This paper presents a new class of gradient methods for distributed machine learning that adaptively skip the gradient calculations to learn with reduced communication and computation. Simple rules are designed to detect slowly-varying gradients and, therefore, trigger the reuse of outdated gradients. The resultant gradient-based algorithms are termed Lazily Aggregated Gradient --- justifying our acronym LAG used henceforth. Theoretically, the merits of this contribution are: i) the convergence rate is the same as batch gradient descent in strongly-convex, convex, and nonconvex smooth cases; and, ii) if the distributed datasets are heterogeneous (quantified by certain measurable constants), the communication rounds needed to achieve a targeted accuracy are reduced thanks to the adaptive reuse of lagged gradients. Numerical experiments on both synthetic and real data corroborate a significant communication reduction compared to alternatives.

A framework is introduced for actively and adaptively solving a sequence of machine learning problems, which are changing in bounded manner from one time step to the next. An algorithm is developed that actively queries the labels of the most informative samples from an unlabeled data pool, and that adapts to the change by utilizing the information acquired in the previous steps. Our analysis shows that the proposed active learning algorithm based on stochastic gradient descent achieves a near-optimal excess risk performance for maximum likelihood estimation. Furthermore, an estimator of the change in the learning problems using the active learning samples is constructed, which provides an adaptive sample size selection rule that guarantees the excess risk is bounded for sufficiently large number of time steps. Experiments with synthetic and real data are presented to validate our algorithm and theoretical results.

Distributed stochastic gradient descent is an important subroutine in distributed learning. A setting of particular interest is when the clients are mobile devices, where two important concerns are communication efficiency and the privacy of the clients. Several recent works have focused on reducing the communication cost or introducing privacy guarantees, but none of the proposed communication efficient methods are known to be privacy preserving and none of the known privacy mechanisms are known to be communication efficient. To this end, we study algorithms that achieve both communication efficiency and differential privacy. For $d$ variables and $n \approx d$ clients, the proposed method uses $O(\log \log(nd))$ bits of communication per client per coordinate and ensures constant privacy. We also extend and improve previous analysis of the \emph{Binomial mechanism} showing that it achieves nearly the same utility as the Gaussian mechanism, while requiring fewer representation bits, which can be of independent interest.

A hyperbolic space has been shown to be more capable of modeling complex networks than a Euclidean space. This paper proposes an explicit update rule along geodesics in a hyperbolic space. The convergence of our algorithm is theoretically guaranteed, and the convergence rate is better than the conventional Euclidean gradient descent algorithm. Moreover, our algorithm avoids the "bias" problem of existing methods using the Riemannian gradient. Experimental results demonstrate the good performance of our algorithm in the \Poincare embeddings of knowledge base data.

Recently, during my coursework, I have been left in awe of the advancements in the field of science in general. In just about a decade, we have completely revolutionized the way we look at the capabilities of machines, the way we build software and much more. Tasks that seemed impossible just a decade ago have become accessible and effortless. Long story short, we have made the machines think!! Sounds cool, isn't it? Artificial Intelligence has truly entered the mainstream consciousness.

Policy evaluation, commonly referred to as value function approximation, is an important and central part in many reinforcement learning (RL) algorithms [27], whose task is to estimate value functions for a fixed policy in a discounted Markov Decision Process (MDP) environment. The value function of each state specifies the accumulated reward an agent would receive in the future by following the fixed policy from that state. Value functions have been widely investigated in RL applications, and it can provide insightful and important information for the agent to obtain an optimal policy, such as important board configurations in Go [24], failure probabilities of large telecommunication networks [9], taxi-out times at large airports [2] and so on. Despite the value functions can be approximated by different ways, the simplest form, linear approximations, are still widely adopted and studied due to their good generalization abilities, relatively efficient computation and solid theoretical guarantees[27, 7, 13, 16]. Temporal Difference (TD) learning is a common approach to this policy evaluation with linear function approximation problem[27]. These typical TD algorithms can be divided into two categories: gradient based methods (e.g., GTD(λ) [28]) and least-square (LS) based methods (e.g., LSTD(λ)[4]). A good survey on these algorithms can be found in [17, 6, 12, 7, 13]. 1 As the development of information technologies, high-dimensional data is widely seen in RL applications [26, 30, 23], which brings serious challenges to design scalable and computationally efficient algorithms for the linear value function approximation problem. To address this practical issue, several approaches have been developed for efficient and effective value function approximation.

When the linear measurements of an instance of low-rank matrix recovery satisfy a restricted isometry property (RIP)---i.e. they are approximately norm-preserving---the problem is known to contain no spurious local minima, so exact recovery is guaranteed. In this paper, we show that moderate RIP is not enough to eliminate spurious local minima, so existing results can only hold for near-perfect RIP. In fact, counterexamples are ubiquitous: we prove that every x is the spurious local minimum of a rank-1 instance of matrix recovery that satisfies RIP. One specific counterexample has RIP constant $\delta=1/2$, but causes randomly initialized stochastic gradient descent (SGD) to fail 12% of the time. SGD is frequently able to avoid and escape spurious local minima, but this empirical result shows that it can occasionally be defeated by their existence. Hence, while exact recovery guarantees will likely require a proof of no spurious local minima, arguments based solely on norm preservation will only be applicable to a narrow set of nearly-isotropic instances.

We consider stochastic gradient descent (SGD) for least-squares regression with potentially several passes over the data. While several passes have been widely reported to perform practically better in terms of predictive performance on unseen data, the existing theoretical analysis of SGD suggests that a single pass is statistically optimal. While this is true for low-dimensional easy problems, we show that for hard problems, multiple passes lead to statistically optimal predictions while single pass does not; we also show that in these hard models, the optimal number of passes over the data increases with sample size. In order to define the notion of hardness and show that our predictive performances are optimal, we consider potentially infinite-dimensional models and notions typically associated to kernel methods, namely, the decay of eigenvalues of the covariance matrix of the features and the complexity of the optimal predictor as measured through the covariance matrix. We illustrate our results on synthetic experiments with non-linear kernel methods and on a classical benchmark with a linear model.

We propose Zeno, a new robust aggregation rule, for distributed synchronous Stochastic Gradient Descent~(SGD) under a general Byzantine failure model. The key idea is to suspect the workers that are potentially malicious, and use a ranking-based preference mechanism. This allows us to generalize beyond past work--in our case, the number of malicious workers can be arbitrarily large, and we use only the weakest assumption on honest workers~(at least one honest worker). We prove the convergence of SGD under these scenarios. Empirical results show that Zeno outperforms existing approaches under various attacks.

Regularized nonlinear acceleration (RNA) is a generic extrapolation scheme for optimization methods, with marginal computational overhead. It aims to improve convergence using only the iterates of simple iterative algorithms. However, so far its application to optimization was theoretically limited to gradient descent and other single-step algorithms. Here, we adapt RNA to a much broader setting including stochastic gradient with momentum and Nesterov's fast gradient. We use it to train deep neural networks, and empirically observe that extrapolated networks are more accurate, especially in the early iterations. A straightforward application of our algorithm when training ResNet-152 on ImageNet produces a top-1 test error of 20.88%, improving by 0.8% the reference classification pipeline. Furthermore, the code runs offline in this case, so it never negatively affects performance.