arXiv subjects

Murat A. Erdogdu

Publications and source records attributed to Murat A. Erdogdu.

49 records · Page 3Linked to original sources

arXiv2021

Convergence Rates of Stochastic Gradient Descent under Infinite Noise Variance

Recent studies have provided both empirical and theoretical evidence illustrating that heavy tails can emerge in stochastic gradient descent (SGD) in various scenarios. Such heavy tails potentially result in iterates with diverging variance, which hinders the use of conventional convergence analysis techniques that rely on the existence of the second-order moments. In this paper, we provide convergence guarantees for SGD under a state-dependent and heavy-tailed noise with a potentially infinite variance, for a class of strongly convex objectives. In the case where the $p$-th moment of the noise exists for some $p\in [1,2)$, we first identify a condition on the Hessian, coined '$p$-positive (semi-)definiteness', that leads to an interesting interpolation between positive semi-definite matrices ($p=2$) and diagonally dominant matrices with non-negative diagonal entries ($p=1$). Under this condition, we then provide a convergence rate for the distance to the global optimum in $L^p$. Furthermore, we provide a generalized central limit theorem, which shows that the properly scaled Polyak-Ruppert averaging converges weakly to a multivariate $α$-stable random vector. Our results indicate that even under heavy-tailed noise with infinite variance, SGD can converge to the global optimum without necessitating any modification neither to the loss function or to the algorithm itself, as typically required in robust statistics. We demonstrate the implications of our results to applications such as linear regression and generalized linear models subject to heavy-tailed data.

math.OC↗

arXiv2020

An Analysis of Constant Step Size SGD in the Non-convex Regime: Asymptotic Normality and Bias

Structured non-convex learning problems, for which critical points have favorable statistical properties, arise frequently in statistical machine learning. Algorithmic convergence and statistical estimation rates are well-understood for such problems. However, quantifying the uncertainty associated with the underlying training algorithm is not well-studied in the non-convex setting. In order to address this shortcoming, in this work, we establish an asymptotic normality result for the constant step size stochastic gradient descent (SGD) algorithm--a widely used algorithm in practice. Specifically, based on the relationship between SGD and Markov Chains [DDB19], we show that the average of SGD iterates is asymptotically normally distributed around the expected value of their unique invariant distribution, as long as the non-convex and non-smooth objective function satisfies a dissipativity property. We also characterize the bias between this expected value and the critical points of the objective function under various local regularity conditions. Together, the above two results could be leveraged to construct confidence intervals for non-convex problems that are trained using the SGD algorithm.

stat.ML↗

arXiv2020

On the Convergence of Langevin Monte Carlo: The Interplay between Tail Growth and Smoothness

We study sampling from a target distribution ${ν_* = e^{-f}}$ using the unadjusted Langevin Monte Carlo (LMC) algorithm. For any potential function $f$ whose tails behave like ${\|x\|^α}$ for ${α\in [1,2]}$, and has $β$-Hölder continuous gradient, we prove that ${\widetilde{\mathcal{O}} \Big(d^{\frac{1}β+\frac{1+β}β(\frac{2}α - \boldsymbol{1}_{\{α\neq 1\}})} ε^{-\frac{1}β}\Big)}$ steps are sufficient to reach the $ε$-neighborhood of a $d$-dimensional target distribution $ν_*$ in KL-divergence. This convergence rate, in terms of $ε$ dependency, is not directly influenced by the tail growth rate $α$ of the potential function as long as its growth is at least linear, and it only relies on the order of smoothness $β$. One notable consequence of this result is that for potentials with Lipschitz gradient, i.e. $β=1$, our rate recovers the best known rate ${\widetilde{\mathcal{O}}(dε^{-1})}$ which was established for strongly convex potentials in terms of $ε$ dependency, but we show that the same rate is achievable for a wider class of potentials that are degenerately convex at infinity. The growth rate $α$ starts to have an effect on the established rate in high dimensions where $d$ is large; furthermore, it recovers the best-known dimension dependency when the tail growth of the potential is quadratic, i.e. ${α= 2}$, in the current setup. Our framework allows for finite perturbations, and any order of smoothness ${β\in(0,1]}$; consequently, our results are applicable to a wide class of non-convex potentials that are weakly smooth and exhibit at least linear tail growth.

stat.ML↗

arXiv2020

Stochastic Runge-Kutta Accelerates Langevin Monte Carlo and Beyond

Sampling with Markov chain Monte Carlo methods often amounts to discretizing some continuous-time dynamics with numerical integration. In this paper, we establish the convergence rate of sampling algorithms obtained by discretizing smooth Itô diffusions exhibiting fast Wasserstein-$2$ contraction, based on local deviation properties of the integration scheme. In particular, we study a sampling algorithm constructed by discretizing the overdamped Langevin diffusion with the method of stochastic Runge-Kutta. For strongly convex potentials that are smooth up to a certain order, its iterates converge to the target distribution in $2$-Wasserstein distance in $\tilde{\mathcal{O}}(dε^{-2/3})$ iterations. This improves upon the best-known rate for strongly log-concave sampling based on the overdamped Langevin equation using only the gradient oracle without adjustment. In addition, we extend our analysis of stochastic Runge-Kutta methods to uniformly dissipative diffusions with possibly non-convex potentials and show they achieve better rates compared to the Euler-Maruyama scheme in terms of the dependence on tolerance $ε$. Numerical studies show that these algorithms lead to better stability and lower asymptotic errors.

stat.ML↗

arXiv2019

Global Non-convex Optimization with Discretized Diffusions

An Euler discretization of the Langevin diffusion is known to converge to the global minimizers of certain convex and non-convex optimization problems. We show that this property holds for any suitably smooth diffusion and that different diffusions are suitable for optimizing different classes of convex and non-convex functions. This allows us to design diffusions suitable for globally optimizing convex and non-convex functions not covered by the existing Langevin theory. Our non-asymptotic analysis delivers computable optimization and integration error bounds based on easily accessed properties of the objective and chosen diffusion. Central to our approach are new explicit Stein factor bounds on the solutions of Poisson equations. We complement these results with improved optimization guarantees for targets other than the standard Gibbs measure.

stat.ML↗

arXiv2019

Convergence Rate of Block-Coordinate Maximization Burer-Monteiro Method for Solving Large SDPs

Semidefinite programming (SDP) with diagonal constraints arise in many optimization problems, such as Max-Cut, community detection and group synchronization. Although SDPs can be solved to arbitrary precision in polynomial time, generic convex solvers do not scale well with the dimension of the problem. In order to address this issue, Burer and Monteiro proposed to reduce the dimension of the problem by appealing to a low-rank factorization and solve the subsequent non-convex problem instead. In this paper, we present coordinate ascent based methods to solve this non-convex problem with provable convergence guarantees. More specifically, we prove that the block-coordinate maximization algorithm applied to the non-convex Burer-Monteiro method globally converges to a first-order stationary point with a sublinear rate without any assumptions on the problem. We further show that this algorithm converges linearly around a local maximum provided that the objective function exhibits quadratic decay. We establish that this condition generically holds when the rank of the factorization is sufficiently large. Furthermore, incorporating Lanczos method to the block-coordinate maximization, we propose an algorithm that is guaranteed to return a solution that provides $1-O(1/r)$ approximation to the original SDP without any assumptions, where $r$ is the rank of the factorization. This approximation ratio is known to be optimal (up to constants) under the unique games conjecture, and we can explicitly quantify the number of iterations to obtain such a solution.

math.OC↗

arXiv2019

Normal Approximation for Stochastic Gradient Descent via Non-Asymptotic Rates of Martingale CLT

We provide non-asymptotic convergence rates of the Polyak-Ruppert averaged stochastic gradient descent (SGD) to a normal random vector for a class of twice-differentiable test functions. A crucial intermediate step is proving a non-asymptotic martingale central limit theorem (CLT), i.e., establishing the rates of convergence of a multivariate martingale difference sequence to a normal random vector, which might be of independent interest. We obtain the explicit rates for the multivariate martingale CLT using a combination of Stein's method and Lindeberg's argument, which is then used in conjunction with a non-asymptotic analysis of averaged SGD proposed in [PJ92]. Our results have potentially interesting consequences for computing confidence intervals for parameter estimation with SGD and constructing hypothesis tests with SGD that are valid in a non-asymptotic sense.

math.ST↗

arXiv2017

Inference in Graphical Models via Semidefinite Programming Hierarchies

Maximum A posteriori Probability (MAP) inference in graphical models amounts to solving a graph-structured combinatorial optimization problem. Popular inference algorithms such as belief propagation (BP) and generalized belief propagation (GBP) are intimately related to linear programming (LP) relaxation within the Sherali-Adams hierarchy. Despite the popularity of these algorithms, it is well understood that the Sum-of-Squares (SOS) hierarchy based on semidefinite programming (SDP) can provide superior guarantees. Unfortunately, SOS relaxations for a graph with $n$ vertices require solving an SDP with $n^{Θ(d)}$ variables where $d$ is the degree in the hierarchy. In practice, for $d\ge 4$, this approach does not scale beyond a few tens of variables. In this paper, we propose binary SDP relaxations for MAP inference using the SOS hierarchy with two innovations focused on computational efficiency. Firstly, in analogy to BP and its variants, we only introduce decision variables corresponding to contiguous regions in the graphical model. Secondly, we solve the resulting SDP using a non-convex Burer-Monteiro style method, and develop a sequential rounding procedure. We demonstrate that the resulting algorithm can solve problems with tens of thousands of variables within minutes, and outperforms BP and GBP on practical problems such as image denoising and Ising spin glasses. Finally, for specific graph types, we establish a sufficient condition for the tightness of the proposed partial SOS relaxation.

stat.ML↗

arXiv2016

Scalable Approximations for Generalized Linear Problems

In stochastic optimization, the population risk is generally approximated by the empirical risk. However, in the large-scale setting, minimization of the empirical risk may be computationally restrictive. In this paper, we design an efficient algorithm to approximate the population risk minimizer in generalized linear problems such as binary classification with surrogate losses and generalized linear regression models. We focus on large-scale problems, where the iterative minimization of the empirical risk is computationally intractable, i.e., the number of observations $n$ is much larger than the dimension of the parameter $p$, i.e. $n \gg p \gg 1$. We show that under random sub-Gaussian design, the true minimizer of the population risk is approximately proportional to the corresponding ordinary least squares (OLS) estimator. Using this relation, we design an algorithm that achieves the same accuracy as the empirical risk minimizer through iterations that attain up to a cubic convergence rate, and that are cheaper than any batch optimization algorithm by at least a factor of $\mathcal{O}(p)$. We provide theoretical guarantees for our algorithm, and analyze the convergence behavior in terms of data dimensions. Finally, we demonstrate the performance of our algorithm on well-known classification and regression problems, through extensive numerical studies on large-scale datasets, and show that it achieves the highest performance compared to several other widely used and specialized optimization algorithms.

stat.ML↗

arXiv2015

Convergence rates of sub-sampled Newton methods

We consider the problem of minimizing a sum of $n$ functions over a convex parameter set $\mathcal{C} \subset \mathbb{R}^p$ where $n\gg p\gg 1$. In this regime, algorithms which utilize sub-sampling techniques are known to be effective. In this paper, we use sub-sampling techniques together with low-rank approximation to design a new randomized batch algorithm which possesses comparable convergence rate to Newton's method, yet has much smaller per-iteration cost. The proposed algorithm is robust in terms of starting point and step size, and enjoys a composite convergence rate, namely, quadratic convergence at start and linear convergence when the iterate is close to the minimizer. We develop its theoretical analysis which also allows us to select near-optimal algorithm parameters. Our theoretical results can be used to obtain convergence rates of previously proposed sub-sampling based algorithms as well. We demonstrate how our results apply to well-known machine learning problems. Lastly, we evaluate the performance of our algorithm on several datasets under various scenarios.

stat.ML↗

arXiv2015

Newton-Stein Method: An optimization method for GLMs via Stein's Lemma

We consider the problem of efficiently computing the maximum likelihood estimator in Generalized Linear Models (GLMs) when the number of observations is much larger than the number of coefficients ($n \gg p \gg 1$). In this regime, optimization algorithms can immensely benefit from approximate second order information. We propose an alternative way of constructing the curvature information by formulating it as an estimation problem and applying a Stein-type lemma, which allows further improvements through sub-sampling and eigenvalue thresholding. Our algorithm enjoys fast convergence rates, resembling that of second order methods, with modest per-iteration cost. We provide its convergence analysis for the general case where the rows of the design matrix are samples from a sub-gaussian distribution. We show that the convergence has two phases, a quadratic phase followed by a linear phase. Finally, we empirically demonstrate that our algorithm achieves the highest performance compared to various algorithms on several datasets.

stat.ML↗

arXiv2015

Flexible results for quadratic forms with applications to variance components estimation

We derive convenient uniform concentration bounds and finite sample multivariate normal approximation results for quadratic forms, then describe some applications involving variance components estimation in linear random-effects models. Random-effects models and variance components estimation are classical topics in statistics, with a corresponding well-established asymptotic theory. However, our finite sample results for quadratic forms provide additional flexibility for easily analyzing random-effects models in non-standard settings, which are becoming more important in modern applications (e.g. genomics). For instance, in addition to deriving novel non-asymptotic bounds for variance components estimators in classical linear random-effects models, we provide a concentration bound for variance components estimators in linear models with correlated random-effects. Our general concentration bound is a uniform version of the Hanson-Wright inequality. The main normal approximation result in the paper is derived using Reinert and Röllin's (2009) embedding technique and multivariate Stein's method with exchangeable pairs.

math.ST↗

arXiv2015

SEISMIC: A Self-Exciting Point Process Model for Predicting Tweet Popularity

Social networking websites allow users to create and share content. Big information cascades of post resharing can form as users of these sites reshare others' posts with their friends and followers. One of the central challenges in understanding such cascading behaviors is in forecasting information outbreaks, where a single post becomes widely popular by being reshared by many users. In this paper, we focus on predicting the final number of reshares of a given post. We build on the theory of self-exciting point processes to develop a statistical model that allows us to make accurate predictions. Our model requires no training or expensive feature engineering. It results in a simple and efficiently computable formula that allows us to answer questions, in real-time, such as: Given a post's resharing history so far, what is our current estimate of its final number of reshares? Is the post resharing cascade past the initial stage of explosive growth? And, which posts will be the most reshared in the future? We validate our model using one month of complete Twitter data and demonstrate a strong improvement in predictive accuracy over existing approaches. Our model gives only 15% relative error in predicting final size of an average information cascade after observing it for just one hour.

cs.SI↗