Bayesian optimization (BO) with preference-based feedback has recently garnered significant attention due to its emerging applications. We refer to this problem as Bayesian Optimization from Human Feedback (BOHF), which differs from conventional BO by learning the best actions from a reduced feedback model, where only the preference between two actions is revealed to the learner at each time step. The objective is to identify the best action using a limited number of preference queries, typically obtained through costly human feedback. Existing work, which adopts the Bradley-Terry-Luce (BTL) feedback model, provides regret bounds for the performance of several algorithms. In this work, within the same framework we develop tighter performance guarantees. Specifically, we derive regret bounds of $\tilde{\mathcal{O}}(\sqrt{\Gamma(T)T})$, where $\Gamma(T)$ represents the maximum information gain—a kernel-specific complexity term—and $T$ is the number of queries. Our results significantly improve upon existing bounds. Notably, for common kernels, we show that the order-optimal sample complexities of conventional BO—achieved with richer feedback models—are recovered. In other words, the same number of preferential samples as scalar-valued samples is sufficient to find a nearly optimal solution.

Achieving sample efficiency in Reinforcement Learning (RL) is primarily hinged on the efficient exploration of the underlying environment, but it is still unknown what are the best exploration strategies in different settings. We consider the reward-free RL problem, which operates in two phases: an exploration phase, where the agent gathers exploration trajectories over episodes irrespective of any predetermined reward function, and a subsequent planning phase, where a reward function is introduced. The agent then utilizes the episodes from the exploration phase to calculate a near-optimal policy. Existing algorithms and sample complexities for reward-free RL are limited to tabular, linear or very smooth function approximations, leaving the problem largely open for more general cases. We consider a broad range of kernel-based function approximations, including non-smooth kernels, and propose an algorithm based on adaptive domain partitioning. We show that our algorithm achieves order-optimal sample complexity for a large class of common kernels, which includes Matérn and Neural Tangent kernels.

Second-order optimization has been shown to accelerate the training of deep neural networks in many applications, often yielding faster progress per iteration on the training loss compared to first-order optimizers. However, the generalization properties of second-order methods are still being debated. Theoretical investigations have proved difficult to carry out outside the tractable settings of heavily simplified model classes - thus, the relevance of existing theories to practical deep learning applications remains unclear. Similarly, empirical studies in large-scale models and real datasets are significantly confounded by the necessity to approximate second-order updates in practice. It is often unclear whether the observed generalization behaviour arises specifically from the second-order nature of the parameter updates, or instead reflects the specific structured (e.g. Kronecker) approximations used or any damping-based interpolation towards first-order updates. Here, we show for the first time that exact Gauss-Newton (GN) updates take on a tractable form in a class of deep reversible architectures that are sufficiently expressive to be meaningfully applied to common benchmark datasets. We exploit this novel setting to study the training and generalization properties of the GN optimizer. We find that exact GN generalizes poorly. In the mini-batch training setting, this manifests as rapidly saturating progress even on the training loss, with parameter updates found to overfit each mini-batch without producing the features that would support generalization to other mini-batches. In contrast to previous work, we show that our experiments run in the feature learning regime, in which the neural tangent kernel (NTK) changes during the course of training. However, changes in the NTK are not associated with any significant change in neural representations, explaining the lack of generalization.

In this work, we propose a method to create domain-sensitive speech recognition models that utilize textual domain information by conditioning its generation on a given text prompt. This is accomplished by fine-tuning a pre-trained, end-to-end model (Whisper) to learn from demonstrations with prompt examples. We show that this ability can be generalized to different domains and even various prompt contexts, with our model gaining a Word Error Rate (WER) reduction of up to 33% on unseen datasets from various domains, such as medical conversation, air traffic control communication, and financial meetings. Considering the limited availability of audio-transcript pair data, we further extend our method to text-only fine-tuning to achieve domain sensitivity as well as domain adaptation. We demonstrate that our text-only fine-tuned model can also attend to various prompt contexts, with the model reaching the most WER reduction of 29% on the medical conversation dataset.

Channel modelling is essential to designing modern wireless communication systems. The increasing complexity of channel modelling and the cost of collecting high-quality wireless channel data have become major challenges. In this paper, we propose a diffusion model based channel sampling approach for rapidly synthesizing channel realizations from limited data. We use a diffusion model with a U Net based architecture operating in the frequency space domain. To evaluate how well the proposed model reproduces the true distribution of channels in the training dataset, two evaluation metrics are used: i) the approximate 2-Wasserstein distance between real and generated distributions of the normalized power spectrum in the antenna and frequency domains and ii) precision and recall metric for distributions. We show that, compared to existing GAN based approaches which suffer from mode collapse and unstable training, our diffusion based approach trains stably and generates diverse and high-fidelity samples from the true channel distribution. We also show that we can pretrain the model on a simulated urban macro-cellular channel dataset and fine-tune it on a smaller, out-of-distribution urban micro-cellular dataset, therefore showing that it is feasible to model real world channels using limited data with this approach.

The field of image generation has made significant progress thanks to the introduction of Diffusion Models, which learn to progressively reverse a given image corruption. Recently, a few studies introduced alternative ways of corrupting images in Diffusion Models, with an emphasis on blurring. However, these studies are purely empirical and it remains unclear what is the optimal procedure for corrupting an image. In this work, we hypothesize that the optimal procedure minimizes the length of the path taken when corrupting an image towards a given final state. We propose the Fisher metric for the path length, measured in the space of probability distributions. We compute the shortest path according to this metric, and we show that it corresponds to a combination of image sharpening, rather than blurring, and noise deblurring. While the corruption was chosen arbitrarily in previous work, our Shortest Path Diffusion (SPD) determines uniquely the entire spatiotemporal structure of the corruption. We show that SPD improves on strong baselines without any hyperparameter tuning, and outperforms all previous Diffusion Models based on image blurring. Furthermore, any small deviation from the shortest path leads to worse performance, suggesting that SPD provides the optimal procedure to corrupt images. Our work sheds new light on observations made in recent works and provides a new approach to improve diffusion models on images and other types of data.

Neural tangent (NT) kernel models have attracted great attention lately, partly for explaining overparameterized neural networks in their limit. In previous work, generalization bounds for overparameterized neural networks were given in expectation. In this work, we prove uniform generalization bounds for NT kernel models by characterizing the information gain of the NT kernel. Our bounds capture the exact error rates based on the smoothness of the activation functions. Our results on information gain can be applied to other problems where overparameterized neural networks are used; e.g., certain reinforcement learning and bandit problems.

Meta-learning models transfer the knowledge acquired from previous tasks to quickly learn new ones. They are trained on benchmarks with a fixed number of data points per task. This number is usually arbitrary and it is unknown how it affects performance at testing. Since labelling of data is expensive, finding the optimal allocation of labels across training tasks may reduce costs. Given a fixed budget of labels, should we use a small number of highly labelled tasks, or many tasks with few labels each? Should we allocate more labels to some tasks and less to others? We show that: 1) If tasks are homogeneous, there is a uniform optimal allocation, whereby all tasks get the same amount of data; 2) At fixed budget, there is a trade-off between number of tasks and number of data points per task, with a unique solution for the optimum; 3) When trained separately, harder task should get more data, at the cost of a smaller number of tasks; 4) When training on a mixture of easy and hard tasks, more data should be allocated to easy tasks. Interestingly, Neuroscience experiments have shown that human visual skills also transfer better from easy tasks. We prove these results mathematically on mixed linear regression, and we show empirically that the same results hold for few-shot image classification on CIFAR-FS and mini-ImageNet. Our results provide guidance for allocating labels across tasks when collecting data for meta-learning.

Recently, successful applications of reinforcement learning to chip placement have emerged. Pretrained models are necessary to improve efficiency and effectiveness. Currently, the weights of objective metrics (e.g., wirelength, congestion, and timing) are fixed during pretraining. However, fixed-weighed models cannot generate the diversity of placements required for engineers to accommodate changing requirements as they arise. This paper proposes flexible multiple-objective reinforcement learning (MORL) to support objective functions with inference-time variable weights using just a single pretrained model. Our macro placement results show that MORL can generate the Pareto frontier of multiple objectives effectively.

Kernel-based models such as kernel ridge regression and Gaussian processes are ubiquitous in machine learning applications for regression and optimization. It is well known that a major downside for kernel-based models is the high computational cost; given a dataset of $n$ samples, the cost grows as $\mathcal{O}(n^3)$. Existing sparse approximation methods can yield a significant reduction in the computational cost, effectively reducing the actual cost down to as low as $\mathcal{O}(n)$ in certain cases. Despite this remarkable empirical success, significant gaps remain in the existing results for the analytical bounds on the error due to approximation. In this work, we provide novel confidence intervals for the Nyström method and the sparse variational Gaussian process approximation method, which we establish using novel interpretations of the approximate (surrogate) posterior variance of the models. Our confidence intervals lead to improved performance bounds in both regression and optimization problems.

Consider the sequential optimization of a continuous, possibly non-convex, and expensive to evaluate objective function $f$. The problem can be cast as a Gaussian Process (GP) bandit where $f$ lives in a reproducing kernel Hilbert space (RKHS). The state of the art analysis of several learning algorithms shows a significant gap between the lower and upper bounds on the simple regret performance. When $N$ is the number of exploration trials and $\gamma_N$ is the maximal information gain, we prove an $\tilde{\mathcal{O}}(\sqrt{\gamma_N/N})$ bound on the simple regret performance of a pure exploration algorithm that is significantly tighter than the existing bounds. We show that this bound is order optimal up to logarithmic factors for the cases where a lower bound on regret is known. To establish these results, we prove novel and sharp confidence intervals for GP models applicable to RKHS elements which may be of broader interest.

Natural conversations are filled with disfluencies.This study investigates if and how BERT understands disfluency with three experiments: (1) a behavioural study using a downstream task, (2) an analysis of sentence embeddings and (3) an analysis of the attention mechanism on disfluency.The behavioural study shows that without fine-tuning on disfluent data, BERT does not suffer significant performance loss when presented disfluent compared to fluent inputs (exp1).Analysis on sentence embeddings of disfluent and fluent sentence pairs reveals that the deeper the layer, the more similar their representation (exp2).This indicates that deep layers of BERT become relatively invariant to disfluency.We pinpoint attention as a potential mechanism that could explain this phenomenon (exp3).Overall, the study suggests that BERT has knowledge of disfluency structure.We emphasise the potential of using BERT to understand natural utterances without disfluency removal.

In this paper, we investigate how to utilize renewable energy harvested by small cells to increase network capacity. Specifically, we try to maximize the average capacity of a small cell with the harvested energy from the environments (e.g., solar and wind) with constraints on energy causality and battery capacity. With the coverage preserved by the umbrella cells, we investigate the potential of capacity improvement by dynamic on-off operations of small cells. We propose a heuristic polynomial-time near-optimal algorithm for joint power control and sleep-awake scheduling of this mixed-integer optimization problem. The capacity obtained by the proposed heuristic algorithm can approach the maximal capacity as long as the small cell can be equipped with a battery with large enough capacity. In the simulations, we demonstrate that our proposed algorithm can increase system capacity by 25%. We find that always attempting to keep a small cell in active state may not be always a good strategy for capacity maximization even when the optimal transmit power allocation is applied.

Energy-efficient base station (BS) operation is a key design goal in green cellular networks. An effective way for energy conservation of BSs is to switch BSs on/off according to the traffic profile. However, such operations may create coverage holes in the network. In this paper, we aim to minimize the total power consumption of the network by switching BSs on/off adaptively while maintaining the network coverage. We find that the BS activation problem for minimal network power consumption with full network coverage preservation is an NP-hard problem. To address the problem, we first derive the optimal cell size for minimizing BS power consumption per unit coverage area and propose a polynomial-time algorithm for energy-efficient BS activation. The simulation results show that our algorithm can approach the minimum network power consumption and adapt to network traffic load under non-uniform traffic load distributions. More importantly, we demonstrate that network densification with small cells for bursting throughput in hot spot areas can also be beneficial in saving network energy during the low traffic load period.

Energy-efficient base station (BS) operation is a key design goal in green cellular networks. The most effective way for energy conservation of BSs is to switch on/off BS adaptively. However, such operations may induce the problem of creating coverage holes in the network. In this paper, we attempt to minimize the total power consumption of the network by switching on and off BSs adaptively while maintaining the network coverage. Specifically, we derive the optimal cell size and determine the optimal number of active BSs for power consumption minimization with network coverage preservation. We also propose a near-optimal polynomial-time heuristic algorithm for energy-efficient BS activation with network coverage preservation. The simulation results show that our algorithm can approach the performance upper bound with only approximately 3dB performance loss in overall network power consumption for a cellular network with various deployment strategies.

Minimization of power consumption is a critical design goal for wireless-relay networks comprised of battery-powered sensor devices. Traditionally, wireless communication is optimized to minimize the total transmission power. In the paper, we further include the detection power and the reception power in order to minimize the overall network power consumption for information exchange in wireless-relay sensor networks (WSNs). Specifically, after the physical attributes of relay nodes are given, we wish to maximize the network lifetime by a proper selection of the transmission range, the sleep period, and the participation density. We propose a random gossip network (RGN) as a framework for our analysis. We find simple rules govern the optimal settings of these three parameters for all-to-all broadcast in the RGN. One key relationship is that the optimal number of nodes broadcasting messages in a time epoch within the transmission range depends only on the path loss exponent and the network dimensionality. This optimal value does not depend on factors such as the physical network size, the reception cost, and the message origination rate. Moreover, it is shown that neither increasing nor decreasing the physical network size will affect the optimal value of these design parameters. The optimal setting for power consumption minimization is a scalable solution. Finally, we demonstrate that our results can be well applied to predict the optimal value of operational parameters for information exchange as well as information dissemination in actual WSNs.

To date there is no practical means to evaluate the true word error probability (WEP) of a given turbo or LDPC code because typical decoders cannot achieve the performance of ML decoding. In this paper, we propose a viable methodology to establish tight bounds on the ML-decoding WEP for these codes through empirical simulation. Our framework centers on the efficient use of multiple-output decoding induced by receiver-generated side information, or gift. At low WEP regime, perturbed decoding can give tight bounds. In high WEP regime, due to the prohibitive complexity of perturbed decoding, we instead pursue other type of gifts. The effectiveness of various types of gifts is investigated in detail. We observe that the complexity of gift-assisted decoding is dominated by the effort to identify partial gifts that can then be further extended. Using bit values as gifts and an algorithm that maximizes the efficiency of identifying valid partial gifts, the ML bounds of turbo and LDPC codes are evaluated. At low WEP regime, our approach successfully yields the ML performance for these codes. Their WEP are shown to be very far from the sphere packing bound. At higher WEP regime, our results indicate that best-performing message-passing decoders underperform an ML decoder by at least 0.2 dB.

Instant messaging (IM) services enable real-time text and multimedia exchange and online presence awareness. Users typically log onto instant messaging services persistently to discover available friends and also to be discovered. However, our analysis shows that the frequency exchange of presence information incurs massive power consumption to mobile devices over cellular or wireless local area networks. Such power consumption penalty can render persistent-instant messaging infeasible for battery-powered mobile devices. In this paper, we propose several solutions to mitigate the power consumption problem. By reducing the network access and keeping mobile devices in the sleep mode as much as possible, these solutions achieve significant power saving. The power consumption of the proposed solutions is derived analytically in this paper and the proposed solutions are implemented using a Jabber-based architecture. Actual power measurement results show that the power consumption of the proposed solutions agrees well with our analysis, and significant power saving can be achieved on mobile handsets with our low power consumption solutions implemented.

Sequences with specific autocorrelation (AC) and cross-correlation (CC) properties are crucial components in radar and wireless communications. In this paper, we derive the theoretical bounds on the AC and CC for OFDM sequences with constraints of spectral nulls, e.g., the mandatory nulls on the DC sub-carrier and guardbands in OFDM systems. The bounds and trade-off limits are provided for the properties of sequences, including the peak AC and CC levels, the cardinality of the sequence set, the sequence length, and the temporal length of the low correlation zone. We also investigate the trade-offs of correlations among sequence sets. The presented trade-off limits can serve as guidelines for applications where the performance measures or design criteria are related to the peak AC and CC levels.

Multiple-input multiple-output (MIMO) technique provides capacity improvement which scales linearly with the number of antennas. However, such improvement cannot be realized in line-of-sight (LOS) environments due to lack of scattering. It had been proposed that if a mobile station can receive signal from multiple repeaters, the artificial multipath could render an equivalent high-rank end-to-end MIMO channel. In this paper, we study whether fixed-location repeaters can induce significant MIMO capacity gain throughout the coverage of a cell in a LOS environment. Our results indicate that, for a wide range of repeater locations and antenna parameters, the majority of the coverage area of a cell can support a high-order capacity scaling factor. We further show that for a capacity scaling factor of 2, the corresponding coverage consists of a small number of simply-connected areas. We present the expressions for the contour of these areas as a function of base station and repeater locations, antenna spacings, and antenna orientations.

Fountain codes are suitable for data transmission over binary erasure channels. When such codes are applied in live broadcast applications, due to the limited bandwidth and delay constraints, it is often convenient to treat the input media stream as a concatenation of short data blocks and to encode each block individually. However, the use of a small block size may result in poor coding efficiency. In this paper, we introduce an approach to enhance the coding efficiency by encoding over progressively more source bits. Our approach can be considered as a generalization of the expanding window fountain code concept. In addition, we provide an error analysis based on state generating functions. The state generating function is useful both as a performance evaluation tool and as a design criterion for the generalized expanding window codes.

Instant messaging (IM) services enable real-time text and multimedia exchange and online presence awareness. Users typically log onto IM services persistently in order to discover available friends and also to be discovered. However, our analysis shows that running a desktop-oriented IM client on a mobile device over a cellular or a wireless LAN network incurs massive power consumption. Such power consumption penalty can render persistent-IM infeasible for battery-powered mobile devices. The reason for the power penalty lies in the fact that IM must communicate with the network frequently to exchange presence information with remote entities. In this paper, we propose several low power consumption solutions for IM on mobile devices. By reducing the network accesses and keeping mobile devices in the sleep mode as much as possible, these solutions achieve significant power saving. We implement the proposed solutions using a Jabber-based architecture. We also present power measurement results under this implementation. The results agree with our analysis very well.

Although yielding the lowest error probability, ML decoding of turbo codes has been considered unrealistic so far because efficient ML decoders have not been discovered. In this paper, we propose the Genetic Decoding Algorithm (GDA) for turbo codes. GDA combines the principles of perturbed decoding and genetic algorithm. In GDA, chromosomes are random additive perturbation noises. A conventional turbo decoder is used to assign fitness values to the chromosomes in the population. After generations of evolution, good chromosomes that correspond to decoded codewords of very good likelihood emerge. GDA can be used as a practical decoder for turbo codes in certain contexts. It is also a natural multiple-output decoder. The most important aspect of GDA, in our opinion, is that one can utilize GDA to empirically determine a lower bound on the error probability with ML decoding. Our results show that, at a word error probability of 10-4, GDA achieves the performance of ML decoding. Using GDA, we establish that an ML decoder only slightly outperforms a MAP-based iterative decoder at this word error probability for the block size we used and the turbo code defined for WCDMA.

In perturbed decoding algorithm (PA), independently perturbed versions of the originally received signal are passed through a decoder in order to obtain several distinct decoder outputs. PA can be used to enhance the performance of a concatenated coded system, especially for those where multiple-output inner decoder is unavailable. A number of questions regarding PA remain unanswered, including how best the original signal is to be perturbed, the distribution of the number of perturbed decodings, and the compatibility of PA with various kinds of error control codes. In this paper, we derive the distribution of the number of perturbed decoding given independent Gaussian perturbations. The dominant terms of the distribution indicate that the complexity of PA is highly dependent on the geometric structure of the error control code. Subsequently, for ML inner decoder, we derive the optimal signal-to-perturbation noise level given the operating SNR and the two-centroid spectrum of the inner code. Computer simulation is conducted. The results are consistent with our analysis.

In this paper, we examine single carrier modulation with frequency domain equalization (SC-FDE) for optical communication systems using intensity modulation with direct detection (IM/DD) in the presence of additive white Gaussian noise. SC-FDE is a technique whose processing is quite similar to that of OFDM. Unlike OFDM, however, SC-FDE is compatible with IM/DD channels because it does not require the transmission of a strong dc. For an IM/DD channel, at high data rate regime, the effect of intersymbol interference (ISI) starts to dominate system performance. In this paper, we examine the use of SC-FDE and pulse position modulation (PPM) to combat ISI for IM/DD channels. Such a system enjoys the average power efficiency of PPM and the low complexity equalization of SC-FDE. We analyze the bit error performance of SC-FDE and compare it with a few known techniques

We consider a concatenated coded system consists of an inner error correcting code and an outer error detecting code. In a conventional decoding scheme, the inner decoder produces the best codeword from its perspective. The best codeword is then checked by the outer decoder. The performance of the concatenated coded system can be improved by having the inner decoder produce not only the most likely candidate but also other highly likely candidates. In this paper, we propose a new algorithm called "perturbed decoding algorithm" (PA). In PA, other highly likely candidate is produced by feeding the inner decoder with slightly perturbed versions of the received signal. The concept of PA is compatible with most combinations of inner code and outer code. Because PA does not require the use of a sophisticated inner decoder, it is straightforward to implement in silicon technology. From our simulation, PA can achieve a performance gain greater than 1 dB

This paper presents an overview of progress in the area of multiple input multiple output (MIMO) space-time coded wireless systems. After some background on the research leading to the discovery of the enormous potential of MIMO wireless links, we highlight the different classes of techniques and algorithms proposed which attempt to realize the various benefits of MIMO including spatial multiplexing and space-time coding schemes. These algorithms are often derived and analyzed under ideal independent fading conditions. We present the state of the art in channel modeling and measurements, leading to a better understanding of actual MIMO gains. Finally, the paper addresses current questions regarding the integration of MIMO links in practical wireless systems and standards.

We investigate the effects of fading correlations in multielement antenna (MEA) communication systems. Pioneering studies showed that if the fades connecting pairs of transmit and receive antenna elements are independently, identically distributed, MEAs offer a large increase in capacity compared to single-antenna systems. An MEA system can be described in terms of spatial eigenmodes, which are single-input single-output subchannels. The channel capacity of an MEA is the sum of capacities of these subchannels. We show that the fading correlation affects the MEA capacity by modifying the distributions of the gains of these subchannels. The fading correlation depends on the physical parameters of MEA and the scatterer characteristics. In this paper, to characterize the fading correlation, we employ an abstract model, which is appropriate for modeling narrow-band Rayleigh fading in fixed wireless systems.

Multiple-antenna systems provide very high capacity compared to single antenna systems in a Rayleigh fading environment. Space-time codes are channel codes designed to exploit this high capacity for multiple-antenna systems without requiring instantaneous channel knowledge at the transmitter. A practical concern for high data rate space-time codes is their decoding complexity. The decoding complexity with ML criterion can be prohibitively large. In this paper, we focus on layered space-time (LST) codes. Two types of LST codes, the horizontally-layered space-time (HLST) codes and the diagonally-layered space-time (DLST) codes, are presented. We analyze the performance of both types of LST codes under slow and fast fading conditions. We conclude that, in a slow fading environment, DLST codes have superior performance over HLST codes. The design criteria for DLST codes are proposed.

Dual antenna-array systems provide very high capacity compared to single antenna systems in a Rayleigh fading environment. If the transmitter does not have the channel state information, to utilize this high capacity, space-time codes must be employed. The diagonally-layered space-time (DLST) architecture is a structure that is capable of providing a high data rate for a low decoding complexity. We analyze the performance of DLST codes with a hard decision-feedback decoder and a soft decision-feedback decoder with iterative decoding. We analyze the error probability performance and propose the criteria for designing the constituent codes.

We examine the use of differential pulse-position modulation (DPPM) for optical communication systems using intensity modulation with direct detection in the presence of additive white Gaussian noise. We present expressions for the error probability and power spectral density of DPPM. We show that for a given bandwidth, DPPM requires significantly less average power than pulse position modulation (PPM). We also examine the performance of DPPM in the presence of multipath intersymbol interference (ISI). We find that the ISI penalties incurred by PPM and DPPM exhibit very similar dependencies upon the channel RMS delay spread. We discuss the use of chip-rate and multichip-rate equalization to combat ISI. Finally, we describe potential problems caused by the nonuniform bit-rate characteristic of DPPM, and we propose several solutions.

Theory of shaping and nonequiprobable signaling, which has been developed for conventional electrical signals, must be modified to treat intensity-modulated (IM) signals. We show that for IM signals, the optimum shape of the constellation bounding region in N-dimensional (N-D) space is an N-D simplex. As N/spl rarr//spl infin/, the maximum achievable shape gain is 1.33 dB (in terms of transmitted power), and the resulting marginal signaling distribution on the one-dimensional (1-D) constituent constellation is exponential. We also investigate the tradeoffs between shaping and its negative consequences, and find that a 1-dB shape gain can be achieved while incurring reasonable increases in peak-to-average power ratio and constellation expansion ratio.