Nguyen, Thanh Huy, Simsekli, Umut, Gurbuzbalaban, Mert, RICHARD, Gaël

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 $\alpha$-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 L\'{e}vy 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.

Kolouri, Soheil, Nadjahi, Kimia, Simsekli, Umut, Badeau, Roland, Rohde, Gustavo

The Wasserstein distance and its variations, e.g., the sliced-Wasserstein (SW) distance, have recently drawn attention from the machine learning community. The SW distance, specifically, was shown to have similar properties to the Wasserstein distance, while being much simpler to compute, and is therefore used in various applications including generative modeling and general supervised/unsupervised learning. In this paper, we first clarify the mathematical connection between the SW distance and the Radon transform. We then utilize the generalized Radon transform to define a new family of distances for probability measures, which we call generalized sliced-Wasserstein (GSW) distances. We further show that, similar to the SW distance, the GSW distance can be extended to a maximum GSW (max-GSW) distance.

Nadjahi, Kimia, Durmus, Alain, Simsekli, Umut, Badeau, Roland

Minimum expected distance estimation (MEDE) algorithms have been widely used for probabilistic models with intractable likelihood functions and they have become increasingly popular due to their use in implicit generative modeling (e.g.\ Wasserstein generative adversarial networks, Wasserstein autoencoders). Emerging from computational optimal transport, the Sliced-Wasserstein (SW) distance has become a popular choice in MEDE thanks to its simplicity and computational benefits. While several studies have reported empirical success on generative modeling with SW, the theoretical properties of such estimators have not yet been established. In this study, we investigate the asymptotic properties of estimators that are obtained by minimizing SW. We first show that convergence in SW implies weak convergence of probability measures in general Wasserstein spaces.

Kolouri, Soheil, Nadjahi, Kimia, Simsekli, Umut, Shahrampour, Shahin

Probability metrics have become an indispensable part of modern statistics and machine learning, and they play a quintessential role in various applications, including statistical hypothesis testing and generative modeling. However, in a practical setting, the convergence behavior of the algorithms built upon these distances have not been well established, except for a few specific cases. In this paper, we introduce a broad family of probability metrics, coined as Generalized Sliced Probability Metrics (GSPMs), that are deeply rooted in the generalized Radon transform. We first verify that GSPMs are metrics. Then, we identify a subset of GSPMs that are equivalent to maximum mean discrepancy (MMD) with novel positive definite kernels, which come with a unique geometric interpretation. Finally, by exploiting this connection, we consider GSPM-based gradient flows for generative modeling applications and show that under mild assumptions, the gradient flow converges to the global optimum. We illustrate the utility of our approach on both real and synthetic problems.

Yılmaz, Kenan Y., Cemgil, Ali T., Simsekli, Umut

We derive algorithms for generalised tensor factorisation (GTF) by building upon the well-established theory of Generalised Linear Models. Our algorithms are general in the sense that we can compute arbitrary factorisations in a message passing framework, derived for a broad class of exponential family distributions including special cases such as Tweedie's distributions corresponding to $\beta$-divergences. By bounding the step size of the Fisher Scoring iteration of the GLM, we obtain general updates for real data and multiplicative updates for non-negative data. The GTF framework is, then extended easily to address the problems when multiple observed tensors are factorised simultaneously. We illustrate our coupled factorisation approach on synthetic data as well as on a musical audio restoration problem.

Durmus, Alain, Simsekli, Umut, Moulines, Eric, Badeau, Roland, RICHARD, Gaël

Stochastic Gradient Markov Chain Monte Carlo (SG-MCMC) algorithms have become increasingly popular for Bayesian inference in large-scale applications. Even though these methods have proved useful in several scenarios, their performance is often limited by their bias. In this study, we propose a novel sampling algorithm that aims to reduce the bias of SG-MCMC while keeping the variance at a reasonable level. Our approach is based on a numerical sequence acceleration method, namely the Richardson-Romberg extrapolation, which simply boils down to running almost the same SG-MCMC algorithm twice in parallel with different step sizes. We illustrate our framework on the popular Stochastic Gradient Langevin Dynamics (SGLD) algorithm and propose a novel SG-MCMC algorithm referred to as Stochastic Gradient Richardson-Romberg Langevin Dynamics (SGRRLD).

Jas, Mainak, Tour, Tom Dupré la, Simsekli, Umut, Gramfort, Alexandre

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.

Birdal, Tolga, Simsekli, Umut, Eken, Mustafa Onur, Ilic, Slobodan

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 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 solutions. We devise theoretical convergence guarantees and extensively evaluate our method on synthetic and real benchmarks. 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. Papers published at the Neural Information Processing Systems Conference.

Fallah, Alireza, Gurbuzbalaban, Mert, Ozdaglar, Asuman, Simsekli, Umut, Zhu, Lingjiong

We study distributed stochastic gradient (D-SG) method and its accelerated variant (D-ASG) for solving decentralized strongly convex stochastic optimization problems where the objective function is distributed over several computational units, lying on a fixed but arbitrary connected communication graph, subject to local communication constraints where noisy estimates of the gradients are available. We develop a framework which allows to choose the stepsize and the momentum parameters of these algorithms in a way to optimize performance by systematically trading off the bias, variance, robustness to gradient noise and dependence to network effects. When gradients do not contain noise, we also prove that distributed accelerated methods can \emph{achieve acceleration}, requiring $\mathcal{O}(\kappa \log(1/\varepsilon))$ gradient evaluations and $\mathcal{O}(\kappa \log(1/\varepsilon))$ communications to converge to the same fixed point with the non-accelerated variant where $\kappa$ is the condition number and $\varepsilon$ is the target accuracy. To our knowledge, this is the first acceleration result where the iteration complexity scales with the square root of the condition number in the context of \emph{primal} distributed inexact first-order methods. For quadratic functions, we also provide finer performance bounds that are tight with respect to bias and variance terms. Finally, we study a multistage version of D-ASG with parameters carefully varied over stages to ensure exact $\mathcal{O}(-k/\sqrt{\kappa})$ linear decay in the bias term as well as optimal $\mathcal{O}(\sigma^2/k)$ in the variance term. We illustrate through numerical experiments that our approach results in practical algorithms that are robust to gradient noise and that can outperform existing methods.

Cemgil, Ali Taylan, Kurutmaz, Mehmet Burak, Yildirim, Sinan, Barsbey, Melih, Simsekli, Umut

We introduce a dynamic generative model, Bayesian allocation model (BAM), which establishes explicit connections between nonnegative tensor factorization (NTF), graphical models of discrete probability distributions and their Bayesian extensions, and the topic models such as the latent Dirichlet allocation. BAM is based on a Poisson process, whose events are marked by using a Bayesian network, where the conditional probability tables of this network are then integrated out analytically. We show that the resulting marginal process turns out to be a Polya urn, an integer valued self-reinforcing process. This urn processes, which we name a Polya-Bayes process, obey certain conditional independence properties that provide further insight about the nature of NTF. These insights also let us develop space efficient simulation algorithms that respect the potential sparsity of data: we propose a class of sequential importance sampling algorithms for computing NTF and approximating their marginal likelihood, which would be useful for model selection. The resulting methods can also be viewed as a model scoring method for topic models and discrete Bayesian networks with hidden variables. The new algorithms have favourable properties in the sparse data regime when contrasted with variational algorithms that become more accurate when the total sum of the elements of the observed tensor goes to infinity. We illustrate the performance on several examples and numerically study the behaviour of the algorithms for various data regimes.