Manjesh K. Hanawal

Divya J. Bajpai, Vivek K. Trivedi, Sohan L. Yadav et al.

Deep Neural Networks (DNNs) have drawn attention because of their outstanding performance on various tasks. However, deploying full-fledged DNNs in resource-constrained devices (edge, mobile, IoT) is difficult due to their large size. To overcome the issue, various approaches are considered, like offloading part of the computation to the cloud for final inference (split computing) or performing the inference at an intermediary layer without passing through all layers (early exits). In this work, we propose combining both approaches by using early exits in split computing. In our approach, we decide up to what depth of DNNs computation to perform on the device (splitting layer) and whether a sample can exit from this layer or need to be offloaded. The decisions are based on a weighted combination of accuracy, computational, and communication costs. We develop an algorithm named SplitEE to learn an optimal policy. Since pre-trained DNNs are often deployed in new domains where the ground truths may be unavailable and samples arrive in a streaming fashion, SplitEE works in an online and unsupervised setup. We extensively perform experiments on five different datasets. SplitEE achieves a significant cost reduction ($>50\%$) with a slight drop in accuracy ($<2\%$) as compared to the case when all samples are inferred at the final layer. The anonymized source code is available at \url{https://anonymous.4open.science/r/SplitEE_M-B989/README.md}.

Hari Narayan N U, Manjesh K. Hanawal, Avinash Bhardwaj

Deep Neural Networks (DNNs) are generally designed as sequentially cascaded differentiable blocks/layers with a prediction module connected only to its last layer. DNNs can be attached with prediction modules at multiple points along the backbone where inference can stop at an intermediary stage without passing through all the modules. The last exit point may offer a better prediction error but also involves more computational resources and latency. An exit point that is `optimal' in terms of both prediction error and cost is desirable. The optimal exit point may depend on the latent distribution of the tasks and may change from one task type to another. During neural inference, the ground truth of instances may not be available and error rates at each exit point cannot be estimated. Hence one is faced with the problem of selecting the optimal exit in an unsupervised setting. Prior works tackled this problem in an offline supervised setting assuming that enough labeled data is available to estimate the error rate at each exit point and tune the parameters for better accuracy. However, pre-trained DNNs are often deployed in new domains for which a large amount of ground truth may not be available. We model the problem of exit selection as an unsupervised online learning problem and use bandit theory to identify the optimal exit point. Specifically, we focus on Elastic BERT, a pre-trained multi-exit DNN to demonstrate that it `nearly' satisfies the Strong Dominance (SD) property making it possible to learn the optimal exit in an online setup without knowing the ground truth labels. We develop upper confidence bound (UCB) based algorithm named UEE-UCB that provably achieves sub-linear regret under the SD property. Thus our method provides a means to adaptively learn domain-specific optimal exit points in multi-exit DNNs. We empirically validate our algorithm on IMDb and Yelp datasets.

Debamita Ghosh, Manjesh K. Hanawal, Nikola Zlatanov

Holographic metasurface transceivers (HMT) is an emerging technology for enhancing the coverage and rate of wireless communication systems. However, acquiring accurate channel state information in HMT-assisted wireless communication systems is critical for achieving these goals. In this paper, we propose an algorithm for learning the optimal phase-shifts at a HMT for the far-field channel model. Our proposed algorithm exploits the structure of the channel gains in the far-field regions and learns the optimal phase-shifts in presence of noise in the received signals. We prove that the probability that the optimal phase-shifts estimated by our proposed algorithm deviate from the true values decays exponentially in the number of pilot signals. Extensive numerical simulations validate the theoretical guarantees and also demonstrate significant gains as compared to the state-of-the-art policies.

Debamita Ghosh, Haseen Rahman, Manjesh K. Hanawal et al.

Millimeter wave (mmWave) communications have a broad spectrum and can support data rates in the order of gigabits per second, as envisioned in 5G systems. However, they cannot be used for long distances due to their sensitivity to attenuation loss. To enable their use in the 5G network, it requires that the transmission energy be focused in sharp pencil beams. As any misalignment between the transmitter and receiver beam pair can reduce the data rate significantly, it is important that they are aligned as much as possible. To find the best transmit-receive beam pair, recent beam alignment (BA) techniques examine the entire beam space, which might result in a large amount of BA latency. Recent works propose to adaptively select the beams such that the cumulative reward measured in terms of received signal strength or throughput is maximized. In this paper, we develop an algorithm that exploits the unimodal structure of the received signal strengths of the beams to identify the best beam in a finite time using pure exploration strategies. Strategies that identify the best beam in a fixed time slot are more suitable for wireless network protocol design than cumulative reward maximization strategies that continuously perform exploration and exploitation. Our algorithm is named Unimodal Bandit for Best Beam (UB3) and identifies the best beam with a high probability in a few rounds. We prove that the error exponent in the probability does not depend on the number of beams and show that this is indeed the case by establishing a lower bound for the unimodal bandits. We demonstrate that UB3 outperforms the state-of-the-art algorithms through extensive simulations. Moreover, our algorithm is simple to implement and has lower computational complexity.

Dmitry Artemasov, Alexander Shmatok, Kirill Andreev et al.

The integration of terahertz communications and ultra-massive multiple-input multiple-output (UM-MIMO) systems in 6G networks is motivated by their ability to enable unprecedented data rates, mitigate spectrum congestion, and enhance overall network performance. However, the enlarged antenna apertures and higher carrier frequencies in these systems increase the Rayleigh distance, causing users to span both the near-field and conventional far-field regions. Accurate spatial precoding thus requires exact channel estimation at the base station - a task made more challenging by the hybrid coexistence of near- and far-field effects and the limited number of digital chains available in hybrid beamforming architectures. In this paper, we propose a block recurrent transformer model to address this challenge. We demonstrate that a single transformer block equipped with state memory can be trained once and then iteratively applied for hybrid-field channel estimation. Furthermore, we train the model such that it generalizes to wireless channels with varying scatterer distances, different numbers of propagation paths, and wideband operation. Simulation results show that the proposed method achieves performance gains of approximately 5 dB and 7.5 dB in normalized mean squared error (NMSE) over state-of-the-art solutions in narrowband and wideband scenarios, respectively.

Manjesh K. Hanawal, Sumit J. Darak

In wireless networks, the rate achieved depends on factors like level of interference, hardware impairments, and channel gain. Often, instantaneous values of some of these factors can be measured, and they provide useful information about the instantaneous rate achieved. For example, higher interference implies a lower rate. In this work, we treat any such measurable quality that has a non-zero correlation with the rate achieved as side-information and study how it can be exploited to quickly learn the channel that offers higher throughput (reward). When the mean value of the side-information is known, using control variate theory we develop algorithms that require fewer samples to learn the parameters and can improve the learning rate compared to cases where side-information is ignored. Specifically, we incorporate side-information in the classical Upper Confidence Bound (UCB) algorithm and quantify the gain achieved in the regret performance. We show that the gain is proportional to the amount of the correlation between the reward and associated side-information. We discuss in detail various side-information that can be exploited in cognitive radio and air-to-ground communication in $L-$band. We demonstrate that correlation between the reward and side-information is often strong in practice and exploiting it improves the throughput significantly.

Rahul Vaze, Manjesh K. Hanawal

We consider a continuous-time multi-arm bandit problem (CTMAB), where the learner can sample arms any number of times in a given interval and obtain a random reward from each sample, however, increasing the frequency of sampling incurs an additive penalty/cost. Thus, there is a tradeoff between obtaining large reward and incurring sampling cost as a function of the sampling frequency. The goal is to design a learning algorithm that minimizes regret, that is defined as the difference of the payoff of the oracle policy and that of the learning algorithm. CTMAB is fundamentally different than the usual multi-arm bandit problem (MAB), e.g., even the single-arm case is non-trivial in CTMAB, since the optimal sampling frequency depends on the mean of the arm, which needs to be estimated. We first establish lower bounds on the regret achievable with any algorithm and then propose algorithms that achieve the lower bound up to logarithmic factors. For the single-arm case, we show that the lower bound on the regret is $Ω((\log T)^2/μ)$, where $μ$ is the mean of the arm, and $T$ is the time horizon. For the multiple arms case, we show that the lower bound on the regret is $Ω((\log T)^2 μ/Δ^2)$, where $μ$ now represents the mean of the best arm, and $Δ$ is the difference of the mean of the best and the second-best arm. We then propose an algorithm that achieves the bound up to constant terms.

Sayan Chatterjee, Manjesh K. Hanawal

Critical role of Internet of Things (IoT) in various domains like smart city, healthcare, supply chain and transportation has made them the target of malicious attacks. Past works in this area focused on centralized Intrusion Detection System (IDS), assuming the existence of a central entity to perform data analysis and identify threats. However, such IDS may not always be feasible, mainly due to spread of data across multiple sources and gathering at central node can be costly. Also, the earlier works primarily focused on improving True Positive Rate (TPR) and ignored the False Positive Rate (FPR), which is also essential to avoid unnecessary downtime of the systems. In this paper, we first present an architecture for IDS based on hybrid ensemble model, named PHEC, which gives improved performance compared to state-of-the-art architectures. We then adapt this model to a federated learning framework that performs local training and aggregates only the model parameters. Next, we propose Noise-Tolerant PHEC in centralized and federated settings to address the label-noise problem. The proposed idea uses classifiers using weighted convex surrogate loss functions. Natural robustness of KNN classifier towards noisy data is also used in the proposed architecture. Experimental results on four benchmark datasets drawn from various security attacks show that our model achieves high TPR while keeping FPR low on noisy and clean data. Further, they also demonstrate that the hybrid ensemble models achieve performance in federated settings close to that of the centralized settings.

Arun Verma, Manjesh K. Hanawal

This paper studies a new variant of the stochastic multi-armed bandits problem where auxiliary information about the arm rewards is available in the form of control variates. In many applications like queuing and wireless networks, the arm rewards are functions of some exogenous variables. The mean values of these variables are known a priori from historical data and can be used as control variates. Leveraging the theory of control variates, we obtain mean estimates with smaller variance and tighter confidence bounds. We develop an upper confidence bound based algorithm named UCB-CV and characterize the regret bounds in terms of the correlation between rewards and control variates when they follow a multivariate normal distribution. We also extend UCB-CV to other distributions using resampling methods like Jackknifing and Splitting. Experiments on synthetic problem instances validate performance guarantees of the proposed algorithms.

Arun Verma, Manjesh K. Hanawal, Arun Rajkumar et al.

We consider the problem of sequentially allocating resources in a censored semi-bandits setup, where the learner allocates resources at each step to the arms and observes loss. The loss depends on two hidden parameters, one specific to the arm but independent of the resource allocation, and the other depends on the allocated resource. More specifically, the loss equals zero for an arm if the resource allocated to it exceeds a constant (but unknown) arm dependent threshold. The goal is to learn a resource allocation that minimizes the expected loss. The problem is challenging because the loss distribution and threshold value of each arm are unknown. We study this setting by establishing its `equivalence' to Multiple-Play Multi-Armed Bandits (MP-MAB) and Combinatorial Semi-Bandits. Exploiting these equivalences, we derive optimal algorithms for our problem setting using known algorithms for MP-MAB and Combinatorial Semi-Bandits. The experiments on synthetically generated data validate the performance guarantees of the proposed algorithms.

Vinod S. Khandkar, Manjesh K. Hanawal

Network middle-boxes often classify the traffic flows on the Internet to perform traffic management or discriminate one traffic against the other. As the widespread adoption of HTTPS protocol has made it difficult to classify the traffic looking into the content field, one of the fields the middle-boxes look for is Server Name Indicator (SNI), which goes in plain text. SNI field contains information about the host and can, in turn, reveal the type of traffic. This paper presents a method to mask the server host identity by encrypting the SNI. We develop a simple method that completes the SSL/TLS connection establishment over two handshakes - the first handshake establishes a secure channel without sharing SNI information, and the second handshake shares the encrypted SNI. Our method makes it mandatory for fronting servers to always accept the handshake request without the SNI and respond with a valid SSL certificate. As there is no modification in already proven SSL/TLS encryption mechanism and processing of handshake messages, the new method enjoys all security benefits of existing secure channel establishment and needs no modification in existing routers/middle-boxes. Using customized client-server over the live Internet, we demonstrate the feasibility of our method. Moreover, the impact analysis shows that the method adheres to almost all SSL/TLS related Internet standards requirements.

Debamita Ghosh, Manjesh K. Hanawal, Nikola Zlatanov

We study wireless power transmission by an energy source to multiple energy harvesting nodes with the aim to maximize the energy efficiency. The source transmits energy to the nodes using one of the available power levels in each time slot and the nodes transmit information back to the energy source using the harvested energy. The source does not have any channel state information and it only knows whether a received codeword from a given node was successfully decoded or not. With this limited information, the source has to learn the optimal power level that maximizes the energy efficiency of the network. We model the problem as a stochastic Multi-Armed Bandits problem and develop an Upper Confidence Bound based algorithm, which learns the optimal transmit power of the energy source that maximizes the energy efficiency. Numerical results validate the performance guarantees of the proposed algorithm and show significant gains compared to the benchmark schemes.

Arun Verma, Manjesh K. Hanawal, Csaba Szepesvári et al.

In this paper, we study Contextual Unsupervised Sequential Selection (USS), a new variant of the stochastic contextual bandits problem where the loss of an arm cannot be inferred from the observed feedback. In our setup, arms are associated with fixed costs and are ordered, forming a cascade. In each round, a context is presented, and the learner selects the arms sequentially till some depth. The total cost incurred by stopping at an arm is the sum of fixed costs of arms selected and the stochastic loss associated with the arm. The learner's goal is to learn a decision rule that maps contexts to arms with the goal of minimizing the total expected loss. The problem is challenging as we are faced with an unsupervised setting as the total loss cannot be estimated. Clearly, learning is feasible only if the optimal arm can be inferred (explicitly or implicitly) from the problem structure. We observe that learning is still possible when the problem instance satisfies the so-called 'Contextual Weak Dominance' (CWD) property. Under CWD, we propose an algorithm for the contextual USS problem and demonstrate that it has sub-linear regret. Experiments on synthetic and real datasets validate our algorithm.

Arun Verma, Manjesh K. Hanawal, Nandyala Hemachandra

Thompson Sampling has generated significant interest due to its better empirical performance than upper confidence bound based algorithms. In this paper, we study Thompson Sampling based algorithm for Unsupervised Sequential Selection (USS) problem. The USS problem is a variant of the stochastic multi-armed bandits problem, where the loss of an arm can not be inferred from the observed feedback. In the USS setup, arms are associated with fixed costs and are ordered, forming a cascade. In each round, the learner selects an arm and observes the feedback from arms up to the selected arm. The learner's goal is to find the arm that minimizes the expected total loss. The total loss is the sum of the cost incurred for selecting the arm and the stochastic loss associated with the selected arm. The problem is challenging because, without knowing the mean loss, one cannot compute the total loss for the selected arm. Clearly, learning is feasible only if the optimal arm can be inferred from the problem structure. As shown in the prior work, learning is possible when the problem instance satisfies the so-called `Weak Dominance' (WD) property. Under WD, we show that our Thompson Sampling based algorithm for the USS problem achieves near optimal regret and has better numerical performance than existing algorithms.

Arun Verma, Manjesh K. Hanawal

In this paper, we study a novel Stochastic Network Utility Maximization (NUM) problem where the utilities of agents are unknown. The utility of each agent depends on the amount of resource it receives from a network operator/controller. The operator desires to do a resource allocation that maximizes the expected total utility of the network. We consider threshold type utility functions where each agent gets non-zero utility if the amount of resource it receives is higher than a certain threshold. Otherwise, its utility is zero (hard real-time). We pose this NUM setup with unknown utilities as a regret minimization problem. Our goal is to identify a policy that performs as `good' as an oracle policy that knows the utilities of agents. We model this problem setting as a bandit setting where feedback obtained in each round depends on the resource allocated to the agents. We propose algorithms for this novel setting using ideas from Multiple-Play Multi-Armed Bandits and Combinatorial Semi-Bandits. We show that the proposed algorithm is optimal when all agents have the same utility. We validate the performance guarantees of our proposed algorithms through numerical experiments.

Debamita Ghosh, Arun Verma, Manjesh K. Hanawal

Recent advances in wireless radio frequency (RF) energy harvesting allows sensor nodes to increase their lifespan by remotely charging their batteries. The amount of energy harvested by the nodes varies depending on their ambient environment, and proximity to the source. The lifespan of the sensor network depends on the minimum amount of energy a node can harvest in the network. It is thus important to learn the least amount of energy harvested by nodes so that the source can transmit on a frequency band that maximizes this amount. We model this learning problem as a novel stochastic Maximin Multi-Armed Bandits (Maximin MAB) problem and propose an Upper Confidence Bound (UCB) based algorithm named Maximin UCB. Maximin MAB is a generalization of standard MAB and enjoys the same performance guarantee as that of the UCB1 algorithm. Experimental results validate the performance guarantees of our algorithm.

Sumit J. Darak, Manjesh K. Hanawal

Next-generation networks are expected to be ultra-dense with a very high peak rate but relatively lower expected traffic per user. For such scenario, existing central controller based resource allocation may incur substantial signaling (control communications) leading to a negative effect on the quality of service (e.g. drop calls), energy and spectrum efficiency. To overcome this problem, cognitive ad-hoc networks (CAHN) that share spectrum with other networks are being envisioned. They allow some users to identify and communicate in `free slots' thereby reducing signaling load and allowing the higher number of users per base stations (dense networks). Such networks open up many interesting challenges such as resource identification, coordination, dynamic and context-aware adaptation for which Machine Learning and Artificial Intelligence framework offers novel solutions. In this paper, we discuss state-of-the-art multi-armed multi-player bandit based distributed learning algorithms that allow users to adapt to the environment and coordinate with other players/users. We also discuss various open research problems for feasible realization of CAHN and interesting applications in other domains such as energy harvesting, Internet of Things, and Smart grids.

Arun Verma, Manjesh K. Hanawal, Nandyala Hemachandra

In medical diagnosis, physicians predict the state of a patient by checking measurements (features) obtained from a sequence of tests, e.g., blood test, urine test, followed by invasive tests. As tests are often costly, one would like to obtain only those features (tests) that can establish the presence or absence of the state conclusively. Another aspect of medical diagnosis is that we are often faced with unsupervised prediction tasks as the true state of the patients may not be known. Motivated by such medical diagnosis problems, we consider a {\it Cost-Sensitive Medical Diagnosis} (CSMD) problem, where the true state of patients is unknown. We formulate the CSMD problem as a feature selection problem where each test gives a feature that can be used in a prediction model. Our objective is to learn strategies for selecting the features that give the best trade-off between accuracy and costs. We exploit the `Weak Dominance' property of problem to develop online algorithms that identify a set of features which provides an `optimal' trade-off between cost and accuracy of prediction without requiring to know the true state of the medical condition. Our empirical results validate the performance of our algorithms on problem instances generated from real-world datasets.

Arun Verma, Manjesh K. Hanawal, Arun Rajkumar et al.

In this paper, we study censored Semi-Bandits, a novel variant of the semi-bandits problem. The learner is assumed to have a fixed amount of resources, which it allocates to the arms at each time step. The loss observed from an arm is random and depends on the amount of resources allocated to it. More specifically, the loss equals zero if the allocation for the arm exceeds a constant (but unknown)threshold that can be dependent on the arm. Our goal is to learn a feasible allocation that minimizes the expected loss. The problem is challenging because the loss distribution and threshold value of each arm are unknown. We study this novel setting by establishing its `equivalence' to Multiple-Play Multi-Armed Bandits(MP-MAB) and Combinatorial Semi-Bandits. Exploiting these equivalences, we derive optimal algorithms for our setting using existing algorithms for MP-MABand Combinatorial Semi-Bandits. Experiments on synthetically generated data validate performance guarantees of the proposed algorithms.

Arun Verma, Manjesh K. Hanawal, Csaba Szepesvári et al.

In many security and healthcare systems, the detection and diagnosis systems use a sequence of sensors/tests. Each test outputs a prediction of the latent state and carries an inherent cost. However, the correctness of the predictions cannot be evaluated since the ground truth annotations may not be available. Our objective is to learn strategies for selecting a test that gives the best trade-off between accuracy and costs in such Unsupervised Sensor Selection (USS) problems. Clearly, learning is feasible only if ground truth can be inferred (explicitly or implicitly) from the problem structure. It is observed that this happens if the problem satisfies the 'Weak Dominance' (WD) property. We set up the USS problem as a stochastic partial monitoring problem and develop an algorithm with sub-linear regret under the WD property. We argue that our algorithm is optimal and evaluate its performance on problem instances generated from synthetic and real-world datasets.

Harshvardhan Tibrewal, Sravan Patchala, Manjesh K. Hanawal et al.

We consider an ad hoc network where multiple users access the same set of channels. The channel characteristics are unknown and could be different for each user (heterogeneous). No controller is available to coordinate channel selections by the users, and if multiple users select the same channel, they collide and none of them receive any rate (or reward). For such a completely decentralized network we develop algorithms that aim to achieve optimal network throughput. Due to lack of any direct communication between the users, we allow each user to exchange information by transmitting in a specific pattern and sense such transmissions from others. However, such transmissions and sensing for information exchange do not add to network throughput. For the wideband sensing and narrowband sensing scenarios, we first develop explore-and-commit algorithms that converge to near-optimal allocation with high probability in a small number of rounds. Building on this, we develop an algorithm that gives logarithmic regret, even when the number of users changes with time. We validate our claims through extensive experiments and show that our algorithms perform significantly better than the state-of-the-art CSM-MAB, dE3 and dE3-TS algorithms.

Sumit J Darak, Manjesh K. Hanawal

Next generation networks are expected to be ultradense and aim to explore spectrum sharing paradigm that allows users to communicate in licensed, shared as well as unlicensed spectrum. Such ultra-dense networks will incur significant signaling load at base stations leading to a negative effect on spectrum and energy efficiency. To minimize signaling overhead, an adhoc approach is being considered for users communicating in the unlicensed and shared spectrums. For such users, decisions need to be completely decentralized as: 1) No communication between users and signaling from the base station is possible which necessitates independent channel selection at each user. A collision occurs when multiple users transmit simultaneously on the same channel, 2) Channel qualities may be heterogeneous, i.e., they are not same across all users, and moreover, are unknown, and 3) The network could be dynamic where users can enter or leave anytime. We develop a multi-armed bandit based distributed algorithm for static networks and extend it for the dynamic networks. The algorithms aim to achieve stable orthogonal allocation (SOC) in finite time and meet the above three constraints with two novel characteristics: 1) Low complexity narrowband radio compared to wideband radio in existing works, and 2) Epoch-less approach for dynamic networks. We establish a convergence of our algorithms to SOC and validate via extensive simulation experiments.

Manjesh K. Hanawal, Sumit J. Darak

We study stochastic multi-armed bandits with many players. The players do not know the number of players, cannot communicate with each other and if multiple players select a common arm they collide and none of them receive any reward. We consider the static scenario, where the number of players remains fixed, and the dynamic scenario, where the players enter and leave at any time. We provide algorithms based on a novel `trekking approach' that guarantees constant regret for the static case and sub-linear regret for the dynamic case with high probability. The trekking approach eliminates the need to estimate the number of players resulting in fewer collisions and improved regret performance compared to the state-of-the-art algorithms. We also develop an epoch-less algorithm that eliminates any requirement of time synchronization across the players provided each player can detect the presence of other players on an arm. We validate our theoretical guarantees using simulation based and real test-bed based experiments.

Manjesh K. Hanawal, Hao Liu, Henghui Zhu et al.

We consider the problem of learning a policy for a Markov decision process consistent with data captured on the state-actions pairs followed by the policy. We assume that the policy belongs to a class of parameterized policies which are defined using features associated with the state-action pairs. The features are known a priori, however, only an unknown subset of them could be relevant. The policy parameters that correspond to an observed target policy are recovered using $\ell_1$-regularized logistic regression that best fits the observed state-action samples. We establish bounds on the difference between the average reward of the estimated and the original policy (regret) in terms of the generalization error and the ergodic coefficient of the underlying Markov chain. To that end, we combine sample complexity theory and sensitivity analysis of the stationary distribution of Markov chains. Our analysis suggests that to achieve regret within order $O(\sqrtε)$, it suffices to use training sample size on the order of $Ω(\log n \cdot poly(1/ε))$, where $n$ is the number of the features. We demonstrate the effectiveness of our method on a synthetic robot navigation example.

Manjesh K. Hanawal, Amir Leshem, Venkatesh Saligrama

We study stochastic linear optimization problem with bandit feedback. The set of arms take values in an $N$-dimensional space and belong to a bounded polyhedron described by finitely many linear inequalities. We provide a lower bound for the expected regret that scales as $Ω(N\log T)$. We then provide a nearly optimal algorithm and show that its expected regret scales as $O(N\log^{1+ε}(T))$ for an arbitrary small $ε>0$. The algorithm alternates between exploration and exploitation intervals sequentially where deterministic set of arms are played in the exploration intervals and greedily selected arm is played in the exploitation intervals. We also develop an algorithm that achieves the optimal regret when sub-Gaussianity parameter of the noise term is known. Our key insight is that for a polyhedron the optimal arm is robust to small perturbations in the reward function. Consequently, a greedily selected arm is guaranteed to be optimal when the estimation error falls below some suitable threshold. Our solution resolves a question posed by Rusmevichientong and Tsitsiklis (2011) that left open the possibility of efficient algorithms with asymptotic logarithmic regret bounds. We also show that the regret upper bounds hold with probability $1$. Our numerical investigations show that while theoretical results are asymptotic the performance of our algorithms compares favorably to state-of-the-art algorithms in finite time as well.