In large-scale distributed learning, security issues have become increasingly important. Particularly in a decentralized environment, some computing units may behave abnormally, or even exhibit Byzantine failures---arbitrary and potentially adversarial behavior. In this paper, we develop distributed learning algorithms that are provably robust against such failures, with a focus on achieving optimal statistical performance. A main result of this work is a sharp analysis of two robust distributed gradient descent algorithms based on median and trimmed mean operations, respectively. We prove statistical error rates for three kinds of population loss functions: strongly convex, non-strongly convex, and smooth non-convex. In particular, these algorithms are shown to achieve order-optimal statistical error rates for strongly convex losses. To achieve better communication efficiency, we further propose a median-based distributed algorithm that is provably robust, and uses only one communication round. For strongly convex quadratic loss, we show that this algorithm achieves the same optimal error rate as the robust distributed gradient descent algorithms.

The concept of sample mean in dynamic time warping (DTW) spaces has been successfully applied to improve pattern recognition systems and generalize centroid-based clustering algorithms. Its existence has neither been proved nor challenged. This article presents sufficient conditions for existence of a sample mean in DTW spaces. The proposed result justifies prior work on approximate mean algorithms, sets the stage for constructing exact mean algorithms, and is a first step towards a statistical theory of DTW spaces.

Finding parameters that minimise a loss function is at the core of many machine learning methods. The Stochastic Gradient Descent algorithm is widely used and delivers state of the art results for many problems. Nonetheless, Stochastic Gradient Descent typically cannot find the global minimum, thus its empirical effectiveness is hitherto mysterious. We derive a correspondence between parameter inference and free energy minimisation in statistical physics. The degree of undersampling plays the role of temperature. Analogous to the energy-entropy competition in statistical physics, wide but shallow minima can be optimal if the system is undersampled, as is typical in many applications. Moreover, we show that the stochasticity in the algorithm has a non-trivial correlation structure which systematically biases it towards wide minima. We illustrate our argument with two prototypical models: image classification using deep learning, and a linear neural network where we can analytically reveal the relationship between entropy and out-of-sample error.

The implicit bias of gradient descent is not fully understood even in simple linear classification tasks (e.g., logistic regression). Soudry et al. (2018) studied this bias on separable data, where there are multiple solutions that correctly classify the data. It was found that, when optimizing monotonically decreasing loss functions with exponential tails using gradient descent, the linear classifier specified by the gradient descent iterates converge to the $L_2$ max margin separator. However, the convergence rate to the maximum margin solution with fixed step size was found to be extremely slow: $1/\log(t)$. Here we examine how the convergence is influenced by using different loss functions and by using variable step sizes. First, we calculate the convergence rate for loss functions with poly-exponential tails near $\exp(-u^{\nu})$. We prove that $\nu=1$ yields the optimal convergence rate in the range $\nu>0.25$. Based on further analysis we conjecture that this remains the optimal rate for $\nu \leq 0.25$, and even for sub-poly-exponential tails --- until loss functions with polynomial tails no longer converge to the max margin. Second, we prove the convergence rate could be improved to $(\log t) /\sqrt{t}$ for the exponential loss, by using aggressive step sizes which compensate for the rapidly vanishing gradients.

Let $A:[0,1]\rightarrow\mathbb{H}_m$ (the space of Hermitian matrices) be a matrix valued function which is low rank with entries in H\"{o}lder class $\Sigma(\beta,L)$. The goal of this paper is to study statistical estimation of $A$ based on the regression model $\mathbb{E}(Y_j|\tau_j,X_j) = \langle A(\tau_j), X_j \rangle,$ where $\tau_j$ are i.i.d. uniformly distributed in $[0,1]$, $X_j$ are i.i.d. matrix completion sampling matrices, $Y_j$ are independent bounded responses. We propose an innovative nuclear norm penalized local polynomial estimator and establish an upper bound on its point-wise risk measured by Frobenius norm. Then we extend this estimator globally and prove an upper bound on its integrated risk measured by $L_2$-norm. We also propose another new estimator based on bias-reducing kernels to study the case when $A$ is not necessarily low rank and establish an upper bound on its risk measured by $L_{\infty}$-norm. We show that the obtained rates are all optimal up to some logarithmic factor in minimax sense. Finally, we propose an adaptive estimation procedure based on Lepski's method and the penalized data splitting technique which is computationally efficient and can be easily implemented and parallelized.

We propose a method for practical downlink coordinated multipoint (DL CoMP) implementation in 4G LTE/LTE-A systems using supervised machine learning. Contributions of this paper are: 1) demonstrating that a support vector machine classifier can learn the optimal conditions at which DL CoMP can be dynamically triggered, 2) improving user throughput in DL CoMP as a result of learning the optimal triggering conditions of DL CoMP, and 3) showing that the machine learning approach is scalable to more than a single macro. The simulation results show an improvement in the pico cell average and edge throughputs and a reduction in the downlink block error rate due to the informed triggering of the multiple radio streams as part of DL CoMP as learned from the support vector machine.

There is no thumb rule of choosing the best kernel. The only solution is Cross-validation. Try several different Kernels, and evaluate their performance metrics such as AUC and select the one with highest AUC. If you want to compare in terms of speed, linear kernels usually compute much faster than radial or polynomial kernels.

We introduce Recurrent Predictive State Policy (RPSP) networks, a recurrent architecture that brings insights from predictive state representations to reinforcement learning in partially observable environments. Predictive state policy networks consist of a recursive filter, which keeps track of a belief about the state of the environment, and a reactive policy that directly maps beliefs to actions, to maximize the cumulative reward. The recursive filter leverages predictive state representations (PSRs) (Rosencrantz and Gordon, 2004; Sun et al., 2016) by modeling predictive state-- a prediction of the distribution of future observations conditioned on history and future actions. This representation gives rise to a rich class of statistically consistent algorithms (Hefny et al., 2018) to initialize the recursive filter. Predictive state serves as an equivalent representation of a belief state. Therefore, the policy component of the RPSP-network can be purely reactive, simplifying training while still allowing optimal behaviour. Moreover, we use the PSR interpretation during training as well, by incorporating prediction error in the loss function. The entire network (recursive filter and reactive policy) is still differentiable and can be trained using gradient based methods. We optimize our policy using a combination of policy gradient based on rewards (Williams, 1992) and gradient descent based on prediction error. We show the efficacy of RPSP-networks under partial observability on a set of robotic control tasks from OpenAI Gym. We empirically show that RPSP-networks perform well compared with memory-preserving networks such as GRUs, as well as finite memory models, being the overall best performing method.

An important metric of users' satisfaction and engagement within on-line streaming services is the user session length, i.e. the amount of time they spend on a service continuously without interruption. Being able to predict this value directly benefits the recommendation and ad pacing contexts in music and video streaming services. Recent research has shown that predicting the exact amount of time spent is highly nontrivial due to many external factors for which a user can end a session, and the lack of predictive covariates. Most of the other related literature on duration based user engagement has focused on dwell time for websites, for search and display ads, mainly for post-click satisfaction prediction or ad ranking. In this work we present a novel framework inspired by hierarchical Bayesian modeling to predict, at the moment of login, the amount of time a user will spend in the streaming service. The time spent by a user on a platform depends upon user-specific latent variables which are learned via hierarchical shrinkage. Our framework enjoys theoretical guarantees, naturally incorporates flexible parametric/nonparametric models on the covariates and is found to outperform state-of- the-art estimators in terms of efficiency and predictive performance on real world datasets.

Deep neural networks (DNNs) often require good regularizers to generalize well. Currently, state-of-the-art DNN regularization techniques consist in randomly dropping units and/or connections on each iteration of the training algorithm. Dropout and DropConnect are characteristic examples of such regularizers, that are widely popular among practitioners. However, a drawback of such approaches consists in the fact that their postulated probability of random unit/connection omission is a constant that must be heuristically selected based on the obtained performance in some validation set. To alleviate this burden, in this paper we regard the DNN regularization problem from a Bayesian inference perspective: We impose a sparsity-inducing prior over the network synaptic weights, where the sparsity is induced by a set of Bernoulli-distributed binary variables with Beta (hyper-)priors over their prior parameters. This way, we eventually allow for marginalizing over the DNN synaptic connectivity for output generation, thus giving rise to an effective, heuristics-free, network regularization scheme. We perform Bayesian inference for the resulting hierarchical model by means of an efficient Black-Box Variational inference scheme. We exhibit the advantages of our method over existing approaches by conducting an extensive experimental evaluation using benchmark datasets.