The lack of reliable observed data at each decision-making device calls for robust decen-tralized coordination algorithms, whose purpose is to minimize the loss with respect to the centralized solutions. The emphasis on acommonperformance goal and the la-tency constraints for inter-device communication requires a different approach from classical device cooperation frameworks. For example, in egoistic game-theoretical ap-proaches [65], the radio devices are conflicting with each other and potential equilibria do notautomaticallytranslate into global network gains. In this case, the imperfect coor-dination which hinder the maximization of the global performance metric arises from the distributed nature of the observed data, based upon which the decisions are made.
The theoretical roots behindone-shotdecentralized coordination can instead be found in the so-called Team Decision theory [66], which is known for a long time and often involves solving a non-trivial distributed functional optimization problem. Yet, the strong development of the computational capabilities in the past decade has opened up new avenues for solving such difficult problems. In the following, we show how the TD formulation unfolds in the context of beam-domain coordination.
Hˆ(k) s1 Hˆ(k)
→vk v
k
Figure 1.1 – Beam decision with distributed information. The k-th device makes its decision on the basis of its own global network state estimateHˆ(k).
1.5.1 Decentralized Beam-Domain Coordination
Let us consider a network with K cooperating devices, which will be instantiated as BSs or UEs in the chapters of this thesis (see Table1.1). We assume that thek-th device adopts thestrategysk :Cm → Sk ⊆Cdk, based on local estimates, whereSkis its deci-sion sub-space, i.e. its beam or precodercodebookin beam-domain coordination (refer to Fig.1.1). The general TD problem, whose goal is to maximize the global network performance metricf :Cm×QK
k=1Cdk →R, can be formulated as follows:
s∗1, . . . , s∗K
= argmax
s1,...,sK EH,Hˆ(1),...,Hˆ(K) h
f s1
Hˆ(1)
, . . . , sK Hˆ(K)
,Hi
, (1.1) where
• H∈Cmis the global state of the network1;
• Hˆ(k)∈Cmis the local estimate ofHwhich is available at thek-th device.
The formulation in (1.1) refers to a static setting where each of theK devices de-signs transmission policies in order to coordinate with the other devices, based on the expectation over thejoint Probability Density Function (PDF)of the actual network state and all local estimates, defined as
pH,Hˆ(1)
,...,Hˆ(K). (1.2)
Thus, the mutual correlation betweenHˆ(1), . . . ,Hˆ(K)and the correlation between these estimates and the actual stateHset a limit to the coordination performance. In particu-lar, the solution to (1.1) depends on the associated information structure, i.e. the nature of the observations made at each device and how such local information relates to the actual global state. In the following, we will introduce and motivate the decentralized information structures that will be considered throughout this thesis.
1We stress thatHcan be either CSI or related to CSI, as e.g. location information (refer to Table1.1).
Chapter 1. Introduction and Motivation
1.5.2 Distributed Information Structures
The information structure underpinning the TD problem in (1.1) describes how the lo-cal information Hˆ(k) available at thek-th device relates to the local estimates at the other devicesHˆ(j), ∀j 6= k, as well as to the actual global state information vectorH.
Note that the configuration in (1.1) precludes explicit interactions between the cooper-ating devices, which is consistent withlow-latencyapplications. In this respect, the large number of devices in future mobile networks represents a favorable circumstance as de-vices that are located closer to each other have greater chance forlow-latencydirect D2D communication (refer to Section1.2.3). Thus, some rounds of information exchanges be-tween the devices can be assumed, leading to specific information configurations and easier, more practical, TD problems. In this section, we first describe an intuitive and tractable information model which consists in considering Gaussian noise-corrupted global information at each device. Then, we describe a hierarchical information setup which arises in networks where some devices are endowed with greater information gathering capabilities.
Distributed Gaussian Information Configuration
The distributed Gaussian configuration assumes that some Gaussian noise with device-dependent covariance matrixΣk∈Cm×mcorrupts the estimateHˆ(k)at thek-th device.
In particular, we can write the estimateHˆ(k)at thek-th device as Hˆ(k)=
q
1−Σ2kH+Σ2kN, ∀k={1, . . . , K}, (1.3) whereN∼ CN(0,Im)is Gaussian-distributed noise.
Under this model, the knowledge of the noise variances at the other devices is suf-ficient to derive the joint PDF in (1.2). In practical scenarios, some devices can have greater sensing and estimation capabilities with respect to other devices, as e.g. high-end devices compared to low-high-end devices. Such cases are well-captured in (1.3), where greater noise variances can be assumed for the low-end devices. Some examples on practical applications of such configuration can be found in [67–69]. In general, this model can be extended toanykind of noise, as for example uniform bounded noise. In Chapter3, we assume a bounded error model for the location side-information, such that the location estimates at each device fall inside a disk around the actual locations.
Hierarchical Distributed Configuration
The hierarchical distributed information structure is obtained when the devices can be ordered such that thek-th device has access to the information – so, the transmission decision – at thej-th device, wherej < k, in addition to its local information. This con-figuration implies that the first device is the least informed one while theK-th device is the most informed one and knows the information at alllower-ranked devices. In a mathematical sense, it means that there exist some functionshk,j:Cm →Cmsuch that
Hˆ(j) =hk,j Hˆ(k)
, ∀j < k. (1.4) The advantage of the hierarchical distributed configuration is that – unlike the infor-mation structure in (1.3) – the devices can follow a chain of strategies where the better informedk-th device can adapt its own strategies to the known strategies taken at the lower-ranked less-informedj-th device, wherej < k, in order to improve the common performance metric. The hierarchical distributed configuration is obtainable through e.g. multi-level quantization schemes [70] or also D2D side-links (refer to Section1.2.3).
A remaining obstacle resides in the fact thek-th device is not able to predict with ex-actitude the strategies of the better informedj-th devices, wherej > k. Low-complex sub-optimal solutions are possible. For instance, thek-th device can assume that the higher-ranked devices have access to the same local informationHˆ(k). Following this approximation, the (hierarchical) TD at thek-th device is obtained as follows:
sHCk ,s¯k+1, . . . ,s¯K
= argmax
sk,...,sK
EH|Hˆ(k) h
f
s∗1, . . . , s∗k−1, sk Hˆ(k)
, . . . , sK Hˆ(k)
,H i . (1.5) Remark1.1. The decisionss¯∗k+1, . . . ,s¯∗Kin (1.5) areauxiliaryvariables which are not used for actual transmission. The higher-ranked j-th device, where j > k, will use more accurate information to derive its own decision.
Under the hierarchical information setting, the TD optimization problem reduces to a conventional robust optimization problem. Indeed, the optimal decision at thek-th device in (1.5) dependssolelyon its local estimateHˆ(k), i.e. the generick-th device do not need to estimate the information and the decisions at the other devices.
The hierarchical configuration has been exploited in [71] to design a robust precoder for distributed Network MIMO. In Chapters4,5and6, the hierarchical setup will prove beneficial for achieving beam-domain coordination withlow-complexity.
Chapter 1. Introduction and Motivation
Cooperation among Nature of the Information Coordination Goal Chapter BSandUE(Fig.1.2a) Location Single-User Beam Alignment 3
K UEs(Fig.1.2b) Out-of-Band and Statistical Multi-User Beam Selection 4,5 K BSs(Fig.1.2c) Beam Index Scheduling in Spectrum Sharing 6 Table 1.1 – The coordination scenarios which are studied throughout this thesis.