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.
New sampling algorithm for non-smooth potentials.
problem Sampling from non-smooth potentials.
method Proximal algorithm based on rejection sampling.
result Achieves better complexity than existing methods.
We prove quantitative convergence rates at which discrete Langevin-like processes converge to the invariant distribution of a related stochastic differential equation. We study the setup where the additive noise can be non-Gaussian and state-dependent and the potential function can be non-convex. We show that the key p…
Study of non-convex potential functions in deep learning with Poincaré inequality.
problem Understanding convergence of stochastic dynamics in non-convex potential landscapes.
method Introduced log-Polyak-Lojasiewicz (log-PL) measures and analyzed their convergence properties.
result Langevin dynamics converges at a rate of O ~ ( 1 / ε ) \tilde{\mathcal{O}}(1/ε) O ~ ( 1/ ε ) for sufficiently small ε ε ε . Lower bounds on queries needed for finding stationary points in non-convex optimization.
problem Finding ε ε ε -stationary points in non-convex stochastic optimization. method Proving lower bounds on the number of queries required by stochastic first-order methods.
result Lower bounds on the number of queries required to find ε ε ε -stationary points are tight and optimal. 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…
Accelerates convergence in global non-convex optimization with reversible diffusion.
problem Global non-convex optimization challenges.
method Utilizes reversible diffusion processes with adaptive diffusion coefficients.
result Accelerated convergence with reduced discretization error.
Shielded LMC samples from non-convex spaces with repulsive drift.
problem Sampling from non-convex spaces with convex holes.
method Combining adaptive temperature and repulsive drift.
result Advantages over unconstrained sampling in constrained spaces.
Linear speedup achieved in non-convex optimization for decentralized systems.
problem Achieving optimal performance in decentralized non-convex optimization.
method Examined the dependence of convergence guarantees on spectral properties of combination policies.
result Linear speedup in saddle-point escape time for symmetric combination policies.
New method improves sampling from non-convex distributions using HFHR dynamics.
problem Sampling from non-log-concave densities with non-convex potential functions.
method Hessian-free high-resolution dynamics (HFHR) with reflection/synchronous coupling.
result HFHR dynamics converges faster than kinetic Langevin dynamics (KLD) for non-convex potentials.
Recent studies have illustrated that stochastic gradient Markov Chain Monte Carlo techniques have a strong potential in non-convex optimization, where local and global convergence guarantees can be shown under certain conditions. By building up on this recent theory, in this study, we develop an asynchronous-parallel s…
This paper tackles global Nash equilibrium in non-convex multi-player games.
problem Challenges in finding global Nash equilibrium due to non-convexity.
method Conjugate transformation and variational inequality formulation to prove existence and design algorithms.
result Designs an ODE-based algorithm with exponential convergence rate and proves its effectiveness in practical scenarios.
LMC algorithm achieves efficient sampling from complex distributions with specific tail behaviors.
problem Sampling from distributions with specific tail behaviors and Hölder continuous gradients.
method Unadjusted Langevin Monte Carlo (LMC) algorithm with analysis of potential function tail growth and smoothness.
result LMC achieves efficient sampling with a rate independent of tail growth for linearly growing tails.
Neural networks are a powerful class of functions that can be trained with simple gradient descent to achieve state-of-the-art performance on a variety of applications. Despite their practical success, there is a paucity of results that provide theoretical guarantees on why they are so effective. Lying in the center of…
Learning with a {\it convex loss} function has been a dominating paradigm for many years. It remains an interesting question how non-convex loss functions help improve the generalization of learning with broad applicability. In this paper, we study a family of objective functions formed by truncating traditional loss f…
Paper tackles fast convergence for non-convex strongly-concave min-max problems.
problem Non-convex strongly-concave min-max problems in deep learning.
method Proximal stage-based method with PL condition for faster convergence.
result Established fast convergence in primal objective gap and duality gap.
This study analyzes AdaGrad's stability and convergence in non-convex optimization.
problem Lack of theoretical analysis for AdaGrad in non-convex optimization.
method Novel stopping time-based techniques from probability theory.
result Established stability and derived convergence rates for AdaGrad.
New PAC-Bayesian bounds for online learning with data streams.
problem Challenges of traditional PAC-Bayesian bounds in dynamic data collection.
method Developed new PAC-Bayesian bounds in online learning framework, using updated regret definition and batch-to-online conversion.
result PAC-Bayesian bounds hold for online learning with dependent data and non-convex losses.
New optimizer G-AdaGrad improves upon AdaGrad for non-convex machine learning problems.
problem Solving non-convex machine learning problems efficiently.
method Proposes a new optimizer G-AdaGrad and analyzes its convergence using state-space models.
result Empirical results show G-AdaGrad performs better than AdaGrad and Adam.
Neural nets solve electric field in non-convex microfluidic devices.
problem Solving differential equations in non-convex geometries.
method Neural network approximation of electric potential and field.
result Deep neural networks outperform shallow networks in accuracy.
This thesis tackles non-convex Bayesian learning via scalable dynamic importance sampling algorithms.
problem Non-convex Bayesian learning problem in deep neural networks.
method Replica exchange Langevin Monte Carlo, control variates method, population-chain replica exchange, scalable dynamic importance sampling.
result Control variates method reduces variance and accelerates convergence in non-convex Bayesian learning.
A new optimization method, BPM, converges linearly in non-convex, non-smooth problems.
problem Non-smooth and non-convex optimization challenges.
method Ball-Proximal Point Method (BPM), inspired by Proximal Point Method (PPM).
result BPM converges linearly and in a finite number of steps in non-convex, non-smooth problems.
The paper provides gradient estimates for Neumann semigroups on manifolds with boundary under unbounded curvature conditions.
problem Gradient estimates for Neumann semigroups on manifolds with boundary under unbounded curvature conditions.
method Establishes Bismut-type formulas and gradient estimates for Feynman--Kac semigroups on Riemannian manifolds with boundary, under geometric conditions formulated in terms of Ricci curvature and second fundamental form.
result Derives pointwise gradient estimates for the Neumann semigroup under variable, possibly unbounded, lower curvature bounds.
A new ODE model explains gradient descent dynamics near edge of stability.
problem Understanding gradient-based training over non-convex landscapes.
method Rod Flow, a new ODE approximation of GD dynamics.
result Rod Flow accurately predicts critical sharpness threshold and self-stabilization in quartic potentials.
We interpret the variational inference of the Stochastic Gradient Descent (SGD) as minimizing a new potential function named the \textit{quasi-potential}. We analytically construct the quasi-potential function in the case when the loss function is convex and admits only one global minimum point. We show in this case th…
Unified distributed SGD improves non-convex optimization for large datasets.
problem Bottleneck in scaling SGD for non-convex functions and large datasets.
method Distributed and parallel implementation of SGD (DPSGD) combining asynchronous distribution and lock-free parallelism.
result DPSGD achieves better convergence rate and speed-up with more cores and workers.
Study on convergence of exponential probability measures with applications to maximum entropy models and SGLD.
problem Characterizing the limit of probability measures with exponential densities as temperature approaches zero.
method Quantitative bounds on Wasserstein distance using geometric measure theory tools.
result Established quantitative convergence results for norm-like potentials under invertibility conditions.
Optimizes binary regression models with gradient ascent-descent methods.
problem Regression problems with binary weights in quantized learning and digital communication.
method Maximin optimization using gradient ascent-descent methods.
result The approach is optimal in linear regression with low noise and robust regression with few outliers.
Neural networks can represent complex piecewise functions efficiently.
problem Representing continuous piecewise affine functions with neural networks.
method Two hidden layers with ReLU activation, O ( p ) O(p) O ( p ) neurons for p p p pieces. result CPA functions can be represented by a neural network with linear size.
This work builds the connection between the regularity theory of optimal transportation map, Monge-Ampère equation and GANs, which gives a theoretic understanding of the major drawbacks of GANs: convergence difficulty and mode collapse. According to the regularity theory of Monge-Ampère equation, if the support of the …
Low-rank matrix is desired in many machine learning and computer vision problems. Most of the recent studies use the nuclear norm as a convex surrogate of the rank operator. However, all singular values are simply added together by the nuclear norm, and thus the rank may not be well approximated in practical problems. …
A distributed algorithm reduces communication complexity for non-convex optimization.
problem Efficiently solving non-convex optimization problems in a distributed setting.
method Parallel Restarted SPIDER algorithm, incorporating SPIDER gradient estimator.
result Achieves optimal communication complexity O ( ε − 1 ) O(ε^{-1}) O ( ε − 1 ) and optimal computation complexity. Improved reSGLD accelerates convergence in non-convex learning problems.
problem Inefficient swaps due to noisy energy estimators in reSGLD.
method Variance reduction for noisy energy estimators, theoretical analysis, and numerical experiments.
result Exponential acceleration in convergence for non-convex learning problems.
Paper analyzes convergence of decentralized algorithms with noise and bias.
problem Finite time convergence analysis of decentralized stochastic approximation schemes.
method Separated iterates into consensual parts and consensus error; bounded consensus error in terms of stationarity.
result Decentralized SA scheme converges at O ( log T / T ) {\cal O}(\log T/ \sqrt{T} ) O ( log T / T ) rate. New algorithms improve sampling from complex distributions.
problem Sampling from high-dimensional target distributions with super-linearly growing potentials.
method Proposed aHOLA and aHOLLA algorithms with non-asymptotic convergence bounds.
result Achieved state-of-the-art rates of convergence in non-convex settings.
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.
Integrates estimation and optimization for uncertain parameters.
problem Optimizing with uncertain parameters whose distributions can be estimated.
method Integrated Conditional Estimation-Optimization (ICEO) framework.
result Asymptotically consistent and provides finite performance guarantees.
We propose a new algorithm to learn a one-hidden-layer convolutional neural network where both the convolutional weights and the outputs weights are parameters to be learned. Our algorithm works for a general class of (potentially overlapping) patches, including commonly used structures for computer vision tasks. Our a…
We present Zeno, a technique to make distributed machine learning, particularly Stochastic Gradient Descent (SGD), tolerant to an arbitrary number of faulty workers. Zeno generalizes previous results that assumed a majority of non-faulty nodes; we need assume only one non-faulty worker. Our key idea is to suspect worke…
Paper studies early-stopped mirror descent for noisy sparse phase retrieval.
problem Recovering a sparse signal from noisy quadratic measurements.
method Early-stopped mirror descent with hyperbolic entropy mirror map.
result Achieves nearly minimax-optimal rate of convergence for k k k -sparse signals. Paper introduces LR to generate synthetic data with privacy protection.
problem Privacy concerns limit the use of sensitive datasets.
method Local Resampler (LR) using k-nearest neighbors algorithm.
result LR effectively mitigates outlier-driven disclosure risks.
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.
This paper explores optimising acquisition functions in Bayesian optimisation.
problem Optimising acquisition functions in Bayesian optimisation is challenging due to their non-convex nature.
method The authors derive compositional forms for acquisition functions and use them to recast maximisation as a compositional optimisation problem.
result The compositional approach to maximising acquisition functions shows empirical advantages across various tasks.
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.
Adaptive algorithms improve performance in non-convex optimization across various scenarios.
problem Improper handling of noise scales, gradient magnitudes, and smoothness in non-convex optimization.
method Design and analysis of noise-adaptive, scale-free, and generalized algorithms.
result Adaptive algorithms achieve optimal rates and performance in diverse optimization settings.
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.
ECP optimizes expensive functions without knowing Lipschitz constant.
problem Optimizing expensive, non-convex functions with unknown Lipschitz constants.
method ECP minimizes evaluations by focusing on potentially optimal regions, eliminating Lipschitz constant estimation.
result Guaranteed no-regret performance and minimax-optimal regret bounds.
CoolMomentum combines momentum and Simulated Annealing for deep learning optimization.
problem Global optimization of non-convex functions in deep learning.
method Discretized Langevin dynamics with Simulated Annealing.
result CoolMomentum achieves high accuracy on Resnet-20 on Cifar-10 and Efficientnet-B0 on Imagenet.