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With the rising demand for high-rate and timely communications, fog radio access networks (F-RANs) offer a promising solution. This work investigates age of information (AoI) performance in F-RANs, consisting of multiple content users (CUs), enhanced remote radio heads (eRRHs), and content providers (CPs). Time-critical CUs need rapid content updates from CPs but cannot communicate directly with them; instead, eRRHs act as intermediaries. CUs decide whether to request content from a CP and which eRRH to send the request to, while eRRHs decide whether to command CPs to update content or use cached content. We study two general classes of policies: (i) oblivious policies, where decision-making is independent of historical information, and (ii) non-oblivious policies, where decisions are influenced by historical information. First, we obtain closed-form expressions for the average AoI of eRRHs under both policy types. Due to the complexity of calculating closed-form expressions for CUs, we then derive general upper bounds for their average AoI. Next, we identify optimal policies for both types. Under both optimal policies, each CU requests content from each CP at an equal rate, consolidating all requests to a single eRRH when demand is low or resources are limited, and distributing requests evenly among eRRHs when demand is high and resources are ample. eRRHs command content from each CP at an equal rate under an optimal oblivious policy, while prioritize the CP with the highest age under an optimal non-oblivious policy. Our numerical results validate these theoretical findings.

We address the challenge of sampling and remote estimation for autoregressive Markovian processes in a multi-hop wireless network with statistically-identical agents. Agents cache the most recent samples from others and communicate over wireless collision channels governed by an underlying graph topology. Our goal is to minimize time-average estimation error and/or age of information with decentralized scalable sampling and transmission policies, considering both oblivious (where decision-making is independent of the physical processes) and non-oblivious policies (where decision-making depends on physical processes). We prove that in oblivious policies, minimizing estimation error is equivalent to minimizing the age of information. The complexity of the problem, especially the multi-dimensional action spaces and arbitrary network topologies, makes theoretical methods for finding optimal transmission policies intractable. We optimize the policies using a graphical multi-agent reinforcement learning framework, where each agent employs a permutation-equivariant graph neural network architecture. Theoretically, we prove that our proposed framework exhibits desirable transferability properties, allowing transmission policies trained on small- or moderate-size networks to be executed effectively on large-scale topologies. Numerical experiments demonstrate that (i) Our proposed framework outperforms state-of-the-art baselines; (ii) The trained policies are transferable to larger networks, and their performance gains increase with the number of agents; (iii) The training procedure withstands non-stationarity even if we utilize independent learning techniques; and, (iv) Recurrence is pivotal in both independent learning and centralized training and decentralized execution, and improves the resilience to non-stationarity in independent learning.

In communication and computation cooperative networks (3CNs), timely computation is crucial but not always guaranteed. There is a strong demand for a computational task to be completed within a given deadline. The time taken involves both processing time, communication time, and the impact of the deadline. However, a measure of such timeliness in 3CNs is lacking. In this paper, we introduce the novel concept, Age of Computing (AoC), to capture computation freshness in 3CNs. We analyze AoC in a line topology consisting of a source, a transmitter, a receiver, and a computational node. Tasks generated by the source are immediately available at the transmitter, where they enter a communication queue. These tasks then pass to the receiver and subsequently to a computation queue at the computational node for processing. Each task has a deadline, requiring completion within this timeframe. AoC is evaluated under two types of deadlines: (i) soft deadline, tasks can be fed back to the source if delayed beyond the deadline, but with additional latency; (ii) hard deadline, tasks delayed beyond the deadline are discarded. Under both deadlines, we derive the AoC formula and a general expression for the time-average AoC. For the first-come, first-serve discipline, we obtain a closed-form expression for the average AoC under the soft deadline and an approximation for the hard deadline. In addition to freshness, we define computation throughput, providing a general expression and an approximation. To explore the relationship between freshness and throughput, we construct an optimization problem and prove that the objective pair is a weakly Pareto-optimal point. Numerical results validate all the theoretical findings. Additionally, they reveal that under the hard deadline, the computation throughput serves as a reliable proxy for the average AoC.

Self-supervised learning (SSL) has shown promising results in various speech and natural language processing applications. However, its efficacy in music information retrieval (MIR) still remains largely unexplored. While previous SSL models pre-trained on music recordings may have been mostly closed-sourced, recent speech models such as wav2vec2.0 have shown promise in music modelling. Nevertheless, research exploring the effectiveness of applying speech SSL models to music recordings has been limited. We explore the music adaption of SSL with two distinctive speech-related models, data2vec1.0 and Hubert, and refer to them as music2vec and musicHuBERT, respectively. We train $12$ SSL models with 95M parameters under various pre-training configurations and systematically evaluate the MIR task performances with 13 different MIR tasks. Our findings suggest that training with music data can generally improve performance on MIR tasks, even when models are trained using paradigms designed for speech. However, we identify the limitations of such existing speech-oriented designs, especially in modelling polyphonic information. Based on the experimental results, empirical suggestions are also given for designing future musical SSL strategies and paradigms.

In the era of extensive intersection between art and Artificial Intelligence (AI), such as image generation and fiction co-creation, AI for music remains relatively nascent, particularly in music understanding. This is evident in the limited work on deep music representations, the scarcity of large-scale datasets, and the absence of a universal and community-driven benchmark. To address this issue, we introduce the Music Audio Representation Benchmark for universaL Evaluation, termed MARBLE. It aims to provide a benchmark for various Music Information Retrieval (MIR) tasks by defining a comprehensive taxonomy with four hierarchy levels, including acoustic, performance, score, and high-level description. We then establish a unified protocol based on 14 tasks on 8 public-available datasets, providing a fair and standard assessment of representations of all open-sourced pre-trained models developed on music recordings as baselines. Besides, MARBLE offers an easy-to-use, extendable, and reproducible suite for the community, with a clear statement on copyright issues on datasets. Results suggest recently proposed large-scale pre-trained musical language models perform the best in most tasks, with room for further improvement. The leaderboard and toolkit repository are published at https://marble-bm.shef.ac.uk to promote future music AI research., Comment: camera-ready version for NeurIPS 2023

Due to Multilingual Neural Machine Translation's (MNMT) capability of zero-shot translation, many works have been carried out to fully exploit the potential of MNMT in zero-shot translation. It is often hypothesized that positional information may hinder the MNMT from outputting a robust encoded representation for decoding. However, previous approaches treat all the positional information equally and thus are unable to selectively remove certain positional information. In sharp contrast, this paper investigates how to learn to selectively preserve useful positional information. We describe the specific mechanism of positional information influencing MNMT from the perspective of linguistics at the token level. We design a token-level position disentangle module (TPDM) framework to disentangle positional information at the token level based on the explanation. Our experiments demonstrate that our framework improves zero-shot translation by a large margin while reducing the performance loss in the supervised direction compared to previous works.

Self-supervised learning (SSL) has recently emerged as a promising paradigm for training generalisable models on large-scale data in the fields of vision, text, and speech. Although SSL has been proven effective in speech and audio, its application to music audio has yet to be thoroughly explored. This is partially due to the distinctive challenges associated with modelling musical knowledge, particularly tonal and pitched characteristics of music. To address this research gap, we propose an acoustic Music undERstanding model with large-scale self-supervised Training (MERT), which incorporates teacher models to provide pseudo labels in the masked language modelling (MLM) style acoustic pre-training. In our exploration, we identified an effective combination of teacher models, which outperforms conventional speech and audio approaches in terms of performance. This combination includes an acoustic teacher based on Residual Vector Quantisation - Variational AutoEncoder (RVQ-VAE) and a musical teacher based on the Constant-Q Transform (CQT). Furthermore, we explore a wide range of settings to overcome the instability in acoustic language model pre-training, which allows our designed paradigm to scale from 95M to 330M parameters. Experimental results indicate that our model can generalise and perform well on 14 music understanding tasks and attain state-of-the-art (SOTA) overall scores., Comment: accepted by ICLR 2024

Previous research has demonstrated that natural language explanations provide valuable inductive biases that guide models, thereby improving the generalization ability and data efficiency. In this paper, we undertake a systematic examination of the effectiveness of these explanations. Remarkably, we find that corrupted explanations with diminished inductive biases can achieve competitive or superior performance compared to the original explanations. Our findings furnish novel insights into the characteristics of natural language explanations in the following ways: (1) the impact of explanations varies across different training styles and datasets, with previously believed improvements primarily observed in frozen language models. (2) While previous research has attributed the effect of explanations solely to their inductive biases, our study shows that the effect persists even when the explanations are completely corrupted. We propose that the main effect is due to the provision of additional context space. (3) Utilizing the proposed automatic perturbed context, we were able to attain comparable results to annotated explanations, but with a significant increase in computational efficiency, 20-30 times faster.

Recent years have witnessed the emergence of textual commonsense knowledge bases, aimed at providing more nuanced and context-rich knowledge. The integration of external commonsense into language models has been shown to be a key enabler in advancing the state-of-the-art for a wide range of NLP tasks. However, incorporating textual commonsense descriptions is computationally expensive, as compared to encoding conventional symbolic knowledge. In this paper, we propose a method to improve its efficiency without modifying the model. We group training samples with similar commonsense descriptions into a single batch, thus reusing the encoded description across multiple samples. One key observation is that the upper bound of batch partitioning can be reduced to the classic {\it graph k-cut problem}. Consequently, we propose a spectral clustering-based algorithm to solve this problem. Extensive experiments illustrate that the proposed batch partitioning approach effectively reduces the computational cost while preserving performance. The efficiency improvement is more pronounced on larger datasets and on devices with more memory capacity, attesting to its practical utility for large-scale applications.

Is there a unified framework for graph-based retrosynthesis prediction? Through analysis of full-, semi-, and non-template retrosynthesis methods, we discovered that they strive to strike an optimal balance between combinability and consistency: \textit{Should atoms be combined as motifs to simplify the molecular editing process, or should motifs be broken down into atoms to reduce the vocabulary and improve predictive consistency?} Recent works have studied several specific cases, while none of them explores different combinability-consistency trade-offs. Therefore, we propose MotifRetro, a dynamic motif editing framework for retrosynthesis prediction that can explore the entire trade-off space and unify graph-based models. MotifRetro comprises two components: RetroBPE, which controls the combinability-consistency trade-off, and a motif editing model, where we introduce a novel LG-EGAT module to dynamiclly add motifs to the molecule. We conduct extensive experiments on USPTO-50K to explore how the trade-off affects the model performance and finally achieve state-of-the-art performance.

The spread of an undesirable contact process, such as an infectious disease (e.g. COVID-19), is contained through testing and isolation of infected nodes. The temporal and spatial evolution of the process (along with containment through isolation) render such detection as fundamentally different from active search detection strategies. In this work, through an active learning approach, we design testing and isolation strategies to contain the spread and minimize the cumulative infections under a given test budget. We prove that the objective can be optimized, with performance guarantees, by greedily selecting the nodes to test. We further design reward-based methodologies that effectively minimize an upper bound on the cumulative infections and are computationally more tractable in large networks. These policies, however, need knowledge about the nodes' infection probabilities which are dynamically changing and have to be learned by sequential testing. We develop a message-passing framework for this purpose and, building on that, show novel tradeoffs between exploitation of knowledge through reward-based heuristics and exploration of the unknown through a carefully designed probabilistic testing. The tradeoffs are fundamentally distinct from the classical counterparts under active search or multi-armed bandit problems (MABs). We provably show the necessity of exploration in a stylized network and show through simulations that exploration can outperform exploitation in various synthetic and real-data networks depending on the parameters of the network and the spread.

The deep learning community has witnessed an exponentially growing interest in self-supervised learning (SSL). However, it still remains unexplored how to build a framework for learning useful representations of raw music waveforms in a self-supervised manner. In this work, we design Music2Vec, a framework exploring different SSL algorithmic components and tricks for music audio recordings. Our model achieves comparable results to the state-of-the-art (SOTA) music SSL model Jukebox, despite being significantly smaller with less than 2% of parameters of the latter. The model will be released on Huggingface(Please refer to: https://huggingface.co/m-a-p/music2vec-v1)

The secondary structure of ribonucleic acid (RNA) is more stable and accessible in the cell than its tertiary structure, making it essential for functional prediction. Although deep learning has shown promising results in this field, current methods suffer from poor generalization and high complexity. In this work, we reformulate the RNA secondary structure prediction as a K-Rook problem, thereby simplifying the prediction process into probabilistic matching within a finite solution space. Building on this innovative perspective, we introduce RFold, a simple yet effective method that learns to predict the most matching K-Rook solution from the given sequence. RFold employs a bi-dimensional optimization strategy that decomposes the probabilistic matching problem into row-wise and column-wise components to reduce the matching complexity, simplifying the solving process while guaranteeing the validity of the output. Extensive experiments demonstrate that RFold achieves competitive performance and about eight times faster inference efficiency than the state-of-the-art approaches. The code and Colab demo are available in (http://github.com/A4Bio/RFold)., Comment: Accepted by ICML 2024

In this paper, we propose a new method for knowledge base completion (KBC): instance-based learning (IBL). For example, to answer (Jill Biden, lived city,? ), instead of going directly to Washington D.C., our goal is to find Joe Biden, who has the same lived city as Jill Biden. Through prototype entities, IBL provides interpretability. We develop theories for modeling prototypes and combining IBL with translational models. Experiments on various tasks confirmed the IBL model's effectiveness and interpretability. In addition, IBL shed light on the mechanism of rule-based KBC models. Previous research has generally agreed that rule-based models provide rules with semantically compatible premises and hypotheses. We challenge this view. We begin by demonstrating that some logical rules represent {\it instance-based equivalence} (i.e. prototypes) rather than semantic compatibility. These are denoted as {\it IBL rules}. Surprisingly, despite occupying only a small portion of the rule space, IBL rules outperform non-IBL rules in all four benchmarks. We use a variety of experiments to demonstrate that rule-based models work because they have the ability to represent instance-based equivalence via IBL rules. The findings provide new insights of how rule-based models work and how to interpret their rules.

This paper details our participation in the Challenges and Applications of Automated Extraction of Socio-political Events from Text (CASE) workshop @ EMNLP 2022, where we take part in Subtask 1 of Shared Task 3. We approach the given task of event causality detection by proposing a self-training pipeline that follows a teacher-student classifier method. More specifically, we initially train a teacher model on the true, original task data, and use that teacher model to self-label data to be used in the training of a separate student model for the final task prediction. We test how restricting the number of positive or negative self-labeled examples in the self-training process affects classification performance. Our final results show that using self-training produces a comprehensive performance improvement across all models and self-labeled training sets tested within the task of event causality sequence classification. On top of that, we find that self-training performance did not diminish even when restricting either positive/negative examples used in training. Our code is be publicly available at https://github.com/Gzhang-umich/1CademyTeamOfCASE., Comment: Paper from CASE workshop at EMNLP 2022

In this paper, we present our approach and empirical observations for Cause-Effect Signal Span Detection -- Subtask 2 of Shared task 3~\cite{tan-etal-2022-event} at CASE 2022. The shared task aims to extract the cause, effect, and signal spans from a given causal sentence. We model the task as a reading comprehension (RC) problem and apply a token-level RC-based span prediction paradigm to the task as the baseline. We explore different training objectives to fine-tune the model, as well as data augmentation (DA) tricks based on the language model (LM) for performance improvement. Additionally, we propose an efficient beam-search post-processing strategy to due with the drawbacks of span detection to obtain a further performance gain. Our approach achieves an average $F_1$ score of 54.15 and ranks \textbf{$1^{st}$} in the CASE competition. Our code is available at \url{https://github.com/Gzhang-umich/1CademyTeamOfCASE}., Comment: paper of CASE workshop in EMNLP 2022

In applications of group testing in networks, e.g. identifying individuals who are infected by a disease spread over a network, exploiting correlation among network nodes provides fundamental opportunities in reducing the number of tests needed. We model and analyze group testing on $n$ correlated nodes whose interactions are specified by a graph $G$. We model correlation through an edge-faulty random graph formed from $G$ in which each edge is dropped with probability $1-r$, and all nodes in the same component have the same state. We consider three classes of graphs: cycles and trees, $d$-regular graphs and stochastic block models or SBM, and obtain lower and upper bounds on the number of tests needed to identify the defective nodes. Our results are expressed in terms of the number of tests needed when the nodes are independent and they are in terms of $n$, $r$, and the target error. In particular, we quantify the fundamental improvements that exploiting correlation offers by the ratio between the total number of nodes $n$ and the equivalent number of independent nodes in a classic group testing algorithm. The lower bounds are derived by illustrating a strong dependence of the number of tests needed on the expected number of components. In this regard, we establish a new approximation for the distribution of component sizes in "$d$-regular trees" which may be of independent interest and leads to a lower bound on the expected number of components in $d$-regular graphs. The upper bounds are found by forming dense subgraphs in which nodes are more likely to be in the same state. When $G$ is a cycle or tree, we show an improvement by a factor of $log(1/r)$. For grid, a graph with almost $2n$ edges, the improvement is by a factor of ${(1-r) \log(1/r)}$, indicating drastic improvement compared to trees. When $G$ has a larger number of edges, as in SBM, the improvement can scale in $n$.

Rules have a number of desirable properties. It is easy to understand, infer new knowledge, and communicate with other inference systems. One weakness of the previous rule induction systems is that they only find rules within a knowledge base (KB) and therefore cannot generalize to more open and complex real-world rules. Recently, the language model (LM)-based rule generation are proposed to enhance the expressive power of the rules. In this paper, we revisit the differences between KB-based rule induction and LM-based rule generation. We argue that, while KB-based methods inducted rules by discovering data commonalities, the current LM-based methods are "learning rules from rules". This limits these methods to only produce "canned" rules whose patterns are constrained by the annotated rules, while discarding the rich expressive power of LMs for free text. Therefore, in this paper, we propose the open rule induction problem, which aims to induce open rules utilizing the knowledge in LMs. Besides, we propose the Orion (\underline{o}pen \underline{r}ule \underline{i}nducti\underline{on}) system to automatically mine open rules from LMs without supervision of annotated rules. We conducted extensive experiments to verify the quality and quantity of the inducted open rules. Surprisingly, when applying the open rules in downstream tasks (i.e. relation extraction), these automatically inducted rules even outperformed the manually annotated rules.

We study how to enhance text representation via textual commonsense. We point out that commonsense has the nature of domain discrepancy. Namely, commonsense has different data formats and is domain-independent from the downstream task. This nature brings challenges to introducing commonsense in general text understanding tasks. A typical method of introducing textual knowledge is continuing pre-training over the commonsense corpus. However, it will cause catastrophic forgetting to the downstream task due to the domain discrepancy. In addition, previous methods of directly using textual descriptions as extra input information cannot apply to large-scale commonsense. In this paper, we propose to use large-scale out-of-domain commonsense to enhance text representation. In order to effectively incorporate the commonsense, we proposed OK-Transformer (\underline{O}ut-of-domain \underline{K}nowledge enhanced \underline{Transformer}). OK-Transformer effectively integrates commonsense descriptions and enhances them to the target text representation. In addition, OK-Transformer can adapt to the Transformer-based language models (e.g. BERT, RoBERTa) for free, without pre-training on large-scale unsupervised corpora. We have verified the effectiveness of OK-Transformer in multiple applications such as commonsense reasoning, general text classification, and low-resource commonsense settings.

The interplay between timeliness and rate efficiency is investigated in packet erasure broadcast channels with feedback. A scheduling framework is proposed in which coding actions, as opposed to users, are scheduled to attain desired tradeoffs between rate and age of information (AoI). This tradeoff is formalized by an upper bound on AoI as a function of the target rate constraints and two lower bounds: one as a function of the communication rate and one as a function of the arrival rate. Simulation results show that (i) coding can be beneficial in reducing AoI in the regime of moderate arrival rates even without rate constraints and the benefit increases with the number of users, and (ii) AoI increases with both the target rate constraint and the arrival rate when either is kept fixed, but decreases with them when they are set to be equal.

Efficient sampling and remote estimation are critical for a plethora of wireless-empowered applications in the Internet of Things and cyber-physical systems. Motivated by such applications, this work proposes decentralized policies for the real-time monitoring and estimation of autoregressive processes over random access channels. Two classes of policies are investigated: (i) oblivious schemes in which sampling and transmission policies are independent of the processes that are monitored, and (ii) non-oblivious schemes in which transmitters causally observe their corresponding processes for decision making. In the class of oblivious policies, we show that minimizing the expected time-average estimation error is equivalent to minimizing the expected age of information. Consequently, we prove lower and upper bounds on the minimum achievable estimation error in this class. Next, we consider non-oblivious policies and design a threshold policy, called error-based thinning, in which each source node becomes active if its instantaneous error has crossed a fixed threshold (which we optimize). Active nodes then transmit stochastically following a slotted ALOHA policy. A closed-form, approximately optimal, solution is found for the threshold as well as the resulting estimation error. It is shown that non-oblivious policies offer a multiplicative gain close to $3$ compared to oblivious policies. Moreover, it is shown that oblivious policies that use the age of information for decision making improve the state-of-the-art at least by the multiplicative factor $2$. The performance of all discussed policies is compared using simulations. The numerical comparison shows that the performance of the proposed decentralized policy is very close to that of centralized greedy scheduling.

In applications of remote sensing, estimation, and control, timely communication is not always ensured by high-rate communication. This work proposes distributed age-efficient transmission policies for random access channels with $M$ transmitters. In the first part of this work, we analyze the age performance of stationary randomized policies by relating the problem of finding age to the absorption time of a related Markov chain. In the second part of this work, we propose the notion of \emph{age-gain} of a packet to quantify how much the packet will reduce the instantaneous age of information at the receiver side upon successful delivery. We then utilize this notion to propose a transmission policy in which transmitters act in a distributed manner based on the age-gain of their available packets. In particular, each transmitter sends its latest packet only if its corresponding age-gain is beyond a certain threshold which could be computed adaptively using the collision feedback or found as a fixed value analytically in advance. Both methods improve age of information significantly compared to the state of the art. In the limit of large $M$, we prove that when the arrival rate is small (below $\frac{1}{eM}$), slotted ALOHA-type algorithms are asymptotically optimal. As the arrival rate increases beyond $\frac{1}{eM}$, while age increases under slotted ALOHA, it decreases significantly under the proposed age-based policies. For arrival rates $\theta$, $\theta=\frac{1}{o(M)}$, the proposed algorithms provide a multiplicative factor of at least two compared to the minimum age under slotted ALOHA (minimum over all arrival rates). We conclude that, as opposed to the common practice, it is beneficial to increase the sampling rate (and hence the arrival rate) and transmit packets selectively based on their age-gain.

This paper investigates the incentives of mobile network operators (MNOs) for acquiring additional spectrum to offer mobile virtual network operators (MVNOs) and thereby inviting competition for a common pool of end users (EUs). We consider interactions between two service providers, an MNO and an MVNO, when the EUs 1) must choose one of them 2) have the option to defect to an outside option should the SP duo offer unsatisfactory access fees or qualities of service. We formulate a multi-stage hybrid of cooperative bargaining and non-cooperative games in which the two SPs jointly determine their spectrum acquisitions, allocations and mutual money flows through the bargaining game, and subsequently individually determine the access fees for the EUs through the non-cooperative game. We identify when the overall equilibrium solutions exist, when it is unique and characterize the equilibrium solutions when they exist. The characterizations are easy to compute, and are in closed form or involve optimizations in only one decision variable. The hybrid framework allows us to determine whether and by how much the different entities benefit due to the cooperation in spectrum acquisition decision.

Age of Information (AoI) is studied in two-user broadcast networks with feedback, and lower and upper bounds are derived on the expected weighted sum AoI of the users. In particular, a class of simple coding actions is considered and within this class, randomized and deterministic policies are devised. Explicit conditions are found for symmetric dependent channels under which coded randomized policies strictly outperform the corresponding uncoded policies. Similar behavior is numerically shown for deterministic policies.

We consider the economics of the interaction between Mobile Virtual Network Operators (MVNOs) and Mobile Network Operators (MNOs). We investigate the incentives of an MNO for offering some of her resources to an MVNO instead of using the resources for her own End-Users (EUs). We consider a market with one MNO and one MVNO, and a continuum of undecided EUs. Two cases for EUs are considered: (i) when EUs need to choose either the MNO or the MVNO, and (ii) when EUs have an outside option. In each of these cases, we consider a non-cooperative framework of sequential game and a cooperative framework of bargaining game. We characterize the Subgame Perfect Nash Equilibria (SPNE) and Nash Bargaining Solution (NBS) of the sequential and bargaining games, respectively. We show that in the non-cooperative framework, SPNE assumes two forms: (i) the MNO invests minimally on her infrastructure and the MVNO leases all the newly invested resources and (ii) the MNO invests more on her infrastructure, the MVNO leases only part of these resources to the MVNO, and the MNO uses the rest herself to attract EUs. Thus, in both, the MNO generates revenue indirectly through the MVNO. In addition to that, in (ii), the MNO generates revenue directly from the EUs. We also prove that in the bargaining framework, the MVNO either reserves all the resources or no resources from the MNO, and the MNO's investments are guided by whether EUs have an outside option. If they don't, then the MNO invests as little as possible on her infrastructure. If they do, then the MNO invests more., Comment: 14 pages, Submitted to IEEE/ACM Transaction on Networking