arXiv subjects

Umut Şimşekli

Publications and source records attributed to Umut Şimşekli.

47 records · Page 3Linked to original sources

arXiv2019

First Exit Time Analysis of Stochastic Gradient Descent Under Heavy-Tailed Gradient Noise

Stochastic gradient descent (SGD) has been widely used in machine learning due to its computational efficiency and favorable generalization properties. Recently, it has been empirically demonstrated that the gradient noise in several deep learning settings admits a non-Gaussian, heavy-tailed behavior. This suggests that the gradient noise can be modeled by using $α$-stable distributions, a family of heavy-tailed distributions that appear in the generalized central limit theorem. In this context, SGD can be viewed as a discretization of a stochastic differential equation (SDE) driven by a Lévy motion, and the metastability results for this SDE can then be used for illuminating the behavior of SGD, especially in terms of `preferring wide minima'. While this approach brings a new perspective for analyzing SGD, it is limited in the sense that, due to the time discretization, SGD might admit a significantly different behavior than its continuous-time limit. Intuitively, the behaviors of these two systems are expected to be similar to each other only when the discretization step is sufficiently small; however, to the best of our knowledge, there is no theoretical understanding on how small the step-size should be chosen in order to guarantee that the discretized system inherits the properties of the continuous-time system. In this study, we provide formal theoretical analysis where we derive explicit conditions for the step-size such that the metastability behavior of the discrete-time system is similar to its continuous-time limit. We show that the behaviors of the two systems are indeed similar for small step-sizes and we identify how the error depends on the algorithm and problem parameters. We illustrate our results with simulations on a synthetic model and neural networks.

stat.ML↗

arXiv2019

Sliced-Wasserstein Flows: Nonparametric Generative Modeling via Optimal Transport and Diffusions

By building upon the recent theory that established the connection between implicit generative modeling (IGM) and optimal transport, in this study, we propose a novel parameter-free algorithm for learning the underlying distributions of complicated datasets and sampling from them. The proposed algorithm is based on a functional optimization problem, which aims at finding a measure that is close to the data distribution as much as possible and also expressive enough for generative modeling purposes. We formulate the problem as a gradient flow in the space of probability measures. The connections between gradient flows and stochastic differential equations let us develop a computationally efficient algorithm for solving the optimization problem. We provide formal theoretical analysis where we prove finite-time error guarantees for the proposed algorithm. To the best of our knowledge, the proposed algorithm is the first nonparametric IGM algorithm with explicit theoretical guarantees. Our experimental results support our theory and show that our algorithm is able to successfully capture the structure of different types of data distributions.

stat.ML↗

arXiv2019

Probabilistic Permutation Synchronization using the Riemannian Structure of the Birkhoff Polytope

We present an entirely new geometric and probabilistic approach to synchronization of correspondences across multiple sets of objects or images. In particular, we present two algorithms: (1) Birkhoff-Riemannian L-BFGS for optimizing the relaxed version of the combinatorially intractable cycle consistency loss in a principled manner, (2) Birkhoff-Riemannian Langevin Monte Carlo for generating samples on the Birkhoff Polytope and estimating the confidence of the found solutions. To this end, we first introduce the very recently developed Riemannian geometry of the Birkhoff Polytope. Next, we introduce a new probabilistic synchronization model in the form of a Markov Random Field (MRF). Finally, based on the first order retraction operators, we formulate our problem as simulating a stochastic differential equation and devise new integrators. We show on both synthetic and real datasets that we achieve high quality multi-graph matching results with faster convergence and reliable confidence/uncertainty estimates.

cs.CV↗

arXiv2019

Bayesian Pose Graph Optimization via Bingham Distributions and Tempered Geodesic MCMC

We introduce Tempered Geodesic Markov Chain Monte Carlo (TG-MCMC) algorithm for initializing pose graph optimization problems, arising in various scenarios such as SFM (structure from motion) or SLAM (simultaneous localization and mapping). TG-MCMC is first of its kind as it unites asymptotically global non-convex optimization on the spherical manifold of quaternions with posterior sampling, in order to provide both reliable initial poses and uncertainty estimates that are informative about the quality of individual solutions. We devise rigorous theoretical convergence guarantees for our method and extensively evaluate it on synthetic and real benchmark datasets. Besides its elegance in formulation and theory, we show that our method is robust to missing data, noise and the estimated uncertainties capture intuitive properties of the data.

cs.CV↗

arXiv2019

Non-Asymptotic Analysis of Fractional Langevin Monte Carlo for Non-Convex Optimization

Recent studies on diffusion-based sampling methods have shown that Langevin Monte Carlo (LMC) algorithms can be beneficial for non-convex optimization, and rigorous theoretical guarantees have been proven for both asymptotic and finite-time regimes. Algorithmically, LMC-based algorithms resemble the well-known gradient descent (GD) algorithm, where the GD recursion is perturbed by an additive Gaussian noise whose variance has a particular form. Fractional Langevin Monte Carlo (FLMC) is a recently proposed extension of LMC, where the Gaussian noise is replaced by a heavy-tailed α-stable noise. As opposed to its Gaussian counterpart, these heavy-tailed perturbations can incur large jumps and it has been empirically demonstrated that the choice of α-stable noise can provide several advantages in modern machine learning problems, both in optimization and sampling contexts. However, as opposed to LMC, only asymptotic convergence properties of FLMC have been yet established. In this study, we analyze the non-asymptotic behavior of FLMC for non-convex optimization and prove finite-time bounds for its expected suboptimality. Our results show that the weak-error of FLMC increases faster than LMC, which suggests using smaller step-sizes in FLMC. We finally extend our results to the case where the exact gradients are replaced by stochastic gradients and show that similar results hold in this setting as well.

math.OC↗

arXiv2018

Asynchronous Stochastic Quasi-Newton MCMC for Non-Convex Optimization

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 stochastic L-BFGS algorithm for non-convex optimization. The proposed algorithm is suitable for both distributed and shared-memory settings. We provide formal theoretical analysis and show that the proposed method achieves an ergodic convergence rate of ${\cal O}(1/\sqrt{N})$ ($N$ being the total number of iterations) and it can achieve a linear speedup under certain conditions. We perform several experiments on both synthetic and real datasets. The results support our theory and show that the proposed algorithm provides a significant speedup over the recently proposed synchronous distributed L-BFGS algorithm.

stat.ML↗

arXiv2017

HAMSI: A Parallel Incremental Optimization Algorithm Using Quadratic Approximations for Solving Partially Separable Problems

We propose HAMSI (Hessian Approximated Multiple Subsets Iteration), which is a provably convergent, second order incremental algorithm for solving large-scale partially separable optimization problems. The algorithm is based on a local quadratic approximation, and hence, allows incorporating curvature information to speed-up the convergence. HAMSI is inherently parallel and it scales nicely with the number of processors. Combined with techniques for effectively utilizing modern parallel computer architectures, we illustrate that the proposed method converges more rapidly than a parallel stochastic gradient descent when both methods are used to solve large-scale matrix factorization problems. This performance gain comes only at the expense of using memory that scales linearly with the total size of the optimization variables. We conclude that HAMSI may be considered as a viable alternative in many large scale problems, where first order methods based on variants of stochastic gradient descent are applicable.

stat.ML↗

arXiv2017

Learning the Morphology of Brain Signals Using Alpha-Stable Convolutional Sparse Coding

Neural time-series data contain a wide variety of prototypical signal waveforms (atoms) that are of significant importance in clinical and cognitive research. One of the goals for analyzing such data is hence to extract such 'shift-invariant' atoms. Even though some success has been reported with existing algorithms, they are limited in applicability due to their heuristic nature. Moreover, they are often vulnerable to artifacts and impulsive noise, which are typically present in raw neural recordings. In this study, we address these issues and propose a novel probabilistic convolutional sparse coding (CSC) model for learning shift-invariant atoms from raw neural signals containing potentially severe artifacts. In the core of our model, which we call $α$CSC, lies a family of heavy-tailed distributions called $α$-stable distributions. We develop a novel, computationally efficient Monte Carlo expectation-maximization algorithm for inference. The maximization step boils down to a weighted CSC problem, for which we develop a computationally efficient optimization algorithm. Our results show that the proposed algorithm achieves state-of-the-art convergence speeds. Besides, $α$CSC is significantly more robust to artifacts when compared to three competing algorithms: it can extract spike bursts, oscillations, and even reveal more subtle phenomena such as cross-frequency coupling when applied to noisy neural time series.

stat.ML↗

arXiv2017

Fractional Langevin Monte Carlo: Exploring Lévy Driven Stochastic Differential Equations for Markov Chain Monte Carlo

Along with the recent advances in scalable Markov Chain Monte Carlo methods, sampling techniques that are based on Langevin diffusions have started receiving increasing attention. These so called Langevin Monte Carlo (LMC) methods are based on diffusions driven by a Brownian motion, which gives rise to Gaussian proposal distributions in the resulting algorithms. Even though these approaches have proven successful in many applications, their performance can be limited by the light-tailed nature of the Gaussian proposals. In this study, we extend classical LMC and develop a novel Fractional LMC (FLMC) framework that is based on a family of heavy-tailed distributions, called $α$-stable Lévy distributions. As opposed to classical approaches, the proposed approach can possess large jumps while targeting the correct distribution, which would be beneficial for efficient exploration of the state space. We develop novel computational methods that can scale up to large-scale problems and we provide formal convergence analysis of the proposed scheme. Our experiments support our theory: FLMC can provide superior performance in multi-modal settings, improved convergence rates, and robustness to algorithm parameters.

stat.CO↗

arXiv2016

Stochastic Quasi-Newton Langevin Monte Carlo

Recently, Stochastic Gradient Markov Chain Monte Carlo (SG-MCMC) methods have been proposed for scaling up Monte Carlo computations to large data problems. Whilst these approaches have proven useful in many applications, vanilla SG-MCMC might suffer from poor mixing rates when random variables exhibit strong couplings under the target densities or big scale differences. In this study, we propose a novel SG-MCMC method that takes the local geometry into account by using ideas from Quasi-Newton optimization methods. These second order methods directly approximate the inverse Hessian by using a limited history of samples and their gradients. Our method uses dense approximations of the inverse Hessian while keeping the time and memory complexities linear with the dimension of the problem. We provide a formal theoretical analysis where we show that the proposed method is asymptotically unbiased and consistent with the posterior expectations. We illustrate the effectiveness of the approach on both synthetic and real datasets. Our experiments on two challenging applications show that our method achieves fast convergence rates similar to Riemannian approaches while at the same time having low computational requirements similar to diagonal preconditioning approaches.

stat.ML↗

arXiv2015

Parallel Stochastic Gradient Markov Chain Monte Carlo for Matrix Factorisation Models

For large matrix factorisation problems, we develop a distributed Markov Chain Monte Carlo (MCMC) method based on stochastic gradient Langevin dynamics (SGLD) that we call Parallel SGLD (PSGLD). PSGLD has very favourable scaling properties with increasing data size and is comparable in terms of computational requirements to optimisation methods based on stochastic gradient descent. PSGLD achieves high performance by exploiting the conditional independence structure of the MF models to sub-sample data in a systematic manner as to allow parallelisation and distributed computation. We provide a convergence proof of the algorithm and verify its superior performance on various architectures such as Graphics Processing Units, shared memory multi-core systems and multi-computer clusters.

stat.ML↗