Mixture of linear regression is well studied in statistics and machine learning, where the data points are generated probabilistically using $k$ linear models. Algorithms like Expectation Maximization (EM) may be used to recover the ground truth regressors for this problem. Recently, in \cite{pal2022learning,ghosh_agnostic} the mixed linear regression problem is studied in the agnostic setting, where no generative model on data is assumed. Rather, given a set of data points, the objective is \emph{fit} $k$ lines by minimizing a suitable loss function. It is shown that a modification of EM, namely gradient EM converges exponentially to appropriately defined loss minimizer even in the agnostic setting. In this paper, we study the problem of \emph{fitting} $k$ parametric functions to given set of data points. We adhere to the agnostic setup. However, instead of fitting lines equipped with quadratic loss, we consider any arbitrary parametric function fitting equipped with a strongly convex and smooth loss. This framework encompasses a large class of problems including mixed linear regression (regularized), mixed linear classifiers (mixed logistic regression, mixed Support Vector Machines) and mixed generalized linear regression. We propose and analyze gradient EM for this problem and show that with proper initialization and separation condition, the iterates of gradient EM converge exponentially to appropriately defined population loss minimizers with high probability. This shows the effectiveness of EM type algorithm which converges to \emph{optimal} solution in the non-generative setup beyond mixture of linear regression.

This work investigates the problem of best arm identification for multi-agent multi-armed bandits. We consider $N$ agents grouped into $M$ clusters, where each cluster solves a stochastic bandit problem. The mapping between agents and bandits is \textit{a priori} unknown. Each bandit is associated with $K$ arms, and the goal is to identify the best arm for each agent under a $\delta$-probably correct ($\delta$-PC) framework, while minimizing sample complexity and communication overhead. We propose two novel algorithms: \emph{Clustering then Best Arm Identification} (\texttt{Cl-BAI}) and \emph{Best Arm Identification then Clustering} (\texttt{BAI-Cl}). \texttt{Cl-BAI} employs a two-phase approach that first clusters agents based on the bandit problems they are learning, followed by identifying the best arm for each cluster. \texttt{BAI-Cl} reverses the sequence by identifying the best arms first and then clustering agents accordingly. Both algorithms exploit the successive elimination framework to ensure computational efficiency and high accuracy. Theoretical analysis establishes $\delta$-PC guarantees for both methods, derives bounds on their sample complexity, and provides a lower bound for the problem class. Moreover, when $M$ is small (a constant), we show that the sample complexity of (a variant of) \texttt{BAI-Cl} is (order-wise) minimax optimal. Experiments on synthetic and real-world (Movie Lens, Yelp) data demonstrates the superior performance of the proposed algorithms in terms of sample and communication efficiency, particularly in settings where $M \ll N$.

In many, if not most, machine learning applications the training data is naturally heterogeneous (e.g. federated learning, adversarial attacks and domain adaptation in neural net training). Data heterogeneity is identified as one of the major challenges in modern day large-scale learning. A classical way to represent heterogeneous data is via a mixture model. In this paper, we study generalization performance and statistical rates when data is sampled from a mixture distribution. We first characterize the heterogeneity of the mixture in terms of the pairwise total variation distance of the sub-population distributions. Thereafter, as a central theme of this paper, we characterize the range where the mixture may be treated as a single (homogeneous) distribution for learning. In particular, we study the generalization performance under the classical PAC framework and the statistical error rates for parametric (linear regression, mixture of hyperplanes) as well as non-parametric (Lipschitz, convex and Hölder-smooth) regression problems. In order to do this, we obtain Rademacher complexity and (local) Gaussian complexity bounds with mixture data, and apply them to get the generalization and convergence rates respectively. We observe that as the (regression) function classes get more complex, the requirement on the pairwise total variation distance gets stringent, which matches our intuition. We also do a finer analysis for the case of mixed linear regression and provide a tight bound on the generalization error in terms of heterogeneity.

Mixed linear regression is a well-studied problem in parametric statistics and machine learning. Given a set of samples, tuples of covariates and labels, the task of mixed linear regression is to find a small list of linear relationships that best fit the samples. Usually it is assumed that the label is generated stochastically by randomly selecting one of two or more linear functions, applying this chosen function to the covariates, and potentially introducing noise to the result. In that situation, the objective is to estimate the ground-truth linear functions up to some parameter error. The popular expectation maximization (EM) and alternating minimization (AM) algorithms have been previously analyzed for this. In this paper, we consider the more general problem of agnostic learning of mixed linear regression from samples, without such generative models. In particular, we show that the AM and EM algorithms, under standard conditions of separability and good initialization, lead to agnostic learning in mixed linear regression by converging to the population loss minimizers, for suitably defined loss functions. In some sense, this shows the strength of AM and EM algorithms that converges to ``optimal solutions'' even in the absence of realizable generative models.

Given a dataset of individuals each described by a covariate vector, a treatment, and an observed outcome on the treatment, the goal of the individual treatment effect (ITE) estimation task is to predict outcome changes resulting from a change in treatment. A fundamental challenge is that in the observational data, a covariate’s outcome is observed only under one treatment, whereas we need to infer the difference in outcomes under two different treatments. Several existing approaches address this issue through training with inferred pseudo-outcomes, but their success relies on the quality of these pseudo-outcomes. We propose PairNet, a novel ITE estimation training strategy that minimizes losses over pairs of examples based on their factual observed outcomes. Theoretical analysis for binary treatments reveals that PairNet is a consistent estimator of ITE risk, and achieves smaller generalization error than baseline models. Empirical comparison with thirteen existing methods across eight benchmarks, covering both discrete and continuous treatments, shows that PairNet achieves significantly lower ITE error compared to the baselines. Also, it is model-agnostic and easy to implement.

This article addresses the problem of detecting, with high probability, the presence of a large class of history-dependent false data injection actuator attacks in cyber-physical systems (CPSs) modeled as stochastic linear time-invariant systems. Among the primary contributions is the introduction of a new detection algorithm or scheme that leverages the fundamental idea of separating or classifying two classes of state trajectories: The first class comprises state trajectories generated by the CPS under attack situations, while the second class includes state trajectories generated under nominal conditions. The existence and non-existence of such a separator are studied and these results are asymptotic. We then establish finite-time guarantees associated with the detection algorithm or scheme. A numerical example is provided to demonstrate the theory.

The problem of saddle-points avoidance for non-convex optimization is quite challenging in large scale distributed learning frameworks. The celebrated cubic-regularized Newton method of Nesterov and Polyak [1] is one of the most elegant algorithms to avoid saddle-points in the standard centralized (non-distributed) setup. In this paper, we analyze the cubic-regularized Newton method in the distributed framework and simultaneously address several practical challenges that naturally arises, such as communication bottleneck and Byzantine attacks. To that end, we propose DISTributed CUbic REgularized Newton's method (DIST-CURE), and obtain convergence guarantees under several settings. We emphasize that the issue of saddle-point avoidance becomes more crucial in the presence of Byzantine machines since rogue machines may create fake local minima near the saddle-points of the loss function (this is known as the saddle-point attack). Being a second order algorithm, the iteration complexity of DIST-CURE is much lower than its first order counterparts, and furthermore we can further compress to achieve communication efficiency. To address the challenge of Byzantine resilience, we employ norm based thresholding on the local solutions. We validate the performance of DIST-CURE with experiments using standard datasets and several types of Byzantine attacks, and obtain an improvement of 25% with respect to first order methods in total iteration complexity. Full Paper: Available at: http://tinyurl.com/3axkfy8w

Online learning in a decentralized two-sided matching markets, where the demand-side (players) compete to match with the supply-side (arms), has received substantial interest because it abstracts out the complex interactions in matching platforms (e.g. UpWork, TaskRabbit). However, past works [1]–[5] assume that the each arm knows their preference ranking over the players (one-sided learning), and each player aim to learn the preference over arms through successive interactions. Moreover, several (impractical) assumptions on the problem are usually made for theoretical tractability such as broadcast player-arm match ( [1], [2], [5]) or serial dictatorship ( [3], [4], [6]). In this paper, we study a decentralized two-sided matching market, where we do not assume that the preference ranking over players are known to the arms apriori. Furthermore, we do not have any structural assumptions on the problem. We propose a multi-phase explore-then-commit type algorithm namely epoch-based CA-ETC (collision avoidance explore then commit) (CA-ETC in short) for this problem that does not require any communication across agents (players and arms) and hence decentralized. We show that for the initial epoch length of To and subsequent epoch-lengths of (for the l-th epoch with E (0,1) as an input parameter to the algorithm), CA-ETC yields a player optimal expected regret of ( ( for the i-th player, where$T$is the learning horizon,$K$is the number of arms and is an appropriately defined problem gap. Furthermore, we propose a blackboard communication based baseline achieving logarithmic regret in$T$.11Appendix at https://bit.ly/ISIT_matchingmarkets

We consider the problem of model selection for the general stochastic contextual bandits under the realizability assumption. We propose a successive refinement based algorithm called Adaptive Contextual Bandit (ACB), that works in phases and successively eliminates model classes that are too simple to fit the given instance. We prove that this algorithm is adaptive, i.e., the regret rate order-wise matches that of any provable contextual bandit algorithm, that needs the knowledge of the true model class. The price of not knowing the correct model class turns out to be only an additive term contributing to the second order term in the regret bound. This cost possess the intuitive property that it becomes smaller as the model class becomes easier to identify, and vice-versa. We also show that a much simpler explore-then-commit (ETC) style algorithm also obtains similar regret bound, despite not knowing the true model class. However, the cost of model selection is higher in ETC as opposed to inACB, as expected. Furthermore, for the special case of linear contextual bandits, we propose specialized algorithms that obtain sharper guarantees compared to the generic setup.

Understanding complex dynamics of two-sided online matching markets, where the demand-side agents compete to match with the supply-side (arms), has recently received substantial interest. To that end, in this paper, we introduce the framework of decentralized two-sided matching market under non stationary (dynamic) environments. We adhere to the serial dictatorship setting, where the demand-side agents have unknown and different preferences over the supply-side (arms), but the arms have fixed and known preference over the agents. We propose and analyze an asynchronous and decentralized learning algorithm, namely Non-Stationary Competing Bandits (NSCB), where the agents play (restrictive) successive elimination type learning algorithms to learn their preference over the arms. The complexity in understanding such a system stems from the fact that the competing bandits choose their actions in an asynchronous fashion, and the lower ranked agents only get to learn from a set of arms, not dominated by the higher ranked agents, which leads toforced exploration. With carefully defined complexity parameters, we characterize thisforced explorationand obtain sub-linear (logarithmic) regret of NSCB. Furthermore, we validate our theoretical findings via experiments.

We present a non-asymptotic lower bound on the spectrum of the design matrix generated by any linear bandit algorithm with sub-linear regret when the action set has well-behaved curvature. Specifically, we show that the minimum eigenvalue of the expected design matrix grows as $\Omega(\sqrt{n})$ whenever the expected cumulative regret of the algorithm is $O(\sqrt{n})$, where $n$ is the learning horizon, and the action-space has a constant Hessian around the optimal arm. This shows that such action-spaces force a polynomial lower bound on the least eigenvalue, rather than a logarithmic lower bound as shown by Lattimore et al. (2017) for discrete (i.e., well-separated) action spaces. Furthermore, while the latter holds only in the asymptotic regime ($n \to \infty$), our result for these “locally rich” action spaces is any-time. Additionally, under a mild technical assumption, we obtain a similar lower bound on the minimum eigen value holding with high probability. We apply our result to two practical scenarios – model selection and clustering in linear bandits. For model selection, we show that an epoch-based linear bandit algorithm adapts to the true model complexity at a rate exponential in the number of epochs, by virtue of our novel spectral bound. For clustering, we consider a multi agent framework where we show, by leveraging the spectral result, that no forced exploration is necessary—the agents can run a linear bandit algorithm and estimate their underlying parameters at once, and hence incur a low regret.

Motivated by the need for communication-efficient distributed learning, we investigate the method for compressing a unit norm vector into the minimum number of bits, while still allowing for some acceptable level of distortion in recovery. This problem has been explored in the rate-distortion/covering code literature, but our focus is exclusively on the "high-distortion" regime. We approach this problem in a worst-case scenario, without any prior information on the vector, but allowing for the use of randomized compression maps. Our study considers both biased and unbiased compression methods and determines the optimal compression rates. It turns out that simple compression schemes are nearly optimal in this scenario. While the results are a mix of new and known, they are compiled in this paper for completeness.

We prove an instance independent (poly) logarithmic regret for stochastic contextual bandits with linear payoff. Previously, in \cite{chu2011contextual}, a lower bound of $\mathcal{O}(\sqrt{T})$ is shown for the contextual linear bandit problem with arbitrary (adversarily chosen) contexts. In this paper, we show that stochastic contexts indeed help to reduce the regret from $\sqrt{T}$ to $\polylog(T)$. We propose Low Regret Stochastic Contextual Bandits (\texttt{LR-SCB}), which takes advantage of the stochastic contexts and performs parameter estimation (in $\ell_2$ norm) and regret minimization simultaneously. \texttt{LR-SCB} works in epochs, where the parameter estimation of the previous epoch is used to reduce the regret of the current epoch. The (poly) logarithmic regret of \texttt{LR-SCB} stems from two crucial facts: (a) the application of a norm adaptive algorithm to exploit the parameter estimation and (b) an analysis of the shifted linear contextual bandit algorithm, showing that shifting results in increasing regret. We have also shown experimentally that stochastic contexts indeed incurs a regret that scales with $\polylog(T)$.

While mixture of linear regressions (MLR) is a well-studied topic, prior works usually do not analyze such models for prediction error. In fact, prediction and loss are not well-defined in the context of mixtures. In this paper, first we show that MLR can be used for prediction where instead of predicting a label, the model predicts a list of values (also known as list-decoding). The list size is equal to the number of components in the mixture, and the loss function is defined to be minimum among the losses resulted by all the component models. We show that with this definition, a solution of the empirical risk minimization (ERM) achieves small probability of prediction error. This begs for an algorithm to minimize the empirical risk for MLR, which is known to be computationally hard. Prior algorithmic works in MLR focus on the realizable setting, i.e., recovery of parameters when data is probabilistically generated by a mixed linear (noisy) model. In this paper we show that a version of the popular expectation minimization (EM) algorithm finds out the best fit lines in a dataset even when a realizable model is not assumed, under some regularity conditions on the dataset and the initial points, and thereby provides a solution for the ERM. We further provide an algorithm that runs in polynomial time in the number of datapoints, and recovers a good approximation of the best fit lines. The two algorithms are experimentally compared.

We address the problem of federated learning (FL) where users are distributed and partitioned into clusters. This setup captures settings where different groups of users have their own objectives (learning tasks) but by aggregating their data with others in the same cluster (same learning task), they can leverage the strength in numbers in order to perform more efficient federated learning. For this new framework of clustered federated learning, we propose the Iterative Federated Clustering Algorithm (IFCA), which alternately estimates the cluster identities of the users and optimizes model parameters for the user clusters via gradient descent. We analyze the convergence rate of this algorithm first in a linear model with squared loss and then for generic strongly convex and smooth loss functions. We show that in both settings, with good initialization, IFCA is guaranteed to converge, and discuss the optimality of the statistical error rate. In particular, for the linear model with two clusters, we can guarantee that our algorithm converges as long as the initialization is slightly better than random. When the clustering structure is ambiguous, we propose to train the models by combining IFCA with the weight sharing technique in multi-task learning. In the experiments, we show that our algorithm can succeed even if we relax the requirements on initialization with random initialization and multiple restarts. We also present experimental results showing that our algorithm is efficient in non-convex problems such as neural networks. We demonstrate the benefits of IFCA over the baselines on several clustered FL benchmarks.

Max-affine regression refers to a model where the unknown regression function is modeled as a maximum of$k$unknown affine functions for a fixed$k \geq 1$. This generalizes linear regression and (real) phase retrieval, and is closely related to convex regression. We study this problem in the high-dimensional setting assuming that$k$is a fixed constant, and focus on the estimation of the unknown coefficients of the affine functions underlying the model. We analyze a natural alternating minimization (AM) algorithm for the non-convex least squares objective when the design is Gaussian. We show that the AM algorithm, when initialized suitably, converges with high probability and at a geometric rate to a small ball around the optimal coefficients. In order to initialize the algorithm, we propose and analyze a combination of a spectral method and a search algorithm in a low-dimensional space, which may be of independent interest. The final rate that we obtain is near-parametric and minimax optimal (up to a polylogarithmic factor) as a function of the dimension, sample size, and noise variance. In that sense, our approach should be viewed as adirectand implementable method of enforcing regularization to alleviate the curse of dimensionality in problems of the convex regression type. Numerical experiments illustrate the sharpness of our bounds in the various problem parameters.

We consider the problem of model selection for two popular stochastic linear bandit settings, and propose algorithms that adapts to the unknown problem complexity. In the first setting, we consider the $K$ armed mixture bandits, where the mean reward of arm $i \in [K]$ is $\mu_i+ ⟨\alpha_{i,t},\theta^* ⟩$, with $\alpha_{i,t} \in \mathbb{R}^d$ being the known context vector and $\mu_i \in [-1,1]$ and $\theta^*$ are unknown parameters. We define $\|\theta^*\|$ as the problem complexity and consider a sequence of nested hypothesis classes, each positing a different upper bound on $\|\theta^*\|$. Exploiting this, we propose Adaptive Linear Bandit (ALB), a novel phase based algorithm that adapts to the true problem complexity, $\|\theta^*\|$. We show that ALB achieves regret scaling of $\widetilde{O}(\|\theta^*\|\sqrt{T})$, where $\|\theta^*\|$ is apriori unknown. As a corollary, when $\theta^*=0$, ALB recovers the minimax regret for the simple bandit algorithm without such knowledge of $\theta^*$. ALB is the first algorithm that uses parameter norm as model section criteria for linear bandits. Prior state of art algorithms achieve a regret of $\widetilde{O}(L\sqrt{T})$, where $L$ is the upper bound on $\|\theta^*\|$, fed as an input to the problem. In the second setting, we consider the standard linear bandit problem (with possibly an infinite number of arms) where the sparsity of $\theta^*$, denoted by $d^* \leq d$, is unknown to the algorithm. Defining $d^*$ as the problem complexity (similar to Foster et. al ’19), we show that ALB achieves $\widetilde{O}(d^*\sqrt{T})$ regret, matching that of an oracle who knew the true sparsity level. This methodology is then extended to the case of finitely many arms and similar results are proven. We further verify through synthetic and real-data experiments that the performance gains are fundamental and not artifacts of mathematical bounds. In particular, we show $1.5-3$x drop in cumulative regret over non-adaptive algorithms.

To address the communication bottleneck problem in distributed optimization within a master-worker framework, we propose LocalNewton, a distributed second-order algorithm with local averaging. In LocalNewton, the worker machines update their model in every iteration by finding a suitable second-order descent direction using only the data and model stored in their own local memory. We let the workers run multiple such iterations locally and communicate the models to the master node only once every few (say $L$) iterations. LocalNewton is highly practical since it requires only one hyperparameter, the number $L$ of local iterations. We use novel matrix concentration based techniques to obtain theoretical guarantees for LocalNewton, and we validate them with detailed empirical evaluation. To enhance practicability, we devise an adaptive scheme to choose $L$, and we show that this reduces the number of local iterations in worker machines between two model synchronizations as the training proceeds, successively refining the model quality at the master. Via extensive experiments using several real-world datasets with AWS Lambda workers and an AWS EC2 master, we show that LocalNewton requires fewer than $60%$ of the communication rounds (between master and workers) and less than $40%$ of the end-to-end running time, compared to state-of-the-art algorithms, to reach the same training loss.

We address the problem of solving mixed random linear equations. In this problem, we have unlabeled observations coming from multiple linear regressions, and each observation corresponds to exactly one of the regression models. The goal is to learn the linear regressors from the observations. Classically, Alternating Minimization (AM) (which may be thought as a variant of Expectation Maximization (EM)) is used to solve this problem. AM iteratively alternates between the estimation of labels and solving the regression problems with the estimated labels. Empirically, it is observed that, for a large variety of non-convex problems including mixed linear regression, AM converges at a much faster rate compared to gradient based algorithms. However, the existing theory suggests similar rate of convergence, failing to capture this empirical behavior. In this paper, we close this gap between theory and practice for the special case of a mixture of $2$ linear regressions. We show that, provided initialized properly, AM enjoys a \emph{super-linear} rate of convergence. To the best of our knowledge, this is the first work that theoretically establishes such rate for AM. Hence, if we want to recover the unknown regressors upto an error (in $\ell_2$ norm) of $\epsilon$, AM only takes $\mathcal{O}(\log \log (1/\epsilon))$ iterations.

We develop a communication-efficient distributed learning algorithm that is robust against Byzantine worker machines. We propose and analyze a distributed gradient-descent algorithm that performs a simple thresholding based on gradient norms to mitigate Byzantine failures. We show the (statistical) error-rate of our algorithm matches that of Yin et al., 2018, which uses more complicated schemes (like coordinate-wise median or trimmed mean). Furthermore, for communication efficiency, we consider a generic class of δ-approximate compressors from Karimireddy et al., 2019, that encompasses sign-based compressors and top-k sparsification. Our algorithm uses compressed gradients and gradient norms for aggregation and Byzantine removal respectively. We establish the statistical error rate of the algorithm for arbitrary (convex or non-convex) smooth loss function. We show that, in certain regime of δ, the rate of convergence is not affected by the compression operation. We have experimentally validated our results and shown good performance in convergence for convex (least-square regression) and non-convex (neural network training) problems.

In this paper, we develop a communication efficient second order distributed Newton-type algorithm. For communication efficiency, we consider a generic class of δ-approximate compressors (Karimireddy et al., 2019), which includes sign-based compression and top-k sparsification. We provide three potential settings where compression can be employed; and provide rate of convergence for smooth objectives. We show that, in the regime where δ is constant, our theoretical convergence rate matches that of a state-of-the-art distributed second order algorithm called DINGO (Crane and Roosta, 2019). This implies that we get the compression for free in this regime. The full paper can be found at https://tinyurl.com/ujnpt4c.

We study the max-affine regression model, where the unknown regression function is modeled as a maximum of a fixed number of affine functions. In recent work [1], we showed that end-to-end parameter estimates were obtainable using this model with an alternating minimization (AM) algorithm provided the covariates (or designs) were normally distributed, and chosen independently of the underlying parameters. In this paper, we show that AM is significantly more robust than the setting of [1]: It converges locally under small-ball design assumptions (which is a much broader class, including bounded log-concave distributions), and even when the underlying parameters are chosen with knowledge of the realized covariates. Once again, the final rate obtained by the procedure is near-parametric and minimax optimal (up to a polylogarithmic factor) as a function of the dimension, sample size, and noise variance. As a by-product of our analysis, we obtain convergence guarantees on a classical algorithm for the (real) phase retrieval problem in the presence of noise under considerably weaker assumptions on the design distribution than was previously known.

We analyze the performance of the LASSO algorithm (basis pursuit, Tibshirani et. al, '96) for a class of structured matrices known as convolutional sparse matrix. Analyzing such matrices is of paramount interest since in many signal processing applications (including computer vision, image and audio processing), a global analysis of the underlying signal often entails understanding the behavior of convolutional sparse matrix. We show that LASSO (ℓ1regularized least squares) with such matrices succeeds under a constraint on local sparsity, as opposed to global sparsity. This conversion from global to local constraint has crucial significance in the above mentioned applications. Under sufficiency conditions like Restricted Eigen-value (RE) and Exact Recovery Coefficient (ERC), we obtain the prediction (in ℓ2norm) error and estimation error rate for LASSO estimator with local sparsity constraints. Furthermore, we obtain an estimation error rate for LASSO estimator in ℓ∞norm under a gaussian noise model. A full version of this paper is accessible at: https://tinyurl.com/rwy2l3o.

We address the problem of Federated Learning (FL) where users are distributed and partitioned into clusters. This setup captures settings where different groups of users have their own objectives (learning tasks) but by aggregating their data with others in the same cluster (same learning task), they can leverage the strength in numbers in order to perform more efficient Federated Learning. We propose a new framework dubbed the Iterative Federated Clustering Algorithm (IFCA), which alternately estimates the cluster identities of the users and optimizes model parameters for the user clusters via gradient descent. We analyze the convergence rate of this algorithm first in a linear model with squared loss and then for generic strongly convex and smooth loss functions. We show that in both settings, with good initialization, IFCA converges at an exponential rate, and discuss the optimality of the statistical error rate. When the clustering structure is ambiguous, we propose to train the models by combining IFCA with the weight sharing technique in multi-task learning. In the experiments, we show that our algorithm can succeed even if we relax the requirements on initialization with random initialization and multiple restarts. We also present experimental results showing that our algorithm is efficient in non-convex problems such as neural networks. We demonstrate the benefits of IFCA over the baselines on several clustered FL benchmarks.

We develop a distributed second order optimization algorithm that is communication-efficient as well as robust against Byzantine failures of the worker machines. We propose an iterative approximate Newton-type algorithm, where the worker machines communicate \emph{only once} per iteration with the central machine. This is in sharp contrast with the state-of-the-art distributed second order algorithms like GIANT \cite{giant}, DINGO\cite{dingo}, where the worker machines send (functions of) local gradient and Hessian sequentially; thus ending up communicating twice with the central machine per iteration. Furthermore, we employ a simple norm based thresholding rule to filter-out the Byzantine worker machines. We establish the linear-quadratic rate of convergence of our proposed algorithm and establish that the communication savings and Byzantine resilience attributes only correspond to a small statistical error rate for arbitrary convex loss functions. To the best of our knowledge, this is the first work that addresses the issue of Byzantine resilience in second order distributed optimization. Furthermore, we validate our theoretical results with extensive experiments on synthetically generated and benchmark LIBSVM \cite{libsvm} data-set and demonstrate convergence guarantees.

We are motivated by the need, in emergency situations, for impromptu (or “as-you-go”) deployment of multihop wireless networks, by human agents or robots (e.g., unmanned aerial vehicles (UAVs)); the agent moves along a line, makes wireless link quality measurements at regular intervals, and makes on-line placement decisions using these measurements. As a first step, we have formulated such deployment along a line as a sequential decision problem. In our earlier work, reported in [1], we proposed two possible deployment approaches: (i) the pure as-you-go approach where the deployment agent can only move forward, and (ii) the explore-forward approach where the deployment agent explores a few successive steps and then selects the best relay placement location among them. The latter was shown to provide better performance (in terms of network cost, network performance, and power expenditure), but at the expense of more measurements and deployment time, which makes explore-forward impractical for quick deployment by an energy constrained agent such as a UAV. Further, since in emergency situations the terrain would be unknown, the deployment algorithm should not require a-priori knowledge of the parameters of the wireless propagation model. In [1], we, therefore, developed learning algorithms for the explore-forward approach. The current paper fills in an important gap by providing deploy-and-learn algorithms for the pure as-you-go approach. We formulate the sequential relay deployment problem as an average cost Markov decision process (MDP), which trades off among power consumption, link outage probabilities, and the number of relay nodes in the deployed network. While the pure as-you-go deployment problem was previously formulated as a discounted cost MDP (see [1]), the discounted cost MDP formulation was not amenable for learning algorithms that are proposed in this paper. In this paper, first we show structural results for the optimal policy corresponding to the average cost MDP, and provide new insights into the optimal policy. Next, by exploiting the special structure of the average cost optimality equation and by using the theory of asynchronous stochastic approximation (in single and two timescale), we develop two learning algorithms that asymptotically converge to the set of optimal policies as deployment progresses. Numerical results show reasonably fast speed of convergence, and hence the model-free algorithms can be useful for practical, fast deployment of emergency wireless networks.

We consider the problem of online learning in misspecified linear stochastic multi-armed bandit problems. Regret guarantees for state-of-the-art linear bandit algorithms such as Optimism in the Face of Uncertainty Linear bandit (OFUL) hold under the assumption that the arms expected rewards are perfectly linear in their features. It is, however, of interest to investigate the impact of potential misspecification in linear bandit models, where the expected rewards are perturbed away from the linear subspace determined by the arms features. Although OFUL has recently been shown to be robust to relatively small deviations from linearity, we show that any linear bandit algorithm that enjoys optimal regret performance in the perfectly linear setting (e.g., OFUL) must suffer linear regret under a sparse additive perturbation of the linear model. In an attempt to overcome this negative result,we define a natural class of bandit models characterized by a non-sparse deviation from linearity. We argue that the OFUL algorithm can fail to achieve sublinear regret even under models that have non-sparse deviation. We finally develop a novel bandit algorithm, comprising a hypothesis test for linearity followed by a decision to use either the OFUL or Upper Confidence Bound (UCB) algorithm. For perfectly linear bandit models, the algorithm provably exhibits OFULs favorable regret performance, while for misspecified models satisfying the non-sparse deviation property, the algorithm avoids the linear regret phenomenon and falls back on UCBs sublinear regret scaling. Numerical experiments on synthetic data, and on recommendation data from the public Yahoo! Learning toRank Challenge dataset, empirically support our findings.

Motivated by the need for impromptu or as-you-go deployment of wireless sensor networks in some situations, we study the problem of optimal sequential deployment of wireless sensors and relays along a line (e.g., a forest trail) of unknown length. Starting from the sink node (e.g., a base station), a ”deployment agent„ walks along the line, stops at equally spaced points (”potential„ relay locations), placing relays at some of these points, until he reaches a location at which the source node (i.e., the sensor) needs to be placed, the objective being to create a multihop wireless relay network between the source and the sink. The deployment agent decides whether to place a relay or not at each of the potential locations, depending upon the link quality measurements to the previously placed relays. In this article, we seek to design efficient deployment algorithms for this class of problems, to achieve the objective of 2-connectivity in the deployed network. We ensure multi-connectivity by allowing each node to communicate with more than one neighbouring node. By proposing a network cost objective that is additive over the deployed relays, we formulate the relay placement problem as a Markov decision process. We provide structural results for the optimal policy and evaluate the performance of the optimal policy via numerical exploration. Computation of such an optimal deployment policy requires a statistical model for radio propagation; we extract this model from the raw data collected via measurements in a forestlike environment. To validate the results obtained from the numerical study, we provide an experimental study of algorithms for 2-connected network deployment.

We are motivated by the problem of impromptu or as-you-go deployment of wireless sensor networks. As an application example, a person, starting from a sink node, walks along a forest trail, makes link quality measurements (with the previously placed nodes) at equally spaced locations, and deploys relays at some of these locations, so as to connect a sensor placed at some a priori unknown point on the trail with the sink node. In this paper, we report our experimental experiences with some as-you-go deployment algorithms. Two algorithms are based on Markov decision process (MDP) formulations, these require a radio propagation model. We also study purely measurement based strategies: one heuristic that is motivated by our MDP formulations, one asymptotically optimal learning algorithm, and one inspired by a popular heuristic. We extract a statistical model of the propagation along a forest trail from raw measurement data, implement the algorithms experimentally in the forest, and compare them. The results provide useful insights regarding the choice of the deployment algorithm and its parameters, and also demonstrate the necessity of a proper theoretical formulation.

The paper presents an elegant approach for designing linear phase low pass digital FIR filter using swarm and evolutionary algorithms. Classical gradient based approaches are not efficient enough for accurate design and thus evolutionary approach is considered to be a better choice. In this paper a hybrid of Genetic Algorithm and Particle Swarm Optimization algorithm with varying neighbourhood topology, namely Genetic Lbest Particle Swarm Optimization with Dynamically Varying Neighbourhood (GLPSO DVN) is used to find the filter coefficients. In this work two objective functions (error metrics) are minimized. The first one is based on stop and pass band ripple and the second one studies the mean square error between the ideal and actual designed filter. The hybrid algorithm is found to produce fitter candidate solution than the classical Lbest PSO. The results are compared with the results obtained by solving the same problem using Lbest PSO (LPSO). It is also observed that GLPSO DVN gives better results than LPSO and as well LPSO DVN.