Many real-world networks are complex dynamical systems, where both local (e.g., changing node attributes) and global (e.g., changing network topology) processes unfold over time. Local dynamics may provoke global changes in the network, and the ability to detect such effects could have profound implications for a number of real-world problems. Most existing techniques focus individually on either local or global aspects of the problem or treat the two in isolation from each other. In this paper we propose a novel network model that simultaneously accounts for both local and global dynamics. To the best of our knowledge, this is the first attempt at modeling and detecting local and global change points on dynamic networks via a unified generative framework. Our model is built upon the popular mixed membership stochastic blockmodels (MMSB) with sparse co-evolving patterns. We derive an efficient stochastic gradient Langevin dynamics (SGLD) sampler for our proposed model, which allows it to scale to potentially very large networks. Finally, we validate our model on both synthetic and real-world data and demonstrate its superiority over several baselines.

Learning to learn by gradient descent by gradient descent, Andrychowicz et al., NIPS 2016 One of the things that strikes me when I read these NIPS papers is just how short some of them are – between the introduction and the evaluation sections you might find only one or two pages! A general form is to start out with a basic mathematical model of the problem domain, expressed in terms of functions. Selected functions are then learned, by reaching into the machine learning toolbox and combining existing building blocks in potentially novel ways. When looked at this way, we could really call machine learning'function learning'. Thinking in terms of functions like this is a bridge back to the familiar (for me at least).

Last year, when I was on a mission to "fix GANs" I had a tendency to focus only on what the loss function is, and completely disregard the issue of how do we actually find a minimum. I reference Marr's three layers of analysis a lot, and I enjoy thinking about problems at the computational level: what is the ultimate goal we do this for? I was convinced GANs were broken at this level: they were trying to optimize for the wrong thing or seek equilibria that don't exist, etc. This is why I enjoyed f-GANs, Wasserstein GANs, instance noise, etc, while being mostly dismissive of attempts to fix things at the optimization level, like DCGAN or improved techniques (Salimans et al. 2016). To my defense, in most of deep learning, the algorithmic level is sorted: stochastic gradient descent.

This work addresses the instability in asynchronous data parallel optimization. It does so by introducing a novel distributed optimizer which is able to efficiently optimize a centralized model under communication constraints. The optimizer achieves this by pushing a normalized sequence of first-order gradients to a parameter server. This implies that the magnitude of a worker delta is smaller compared to an accumulated gradient, and provides a better direction towards a minimum compared to first-order gradients, which in turn also forces possible implicit momentum fluctuations to be more aligned since we make the assumption that all workers contribute towards a single minima. As a result, our approach mitigates the parameter staleness problem more effectively since staleness in asynchrony induces (implicit) momentum, and achieves a better convergence rate compared to other optimizers such as asynchronous easgd and dynsgd, which we show empirically.

The Hessian-vector product has been utilized to find a second-order stationary solution with strong complexity guarantee (e.g., almost linear time complexity in the problem's dimensionality). In this paper, we propose to further reduce the number of Hessian-vector products for faster non-convex optimization. Previous algorithms need to approximate the smallest eigen-value with a sufficient precision (e.g., $\epsilon_2\ll 1$) in order to achieve a sufficiently accurate second-order stationary solution (i.e., $\lambda_{\min}(\nabla^2 f(\x))\geq -\epsilon_2)$. In contrast, the proposed algorithms only need to compute the smallest eigen-vector approximating the corresponding eigen-value up to a small power of current gradient's norm. As a result, it can dramatically reduce the number of Hessian-vector products during the course of optimization before reaching first-order stationary points (e.g., saddle points). The key building block of the proposed algorithms is a novel updating step named the NCG step, which lets a noisy negative curvature descent compete with the gradient descent. We show that the worst-case time complexity of the proposed algorithms with their favorable prescribed accuracy requirements can match the best in literature for achieving a second-order stationary point but with an arguably smaller per-iteration cost. We also show that the proposed algorithms can benefit from inexact Hessian by developing their variants accepting inexact Hessian under a mild condition for achieving the same goal. Moreover, we develop a stochastic algorithm for a finite or infinite sum non-convex optimization problem. To the best of our knowledge, the proposed stochastic algorithm is the first one that converges to a second-order stationary point in {\it high probability} with a time complexity independent of the sample size and almost linear in dimensionality.

When using stochastic gradient descent to solve large-scale machine learning problems, a common practice of data processing is to shuffle the training data, partition the data across multiple machines if needed, and then perform several epochs of training on the re-shuffled (either locally or globally) data. The above procedure makes the instances used to compute the gradients no longer independently sampled from the training data set. Then does the distributed SGD method have desirable convergence properties in this practical situation? In this paper, we give answers to this question. First, we give a mathematical formulation for the practical data processing procedure in distributed machine learning, which we call data partition with global/local shuffling. We observe that global shuffling is equivalent to without-replacement sampling if the shuffling operations are independent. We prove that SGD with global shuffling has convergence guarantee in both convex and non-convex cases. An interesting finding is that, the non-convex tasks like deep learning are more suitable to apply shuffling comparing to the convex tasks. Second, we conduct the convergence analysis for SGD with local shuffling. The convergence rate for local shuffling is slower than that for global shuffling, since it will lose some information if there's no communication between partitioned data. Finally, we consider the situation when the permutation after shuffling is not uniformly distributed (insufficient shuffling), and discuss the condition under which this insufficiency will not influence the convergence rate. Our theoretical results provide important insights to large-scale machine learning, especially in the selection of data processing methods in order to achieve faster convergence and good speedup. Our theoretical findings are verified by extensive experiments on logistic regression and deep neural networks.

We consider in this work a system of two stochastic differential equations named the perturbed compositional gradient flow. By introducing a separation of fast and slow scales of the two equations, we show that the limit of the slow motion is given by an averaged ordinary differential equation. We then demonstrate that the deviation of the slow motion from the averaged equation, after proper rescaling, converges to a stochastic process with Gaussian inputs. This indicates that the slow motion can be approximated in the weak sense by a standard perturbed gradient flow or the continuous-time stochastic gradient descent algorithm that solves the optimization problem for a composition of two functions. As an application, the perturbed compositional gradient flow corresponds to the diffusion limit of the Stochastic Composite Gradient Descent (SCGD) algorithm for minimizing a composition of two expected-value functions in the optimization literatures. For the strongly convex case, such an analysis implies that the SCGD algorithm has the same convergence time asymptotic as the classical stochastic gradient descent algorithm. Thus it validates the effectiveness of using the SCGD algorithm in the strongly convex case.

Recent work has demonstrated the effectiveness of gradient descent for directly recovering the factors of low-rank matrices from random linear measurements in a globally convergent manner when initialized properly. However, the performance of existing algorithms is highly sensitive in the presence of outliers that may take arbitrary values. In this paper, we propose a truncated gradient descent algorithm to improve the robustness against outliers, where the truncation is performed to rule out the contributions of samples that deviate significantly from the {\em sample median} of measurement residuals adaptively in each iteration. We demonstrate that, when initialized in a basin of attraction close to the ground truth, the proposed algorithm converges to the ground truth at a linear rate for the Gaussian measurement model with a near-optimal number of measurements, even when a constant fraction of the measurements are arbitrarily corrupted. In addition, we propose a new truncated spectral method that ensures an initialization in the basin of attraction at slightly higher requirements. We finally provide numerical experiments to validate the superior performance of the proposed approach.

We propose learning deep models that are monotonic with respect to a user-specified set of inputs by alternating layers of linear embeddings, ensembles of lattices, and calibrators (piecewise linear functions), with appropriate constraints for monotonicity, and jointly training the resulting network. We implement the layers and projections with new computational graph nodes in TensorFlow and use the ADAM optimizer and batched stochastic gradients. Experiments on benchmark and real-world datasets show that six-layer monotonic deep lattice networks achieve state-of-the art performance for classification and regression with monotonicity guarantees.

In this paper, we develop a new accelerated stochastic gradient method for efficiently solving the convex regularized empirical risk minimization problem in mini-batch settings. The use of mini-batches is becoming a golden standard in the machine learning community, because mini-batch settings stabilize the gradient estimate and can easily make good use of parallel computing. The core of our proposed method is the incorporation of our new "double acceleration" technique and variance reduction technique. We theoretically analyze our proposed method and show that our method much improves the mini-batch efficiencies of previous accelerated stochastic methods, and essentially only needs size $\sqrt{n}$ mini-batches for achieving the optimal iteration complexities for both non-strongly and strongly convex objectives, where $n$ is the training set size. Further, we show that even in non-mini-batch settings, our method achieves the best known convergence rate for both non-strongly and strongly convex objectives.