Structurebased learning in wireless networks via sparse approximation
 Marco Levorato^{1, 2}Email author,
 Urbashi Mitra^{1} and
 Andrea Goldsmith^{2}
DOI: 10.1186/168714992012278
© Levorato et al.; licensee Springer. 2012
Received: 16 February 2012
Accepted: 18 July 2012
Published: 30 August 2012
Abstract
A novel framework for the online learning of expected costtogo functions characterizing wireless networks performance is proposed. The framework is based on the observation that wireless protocols induce structured and correlated behavior of the finite state machine (FSM) modeling the operations of the network. As a result, a significant dimension reduction can be achieved by projecting the costtogo function on a graph wavelet basis set capturing typical substructures in the graph associated with the FSM. Sparse approximation with random projection is then used to identify a concise set of coefficients representing the costtogo function in the wavelet domain. This Compressed Sensing (CS) approach enables a considerable reduction in the number of observations needed to achieve an accurate estimate of the costtogo function. The proposed method is characterized via stability analysis. In particular, we prove that the standard CS approach of the Least Angle Selection and Shrinkage Operator (LASSO) will not provide stability. We also determine a connection between the structure of the FSM induced by the wireless protocols and the restricted isometry property of the effective projection matrix. Simulation results of our approximation method show that 15 wavelet functions can accurately represent a costtogo function defined on a state space of 2000 states. Moreover, the number of statecost observations needed to estimate the costtogo function is orders of magnitude smaller than that required by traditional online learning techniques.
Introduction
Given the recent explosion in the number and types of wireless devices, new design and optimization paradigms are needed to effectively manage the complex and heterogeneous nature of modern wireless networks. We propose a novel approach for the online learning of costtogo functions in networks modeled via large finite state machines (FSM). Typical cost functions measure performance metrics such as throughput, packet delivery probability and delay. Costtogo functions measure the expected longterm cost incurred by the network from any state of the FSM. Estimation of costtogo functions is instrumental for the optimization of network control strategies. Our estimation approach is based on the observation that wireless networking protocols induce a structured behavior of the FSM, enabling dimension reduction of its state space via waveletprojection and compressed sensinglike techniques. The sparse approximation approach proposed herein considerably reduces the length of the trajectory of the FSM required to achieve an accurate estimate of the costtogo function compared to traditional learning techniques.
Markov models have been widely used for the analysis and optimization of wireless networks[1–9]. In one of the earliest works on protocol modeling[1], a Markov chain is proposed to analyze the saturation throughput of IEEE 802.11 medium access control. The FSM models the backoff countdown counter controlling channel sensing and access of a wireless terminal and the retransmission index of the packet under transmission. In general, the Markov chains defined in these models track the logical state of the wireless protocols (e.g., the retransmission index of the packet being transmitted, the number of packets in the buffer and the backoff counter) as well as environmental variables (e.g., the channel state).
The online optimization of control strategies based on these models requires the estimation of costtogo functions from a samplepath of statecost observations[10–12]. However, the immense size of the state space of FSMs associated with practical wireless networks limits the applicability of traditional online learning techniques to toy networks and extremely simple case studies. In fact, the estimation of costtogo functions in traditional online learning (e.g., Qlearning and Reinforcement learning[10–12]) requires sufficient observation of a samplepath such that it hits all the states of the FSM a large number of times. Approximations of costtogo functions[13, 14] are generally based on oversimplified models and thus cannot be accurately used in general practical networks. For instance, the fluid approximation proposed in[14] is based on the assumption that the costtogo function is smooth in the state space of the FSM, meaning that only small variations of its value computed in neighboring states are allowed. This assumption is suitable for simple cases (e.g., buffer models and cost functions modeling buffer congestion), but does not hold for more complex FSM models and general cost functions.
This work provides the following contributions: 1) we present a framework based on CS for the approximation of costtogo functions; 2) we analyze the structure of the FSMs modeling wireless networks based on their decomposition into fundamental components; 3) we connect the structure of the FSM to the Restricted Isometry properties of the effective projection matrix; 4) we analyze the stability of CS in this context via perturbation analysis; 5) we present a methodology for the use of Diffusion Wavelets (DW) in online learning; 6) we present numerical results illustrating the performance of the proposed approach.
We characterize the performance of sparse approximation applied to the estimation problem addressed herein in terms of the minimum number of states that need to be observed to achieve an accurate estimate of the costtogo function. Our analysis is based on the decomposition of the FSM in fundamental structures we refer to as subchains. The transition matrix associated with the individual subchains is analyzed to measure the incoherence^{a} of the overall transition matrix, which is exploited to determine the conditions under which the restricted isometry property (RIP)[16] holds for our effective random projection matrix.
Note that whereas most prior work on sparse approximation focuses on static scenarios, the framework considered in this article addresses the problem of learning in dynamical systems. The inclusion of states visited a small number of times in the samplepath of the FSM results in instability of the estimation algorithm. To reduce this effect, we use the LS CS algorithm proposed in[15]. LS CS correlates the output of the sparse approximation algorithm by constraining variations in the support of the solution.
Relevant to the approach proposed herein, Mahadevan et al. in[17] proposed the use of DW[18] as a projection basis for the sparse approximation of costtogo functions. In[17], offline estimation of the costtogo function is considered, however, no performance analysis is undertaken. In contrast, we examine online learning and we provide a detailed analysis to assess the performance of sparse approximation applied to Markov models of wireless networks. Compressed sensingbased techniques have been previously applied to estimation problems in networks[19–24]. These works address graphs related to the physical connectivity of the network, where nodes are terminals and links are specific wired or wireless links or modeled by undirected graphs. We address the fundamentally different problem of estimating functions defined on the state space of the FSM, i.e., the logical graph of the wireless network, modeling the temporal evolution of the network from a small number of state observations.
Numerical results for a case of interest show that a small number of graph wavelets (∼15) are sufficient to accurately approximate a costtogo function defined on a state space of approximately 2000 states. Moreover, the proposed algorithm can estimate the costtogo function by observing a trajectory of the state of the FSM visiting a small subset of states in the state space.
The rest of this article is organized as follows. Section ‘System model and problem formulation’ describes the model of the network and defines the estimation problem. The sparse learning algorithm is presented in Section ‘Sparse estimation of cost functions’. Section ‘Structure of the graph’ proposes the decomposition of the overall graph into subchains and analyzes the properties of the transition probability matrix. Section ‘Perturbation analysis and performance bounds’ discusses the stability of sparse approximation applied in our context and characterizes the performance of the learning algorithm in terms of how the number of state observations grows with the network size. Numerical results are presented in Section ‘Numerical results’. Section ‘Conclusions’ concludes the article. The proof of the stated theorems are in Appendices Appendix 1 and Appendix 2.
System model and problem formulation
where E[·] denotes expectation and γ ∈ (0,1) is the discount factor, is the expected discounted longterm cost. This function is also known as the costtogo function and is central to DP and optimal control[10].
is the τstep transition from state s to s^{′}.^{c} Consider the graph associated with the FSM, where vertices are states in$\mathcal{S}$ and edges are statetransitions with nonzero probability. The temporal distance τ in the evolution of the FSM translates into some number of hops in the graphical representation. Starting from a vertex s,$\overline{c}\left(s\right)$ is computed by sequentially summing the discounted and weighed cost of the reachable vertices for an increasing number of hops in the graph. The representation as a graph of the temporal evolution of the network is the key for the sparse learning algorithm proposed herein.
In online learning, the function$\overline{c}$ is estimated from a samplepath of statecost observations. The samplepath O_{ T } of observations up to time T includes the sequence of states {S(0),S(1),…,S(T)} and state transition costs {c(S(0),S(1)),…,c(S(T−1),S(T))}. Denote by$\overline{\mathbf{c}}$ the vector collecting the expected longterm cost$\overline{c}\left(s\right)$ for all$s\in \mathcal{S}$, i.e.,$\overline{\mathbf{c}}=\left[c\right(1),c(2),\dots ,c(N\left)\right]$.^{d} The objective is to build an estimator of$\overline{\mathbf{c}}$ based on the observations O_{ T } minimizing a distance metric such as$\parallel {\widehat{\mathbf{c}}}_{T}\overline{\mathbf{c}}{\parallel}_{2}^{2}$, where${\widehat{\mathbf{c}}}_{T}$ is the estimate at time T.
The main challenge to achieve an accurate estimation of$\overline{\mathbf{c}}$ in wireless networks is the enormous size of the state space$\mathcal{S}$ underlying the associated FSM. In fact, in traditional online learning an accurate estimation of$\overline{\mathbf{c}}$ requires a samplepath of observations where each state in$\mathcal{S}$ is visited a considerable number of times. In fact, all the allowed state transitions need to be observed a sufficient number of times to estimate their probability.
Sparse estimation of cost functions
We now present an algorithm for the online learning of costtogo functions in wireless networks from the observation of a statecost trajectory of the associated FSM. The baseline observation is that networking protocols induce a structured behavior of the network, which is reflected in a structured graph associated with the FSM. Thus, every statecost observation conveys information about multiple states due to the correlated behavior of the network. As a result, we can propose an algorithm to estimate$\overline{\mathbf{c}}$ exploiting this structure from fewer observations than in traditional learning. The algorithm is composed of three elements:

observation: the transition probabilities and cost function c(·) are estimated by observing a statecost samplepath;

projection:$\overline{\mathbf{c}}$ is projected onto a graph wavelet basis set capturing typical structures in the graph;

sparse estimation of$\overline{\mathbf{c}}$: a sparse estimation algorithm is used to identify a concise set of basis functions providing the best fit with the estimated transition probabilities and cost function.
We define the N × N matrix P to be the probability transition matrix where P[s,s^{′}] = p(s,s^{′}) as in Equation (1).
where 1(·) is the indicator function. More refined estimators can be employed to reduce the sampling rate[25].
The estimates$\widehat{\mathbf{P}}\left(T\right)$ and$\widehat{\mathbf{c}}\left(T\right)$ may be very noisy and incomplete estimates of P and c even for T≫N. In fact, an accurate estimation of the transition probabilities from a state s in$\mathcal{S}$ and the cost function c(s) may require a considerable number of visits to s. For asymptotically large T, the average number of times the FSM is in state s is TΠ(s), where$\Pi \left(s\right)=\underset{t\to \infty}{lim}P\left(S\right(t)=sS\left(0\right)={s}_{0})$ is the steadystate probability of s. Note that the average steadystate probability of the states in$\mathcal{S}$ is 1/N, and thus the average number of visits to a state is T/N. However, in a finitelength samplepath, the trajectory of the state of the FSM may remain confined in a region of$\mathcal{S}$ even for lengths T larger than N and the number of states visited may be much smaller than T. Thus, due to the large size of the state space of FSMs modeling wireless networks, the number of observations needed to achieve an accurate estimation of the costtogo function is generally enormous, and it may be larger than the coherence time of the network, meaning that the statistics of the stochastic process modeling the operation of the network may change before the learning process achieves a meaningful estimate of$\overline{\mathbf{c}}$.
To cope with this issue we exploit the fact that FSMs modeling the operations of wireless networks and their associated graphs present a very regular connectivity structure and the transition probabilities are determined by a limited set of parameters, e.g., packet arrival probability in the buffer of the nodes and packet failure (see Section ‘Structure of the graph’). By regular, we mean that the connectivity structure from many nodes of the graph to their 1hop neighbors is similar. Thus, the representation of the graph provided by the transition matrix is intrinsically redundant and trajectories of the network on the graph can thus presumably be described by a small number of functions capturing typical substructures of the graph. We observe that these substructures involve neighborhoods of states at different numbers of hops, corresponding to different temporal distances between observations in the samplepath.
The symmetrization step is required as DWs presume symmetric diffusion operators. The design of wavelet functions tailored to the compression of directed graphs will further improve the performance of the algorithm proposed herein.
where$\widehat{\mathbf{B}}\left(T\right)=\mathbf{R}\left(T\right)(\mathbf{I}\gamma \widehat{\mathbf{P}}(T\left)\right)$ and R(T) is a random projection matrix. The ℓ1 regularization term λ∥x∥_{1}is a sparsitypromoting term, meaning that the least significant coefficients in x are pushed toward zero.
In the Compressed Sensing (CS) literature, the matrix$\widehat{\mathbf{B}}\left(T\right)$ and W in the above equation are generally referred to as the sensing and representation matrices, respectively. Note that the elements in the rows of$\widehat{\mathbf{P}}\left(T\right)$ and$\widehat{\mathbf{c}}\left(T\right)$ corresponding to states not visited in the trajectory O_{ T }are set to zero and can be eliminated in the projection.
Structure of the graph
Wireless networking protocols induce a very structured temporal evolution of the network, and, thus, a very structured graph associated with the FSM. This structure is the key to show some general properties of the transition matrix P that determines the performance of the sparse reconstruction in terms of the minimum number of states that needs to be included in Equation (11) to achieve an accurate reconstruction. Our analysis is based on the decomposition of the overall graph into smaller graphs, which we refer to as subchains. The good incoherence properties of the transition matrices associated with the subchains are reflected in good incoherence of the overall transition matrix and, thus, result in good performance of the sparse reconstruction.
The decomposition into subchains of the complex graph associated with the FSM modeling the temporal evolution of wireless networks results from the observation that the state of the network is the collection of many individual descriptors tracking counters and variables associated with the functioning of protocols and the environment. The temporal evolution of each individual descriptor follows simple rules that can be easily analyzed to retrieve properties of the overall graph. We then define S(t) = {S_{1}(t),…,S_{ D }(t)}, where S_{ d }(t) is the state of the d th subchain at time t. We denote by${\mathcal{S}}_{d}$ the state space of the d th subchain, with$\left{\mathcal{S}}_{d}\right={N}_{d}$.
The subchains track the evolution of the individual components of the state space. Although in the overall FSM the transition probabilities are a function of the overall state of the network, the connectivity structure of the subchains is preserved in the overall FSM. In fact, the state transition from$s=\{{s}_{1},\dots ,{s}_{D}\}\in \mathcal{S}$ to${s}^{\prime}=\{{s}_{1}^{\prime},\dots ,{s}_{D}^{\prime}\}\in \mathcal{S}$ in the overall FSM is allowed^{f} only if the state transition from s_{ d }to${s}_{d}^{\prime}$ is allowed in the corresponding subchain, for all d = 1,…,D. Thus, the properties of the connectivity structure of the subchains are inherited by the overall graph.
In stochastic models for wireless networks two classes of subchains can be identified:

Counterlike subchains (see, Figure2a,b: the FSM is associated with a counter. The value of the counter increments/decrements until it is reset to a given value. Examples of counterlike subchains are: the number of retransmissions of a packet in ARQ protocols, the backoff timers in DCF and the transmission windows and timers in TCP. This class can be further divided into forward counters (Figure2a and backward counters (Figure2b, depending on whether the counter is incremented or decremented until being reset to a predefined value;

Random walk subchains (see, Figure2c): the value of the descriptor variable is subject to random, but constrained, increments and decrements. Examples of random walk subchains are channel state descriptors and variables tracking the number of packets in a buffer.
For instance, in the pioneering work[1], the Markov chain used to analyze the network is the composition of a random walk and a counterlike subchain. It can be observed that counterlike and random walk subchains present a very local and regular connectivity structure. By local, we mean that every state connects to a small neighborhood of states. Regularity implies that states connect to 1hop neighbors in a similar fashion. For instance, in counterlike subchains, states connect to the state corresponding to a reset counter and the state associated with an incremented or decremented value (possibly plus a selfloop). As a result, the overall graph is regular and local. This property is instrumental towards having an efficient compression in the wavelet domain, meaning that only a limited number of notable substructures is needed to model the temporal evolution of the state of the network.
where i_{ d }, j_{ d }, and k_{ d } are the state of the d th subchain in the states associated with states i, j, and k, respectively.
We remark that the value of the transition probability p_{ d }(i_{ d },j_{ d }) is the inverse of the number of outgoing links, i.e., allowed transitions, from i_{ d }. The average inner product${E}_{{i}_{d},{j}_{d}}\left[\sum _{{k}_{d}=1}^{{N}_{d}}{p}_{d}({k}_{d},{i}_{d}){p}_{d}({k}_{d},{j}_{d})\right]$ is calculated by sequentially considering all the columns indexed by i_{ d }= 1,…,N_{ d }−1 and computing the product with the columns indexed by j_{ d }> i_{ d }.
The average inner products of the subchains decrease on average with the number of states of the associated FSM. Although in the general case the probability of transition from a state to its neighbors may be much different from that provided by the natural random walk associated with the graph structure, the locality and regularity of the structure of the subchains cause the average overlap of the sets${\mathcal{S}}_{d}^{\leftarrow}\left({j}_{d}\right)$ to vanish as N_{ d }increases. Thus, sufficiently large subchains are associated with incoherent transition matrices.
where C is a positive constant smaller than 1. We observe that each of these means can be expressed as$O\left(\frac{1}{{N}_{d}}\right)+{\alpha}_{d}$, where α_{ d }= 1/4,2/3,1/(2ℓ + 1) are for the backward counter, forward counter and random walk, respectively. The fact that α_{ d }< 1 for all the subchains is critical for the performance analysis of the CS approach.
Perturbation analysis and performance bounds
In this section, we characterize the performance of the sparse approximation of the costtogo function proposed herein. We first discuss the stability of the solution of (11) and then determine how much compression is possible to ensure good reconstruction of the value function v. The number of observations required for good reconstruction directly translates to the learning rate of our proposed algorithm. An exact analysis of the transition matrix is challenging; however, by exploiting the average behavior of several key structures, we can determine the relationship between the minimum number of observations for this compressed sensing problem and the size of the logical graph.
Perturbation analysis
We discuss in this section how estimation noise in the sensing matrix$\widehat{\mathbf{B}}\left(T\right)=(\mathbf{I}\gamma \widehat{\mathbf{P}}(T\left)\right)$ affects the stability of the reconstruction provided by (11) as new states are visited by the samplepath. We show that the inclusion in (11) of a row in$\widehat{\mathbf{c}}\left(T\right)$ and$\widehat{\mathbf{P}}\left(T\right)$ associated with a state hit by the FSM a small number of times may result in a dramatic change of x^{∗}(T) with respect to x^{∗}(T−1). As we are looking for the fixed point solution of the operator Ω, instability and large variations of the sparse solution are highly undesirable. In order to improve the stability of the algorithm, in the numerical results presented in Section ‘Numerical results’ we use the LS CS algorithm proposed in[15].
where$\mathrm{\Delta}\widehat{\mathbf{A}}\left(T\right)=\left[{\widehat{\mathbf{a}}}_{1}\right(T),{\widehat{\mathbf{a}}}_{2}(T),\dots ,{\widehat{\mathbf{a}}}_{m}(T\left)\right]$,$\mathcal{U}$ is the set of perturbation matrices$\mathcal{U}=\left\{\mathrm{\Delta}\widehat{\mathbf{A}}\right(T):\parallel {\widehat{\mathbf{a}}}_{i}(T){\parallel}_{2}\le \lambda ,i=1,\dots ,m\}$ and m is the number of columns of$\mathrm{\Delta}\widehat{\mathbf{A}}\left(T\right)$. We drop the dependence on T of the estimated matrices and vectors. Thus, the vector x is optimized for a worst case perturbation whose range is determined by the parameter λ, that is, the algorithm is robust to perturbations. Interestingly, the larger the value of λ, the sparser the output vector x, but also the larger the set of perturbations considered. In[27], it was also shown that LASSO is not stable, meaning that small variations of the sensing matrix$\widehat{\mathbf{A}}$ may lead to significantly different output vectors.
The following addresses the instability issues in the solution to the RR problem. In particular, the theorem shows that the inclusion of a new sample may result in suboptimal solutions to the RR problem. Moreover, due to the equivalence between LASSO and the RR problem, the same instability result applies to LASSO as well.
Theorem 1
where${\mathrm{\Gamma}}_{k}({\widehat{\mathbf{A}}}^{\prime},{I}_{{\mathbf{x}}^{\ast}})={({\widehat{\mathbf{A}}}_{k}^{\prime})}^{T}\mathbf{M}{({\mathbf{M}}^{T}\mathbf{M})}^{1}\mathbf{M}{\widehat{\mathbf{A}}}_{k}^{\prime}$, and$\mathbf{M}=[{\widehat{\mathbf{A}}}_{{\mathcal{I}}_{{\mathbf{x}}^{\ast}}\setminus \{k\}}^{\prime},\mathbf{c}]$ then${I}_{{\mathbf{y}}^{\ast}}\ne {I}_{{\mathbf{x}}^{\ast}}$. Thus, the support of the solution of LASSO changes if a new state meeting the hypothesis is added to the Bellman residual.
The proof of the theorem is in Appendix 1.
Minimum number of observations
where R is a random matrix to be defined in the sequel and R(T) is the submatrix formed by retaining the columns of R indexed by states hit in the observation interval T. If the matrix R(T)(I−γ P)W satisfies the socalled restricted isometry property, defined below, then the squared error between x and$\widehat{\mathbf{x}}$ achieved by the Dantzig selector[28] can be bounded; in particular, there are guarantees on correctly estimating the locations of the nonzero components of x and thus the squared error is proportional to the number of nonzero components and the noise variance. Comparable analysis can be made for LASSO[29, 30]. We focus on properties of R(T)(I−γ P) recognizing that projecting onto an orthonormal W would be an isometric operation. We note that a negative result regarding RIP would call the use of our approach into question. However, a positive RIP result suggests that our method work. Analysis of LS CS also relies on RIP parameters.
Our proof exploits arguments from[31] with appropriate tailoring to our framework. We begin with the definition of the properties we wish to show.
Definition 1
holds for all$x\in {\mathbb{R}}^{N}$ having no more than S nonzero entries. Note that B is a K × N matrix. RIP implies that B is approximately an isometry for Ssparse signals.
We have the following result,
Theorem 2
if${K}^{2}\ge \frac{192\phantom{\rule{.3em}{0ex}}log\phantom{\rule{.3em}{0ex}}n{S}^{2}}{{\delta}_{S}^{2}64{c}_{1}}$ and${c}_{1}\ge \frac{{\delta}_{S}^{2}}{64}$. The matrix P is the transition probability matrix for a Markov chain formed by concatenating forward and backward counterlike and random walk subchains and γ < 1. The proof of Theorem 2 can be found in Appendix 2.
We observe that this result states that if the number of observations K is of order$O\left({S}^{2}\sqrt{n\phantom{\rule{.3em}{0ex}}log\phantom{\rule{.3em}{0ex}}n}\right)$ then RIP is satisfied with high probability as the network grows large. We contrast this with the more typical results seen in say channel estimation problems where the order is O(S^{2} log n). We remark that our result on the RIP is not limited to LASSO, but leads to the more general conclusion that sparse estimation algorithms can be used to approximate costtogo functions of wireless networks. Furthermore, the proof of Theorem 2 shows that an arbitrary concatenation of subchains does not affect the RIP property in the limit of large wireless networks.
Numerical results
In this section, we present numerical results for an example of a wireless network to demonstrate the potential of the compressed sensing approach. We consider a wireless network where terminals store packets in a finite buffer of size Q and employ Automatic Retransmission reQuest (ARQ) to improve the delivery rate of packets. Time is divided in slots of fixed duration. For the sake of simplicity we assume that the transmission of a packet occurs in the duration of a time slot and that channel coefficients in the various slots are i.i.d.. Terminals with a nonempty buffer access the channel in a time slot with fixed probability equal to α. The failure probability of a packet transmitted by a terminal is a function of the set of terminals concurrently transmitting in the same slot. Packet arrival in the buffer of the terminals is modeled according to a Poisson process of intensity σ.
The cost function c measures the normalized cost in terms of throughput loss with respect to the saturation throughput achieved by the terminals in the absence of interference. In particular, the cost function is defined as the sum for all the terminals of one minus the failure probability of the transmitted packets. Idleness is assigned a cost equal to 1.
For Q = 5 and F = 4 and 2 terminals the size of the state space is 1681. The transition matrix P is used to compute P_{ symm } defined in Equation (9) and the associated set of DW functions W[18]. DW basis sets are overcomplete. In order to keep complexity low, the columns of W are subsampled. In particular, we select 400 wavelet functions at different time scales.
The proposed algorithm achieves a considerable accuracy in the estimation of$\overline{\mathbf{c}}$ after a very short number of statecost observations, whereas standard learning converges slowly to$\overline{\mathbf{c}}$. Moreover, the solution is extremely stable and the LS CSbased algorithm appears to be very robust to estimation noise.
Conclusions
A novel framework for the online estimation of costtogo functions in wireless networks was proposed. We showed that the inherent regular and local structure of the graph associated with the FSM modeling the operations of wireless networks enables the sparse representation of costtogo functions. Our analysis, based on the decomposition of the overall graph in fundamental smaller structures, connects the structure of the FSM to the RIP of the transition probability matrix. Numerical results show that sparse approximation and projection onto DW basis sets enable a considerable reduction in the number of observations needed to estimate costtogo functions in wireless networks, and have the potential to make online learning practical in this context.
Endnotes
^{a}The incoherence of the transition matrix is connected to the magnitude of the inner products of its columns.
^{b}Note that c(s,s^{′}) can be generalized to be a random variable. In this case the expectation is over all the possible values of c(s,s^{′}).
^{c}Control can be included in the model by defining statistics and cost functions conditioned on a control action.
^{d}The indexing in the vector is based on a univocal map between$\mathcal{S}$ and {1,2,3,…,N}.
^{e}Note that this assumption does not reduce the applicability of the proposed algorithm. In fact, the connectivity structure of the transition matrix is determined by standard protocols that are shared and known by all the nodes.
^{f}By allowed, we means that the state transition has probability equal to zero for any set of parameters.
^{g}We note that numerical evaluations of incoherence for many typical Markov chains has revealed that incoherence holds on average.
^{h}In this analysis we assume that W is an orthonormal set of basis functions. We are aware that DW are overcomplete and, thus, W is not an orthonormal basis set. The design of orthonormal wavelet basis tailored to FSMs modeling wireless networks is an important research direction.
Appendix 1
Proof of Theorem 1
Therefore, if the conditions (34) and (35) hold, then due to Theorem 5 in[27], we have y k∗ = 0 and${I}_{{\mathbf{x}}^{\ast}}\ne {I}_{{\mathbf{y}}^{\ast}}$.
Appendix 2
Proof of Theorem 2
In this appendix, we prove the result on the minimum number of observed states needed for perfect reconstruction of$\overline{\mathbf{c}}$. We first state the following lemma:
Lemma 1
For the sake of exposition we assume that R(T) is an K × n matrix whose components are drawn i.i.d. from a binary distribution i.e.${\mathbf{R}}_{\mathit{\text{ik}}}=\pm \sqrt{\frac{1}{K}}$ with probability$\frac{1}{2}$; thus we have that the R(T)_{ ik } are zero mean and$\mathbf{E}\left[{\mathbf{R}}_{H}^{T}\mathbf{R}(T)\right]=\mathbf{I}$. Other properly constrained distributions for R(T) can be handled. The other matrices specified in (50) are square and of dimension n × n.
where b_{ i } is the i th column of B and$\mathbf{B}\doteq \mathbf{I}\gamma \mathbf{P}$. The equality (a) follows from the independence of the probability transition matrix P and the random projection matrix R. Equations (52) and (54) follow from Equations (13) and (14)–(16), (18)–(20). The next needed inequality is,
Lemma 2
Equation (60) can be manipulated to yield the following relationship between the number of samples K and the size of the logical network, n: the RIP holds with high probability if${K}^{2}\ge \frac{192\phantom{\rule{.3em}{0ex}}log\phantom{\rule{.3em}{0ex}}n{S}^{2}}{{\delta}_{S}^{2}64{c}_{1}}$, where c_{1} is constant selected to ensure that the denominator of the previous expression is positive and the Theorem is shown.
Declarations
Acknowledgements
The study was supported by AFOSR under grants FA9550080480 and FA95501210215 and by the National Science Foundation (NSF) under Grant CCF0917343.
Authors’ Affiliations
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