergence theory and algorithms for solving a class of non-convex non-concave min-max saddle-point problems, whose objective function is weakly convex in the variables of minimization and weakly concave in the variables of maximization. It has many important applications in mach…

Study on polyhedra rigidity, finding non-existence of flexible weakly convex decomposable polyhedra.

The study compares spectral volumes of manifolds with weakly convex boundaries.

A submanifold $M^m$ of a Euclidean space $R^{m+p}$ is said to have harmonic mean curvature vector field if $Δ\vec{H}=0$, where $\vec{H}$ is the mean curvature vector field of $M\hookrightarrow R^{m+p}$ and $Δ$ is the rough Laplacian on $M$. There is a conjecture named after Bangyen Chen which states that submanifolds o…

Gradient descent performs well on weakly convex losses, offering generalization guarantees.

We introduce a generic scheme to solve nonconvex optimization problems using gradient-based algorithms originally designed for minimizing convex functions. Even though these methods may originally require convexity to operate, the proposed approach allows one to use them on weakly convex objectives, which covers a larg…

HF-opt uses Hamiltonian dynamics to optimize functions, achieving accelerated rates with randomized integration time.

In this paper we consider three-manifolds with weakly umbilic boundary (the Second Fundamental form of the boundary is a constant multiple of the metric). We show that if the initial manifold has positive Ricci curvature and the boundary is convex (nonnegative Second Fundamental form), its metric can be deformed via th…

Standard stochastic optimization methods are brittle, sensitive to stepsize choices and other algorithmic parameters, and they exhibit instability outside of well-behaved families of objectives. To address these challenges, we investigate models for stochastic minimization and learning problems that exhibit better robu…

Paper proposes an algorithm for sampling from complex mixture distributions without requiring smoothness.

Localization of chest pathologies in chest X-ray images is a challenging task because of their varying sizes and appearances. We propose a novel weakly supervised method to localize chest pathologies using class aware deep multiscale feature learning. Our method leverages intermediate feature maps from CNN layers at di…

New sampling algorithm for non-smooth potentials.

We study the question of whether parallelization in the exploration of the feasible set can be used to speed up convex optimization, in the local oracle model of computation. We show that the answer is negative for both deterministic and randomized algorithms applied to essentially any of the interesting geometries and…

Strict convexity proven for certain self-expanders in high dimensions.

Paper proposes a weak approximation of reflection coupling for non-convex optimization.

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…

A new biased gradient descent method for conditional stochastic optimization.

Properties of two classes of generally convex sets in the n-dimentional real Euclidean space, called m-semiconvex and weakly m-semiconvex, 1<=m<n, are investigated in the present work. In particular, it is established that an open set with smooth boundary in the plan which is weakly 1-semiconvex but not 1-semiconvex co…

Novel algorithm accelerates PnP methods for image deblurring and super-resolution.

We consider compact convex hypersurfaces contracting by functions of their curvature. Under the mean curvature flow, uniformly convex smooth initial hypersurfaces evolve to remain smooth and uniformly convex, and contract to points after finite time. The same holds if the initial data is only weakly convex or non-smoot…

We introduce the cutting construction of possibly non-compact symplectic toric manifolds, in particular, toric symplectic cones that correspond to a weakly convex good cone. Since the symplectization of a toric contact manifold is a toric symplectic cone, we can also construct toric contact manifolds that correspond to…

We study convex polyhedra in $\mathbb{R}\mathbb{P}^3$ with all their vertices on a sphere. We do not require, in particular, that the polyhedra lie in the interior of the sphere, hence the term "weakly inscribed". Such polyhedra can be interpreted as ideal polyhedra, if we regard $\mathbb{R}\mathbb{P}^3$ as a combinati…