We study convergence properties of Stochastic Gradient Descent (SGD) for convex objectives without assumptions on smoothness or strict convexity. We consider the question of establishing that with high probability the objective evaluated at the candidate minimizer returned by SGD is close to the minimal value of the objective. We compare this result concerning the final candidate minimzer (i.e. the final model parameters learned after all gradient steps) to the online learning techniques of [Zin03] that take a rolling average of the model parameters at the different steps of SGD.

Large models are prevalent in modern machine learning scenarios, including deep learning, recommender systems, etc., which can have millions or even billions of parameters. Parallel algorithms have become an essential solution technique to many large-scale machine learning jobs. In this paper, we propose a model parallel proximal stochastic gradient algorithm, AsyB-ProxSGD, to deal with large models using model parallel blockwise updates while in the meantime handling a large amount of training data using proximal stochastic gradient descent (ProxSGD). In our algorithm, worker nodes communicate with the parameter servers asynchronously, and each worker performs proximal stochastic gradient for only one block of model parameters during each iteration. Our proposed algorithm generalizes ProxSGD to the asynchronous and model parallel setting. We prove that AsyB-ProxSGD achieves a convergence rate of $O(1/\sqrt{K})$ to stationary points for nonconvex problems under \emph{constant} minibatch sizes, where $K$ is the total number of block updates. This rate matches the best-known rates of convergence for a wide range of gradient-like algorithms. Furthermore, we show that when the number of workers is bounded by $O(K^{1/4})$, we can expect AsyB-ProxSGD to achieve linear speedup as the number of workers increases. We implement the proposed algorithm on MXNet and demonstrate its convergence behavior and near-linear speedup on a real-world dataset involving both a large model size and large amounts of data.

Logistic regression is one of the most popular methods in binary classification, wherein estimation of model parameters is carried out by solving the maximum likelihood (ML) optimization problem, and the ML estimator is defined to be the optimal solution of this problem. It is well known that the ML estimator exists when the data is non-separable, but fails to exist when the data is separable. First-order methods are the algorithms of choice for solving large-scale instances of the logistic regression problem. In this paper, we introduce a pair of condition numbers that measure the degree of non-separability or separability of a given dataset in the setting of binary classification, and we study how these condition numbers relate to and inform the properties and the convergence guarantees of first-order methods. When the training data is non-separable, we show that the degree of non-separability naturally enters the analysis and informs the properties and convergence guarantees of two standard first-order methods: steepest descent (for any given norm) and stochastic gradient descent. Expanding on the work of Bach, we also show how the degree of non-separability enters into the analysis of linear convergence of steepest descent (without needing strong convexity), as well as the adaptive convergence of stochastic gradient descent. When the training data is separable, first-order methods rather curiously have good empirical success, which is not well understood in theory. In the case of separable data, we demonstrate how the degree of separability enters into the analysis of $\ell_2$ steepest descent and stochastic gradient descent for delivering approximate-maximum-margin solutions with associated computational guarantees as well. This suggests that first-order methods can lead to statistically meaningful solutions in the separable case, even though the ML solution does not exist.

We propose Lomax delegate racing (LDR) to explicitly model the mechanism of survival under competing risks and to interpret how the covariates accelerate or decelerate the time to event. LDR explains non-monotonic covariate effects by racing a potentially infinite number of sub-risks, and consequently relaxes the ubiquitous proportional-hazards assumption which may be too restrictive. Moreover, LDR is naturally able to model not only censoring, but also missing event times or event types. For inference, we develop a Gibbs sampler under data augmentation for moderately sized data, along with a stochastic gradient descent maximum a posteriori inference algorithm for big data applications. Illustrative experiments are provided on both synthetic and real datasets, and comparison with various benchmark algorithms for survival analysis with competing risks demonstrates distinguished performance of LDR.

In this paper, we provide an overview of first-order and second-order variants of the gradient descent methods commonly used in machine learning. We propose a general framework in which 6 of these methods can be interpreted as different instances of the same approach. These methods are the vanilla gradient descent, the classical and generalized Gauss-Newton methods, the natural gradient descent method, the gradient covariance matrix approach, and Newton's method. Besides interpreting these methods within a single framework, we explain their specificities and show under which conditions some of them coincide. Machine learning generally amounts to solving an optimization problem where a loss function has to be minimized.

Large-scale machine learning training, in particular distributed stochastic gradient descent, needs to be robust to inherent system variability such as node straggling and random communication delays. This work considers a distributed training framework where each worker node is allowed to perform local model updates and the resulting models are averaged periodically. We analyze the true speed of error convergence with respect to wall-clock time (instead of the number of iterations), and analyze how it is affected by the frequency of averaging. Stochastic gradient descent (SGD) is the backbone of stateof-the-art supervised learning, which is revolutionizing inference and decision-making in many diverse applications. Classical SGD was designed to be run on a single computing node, and its error-convergence with respect to the number of iterations has been extensively analyzed and improved via accelerated SGD methods. Due to the massive training data-sets and neural network architectures used today, it has became imperative to design distributed SGD implementations, where gradient computation and aggregation is parallelized across multiple worker nodes. Although parallelism boosts the amount of data processed per iteration, it exposes SGD to unpredictable node slowdown and communication delays stemming from variability in the computing infrastructure. Thus, there is a critical need to make distributed SGD fast, yet robust to system variability.

Machine learning (ML) training algorithms often possess an inherent self-correcting behavior due to their iterative-convergent nature. Recent systems exploit this property to achieve adaptability and efficiency in unreliable computing environments by relaxing the consistency of execution and allowing calculation errors to be self-corrected during training. However, the behavior of such systems are only well understood for specific types of calculation errors, such as those caused by staleness, reduced precision, or asynchronicity, and for specific types of training algorithms, such as stochastic gradient descent. In this paper, we develop a general framework to quantify the effects of calculation errors on iterative-convergent algorithms and use this framework to design new strategies for checkpoint-based fault tolerance. Our framework yields a worst-case upper bound on the iteration cost of arbitrary perturbations to model parameters during training. Our system, SCAR, employs strategies which reduce the iteration cost upper bound due to perturbations incurred when recovering from checkpoints. We show that SCAR can reduce the iteration cost of partial failures by 78% - 95% when compared with traditional checkpoint-based fault tolerance across a variety of ML models and training algorithms.

Modern machine learning focuses on highly expressive models that are able to fit or interpolate the data completely, resulting in zero training loss. For such models, we show that the stochastic gradients of common loss functions satisfy a strong growth condition. Under this condition, we prove that constant step-size stochastic gradient descent (SGD) with Nesterov acceleration matches the convergence rate of the deterministic setting for both convex and strongly-convex functions. In the non-convex setting, this condition implies that SGD can find a first-order stationary point as efficiently as full gradient descent. Under interpolation, we also show that all smooth loss functions with a finite-sum structure satisfy a weaker growth condition. Given this weaker condition, we prove that SGD with a constant step-size attains the deterministic convergence rate in both the strongly-convex and convex settings. Under additional assumptions, the above results enable us to prove an O(1/k^2) mistake bound for $k$ iterations of a stochastic perceptron algorithm using the squared-hinge loss. Finally, we validate our theoretical findings with experiments on synthetic and real datasets.

We propose a population-based Evolutionary Stochastic Gradient Descent (ESGD) framework for optimizing deep neural networks. ESGD combines SGD and gradient-free evolutionary algorithms as complementary algorithms in one framework in which the optimization alternates between the SGD step and evolution step to improve the average fitness of the population. With a back-off strategy in the SGD step and an elitist strategy in the evolution step, it guarantees that the best fitness in the population will never degrade. In addition, individuals in the population optimized with various SGD-based optimizers using distinct hyper-parameters in the SGD step are considered as competing species in a coevolution setting such that the complementarity of the optimizers is also taken into account. The effectiveness of ESGD is demonstrated across multiple applications including speech recognition, image recognition and language modeling, using networks with a variety of deep architectures.

To improve the off-sample generalization of classical procedures minimizing the empirical risk under potentially heavy-tailed data, new robust learning algorithms have been proposed in recent years, with generalized median-of-means strategies being particularly salient. These procedures enjoy performance guarantees in the form of sharp risk bounds under weak moment assumptions on the underlying loss, but typically suffer from a large computational overhead and substantial bias when the data happens to be sub-Gaussian, limiting their utility. In this work, we propose a novel robust gradient descent procedure which makes use of a smoothed multiplicative noise applied directly to observations before constructing a sum of soft-truncated gradient coordinates. We show that the procedure has competitive theoretical guarantees, with the major advantage of a simple implementation that does not require an iterative sub-routine for robustification. Empirical tests reinforce the theory, showing more efficient generalization over a much wider class of data distributions.