We propose accelerated randomized coordinate descent algorithms for stochastic optimization and online learning. Our algorithms have significantly less per-iteration complexity than the known accelerated gradient algorithms. The proposed algorithms for online learning have better regret performance than the known randomized online coordinate descent algorithms. Furthermore, the proposed algorithms for stochastic optimization exhibit as good convergence rates as the best known randomized coordinate descent algorithms. We also show simulation results to demonstrate performance of the proposed algorithms.
More recently, there has been interest in machine learning settings in which training data and features are distributed across nodes of a computing cluster, or more generally, of a network. Nathan and Klabjan @cite_15 proposed an algorithm where nodes parallelly update (possibly overlapping) blocks of feature parameters based on locally available data samples; this was seen as a combination of SVRG and block coordinate descent. @cite_18 studied algorithms for nodes connected through a network; this scenario was referred to as federated learning.
@cite_15 @cite_18
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As the size of modern data sets exceeds the disk and memory capacities of a single computer, machine learning practitioners have resorted to parallel and distributed computing. Given that optimization is one of the pillars of machine learning and predictive modeling, distributed optimization methods have recently garnered ample attention in the literature. Although previous research has mostly focused on settings where either the observations, or features of the problem at hand are stored in distributed fashion, the situation where both are partitioned across the nodes of a computer cluster (doubly distributed) has barely been studied. In this work we propose two doubly distributed optimization algorithms. The first one falls under the umbrella of distributed dual coordinate ascent methods, while the second one belongs to the class of stochastic gradient coordinate descent hybrid methods. We conduct numerical experiments in Spark using real-world and simulated data sets and study the scaling properties of our methods. Our empirical evaluation of the proposed algorithms demonstrates the out-performance of a block distributed ADMM method, which, to the best of our knowledge is the only other existing doubly distributed optimization algorithm. We introduce a new and increasingly relevant setting for distributed optimization in machine learning, where the data defining the optimization are unevenly distributed over an extremely large number of nodes. The goal is to train a high-quality centralized model. We refer to this setting as Federated Optimization. In this setting, communication efficiency is of the utmost importance and minimizing the number of rounds of communication is the principal goal. A motivating example arises when we keep the training data locally on users' mobile devices instead of logging it to a data center for training. In federated optimziation, the devices are used as compute nodes performing computation on their local data in order to update a global model. We suppose that we have extremely large number of devices in the network --- as many as the number of users of a given service, each of which has only a tiny fraction of the total data available. In particular, we expect the number of data points available locally to be much smaller than the number of devices. Additionally, since different users generate data with different patterns, it is reasonable to assume that no device has a representative sample of the overall distribution. We show that existing algorithms are not suitable for this setting, and propose a new algorithm which shows encouraging experimental results for sparse convex problems. This work also sets a path for future research needed in the context of optimization. Emerging technologies and applications including Internet of Things (IoT), social networking, and crowd-sourcing generate large amounts of data at the network edge. Machine learning models are often built from the collected data, to enable the detection, classification, and prediction of future events. Due to bandwidth, storage, and privacy concerns, it is often impractical to send all the data to a centralized location. In this paper, we consider the problem of learning model parameters from data distributed across multiple edge nodes, without sending raw data to a centralized place. Our focus is on a generic class of machine learning models that are trained using gradient-descent based approaches. We analyze the convergence rate of distributed gradient descent from a theoretical point of view, based on which we propose a control algorithm that determines the best trade-off between local update and global parameter aggregation to minimize the loss function under a given resource budget. The performance of the proposed algorithm is evaluated via extensive experiments with real datasets, both on a networked prototype system and in a larger-scale simulated environment. The experimentation results show that our proposed approach performs near to the optimum with various machine learning models and different data distributions. We propose a new scheme for content distribution of large files that is based on network coding. With network coding, each node of the distribution network is able to generate and transmit encoded blocks of information. The randomization introduced by the coding process eases the scheduling of block propagation, and, thus, makes the distribution more efficient. This is particularly important in large unstructured overlay networks, where the nodes need to make block forwarding decisions based on local information only. We compare network coding to other schemes that transmit unencoded information (i.e. blocks of the original file) and, also, to schemes in which only the source is allowed to generate and transmit encoded packets. We study the performance of network coding in heterogeneous networks with dynamic node arrival and departure patterns, clustered topologies, and when incentive mechanisms to discourage free-riding are in place. We demonstrate through simulations of scenarios of practical interest that the expected file download time improves by more than 20-30 with network coding compared to coding at the server only and, by more than 2-3 times compared to sending unencoded information. Moreover, we show that network coding improves the robustness of the system and is able to smoothly handle extreme situations where the server and nodes leave the system. Distributed cloud applications, including batch processing, streaming, and scale-out databases, generate a significant amount of network traffic and a considerable fraction of their runtime is due to network activity. This paper initiates the study of deterministic algorithms for collocating frequently communicating nodes in a distributed networked systems in an online fashion. In particular, we introduce the Balanced RePartitioning (BRP) problem: Given an arbitrary sequence of pairwise communication requests between n nodes, with patterns that may change over time, the objective is to dynamically partition the nodes into ( ) clusters, each of size k, at a minimum cost. Every communication request needs to be served: if the communicating nodes are located in the same cluster, the request is served locally, at cost 0; if the nodes are located in different clusters, the request is served remotely using inter-cluster communication, at cost 1. The partitioning can be updated dynamically (i.e., repartitioned), by migrating nodes between clusters at cost ( ) per node migration. The goal is to devise online algorithms which find a good trade-off between the communication and the migration cost, i.e., “rent” or “buy”, while maintaining partitions which minimize the number of inter-cluster communications. BRP features interesting connections to other well-known online problems. In particular, we show that scenarios with ( =2 ) generalize online paging, and scenarios with (k=2 ) constitute a novel online version of maximum matching. We consider settings both with and without cluster-size augmentation. Somewhat surprisingly (and unlike online paging), we prove that any deterministic online algorithm has a competitive ratio of at least k, even with augmentation. Our main technical contribution is an (O(k k ) )-competitive deterministic algorithm for the setting with (constant) augmentation. This is attractive as, in contrast to ( ), k is likely to be small in practice. For the case of matching ( (k=2 )), we present a constant competitive algorithm that does not rely on augmentation. We present a new approach to scalable training of deep learning machines by incremental block training with intra-block parallel optimization to leverage data parallelism and blockwise model-update filtering to stabilize learning process. By using an implementation on a distributed GPU cluster with an MPI-based HPC machine learning framework to coordinate parallel job scheduling and collective communication, we have trained successfully deep bidirectional long short-term memory (LSTM) recurrent neural networks (RNNs) and fully-connected feed-forward deep neural networks (DNNs) for large vocabulary continuous speech recognition on two benchmark tasks, namely 309-hour Switchboard-I task and 1,860-hour "Switch-board+Fisher" task. We achieve almost linear speedup up to 16 GPU cards on LSTM task and 64 GPU cards on DNN task, with either no degradation or improved recognition accuracy in comparison with that of running a traditional mini-batch based stochastic gradient descent training on a single GPU. The clustering coefficient of a node in a social network is a fundamental measure that quantifies how tightly-knit the community is around the node. Its computation can be reduced to counting the number of triangles incident on the particular node in the network. In case the graph is too big to fit into memory, this is a non-trivial task, and previous researchers showed how to estimate the clustering coefficient in this scenario. A different avenue of research is to to perform the computation in parallel, spreading it across many machines. In recent years MapReduce has emerged as a de facto programming paradigm for parallel computation on massive data sets. The main focus of this work is to give MapReduce algorithms for counting triangles which we use to compute clustering coefficients. Our contributions are twofold. First, we describe a sequential triangle counting algorithm and show how to adapt it to the MapReduce setting. This algorithm achieves a factor of 10-100 speed up over the naive approach. Second, we present a new algorithm designed specifically for the MapReduce framework. A key feature of this approach is that it allows for a smooth tradeoff between the memory available on each individual machine and the total memory available to the algorithm, while keeping the total work done constant. Moreover, this algorithm can use any triangle counting algorithm as a black box and distribute the computation across many machines. We validate our algorithms on real world datasets comprising of millions of nodes and over a billion edges. Our results show both algorithms effectively deal with skew in the degree distribution and lead to dramatic speed ups over the naive implementation. Complex systems in nature and in society are often represented as networks, describing the rich set of interactions between objects of interest. Many deterministic and probabilistic clustering methods have been developed to analyze such structures. Given a network, almost all of them partition the vertices into disjoint clusters, according to their connection profile. However, recent studies have shown that these techniques were too restrictive and that most of the existing networks contained overlapping clusters. To tackle this issue, we present in this paper the Overlapping Stochastic Block Model. Our approach allows the vertices to belong to multiple clusters, and, to some extent, generalizes the well-known Stochastic Block Model [Nowicki and Snijders (2001)]. We show that the model is generically identifiable within classes of equivalence and we propose an approximate inference procedure, based on global and local variational techniques. Using toy data sets as well as the French Political Blogosphere network and the transcriptional network of Saccharomyces cerevisiae, we compare our work with other approaches. We study the composite minimization problem where the objective function is the sum of two convex functions: one is the sum of a finite number of strongly convex and smooth functions, and the other is a general convex function that is non-differentiable. Specifically, we consider the case where the non-differentiable function is block separable and admits a simple proximal mapping for each block. This type of composite optimization is common in many data mining and machine learning problems, and can be solved by block coordinate descent algorithms. We propose an accelerated stochastic block coordinate descent (ASBCD) algorithm, which incorporates the incrementally averaged partial derivative into the stochastic partial derivative and exploits optimal sampling. We prove that ASBCD attains a linear rate of convergence. In contrast to uniform sampling, we reveal that the optimal non-uniform sampling can be employed to achieve a lower iteration complexity. Experimental results on different large-scale real data sets support our theory. Networks or graphs can easily represent a diverse set of data sources that are characterized by interacting units or actors. Social ne tworks, representing people who communicate with each other, are one example. Communities or clusters of highly connected actors form an essential feature in the structure of several empirical networks. Spectral clustering is a popular and computationally feasi ble method to discover these communities. The Stochastic Block Model (, 1983) is a social network model with well defined communities; each node is a member of one community. For a network generated from the Stochastic Block Model, we bound the number of nodes "misclus- tered" by spectral clustering. The asymptotic results in th is paper are the first clustering results that allow the number of clusters in the model to grow with the number of nodes, hence the name high-dimensional. In order to study spectral clustering under the Stochastic Block Model, we first show that under the more general latent space model, the eigenvectors of the normalized graph Laplacian asymptotically converge to the eigenvectors of a "population" normal- ized graph Laplacian. Aside from the implication for spectral clustering, this provides insight into a graph visualization technique. Our method of studying the eigenvectors of random matrices is original. AMS 2000 subject classifications: Primary 62H30, 62H25; secondary 60B20. Distributed cryptographic protocols such as Bitcoin and Ethereum use a data structure known as the block chain to synchronize a global log of events between nodes in their network. Blocks, which are batches of updates to the log, reference the parent they are extending, and thus form the structure of a chain. Previous research has shown that the mechanics of the block chain and block propagation are constrained: if blocks are created at a high rate compared to their propagation time in the network, many conflicting blocks are created and performance suffers greatly. As a result of the low block creation rate required to keep the system within safe parameters, transactions take long to securely confirm, and their throughput is greatly limited. Distributed averaging describes a class of network algorithms for the decentralized computation of aggregate statistics. Initially, each node has a scalar data value, and the goal is to compute the average of these values at every node (the so-called average consensus problem). Nodes iteratively exchange information with their neighbors and perform local updates until the value at every node converges to the initial network average. Much previous work has focused on algorithms where each node maintains and updates a single value; every time an update is performed, the previous value is forgotten. Convergence to the average consensus is achieved asymptotically. The convergence rate is fundamentally limited by network connectivity, and it can be prohibitively slow on topologies such as grids and random geometric graphs, even if the update rules are optimized. In this paper, we provide the first theoretical demonstration that adding a local prediction component to the update rule can significantly improve the convergence rate of distributed averaging algorithms. We focus on the case where the local predictor is a linear combination of the node's current and previous values (i.e., two memory taps), and our update rule computes a combination of the predictor and the usual weighted linear combination of values received from neighboring nodes. We derive the optimal mixing parameter for combining the predictor with the neighbors' values, and conduct a theoretical analysis of the improvement in convergence rate that can be achieved using this acceleration methodology. For a chain topology on N nodes, this leads to a factor of N improvement over standard consensus, and for a two-dimensional grid, our approach achieves a factor of ?N improvement. Federated Learning is a machine learning setting where the goal is to train a high-quality centralized model while training data remains distributed over a large number of clients each with unreliable and relatively slow network connections. We consider learning algorithms for this setting where on each round, each client independently computes an update to the current model based on its local data, and communicates this update to a central server, where the client-side updates are aggregated to compute a new global model. The typical clients in this setting are mobile phones, and communication efficiency is of the utmost importance. In this paper, we propose two ways to reduce the uplink communication costs: structured updates, where we directly learn an update from a restricted space parametrized using a smaller number of variables, e.g. either low-rank or a random mask; and sketched updates, where we learn a full model update and then compress it using a combination of quantization, random rotations, and subsampling before sending it to the server. Experiments on both convolutional and recurrent networks show that the proposed methods can reduce the communication cost by two orders of magnitude. Prediction markets are economic mechanisms for aggregating information about future events through sequential interactions with traders. The pricing mechanisms in these markets are known to be related to optimization algorithms in machine learning and through these connections we have some understanding of how equilibrium market prices relate to the beliefs of the traders in a market. However, little is known about rates and guarantees for the convergence of these sequential mechanisms, and two recent papers cite this as an important open question. In this paper we show how some previously studied prediction market trading models can be understood as a natural generalization of randomized coordinate descent which we call randomized subspace descent (RSD). We establish convergence rates for RSD and leverage them to prove rates for the two prediction market models above, answering the open questions. Our results extend beyond standard centralized markets to arbitrary trade networks. Deep Learning has recently become hugely popular in machine learning, providing significant improvements in classification accuracy in the presence of highly-structured and large databases. Researchers have also considered privacy implications of deep learning. Models are typically trained in a centralized manner with all the data being processed by the same training algorithm. If the data is a collection of users' private data, including habits, personal pictures, geographical positions, interests, and more, the centralized server will have access to sensitive information that could potentially be mishandled. To tackle this problem, collaborative deep learning models have recently been proposed where parties locally train their deep learning structures and only share a subset of the parameters in the attempt to keep their respective training sets private. Parameters can also be obfuscated via differential privacy (DP) to make information extraction even more challenging, as proposed by Shokri and Shmatikov at CCS'15. Unfortunately, we show that any privacy-preserving collaborative deep learning is susceptible to a powerful attack that we devise in this paper. In particular, we show that a distributed, federated, or decentralized deep learning approach is fundamentally broken and does not protect the training sets of honest participants. The attack we developed exploits the real-time nature of the learning process that allows the adversary to train a Generative Adversarial Network (GAN) that generates prototypical samples of the targeted training set that was meant to be private (the samples generated by the GAN are intended to come from the same distribution as the training data). Interestingly, we show that record-level DP applied to the shared parameters of the model, as suggested in previous work, is ineffective (i.e., record-level DP is not designed to address our attack). We construct a connected network of 3.9 million nodes from mobile phone call records, which can be regarded as a proxy for the underlying human communication network at the societ al level. We assign two weights on each edge to reflect the strength of social interaction, which are the aggregate call duration and the cumulative number of calls placed between the individuals over a period of 18 weeks. We present a detailed analysis of this weighted network by examining its degree, strength, and weight distributions, as well as its topological assortativity and weighted assortativity, clustering and weighted clustering, together with correlations between these quantities. We give an account of motif intensity and coherence distributions and compare them to a randomized reference system. We also use the concept of link overlap to measure the number of common neighbours any two adjacent nodes have, which serves as a useful local measure for identifying the interconnectedness of communities. We report a positive correlation between the overlap and weight of a link, thus providing In this paper we develop and analyze Hydra: HYbriD cooRdinAte descent method for solving loss minimization problems with big data. We initially partition the coordinates (features) and assign each partition to a different node of a cluster. At every iteration, each node picks a random subset of the coordinates from those it owns, independently from the other computers, and in parallel computes and applies updates to the selected coordinates based on a simple closed-form formula. We give bounds on the number of iterations sufficient to approximately solve the problem with high probability, and show how it depends on the data and on the partitioning. We perform numerical experiments with a LASSO instance described by a 3TB matrix. We describe a class of algorithms for amortized inference in Bayesian networks. In this setting, we invest computation upfront to support rapid online inference for a wide range of queries. Our approach is based on learning an inverse factorization of a model's joint distribution: a factorization that turns observations into root nodes. Our algorithms accumulate information to estimate the local conditional distributions that constitute such a factorization. These stochastic inverses can be used to invert each of the computation steps leading to an observation, sampling backwards in order to quickly find a likely explanation. We show that estimated inverses converge asymptotically in number of (prior or posterior) training samples. To make use of inverses before convergence, we describe the Inverse MCMC algorithm, which uses stochastic inverses to make block proposals for a Metropolis-Hastings sampler. We explore the efficiency of this sampler for a variety of parameter regimes and Bayes nets. This paper proposes parallel and distributed algorithms for solving very large-scale sparse optimization problems on computer clusters and clouds. Modern datasets usually have a large number of features or training samples, and they are usually stored in a distributed manner. Motivated by the need of solving sparse optimization problems with large datasets, we propose two approaches including (i) distributed implementations of prox-linear algorithms and (ii) GRock, a parallel greedy block coordinate descent method. Different separability properties of the objective terms in the problem enable different data distributed schemes along with their corresponding algorithm implementations. We also establish the convergence of GRock and explain why it often performs exceptionally well for sparse optimization. Numerical results on a computer cluster and Amazon EC2 demonstrate the efficiency and elasticity of our algorithms. Community detection is an important technique to understand structures and patterns in complex networks. Recently, overlapping community detection becomes a trend due to the ubiquity of overlapping and nested communities in real world. However, existing approaches have ignored the use of implicit link preference information, i.e., links can reflect a node's preference on the targets of connections it wants to build. This information has strong impact on community detection since a node prefers to build links with nodes inside its community than those outside its community. In this paper, we propose a preference-based nonnegative matrix factorization (PNMF) model to incorporate implicit link preference information. Unlike conventional matrix factorization approaches, which simply approximate the original adjacency matrix in value, our model maximizes the likelihood of the preference order for each node by following the intuition that a node prefers its neighbors than other nodes. Our model overcomes the indiscriminate penalty problem in which non-linked pairs inside one community are equally penalized in objective functions as those across two communities. We propose a learning algorithm which can learn a node-community membership matrix via stochastic gradient descent with bootstrap sampling. We evaluate our PNMF model on several real-world networks. Experimental results show that our model outperforms state-of-the-art approaches and can be applied to large datasets.