In this paper, we theoretically investigate the effects of noisy labels in offline alignment, with a focus on the interplay between privacy and robustness against adversarial corruption. Specifically, under linear modeling assumptions, we present a unified analysis covering both reinforcement learning from human feedback (RLHF) and direct preference optimization (DPO) under different privacy-corruption scenarios, such as Local differential privacy-then-Corruption (LTC), where human preference labels are privatized before being corrupted by an adversary, and Corruption-then-Local differential privacy (CTL), where labels are corrupted before privacy protection. Our analysis leverages a reduction framework that reduces the offline alignment problem under linear modeling assumptions to parameter estimation in logistic regression. This framework allows us to establish an interesting separation result between LTC and CTL, demonstrating that LTC presents a greater challenge than CTL in offline alignment, even under linear models. As important by-products, our findings also advance the state-of-the-art theoretical results in offline alignment under privacy-only or corruption-only scenarios.

In this paper, we theoretically study the offline alignment of language models with human preference feedback, under both preference label corruption and privacy protections. To this end, we propose a variant of $\chi$PO – Square$\chi$PO, which is a simple one-line change of $\chi$PO with the standard log-loss replaced by a new square loss over probability. Thanks to the inherent nice properties of this new loss, we have advanced the state-of-the-art of differentially private and robust alignment. Specifically, for the local model of label privacy, Square$\chi$PO is the first one that attains optimal rate based on single-policy concentrability even with general function approximations. It also gives the first result under the central model of privacy protection over both prompts (responses) and labels. On the robustness side against Huber label corruption, Square$\chi$PO is the first alignment method that has a meaningful theoretical guarantee under general function approximations. More importantly, Square$\chi$PO can address privacy protection and corruption simultaneously, where an interesting separation is observed, implying that the order of privacy and corruption matters. Furthermore, we show that Square$\chi$PO can also be easily extended to handle the scenario of the general preference model with state-of-the-art guarantees under corruption and privacy. Last but not least, all of our theoretical guarantees enjoy a unified analysis, building upon a new result on the generalization error bounds of least-square regression under corruption and privacy constraints, which we believe is of independent interest to the community.

Network slicing plays a crucial role in realizing 5G/6G advances, enabling diverse Service Level Agreement (SLA) requirements related to latency, throughput, and reliability. Since network slices are deployed end-to-end (E2E), across multiple domains including access, transport, and core networks, it is essential to efficiently decompose an E2E SLA into domain-level targets, so that each domain can provision adequate resources for the slice. However, decomposing SLAs is highly challenging due to the heterogeneity of domains, dynamic network conditions, and the fact that the SLA orchestrator is oblivious to the domain's resource optimization. In this work, we propose Odin, a Bayesian Optimization-based solution that leverages each domain's online feedback for provably-efficient SLA decomposition. Through theoretical analyses and rigorous evaluations, we demonstrate that Odin's E2E orchestrator can achieve up to 45% performance improvement in SLA satisfaction when compared with baseline solutions whilst reducing overall resource costs even in the presence of noisy feedback from the individual domains.

Policy optimization (PO) is a cornerstone of modern reinforcement learning (RL), with diverse applications spanning robotics, healthcare, and large language model training. The increasing deployment of PO in sensitive domains, however, raises significant privacy concerns. In this paper, we initiate a theoretical study of differentially private policy optimization, focusing explicitly on its sample complexity. We first formalize an appropriate definition of differential privacy (DP) tailored to PO, addressing the inherent challenges arising from on-policy learning dynamics and the subtlety involved in defining the unit of privacy. We then systematically analyze the sample complexity of widely-used PO algorithms, including policy gradient (PG), natural policy gradient (NPG) and more, under DP constraints and various settings, via a unified framework. Our theoretical results demonstrate that privacy costs can often manifest as lower-order terms in the sample complexity, while also highlighting subtle yet important observations in private PO settings. These offer valuable practical insights for privacy-preserving PO algorithms.

Learning from preference-based feedback has recently gained considerable traction as a promising approach to align generative models with human interests. Instead of relying on numerical rewards, the generative models are trained using reinforcement learning with human feedback (RLHF). These approaches first solicit feedback from human labelers typically in the form of pairwise comparisons between two possible actions, then estimate a reward model using these comparisons, and finally employ a policy based on the estimated reward model. An adversarial attack in any step of the above pipeline might reveal private and sensitive information of human labelers. In this work, we adopt the notion of \emph{label differential privacy} (DP) and focus on the problem of reward estimation from preference-based feedback while protecting privacy of each individual labelers. Specifically, we consider the parametric Bradley-Terry-Luce (BTL) model for such pairwise comparison feedback involving a latent reward parameter $\theta^* \in \mathbb{R}^d$. Within a standard minimax estimation framework, we provide tight upper and lower bounds on the error in estimating $\theta^*$ under both \emph{local} and \emph{central} models of DP. We show, for a given privacy budget $\epsilon$ and number of samples $n$, that the additional cost to ensure label-DP under local model is $\Theta \big(\frac{1}{ e^\epsilon-1}\sqrt{\frac{d}{n}}\big)$, while it is $\Theta\big(\frac{\sqrt{d}}{\epsilon n} \big)$ under the weaker central model. We perform simulations on synthetic data that corroborate these theoretical results.

We study the constrained Markov decision processes (CMDPs), in which an agent aims to maximize the expected cumulative reward subject to a constraint on the expected total value of a utility function. Existing approaches have primarily focused on \emph{soft} constraint violation, which allows compensation across episodes, making it easier to satisfy the constraints. In contrast, we consider a stronger \emph{hard} constraint violation metric, where only positive constraint violations are accumulated. Our main result is the development of the \emph{first model-free}, \emph{simulator-free} algorithm that achieves a sub-linear regret and a sub-linear hard constraint violation simultaneously, even in \emph{large-scale} systems. In particular, we show that $\tilde{\mathcal{O}}(\sqrt{d^3H^4K})$ regret and $\tilde{\mathcal{O}}(\sqrt{d^3H^4K})$ hard constraint violation bounds can be achieved, where $K$ is the number of episodes, $d$ is the dimension of the feature mapping, $H$ is the length of the episode. Our results are achieved via novel adaptations of the primal-dual LSVI-UCB algorithm, i.e., it searches for the dual variable that balances between regret and constraint violation within every episode, rather than updating it at the end of each episode. This turns out to be crucial for our theoretical guarantees when dealing with hard constraint violations.

We consider cross-silo federated linear contextual bandit (LCB) problem under differential privacy, where multiple silos interact with their respective local users and communicate via a central server to realize collaboration without sacrificing each user's privacy. We identify three issues in the state-of-the-art~\citep{dubey2020differentially}: (i) failure of claimed privacy protection, (ii) incorrect regret bound due to noise miscalculation and (iii) ungrounded communication cost. To resolve these issues, we take a two-step approach. First, we design an algorithmic framework consisting of a generic federated LCB algorithm and flexible privacy protocols. Then, leveraging the proposed framework, we study federated LCBs under two different privacy constraints. We first establish privacy and regret guarantees under silo-level local differential privacy, which fix the issues present in state-of-the-art algorithm. To further improve the regret performance, we next consider shuffle model of differential privacy, under which we show that our algorithm can achieve nearly ``optimal'' regret without a trusted server. We accomplish this via two different schemes -- one relies on a new result on privacy amplification via shuffling for DP mechanisms and another one leverages the integration of a shuffle protocol for vector sum into the tree-based mechanism, both of which might be of independent interest. Finally, we support our theoretical results with numerical evaluations over contextual bandit instances generated from both synthetic and real-life data.

We revisit the problem of federated learning (FL) with private data from people who do not trust the server or other silos/clients. In this context, every silo (e.g. hospital) has data from several people (e.g. patients) and needs to protect the privacy of each person's data (e.g. health records), even if the server and/or other silos try to uncover this data. Inter-Silo Record-Level Differential Privacy (ISRL-DP) prevents each silo's data from being leaked, by requiring that silo $i$'s *communications* satisfy item-level differential privacy. Prior work (Lowy & Razaviyayn, 2023a) characterized the optimal excess risk bounds for ISRL-DP algorithms with *homogeneous* (i.i.d.) silo data and convex loss functions. However, two important questions were left open: 1) Can the same excess risk bounds be achieved with *heterogeneous* (non-i.i.d.) silo data? 2) Can the optimal risk bounds be achieved with *fewer communication rounds*? In this paper, we give positive answers to both questions. We provide novel ISRL-DP FL algorithms that achieve the optimal excess risk bounds in the presence of heterogeneous silo data. Moreover, our algorithms are more *communication-efficient* than the prior state-of-the-art. For smooth loss functions, our algorithm achieves the *optimal* excess risk bound and has *communication complexity that matches the non-private lower bound*. Additionally, our algorithms are more *computationally efficient* than the previous state-of-the-art.

In this paper, we study the multi-armed bandit problem in the best-of-both-worlds (BOBW) setting with heavy-tailed losses, where the losses can be negative and unbounded but have $(1+v)$-th raw moments bounded by $u^{1+v}$ for some known $u>0$ and $v\in(0,1]$. Specifically, we consider the BOBW setting where the underlying environment could be either (oblivious) adversarial (i.e., the loss distribution can change arbitrarily over time) or stochastic (i.e., the loss distribution is fixed over time) and is unknown to the decision-maker a prior, and propose an algorithm that achieves a $T^{\frac{1}{1+v}}$-type worst-case (pseudo-)regret in the adversarial regime and a $\log T$-type gap-dependent regret in the stochastic regime, where $T$ is the time horizon. Compared to the state-of-the-art results, our algorithm offers stronger \emph{high-probability} regret guarantees rather than expected regret guarantees, and more importantly, relaxes a strong technical assumption on the loss distribution. This assumption is needed even for the weaker expected regret obtained in the literature and is generally hard to verify in practice. As a byproduct, relaxing this assumption leads to the first near-optimal regret result for heavy-tailed bandits with Huber contamination in the adversarial regime, in contrast to all previous works focused on the (easier) stochastic regime. Our result also implies a high-probability BOBW regret guarantee when the bounded true losses are protected with pure Local Differential Privacy (LDP), while the existing work ensures the (weaker) \emph{approximate} LDP with the regret bounds in expectation only.

We study model-free reinforcement learning (RL) algorithms in episodic non-stationary constrained Markov decision processes (CMDPs), in which an agent aims to maximize the expected cumulative reward subject to a cumulative constraint on the expected utility (cost). In the non-stationary environment, the reward, utility functions, and the transition kernels can vary arbitrarily over time as long as the cumulative variations do not exceed certain variation budgets. We propose the first model-free, simulator-free RL algorithms with sublinear regret and zero constraint violation for non-stationary CMDPs in both tabular and linear function approximation settings with provable performance guarantees. Our results on regret bound and constraint violation for the tabular case match the corresponding best results for stationary CMDPs when the total budget is known. Additionally, we present a general framework for addressing with the well-known challenges associated with analyzing non-stationary CMDPs, without requiring prior knowledge of the variation budget. We apply the approach for both tabular and linear approximation settings.

In this paper, we study the role of feedback in online learning with switching costs. It has been shown that the minimax regret is $\widetilde{\Theta}(T^{2/3})$ under bandit feedback and improves to $\widetilde{\Theta}(\sqrt{T})$ under full-information feedback, where $T$ is the length of the time horizon. However, it remains largely unknown how the amount and type of feedback generally impact regret. To this end, we first consider the setting of bandit learning with extra observations; that is, in addition to the typical bandit feedback, the learner can freely make a total of $B_{\mathrm{ex}}$ *extra observations*. We fully characterize the minimax regret in this setting, which exhibits an interesting *phase-transition phenomenon*: when $B_{\mathrm{ex}} = O(T^{2/3})$, the regret remains $\widetilde{\Theta}(T^{2/3})$, but when $B_{\mathrm{ex}} = \Omega(T^{2/3})$, it becomes $\widetilde{\Theta}(T/\sqrt{B_{\mathrm{ex}}})$, which improves as the budget $B_{\mathrm{ex}}$ increases. To design algorithms that can achieve the minimax regret, it is instructive to consider a more general setting where the learner has a budget of $B$ *total* observations. We fully characterize the minimax regret in this setting as well and show that it is $\widetilde{\Theta}(T/\sqrt{B})$, which scales smoothly with the total budget $B$. Furthermore, we propose a generic algorithmic framework, which enables us to design different learning algorithms that can achieve matching upper bounds for both settings based on the amount and type of feedback. One interesting finding is that while bandit feedback can still guarantee optimal regret when the budget is relatively limited, it no longer suffices to achieve optimal regret when the budget is relatively large.

In this paper we study the problem of (finite horizon tabular) Markov decision processes (MDPs) with heavy-tailed rewards under the constraint of differential privacy (DP). Compared with the previous studies for private reinforcement learning that typically assume rewards are sampled from some bounded or sub-Gaussian distributions to ensure DP, we consider the setting where reward distributions have only finite $(1+v)$-th moments with some $v \in (0,1]$. By resorting to robust mean estimators for rewards, we first propose two frameworks for heavy-tailed MDPs, i.e., one is for value iteration and another is for policy optimization. Under each framework, we consider both joint differential privacy (JDP) and local differential privacy (LDP) models. Based on our frameworks, we provide regret upper bounds for both JDP and LDP cases, and show that the moment of distributions and privacy budget have significant impact on regrets. Finally, we establish a lower bound of regret minimization for heavy-tailed MDPs in JDP model by reducing it to the instance-independent lower bound of heavy-tailed multi-armed bandits in DP model. We also show the lower bound for the problem in LDP by adopting some private minimax methods. Our results reveal that there are fundamental differences between the problem of private RL with sub-Gaussian and that with heavy-tailed rewards.

We study private and robust multi-armed bandits (MABs), where the agent receives Huber's contaminated heavy-tailed rewards and meanwhile needs to ensure differential privacy. We consider both the finite $k$-th raw moment and the finite $k$-th central moment settings for heavy-tailed rewards distributions with $k\ge 2$. We first present its minimax lower bound, characterizing the information-theoretic limit of regret with respect to privacy budget, contamination level, and heavy-tailedness. Then, we propose a meta-algorithm that builds on a private and robust mean estimation sub-routine \texttt{PRM} that essentially relies on reward truncation and the Laplace mechanism. For the above two different heavy-tailed settings, we give corresponding schemes of \texttt{PRM}, which enable us to achieve nearly-optimal regrets. Moreover, our two proposed truncation-based or histogram-based \texttt{PRM} schemes achieve the optimal trade-off between estimation accuracy, privacy and robustness. Finally, we support our theoretical results and show the effectiveness of our algorithms with experimental studies.

We study regret minimization in finite horizon tabular Markov decision processes (MDPs) under the constraints of differential privacy (DP). This is motivated by the widespread applications of reinforcement learning (RL) in real-world sequential decision making problems, where protecting users' sensitive and private information is becoming paramount. We consider two variants of DP -- joint DP (JDP), where a centralized agent is responsible for protecting users' sensitive data and local DP (LDP), where information needs to be protected directly on the user side. We first propose two general frameworks -- one for policy optimization and another for value iteration -- for designing private, optimistic RL algorithms. We then instantiate these frameworks with suitable privacy mechanisms to satisfy JDP and LDP requirements, and simultaneously obtain sublinear regret guarantees. The regret bounds show that under JDP, the cost of privacy is only a lower order additive term, while for a stronger privacy protection under LDP, the cost suffered is multiplicative. Finally, the regret bounds are obtained by a unified analysis, which, we believe, can be extended beyond tabular MDPs.

In this paper, we consider the Gaussian process (GP) bandit optimization problem in a non-stationary environment. To capture external changes, the black-box function is allowed to be time-varying within a reproducing kernel Hilbert space (RKHS). To this end, we develop WGP-UCB, a novel UCB-type algorithm based on weighted Gaussian process regression. A key challenge is how to cope with infinite-dimensional feature maps. To that end, we leverage kernel approximation techniques to prove a sublinear regret bound, which is the first (frequentist) sublinear regret guarantee on weighted time-varying bandits with general nonlinear rewards. This result generalizes both non-stationary linear bandits and standard GP-UCB algorithms. Further, a novel concentration inequality is achieved for weighted Gaussian process regression with general weights. We also provide universal upper bounds and weight-dependent upper bounds for weighted maximum information gains. These results are of independent interest for applications such as news ranking and adaptive pricing, where weights can be adopted to capture the importance or quality of data. Finally, we conduct experiments to highlight the favorable gains of the proposed algorithm in many cases when compared to existing methods.

Differential privacy (DP) has been recently introduced to linear contextual bandits to formally address the privacy concerns in its associated personalized services to participating users (e.g., recommendations). Prior work largely focus on two trust models of DP – the central model, where a central server is responsible for protecting users’ sensitive data, and the (stronger) local model, where information needs to be protected directly on users’ side. However, there remains a fundamental gap in the utility achieved by learning algorithms under these two privacy models, e.g., if all users are unique within a learning horizon $T$, $\widetilde{O}(\sqrt{T})$ regret in the central model as compared to $\widetilde{O}(T^{3/4})$ regret in the local model. In this work, we aim to achieve a stronger model of trust than the central model, while suffering a smaller regret than the local model by considering recently popular shuffle model of privacy. We propose a general algorithmic framework for linear contextual bandits under the shuffle trust model, where there exists a trusted shuffler – in between users and the central server– that randomly permutes a batch of users data before sending those to the server. We then instantiate this framework with two specific shuffle protocols – one relying on privacy amplification of local mechanisms, and another incorporating a protocol for summing vectors and matrices of bounded norms. We prove that both these instantiations lead to regret guarantees that significantly improve on that of the local model, and can potentially be of the order $\widetilde{O}(T^{3/5})$ if all users are unique. We also verify this regret behavior with simulations on synthetic data. Finally, under the practical scenario of non-unique users, we show that the regret of our shuffle private algorithm scale as $\widetilde{O}(T^{2/3})$, which matches what the central model could achieve in this case.

We study a stochastic bandit problem with a general unknown reward function and a general unknown constraint function. Both functions can be non-linear (even non-convex) and are assumed to lie in a reproducing kernel Hilbert space (RKHS) with a bounded norm. This kernelized bandit setup strictly generalizes standard multi-armed bandits and linear bandits. In contrast to safety-type hard constraints studied in prior works, we consider soft constraints that may be violated in any round as long as the cumulative violations are small, which is motivated by various practical applications. Our ultimate goal is to study how to utilize the nature of soft constraints to attain a finer complexity-regret-constraint trade-off in the kernelized bandit setting. To this end, leveraging primal-dual optimization, we propose a general framework for both algorithm design and performance analysis. This framework builds upon a novel sufficient condition, which not only is satisfied under general exploration strategies, including \emph{upper confidence bound} (UCB), \emph{Thompson sampling} (TS), and new ones based on \emph{random exploration}, but also enables a unified analysis for showing both sublinear regret and sublinear or even zero constraint violation. We demonstrate the superior performance of our proposed algorithms via numerical experiments based on both synthetic and real-world datasets. Along the way, we also make the first detailed comparison between two popular methods for analyzing constrained bandits and Markov decision processes (MDPs) by discussing the key difference and some subtleties in the analysis, which could be of independent interest to the communities.

We study the constrained reinforcement learning problem, in which an agent aims to maximize the expected cumulative reward subject to a constraint on the expected total value of a utility function. In contrast to existing model-based approaches or model-free methods accompanied with a `simulator’, we aim to develop the first \emph{model-free}, \emph{simulator-free} algorithm that achieves a sublinear regret and a sublinear constraint violation even in \emph{large-scale} systems. To this end, we consider the episodic constrained Markov decision processes with linear function approximation, where the transition dynamics and the reward function can be represented as a linear function of some known feature mapping. We show that $\tilde{\mathcal{O}}(\sqrt{d^3H^3T})$ regret and $\tilde{\mathcal{O}}(\sqrt{d^3H^3T})$ constraint violation bounds can be achieved, where $d$ is the dimension of the feature mapping, $H$ is the length of the episode, and $T$ is the total number of steps. Our bounds are attained without explicitly estimating the unknown transition model or requiring a simulator, and they depend on the state space only through the dimension of the feature mapping. Hence our bounds hold even when the number of states goes to infinity. Our main results are achieved via novel adaptations of the standard LSVI-UCB algorithms. In particular, we first introduce primal-dual optimization into the LSVI-UCB algorithm to balance between regret and constraint violation. More importantly, we replace the standard greedy selection with respect to the state-action function with a soft-max policy. This turns out to be key in establishing uniform concentration (a critical step for provably efficient model-free exploration) for the constrained case via its approximation-smoothness trade-off. Finally, we also show that one can achieve an even zero constraint violation for large enough $T$ by trading the regret a little bit but still maintaining the same order with respect to $T$.

To fully utilize the abundant spectrum resources in millimeter wave (mmWave), Beam Alignment (BA) is necessary for large antenna arrays to achieve large array gains. In practical dynamic wireless environments, channel modeling is challenging due to time-varying and multipath effects. In this paper, we formulate the beam alignment problem as a nonstationary online learning problem with the objective to maximize the received signal strength under interference constraint. In particular, we employ the non-stationary kernelized bandit to leverage the correlation among beams and model the complex beamforming and multipath channel functions. Furthermore, to mitigate interference to other user equipment, we leverage the primal-dual method to design a constrained UCB-type kernelized bandit algorithm. Our theoretical analysis indicates that the proposed algorithm can adaptively adjust the beam in time-varying environments, such that both the cumulative regret of the received signal and constraint violations have sublinear bounds with respect to time. This result is of independent interest for applications such as adaptive pricing and news ranking. In addition, the algorithm assumes the channel is a black-box function and does not require any prior knowledge for dynamic channel modeling, and thus is applicable in a variety of scenarios. We further show that if the information about the channel variation is known, the algorithm will have better theoretical guarantees and performance. Finally, we conduct simulations to highlight the effectiveness of the proposed algorithm.

In this paper, we study the problem of global reward maximization with only partial distributed feedback. This problem is motivated by several real-world applications (e.g., cellular network configuration, dynamic pricing, and policy selection) where an action taken by a central entity influences a large population that contributes to the global reward. However, collecting such reward feedback from the entire population not only incurs a prohibitively high cost, but often leads to privacy concerns. To tackle this problem, we consider differentially private distributed linear bandits, where only a subset of users from the population are selected (called clients) to participate in the learning process and the central server learns the global model from such partial feedback by iteratively aggregating these clients’ local feedback in a differentially private fashion. We then propose a unified algorithmic learning framework, called differentially private distributed phased elimination (DP-DPE), which can be naturally integrated with popular differential privacy (DP) models (including central DP, local DP, and shuffle DP). Furthermore, we prove that DP-DPE achieves both sublinear regret and sublinear communication cost. Interestingly, DP-DPE also achieves privacy protection “for free” in the sense that the additional cost due to privacy guarantees is a lower-order additive term. Finally, we conduct simulations to corroborate our theoretical results and demonstrate the effectiveness of DP-DPE.

Motivated by the increasing concern about privacy in nowadays data-intensive online learning systems, we consider a black-box optimization in the nonparametric Gaussian process setting with local differential privacy (LDP) guarantee. Specifically, the rewards from each user are further corrupted to protect privacy and the learner only has access to the corrupted rewards to minimize the regret. We first derive the regret lower bounds for any LDP mechanism and any learning algorithm. Then, we present three almost optimal algorithms based on the GP-UCB framework and Laplace DP mechanism. In this process, we also propose a new Bayesian optimization (BO) method (called MoMA-GP-UCB) based on median-of-means techniques and kernel approximations, which complements previous BO algorithms under heavy-tailed payoffs with reduced complexity. Further, empirical comparisons of different algorithms on both synthetic and real-world datasets highlight the superior performance of MoMA-GP-UCB in both private and non-private scenarios.

In this paper we study the problem of regret minimization in reinforcement learning (RL) under differential privacy constraints. This work is motivated by the wide range of RL applications for providing personalized service, where privacy concerns are becoming paramount. In contrast to previous works, we take the first step towards non-tabular RL settings, while providing a rigorous privacy guarantee. In particular, we consider the adaptive control of differentially private linear quadratic (LQ) systems. We develop the first private RL algorithm, Private-OFU-RL which is able to attain a sub-linear regret while guaranteeing privacy protection. More importantly, the additional cost due to privacy is only on the order of$\frac{\ln(1/\delta)^{1/4}}{\varepsilon^{1/2}}$given privacy parameters$\varepsilon, \delta > 0$. Through this process, we also provide a general procedure for adaptive control of LQ systems under changing regularizers, which not only generalizes previous non-private controls, but also serves as the basis for general private controls.

In this paper, we investigate the energy efficiency (EE) maximization in amplify-and-forward (AF) MIMO two-way relay channel (TWRC) combined with relay antenna selection (AS). An iterative energy efficient AS algorithm is proposed to jointly select the active receive and transmit antennas at the relay, as well as optimize the transmission power of the sources and relay. Specifically, the AS at each iteration is based on a derived closed-form iterative equation of EE, which guides us to select a pair of receive and transmit relay antennas that achieves the largest increment of EE under an initial transmission power. After the AS of each iteration, a power adaptation is immediately adopted where we calculate the optimal transmission power using fractional programming and set it as the initial one for the AS of the next iteration. Simulation results show that our proposed scheme achieves nearly the same performance of exhaustive search while with significantly reduced complexity. Moreover, it is capable of simultaneously improving EE and reducing the transmission power.

As the growth of wireless communications is accompanied by increased energy consumption, energy-efficient communication is becoming imperative. This paper will discuss the energy efficiency of MIMO systems with antenna selection. The optimal method for the maximization of energy efficiency is exhaustive search. To address the problem, an iterative algorithm which includes the transmit antenna selection and power adaptation is proposed. It is based on the iterative property of the energy efficiency, which is derived in this paper. This property guides us to select the antenna that achieves the largest energy efficiency increment at each step. There is also a power adaptation for each step where we calculate the optimal transmission power and then set it as the initial one for the next step. Moreover, one asymptotic property exists in the proposed algorithm, which states that the antenna selection and power adaptation can be decoupled in high and low SNR regimes. This fact reduces the complexity further and enables us to achieve the optimal performance with a greater probability. Simulation results show that the proposed algorithm achieves near-optimal performance in all the SNR regimes and has a remarkable gain over the no selection scheme in the energy efficiency and transmission power.