New sampling method guarantees approximate first-order stationary points for non-convex functions.
problem Sampling from non-log-concave densities with non-convex potential functions.
method Averaged Langevin Monte Carlo with complexity analysis.
result Langevin Monte Carlo outputs a sample with ε-relative Fisher information after O(L²d²/ε²) iterations.
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.
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.
New methods improve convergence in non-convex non-smooth learning problems.
problem Sparse learning from high-dimensional data with non-convex, non-smooth regularizers.
method Stochastic proximal gradient methods with arbitrary sampling.
result Independent sampling improves performance over uniform sampling.
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.
New sampler tackles complex discrete energy landscapes efficiently.
problem Stagnation in gradient-based discrete samplers for non-convex settings.
method DREXEL sampler with Replica Exchange and Adjusted Metropolis.
result Proves samplers satisfy detailed balance and converge to target distribution.
New findings show sampling vs optimization are incomparable in non-convex cases.
problem Comparing sampling and optimization in non-convex Bayesian learning.
method Simpler and stronger separation of sampling and optimization.
result Provable incomparability of sampling and optimization in non-convex cases.
New SGD method uses adaptive sampling to converge faster in non-convex problems.
problem Non-convex optimization problems with noisy gradients.
method Adaptive coordinate sampling in stochastic gradient descent (SGD).
result Almost sure convergence and non-asymptotic bounds established.
Safe sample screening improves RSVM performance without sacrificing accuracy.
problem Improving RSVM performance under noisy conditions.
method Proposed two safe sample screening rules based on CCCP framework for RSVM.
result Significant reduction in computational time for RSVM.
Meta-learning can perform well on non-convex models even with few samples, contrary to convex models.
problem Understanding the sample complexity of meta-learning for non-convex models.
method Constructing a simple meta-learning instance and analyzing the training dynamics of Reptile and multi-task representation learning.
result Meta-learning can achieve new task sample complexity of O ( 1 ) \mathcal{O}(1) O ( 1 ) for non-convex models, unlike convex models which require Ω ( d ) Ω(d) Ω ( d ) samples. New analysis for sampling from non-convex distributions with dependent data.
problem Sampling from non-logconcave distributions in stochastic optimization.
method Stochastic Gradient Langevin Dynamics (SGLD) with dependent data streams.
result Sharper and uniform convergence estimates in L 1 L^1 L 1 -Wasserstein distance. 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 algorithm reduces sample and communication complexity for non-convex optimization problems.
problem Decentralized non-convex optimization with high sample sizes and communication costs.
method D-GET: joint gradient estimation and tracking for decentralized learning.
result Achieves improved sample and communication complexities for non-convex problems.
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 i l d e O ( ε − 3 ) ilde{\mathcal{O}}(ε^{-3}) i l d e O ( ε − 3 ) . We consider the minimization of non-convex functions that typically arise in machine learning. Specifically, we focus our attention on a variant of trust region methods known as cubic regularization. This approach is particularly attractive because it escapes strict saddle points and it provides stronger convergence gu…
Unified analysis improves SAM for non-convex optimization.
problem Improving generalization in machine learning models.
method Sharpness-aware minimization (SAM) and Unified SAM.
result Unified SAM provides convergence guarantees under relaxed assumptions.
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.
Paper develops robust SGLD for solving non-convex DRO problems.
problem Solving non-convex distributionally robust optimisation problems with adversarially corrupted samples.
method Developed a Stochastic Gradient Langevin Dynamics (SGLD) algorithm with non-asymptotic convergence bounds.
result The robust SGLD estimator outperforms vanilla SGLD in terms of test accuracy.
A new algorithm, Regular Tree Search, tackles non-convex simulation optimization problems.
problem Non-convex objective functions in simulation optimization.
method Integrates adaptive sampling with recursive partitioning of the search space.
result Proves global convergence and reliably identifies the global optimum.
This paper tackles non-convex phase retrieval with structured assumptions.
problem Phase retrieval with limited measurements and structure assumptions.
method Non-convex approaches with sample complexity guarantees.
result Sample-efficient recovery with structured signals/images.
A new method improves SVI for high-dimensional, poorly-conditioned distributions.
problem Challenges in existing SVI methods for high-dimensional, poorly-conditioned distributions.
method Trust-region optimization approach leveraging conditional independences and second-order information.
result Superior numerical performance and better scalability in high-dimensional distributions.
New bounds for SGLD show error decreases with more data.
problem Establishing generalization error bounds for SGLD in non-convex settings.
method Using dissipativity, smoothness, and uniform stability, time-independent bounds are derived.
result Error bounds decay to zero as sample size increases.
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.
Improved bounds for MALA in non-convex sampling problems.
problem Sampling from non-convex distributions in high dimensions.
method Metropolis-adjusted Langevin algorithm (MALA) with improved bounds.
result MALA is faster than competitors in many challenging scenarios.
Paper proposes DP-SGD and DP-NSGD for differentially private non-convex optimization.
problem Mitigating privacy risks in large model learning.
method Clip or normalize per-sample gradients and add noise for differential privacy.
result Achieved convergence rate of gradient norm for non-convex optimization.
New algorithm finds local minima in non-convex problems efficiently.
problem Finding local minima in non-convex finite-sum minimization problems.
method Stochastic Trust Region (STR) algorithm combining inexact gradient and Hessian estimation.
result STR finds ( ε , ε ) (ε, \sqrtε) ( ε , ε ) -approximate local minimum with improved efficiency. 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.
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.
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.
Novel bounds for SGLD show generalization error decreases with more samples.
problem Understanding the generalization error of SGLD in non-convex optimization.
method Information-theoretic approach focusing on Kullback-Leibler divergence and sub-exponential loss function.
result Time-independent generalization bounds for SGLD, independent of step size and number of iterations.
Study analyzes FLMC for non-convex optimization with finite-time bounds.
problem Non-convex optimization challenges in machine learning.
method Fractional Langevin Monte Carlo (FLMC) with α-stable noise.
result Finite-time bounds for expected suboptimality of FLMC.
Subspace learning and matrix factorization problems have great many applications in science and engineering, and efficient algorithms are critical as dataset sizes continue to grow. Many relevant problem formulations are non-convex, and in a variety of contexts it has been observed that solving the non-convex problem d…
We investigate 1) the rate at which refined properties of the empirical risk---in particular, gradients---converge to their population counterparts in standard non-convex learning tasks, and 2) the consequences of this convergence for optimization. Our analysis follows the tradition of norm-based capacity control. We p…
Paper develops momentum schemes with variance reduction for non-convex composition optimization.
problem Lack of convergence guarantee and efficient momentum design in existing algorithms.
method Develops various momentum schemes with SPIDER-based variance reduction.
result Achieves near-optimal sample complexity and linear convergence rate.
We consider variants of trust-region and cubic regularization methods for non-convex optimization, in which the Hessian matrix is approximated. Under mild conditions on the inexact Hessian, and using approximate solution of the corresponding sub-problems, we provide iteration complexity to achieve ε ε ε -approximate seco…
SGD converges to global minimum for structured non-convex functions.
problem Optimizing non-convex functions using SGD with slow convergence rates.
method Convergence theorems for SGD on structured non-convex functions, including Quasar and PL conditions.
result SGD converges to global minimum for specific non-convex functions under certain conditions.
Paper proposes a weak approximation of reflection coupling for non-convex optimization.
problem Non-convex optimization problems with different drift terms.
method Proposes an approximate reflection coupling (ARC) for stochastic differential equations (SDEs).
result ARC converges weakly to the reflection coupling and can be applied to non-convex optimization.
Online SGD achieves consistent estimation in high-dimensional non-convex inference tasks.
problem Consistent estimation in high-dimensional non-convex optimization problems.
method Online stochastic gradient descent (SGD) on non-convex losses.
result Nearly sharp thresholds for sample complexity in high-dimensional settings.
Study on policy testing in MDPs with lower bounds and new algorithm.
problem Deciding if policy value exceeds a threshold with limited samples.
method Derived lower bound, proposed new algorithm, reformulated problem, used policy optimization in reversed MDP.
result New algorithm outperforms existing methods in policy testing.
Improved DP algorithms for non-convex optimization with tighter generalization bounds.
problem Private stochastic non-convex optimization in high-dimensional spaces.
method Differential privacy techniques, including adaptive algorithms like DP RMSProp and DP Adam, combined with adaptive data analysis.
result Achieved a sharper rate of p 4 / n \sqrt[4]{p}/\sqrt{n} 4 p / n for population loss, improving upon previous bounds. Kernelized convex clustering handles non-linear and non-convex data.
problem Lack of effective clustering methods for non-linear and non-convex data.
method Kernelized convex clustering in RKHS.
result Superior performance compared to state-of-the-art techniques.
Paper analyzes Greedy-GQ for reinforcement learning with Markovian noise.
problem Analyzing Greedy-GQ for reinforcement learning with Markovian noise.
method Develops finite-sample analysis for Greedy-GQ with linear function approximation under Markovian noise.
result Provides theoretical justification for choosing stepsizes for faster convergence.
Most high-dimensional estimation and prediction methods propose to minimize a cost function (empirical risk) that is written as a sum of losses associated to each data point. In this paper we focus on the case of non-convex losses, which is practically important but still poorly understood. Classical empirical process …
This work studies the strong duality of non-convex matrix factorization problems: we show that under certain dual conditions, these problems and its dual have the same optimum. This has been well understood for convex optimization, but little was known for non-convex problems. We propose a novel analytical framework an…
Faster, better sparse model estimation for large datasets.
problem Sparse model estimation for large datasets with millions of samples and features.
method Coordinate descent, working sets, Anderson acceleration.
result Significantly faster and more efficient than state-of-the-art algorithms.
Two algorithms find optimal points in decentralized optimization.
problem Decentralized non-convex stochastic optimization with composite objective functions.
method Prox-DASA and Prox-DASA-GT algorithms for finding ε-stationary points.
result Achieves comparable complexity without large batch sizes or complex per-iteration operations.
New framework for robust hypothesis testing using Sinkhorn uncertainty sets.
problem Non-convex robust hypothesis testing problem.
method Exact mixed-integer exponential conic reformulation and convex approximation.
result Satisfactory testing performance and computational efficiency.
New algorithm guarantees optimal convergence rate for stochastic optimization.
problem Optimal convergence rate for stochastic optimization algorithms.
method Regularized versions of Minimization by Incremental Surrogate Optimization (MISO) with arbitrary recurrent data sampling.
result Expected optimality gap converges at O ( n − 1 / 2 ) O(n^{-1/2}) O ( n − 1/2 ) under general recurrent sampling schemes.