Student Member, IEEE Jin Danqi

Senior Member, IEEE Jie Chen

Senior Member, IEEE Jingdong Chen

Fellow, IEEE Ali H Sayed

Convex Combination of Diffusion Strategies over Networks

Keywords: Distributed optimization, diffusion strategy, convex combination, adaptive fusion strategy, performance analysis

Combining diffusion strategies with complementary properties enables enhanced performance when they can be run simultaneously. In this paper, we propose two convex combination schemes, the power-normalized one and the sign-regressor one. Without loss of generality, theoretical investigations are focused on the former. An analysis of universality shows that it cannot perform worse than any of its component strategies in terms of the excess mean-square-error (EMSE) at steady-state. A theoretical analysis of stability also reveals that it is more stable than affine combination schemes. Next, several adjustments are proposed to further improve the performance of convex combination schemes. Finally, simulation results are presented to demonstrate their effectiveness as well as the accuracy of the theoretical results.

I. INTRODUCTION

Model and parameter selection problems are ubiquitous and challenging in signal processing and machine learning. In most situations, selecting an optimal model structure is a difficult task and requires deep knowledge of the problem domain. Instead, one can resort to training a collection of models and combining them in a manner that enhances performance. Combination strategies have already been successfully considered for traditional adaptive filters [START_REF] Arenas-Garcia | Mean-square performance of a convex combination of two adaptive filters[END_REF], [START_REF] Arenas-Garcia | Plant identification via adaptive combination of transversal filters[END_REF], in multi-kernel learning [START_REF] Alain | Simple MKL[END_REF], as well as with deep neural network structures [START_REF] Szegedy | Going deeper with convolutions[END_REF].

An inspection of the existing literature on diffusion strategies reveals that they include diffusion LMS [START_REF] Lopes | Diffusion least-mean squares over adaptive networks: Formulation and performance analysis[END_REF], [START_REF] Cattivelli | Diffusion LMS strategies for distributed estimation[END_REF], diffusion APA [START_REF] Li | Distributed adaptive estimation based on the APA algorithm over diffusion networks with changing topology[END_REF], diffusion Kalman filter [START_REF] Cattivelli | Diffusion strategies for distributed Kalman filtering and smoothing[END_REF], diffusion RLS [START_REF] Cattivelli | Diffusion recursive leastsquares for distributed estimation over adaptive networks[END_REF], and others [START_REF] Liu | Diffusion sparse least-mean squares over networks[END_REF]- [START_REF] Nassif | Proximal multitask learning over networks with sparsity-inducing coregularization[END_REF], in addition to multi-task learning counterparts [START_REF] Chen | Multitask diffusion adaptation over networks[END_REF]- [START_REF] Chen | Chapter # -Multitask learning over adaptive networks with grouping strategies[END_REF]. These different strategies perform well in the conditions under which they were derived; some deliver better performance than others depending on the underlying model for the data. In this paper, we show how to take advantage of multiple schemes by combining them in a way that can lead to enhanced performance. We introduce several combination strategies and study how performance improvements occur.

Convex and affine combinations are two useful schemes for fusing adaptive schemes with different adaptation gains [START_REF] Martinez-Ramon | An adaptive combination of adaptive filters for plant identification[END_REF]- [START_REF] Arenas-Garcia | Combinations of adaptive filters: Performance and convergence properties[END_REF] or complementary capabilities [START_REF] Das | Sparse adaptive filtering by an adaptive convex combination of the LMS and the ZA-LMS algorithms[END_REF]- [START_REF] Das | A block-based convex combination of NLMS and ZA-NLMS for identifying sparse systems with variable sparsity[END_REF]. In [START_REF] Jin | Affine combination of diffusion strategies over networks[END_REF], we have thoroughly studied the affine combination of diffusion strategies over networks. Its universality at steady-state was established, and its stochastic behavior was analyzed in the mean and mean-square sense. Though affine combination can provide good performance, convex combination scheme is often preferred for stand-alone adaptive filters since it has a A preliminary version of this work appeared in [START_REF] Jin | Convex combination of diffusion strategies over distributed networks[END_REF]. wider stability range [START_REF] Arenas-Garcia | Combinations of adaptive filters: Performance and convergence properties[END_REF]. The aim of this paper is to determine if these conclusions still hold for convex combinations of diffusion strategies compared to affine combinations ones. This question is particularly challenging in the context of adaptive networks due to the multiple agents that can interact. The contributions of this work are summarized as follows:

1) Convex combinations of diffusion strategies are introduced to address the model and parameter selection problem within the context of distributed estimation over adaptive networks. In particular, two convex combiners are considered, the power-normalized one and the signregressor one. 2) A theoretical analysis of the convex power-normalized combiner is conducted to illustrate universality at steady state, and to derive conditions that ensure the mean and mean-square stabilities of this scheme. Theoretical results reveal that, compared to affine combination, convex power-normalized combiners are more stable. 3) Extensions are discussed to further improve the performance of convex combination schemes. 4) The computation and communication complexities of convex combiners are discussed, and a comparison of convex and affine combination schemes is provided.

Though the convex combination algorithm shares a similar form with the affine combination strategy studied in [START_REF] Jin | Affine combination of diffusion strategies over networks[END_REF], the universality and behavior analyses require original manipulations. Similar steps are voluntarily omitted in the presentation to avoid redundancy and focus on the main differences. The result shows that, as long as each component strategy is stable, any diffusion network with convex power-normalized combination scheme will be stable. This differs from [START_REF] Jin | Affine combination of diffusion strategies over networks[END_REF] where additional conditions are required to ensure the stability of affine combiners.

The paper is organized as follows. Signal model and diffusion LMS algorithm are presented in Section II. Section III presents the convex combination framework and introduces two strategies to adapt the convex combination coefficients. Section IV analyzes the theoretical performance of the convex power-normalized combiner with two diffusion LMS. Extensions of this scheme are discussed to further improve the performance in Section V. Discussions about the computation and communication complexities are provided in Section VI, as well as a comparison with the affine combination scheme. Simulation results are provided in Section VII. Section VIII concludes this work.

Notation. Normal font x denotes scalars. Boldface lower-cased letters x and capital letters X denote column vectors and matrices, respectively. The superscript (•)

> denotes the transpose operator. The inverse of a matrix is denoted by (•) 1 . The mathematical expectation is denoted by E{•}. The operator [ • ]| b a truncates its argument with lower bound a and upper bound b. The operator diag{•} takes the diagonal elements of its matrix argument, or generates a diagonal matrix from its vector argument. I N and 0 N denote identity matrix and zero matrix of size N ⇥ N , respectively. All-one vector of length N is denoted by N . N k denotes the neighbors of node k, including k.

II. SIGNAL MODEL AND DIFFUSION LMS

Consider a connected network consisting of N nodes. The problem is to estimate an unknown parameter vector w ? k of length L ⇥ 1 at each agent k. Agent k has access to temporal measurement sequences {d k,n , u k,n }, where d k,n denotes a reference signal, and u k,n is an L ⇥ 1 regression vector with positive definite covariance matrix R u,k . The data at agent k and time instant n are related according to the linear model:

d k,n = u > k,n w ? k + z k,n , (1)
where z k,n is an additive noise satisfying Assumption 1 below. Note that the network operates in the so-called multi-task setting when the unknown parameter vectors w ? k differ from each other. The single-task setting usually considered in the literature is a special case of the multi-task one considered in this paper. It is obtained by setting w ? 1 = w ? 2 = • • • = w ? N . Assumption 1: z k,n is a zero-mean, stationary, independent and identically distributed (i.i.d.) additive noise with variance 2 z,k , and independent of any other signals. A1 1 is widely adopted in the literature of adaptive filters and distributed online learning over networks. To determine w ?

k , we consider the MSE cost at agent k:

J k (w) = E |d k,n u > k,n w| 2 . (2)
Clearly, J k (w) is minimized at w ? k . For single-task problems, each agent in the network estimates the same parameter vector, while for multi-task problems, agents may estimate distinct parameter vectors.

Diffusion LMS was derived in [START_REF] Lopes | Diffusion least-mean squares over adaptive networks: Formulation and performance analysis[END_REF], [START_REF] Cattivelli | Diffusion LMS strategies for distributed estimation[END_REF], [START_REF] Sayed | Diffusion adaptation over networks[END_REF], [START_REF] Sayed | Adaptive networks[END_REF] to minimize the global cost defined by:

J glob (w) = N X k=1 J k (w) (3)
in a cooperative manner. The general diffusion LMS algorithm is given by:

8 > > > > > > > < > > > > > > > : k,n = X `2N k a 1,`k w `,n k,n+1 = k,n + µ k X `2N k c `k u `,n (d `,n u > `,n k,n ) w k,n+1 = X `2N k a 2,`k `,n+1 (4)
1 In this paper, we adopt the acronym 'A' for 'Assumption'.

where the nonnegative coefficients {a 1,`k }, {a 2,`k } and {c `k} are (`, k)-th entries of two left stochastic matrices A 1 , A 2 and a right stochastic matrix C, respectively, satisfying:

A > 1 N = N , A > 2 N = N , C N = N , (5)
a 1,`k = 0, a 2,`k = 0, c `k = 0 if `/ 2 N k . (6)
Setting A 1 = I or A 2 = I leads to the adapt-then-combine (ATC) and the combine-then-adapt (CTA) diffusion strategy, respectively. Note that algorithm (4) can be used to solve both single-task problems [START_REF] Sayed | Adaptation, Learning, and Optimization over Networks[END_REF] and multi-task problems [START_REF] Chen | Diffusion LMS over multitask networks[END_REF] by selecting appropriate matrices A 1 and A 2 .

III. CONVEX COMBINATION FRAMEWORK

As illustrated in Fig. 1, the convex combination framework consists of two concurrent layers, namely, a diffusion strategy layer and a combination layer. In the diffusion strategy layer, the network simultaneously runs M candidate diffusion strategies, resulting in M groups of estimates for the optimal weight vector. We shall consider, without lack of generality, the case M = 2 in the rest of the paper.

Diffusion strategy S (i) parameters are

A (i) 1 , A (i) 2 , C (i) , µ (i) k for i = 1, 2,
where the superscript (i) is the indicator for the i-th component strategy. Based on the signal model (1), we define the estimates of the reference signal, the a priori output estimation error, and the a priori estimation error, as follows:

y (i) k,n , u > k,n w (i) k,n (7) e (i)
k,n , d k,n u > k,n w (i) k,n (8)
ẽ(i) k,n , u > k,n (w ? k w (i) k,n ). (9)
respectively, at each agent k and time instant n. The input of the combination layer consists of the two outputs from the diffusion strategy layer. By assigning convex combination coefficients:

(1)

k,n , k,n (10)
(2)

k,n , 1 k,n (11)
to the two component strategies at agent k and time instant n, we obtain the overall weight estimation w k,n at the combination layer for agent k:

w k,n = 2 X i=1 (i) k,n w (i) k,n = k,n w (1) k,n + (1 k,n )w (2) k,n . (12)
Convex constraint on

(i) k,n requires that (i) k,n 2 [0, 1]
. By using model [START_REF] Jin | Convex combination of diffusion strategies over distributed networks[END_REF] and equations ( 7)- [START_REF] Vlaski | Diffusion stochastic optimization with non-smooth regularizers[END_REF], we obtain the following relation between the overall quantities at the combination layer and the corresponding quantities at the diffusion strategy layer, at each node k and time instant n:

x k,n = 2 X i=1 (i) k,n x (i) k,n (13)
where the quantities x of the reference signal in [START_REF] Cattivelli | Diffusion LMS strategies for distributed estimation[END_REF], the error signals in [START_REF] Li | Distributed adaptive estimation based on the APA algorithm over diffusion networks with changing topology[END_REF], and the a prior estimation error in [START_REF] Cattivelli | Diffusion strategies for distributed Kalman filtering and smoothing[END_REF], respectively.

The goal of the combination layer is to learn which diffusion strategy performs better at each time instant and each agent, and to assign them with weights in order to optimize the overall network performance. The problem then reduces to designing a strategy to adjust k,n . To avoid using hard-thresholding operators to satisfy the convex combination requirement, we introduce an auxiliary variable ↵ k,n to reparameterize k,n as follows:

(1)

k,n = k,n = 1 1 + e ↵ k,n (14)
(2)

k,n = 1 k,n . (15)
Then, we adjust ↵ k,n by minimizing the MSE at the combination layer, which is defined by:

J MSE n = 1 2 N X k=1 E e 2 k,n . (16)
Adaptation of ↵ k,n can be conducted by performing stochastic gradient descent on [START_REF] Chen | Multitask diffusion adaptation over networks with common latent representations[END_REF], that is,

↵ k,n+1 = ↵ k,n v 0 ↵ k @J MSE n @↵ k,n ⇡ ↵ k,n + v 0 ↵ k k,n (1 k,n )e k,n u > k,n (w (1) k,n w (2) k,n ) (17)
with v 0 ↵ k a positive step-size. On the one hand, observe that iteration [START_REF] Zhao | Distributed clustering and learning over networks[END_REF] will stop if ↵ k,n is allowed to have unbounded growth or decline, since k,n will get close to 0 or 1 and will make the term k,n (1 k,n ) being 0. Then, to ensure continuous learning, we propose restricting ↵ k,n to be within the interval [ ↵ + , ↵ + ] [START_REF] Arenas-Garcia | Combinations of adaptive filters: Performance and convergence properties[END_REF]. On the other hand, in order to compensate the effect of large fluctuations in the power of [u > k,n (w

(1) k,n w (2)
k,n )] at the adaptation level, we propose normalizing the step-sizes v 0 ↵ k in the following two ways.

1) Convex power-normalized scheme:

Setting v 0 ↵ k = v↵ k "+p k,n
with v ↵ k the initial step-size, we obtain the convex powernormalized scheme. In this expression, parameter " is a small positive constant to avoid dividing by zero, and p k,n is an estimate of the power of u > k,n (w

(1) k,n w (2)
k,n ) calculated as:

p k,n = ⌘ p k,n 1 + (1 ⌘) ⇥ u > k,n (w (1) k,n w (2) k,n ) ⇤2 , (18)
with 0 ⌧ ⌘ < 1 a temporal smoothing factor.

2) Convex sign-regressor scheme: To save computation and storage resources in evaluating p k,n in the power-normalized scheme, we introduce another normalization. Setting

v 0 ↵ k = v ↵ k |u > k,n (w (1) k,n w (2)
k,n )| results in the convex sign-regressor scheme given by:

↵ k,n+1 = ↵ k,n + v ↵ k k,n (1 k,n )e k,n sgn{u > k,n (w (1) k,n w (2) k,n )} (19)
where sgn{x} is the sign function. Compared with the powernormalized scheme, [START_REF] Gogineni | Improving the performance of multitask diffusion APA via controlled inter-cluster cooperation[END_REF] requires only evaluating the sign function.

IV. THEORETICAL ANALYSIS OF CONVEX POWER-NORMALIZED SCHEME

Due to space limitation and technical complexity, we shall only conduct the theoretical analysis for the power-normalized scheme. The theoretical analysis of the sign-regressor scheme can be derived by following the same routine. However, other techniques for dealing with the nonlinear sign function may be required.

To facilitate the theoretical analysis, we shall now introduce several assumptions and approximations as in [START_REF] Jin | Affine combination of diffusion strategies over networks[END_REF]. Assumption 2: The regression vector u k,n , generated from a zero-mean random process, is temporally stationary, white (over n) and spatially independent (over k) with positive definite covariance matrix R u,k = E{u k,n u > k,n }. Approximation 1: At steady state, k,n is statistically independent of ẽ(i) k,n and p k,n . Approximation 2: For a large enough temporal smoothing factor ⌘, p k,n is statistically independent of u > k,n w

(i) k,n , that is, it is independent of ẽ(i) k,n . Approximation 3: At each time instant n, k,n is statistically independent of w (i) k,n for i = 1, 2. Approximation 4: n+1 is statistically independent of B (i) n , v (i) n , g (i)
n and r (i) n for i = 1, 2, where n+1 is defined further ahead in [START_REF] Das | A block-based convex combination of NLMS and ZA-NLMS for identifying sparse systems with variable sparsity[END_REF], v (i) n is defined in [START_REF] Sayed | Adaptive networks[END_REF] are jointly Gaussian distributed, which implies [START_REF] Papoulis | Probability, Random Variables, and Stochastic Processes[END_REF]:

E (e e (i) k,n ) 4 = 3 ⇥ J (i) ex,k,n ⇤2 , i = 1, 2, (20)
E (e e

(1)

k,n ) 3 (e e (2)
k,n ) 1 = 3 J (1) ex,k,n J (1,2) ex,k,n , (21)
E (e e

(1)

k,n ) 1 (e e (2)
k,n ) 3 = 3 J (1,2) ex,k,n J (2) ex,k,n , (22)
E (e e

(1)

k,n ) 2 (e e (2)
k,n ) 2 = 2 ⇥ J (1,2) ex,k,n ⇤2 + J (1) ex,k,n J (2) ex,k,n , (23)
where

J (i) ex,k,n and J (1,2)
ex,k,n are defined further in [START_REF] Nassif | Diffusion LMS for multitask problems with local linear equality constraints[END_REF] and ( 26). Approximation 7: At steady state when n ! 1, the variance of k,n is small.

Although not true in general, these assumptions and approximations are usually adopted to simplify the derivation without constraining the conclusions. Specifically, there are several results in the literature showing that performance results obtained under A2 match well with actual performance when step-sizes are sufficiently small [START_REF] Sayed | Adaptation, Learning, and Optimization over Networks[END_REF], [START_REF] Sayed | Adaptive Filters[END_REF]. As discussed in [START_REF] Arenas-Garcia | Mean-square performance of a convex combination of two adaptive filters[END_REF], approximation Ap1 2 is reasonable when adopting a decaying step-size v ↵ k , and is more justified when ↵ k,n+1 approaches ↵ + or ↵ + , in which case k,n (1 k,n ) of ( 17) tends to zero. Ap2 is justified when using a large temporal smoothing factor ⌘. Though actually not hold, Ap3 and Ap4 do not affect the theoretical results heavily, as illustrated in simulation results. Ap5 is widely adopted in the analysis of convex combinations of filters [START_REF] Nascimento | A transient analysis for the convex combination of adaptive filters[END_REF], [START_REF] Silva | A transient analysis for the convex combination of two adaptive filters with transfer of coefficients[END_REF], and it coincides with simulation result that ↵ k,n converges slowly compared to the variation of u k,n , thus to the variation of ẽ(i) k,n . Ap6 is also adopted and examined in the analysis of combinations of adaptive filters [START_REF] Candido | Transient and steady-state analysis of the affine combination of two adaptive filters[END_REF], [START_REF] Nascimento | A transient analysis for the convex combination of adaptive filters[END_REF]. Ap7 has been adopted in the analysis of adaptive filters [START_REF] Arenas-Garcia | Mean-square performance of a convex combination of two adaptive filters[END_REF], and it is reasonable in that at steady state, k,n will converge to a fixed value or fluctuate within a small neighborhood of this value. Thus it is reasonable to assume that the variance of k,n is small. We shall challenge these assumptions and approximations in the simulations.

A. Universality at steady state

The EMSE of component strategy S (i) and that after combination at node k and time instant n are defined by:

J (i) ex,k,n , E{(ẽ (i) k,n ) 2 } (24) J ex,k,n , E [ k,n ẽ(1) k,n + (1 k,n )ẽ (2) k,n ] 2 , (25)
respectively. Let us also introduce the cross-EMSE defined as:

J (1,2)
ex,k,n , E{(ẽ

(1) k,n ẽ(2) k,n } (26)
2 In this paper, we adopt the acronym 'Ap' for 'Approximation'.

The EMSEs of the entire network, for component strategy S (i) and after combination, are respectively defined by:

J (i) ex,net,n , N X k=1 J (i) ex,k,n for i = 1, 2 (27)
J ex,net,n , N X k=1 J ex,k,n . (28)
By taking the limit as n ! 1, we obtain the corresponding values at steady state:

J (i) ex,k,1 , J ex,k,1 , J (i) ex,
ex,net,1 }.

Further, when

J (i) ex,k,1 > J (1,2) ex,k,1 for i = 1, 2 and ¯ k,1 defined in (65) satisfies ¯ k,1 2 1 ✓ + k , ✓ + k
for some node k, there is an improvement in the EMSE after combination, such that:

J ex,net,1 < min{J (1) ex,net,1 , J (2) ex,net,1 }. (30)
Proof: See Appendix A. ⌅ Theorem 1 is meaningful in that it shows that, through local combination at each node, the convex power-normalized scheme reaches the minimal EMSE of the two component strategies, and can even do better.

B. Mean weight and mean-square behaviors analysis

The mean weight and mean-square behavior analysis follows a similar routine as in [START_REF] Jin | Affine combination of diffusion strategies over networks[END_REF] for affine combination schemes. To avoid redundancy, we provide the main results without proving them to better highlight the main differences.

Define the following block vectors:

w ? , col{w ? 1 , • • • , w ? N } (31)
w (i) n , col{w (i) 1,n , • • • , w (i) N,n } (32)
w n , col{w 1,n , • • • , w N,n }, (33)
where w ? is the block optimum weight vector, w (i) n and w n are the block weight estimate of component strategy S (i) and after combination at time instant n, respectively. Using definitions ( 12) and ( 32), [START_REF] Das | A convex combination of NLMS and ZA-NLMS for identifying systems with variable sparsity[END_REF], we arrive at:

w n = n w (1) n + (I NL n )w (2) n , (34)
where

n , diag{ 1,n , • • • , N,n } ⌦ I L (35)
with symbol ⌦ denoting the Kronecker product. The weight error vector of node k for component strategy S (i) and that for combination layer are defined by:

v (i) k,n , w (i) k,n w ? k (36) v k,n , w k,n w ? k , (37)
respectively. By stacking v

(i)
k,n and v k,n over the entire network into block vectors, we have:

v (i) n , col{v (i) 1,n , • • • , v (i) N,n } (38) v n , col{v 1,n , • • • , v N,n }. (39)
Using ( 31)-( 39), we have:

v n = n v (1) n + (I NL n )v (2) n . (40)
1) Mean weight behavior analysis Under approximation Ap3 and using [START_REF] Papoulis | Probability, Random Variables, and Stochastic Processes[END_REF], the mean weight behavior at the combination layer satisfies:

E{v n+1 } = E n+1 E{v (1) n+1 } + E I NL n+1 E{v (2) n+1 }. (41)
We need to evaluate the iteration of E v

(i)
n+1 to analyze the mean weight behavior of v n+1 ; see Appendix B. Based on Appendix B, we obtain the following result.

Theorem 2 (Stability in the mean): Assume data model (1), assumptions A1, A2 and approximation Ap3 hold. Then for any initial conditions, the network with power-normalized scheme [START_REF] Zhao | Distributed clustering and learning over networks[END_REF] asymptotically converges in the mean if the step sizes in the network are chosen to satisfy:

0 < µ (i) k < 2 max {R (i) k } , k = 1, • • • , N and i = 1, 2 (42)
with max {•} denoting the largest eigenvalue of its matrix argument. The asymptotic bias at steady state is given by: 2) . (43)

E v 1 = E 1 I NL B (1) 1 r (1) (I NL E 1 ) I NL B (2) 1 r (
Proof: On the basis of Appendix B, arguments run along the lines of [START_REF] Jin | Affine combination of diffusion strategies over networks[END_REF]Appendix C] where affine combinations of two diffusion LMS are considered. ⌅ Unlike the proof in [START_REF] Jin | Affine combination of diffusion strategies over networks[END_REF] where we need to impose an additional condition on the step-size v ↵ k of the combination layer to ensure that matrix E n+1 is bounded, we do not have such condition in the current work for convex powernormalized scheme. In that way, convex combination schemes are more stable than affine ones in the mean sense.

2) Mean-square behavior analysis

The concept of mean-square stability is one of the most attractive ones within the large branch of stability analysis, and is widely adopted in the analyses of adaptive filters [START_REF] Sayed | Adaptive Filters[END_REF], [START_REF] Widrow | Adaptive Signal Processing[END_REF], [START_REF] Haykin | Adaptive Filter Theory[END_REF] and adaptive networks [START_REF] Sayed | Diffusion adaptation over networks[END_REF]- [START_REF] Sayed | Adaptation, Learning, and Optimization over Networks[END_REF]. Indeed, requiring only mean stability for adaptive filters and adaptive networks is not fully satisfactory since a filter can converge in the mean sense but may fluctuate around its mean value. Mean-square stability analyses provide complementary tools for understanding and predicting filters behavior.

We need to evaluate the evolution of E kv n+1 k 2 ⌃ over time, where ⌃ denotes an arbitrary positive semi-definite matrix, and

kxk 2 ⌃ , x > ⌃x. The evolution of E kv n+1 k 2 ⌃
is depicted in Appendix C. On the basis of Appendix C, we obtain the following theorem.

Theorem 3 (Mean-square stability): Assume data model (1), assumptions A1, A2 and approximations Ap3, Ap4 hold. Assume further that step-sizes µ

(i)
k are sufficiently small such that condition (42) is satisfied and approximations (93), (100) are justified by ignoring higher powers of step-size. Then for any initial conditions, any network with doubly stochastic matrices

A (i) 1 , A (i)
2 , i.e., both columns and rows add up to one, and power-normalized scheme (17) is mean-square stable for sufficiently small step-sizes satisfying condition [START_REF] Nascimento | A transient analysis for the convex combination of adaptive filters[END_REF].

Proof: On the basis of the Appendix C, the proof of this theorem follows the same routine as in [START_REF] Jin | Affine combination of diffusion strategies over networks[END_REF]Appendix E]. We omit it to avoid redundancy. ⌅ Unlike the proof in [START_REF] Jin | Affine combination of diffusion strategies over networks[END_REF] where we impose an additional condition on the step-size v ↵ k of the combination layer to ensure that matrix E > n+1 ⌦ > n+1 is bounded, we do not have such condition here since it is bounded with convex power-normalized scheme.

Corollary 1 (Transient MSD): Using (87) with ⌃ = 1 N I NL , we evaluate the mean-square-deviation (MSD) learning curve of the entire network, defined by

⇠ n+1 , E kv n+1 k 2 1 N I NL .
All terms on the RHS of (87) can be evaluated recursively. See Appendix D for the explicit expressions of these recursions.

Corollary 2 (Steady-state MSD): For sufficiently small stepsizes satisfying condition [START_REF] Nascimento | A transient analysis for the convex combination of adaptive filters[END_REF] to ensure stabilities in the mean and mean-square sense of the power-normalized scheme, the steady-state MSD is provided by (107) in Appendix E.

Proof: Following the same routine as in [36, Appendix G] leads to the result. ⌅ Corollaries 1 and 2 characterize the transient and steadystate MSD of the convex power-normalized scheme. These results help to tune the model parameters in practice.

C. Mean and mean-square behaviors of k,n

To analyze the mean and mean-square behaviors of the power-normalized scheme, we need to study the mean behaviors of n and > n n . They are obtained by characterizing the mean and mean-square behaviors of k,n since n is diagonal. Following the lines in [START_REF] Nascimento | A transient analysis for the convex combination of adaptive filters[END_REF], [START_REF] Silva | A transient analysis for the convex combination of two adaptive filters with transfer of coefficients[END_REF], we can conduct a theoretical analysis for k,n by using a first-order Taylor series expansion. The derivations are depicted in Appendix F and Appendix G. We arrive at the following two theorems. We shall now introduce several extensions to further improve the performance of convex combiners, and generalize them to multiple strategies.

A. Performance improvements

Several adjustments can be implemented to further improve the performance of combination schemes in certain situations. We shall now extend the adjustment strategies already proposed for combinations of adaptive filters [START_REF] Arenas-Garcia | Plant identification via adaptive combination of transversal filters[END_REF], [START_REF] Arenas-Garcia | Combinations of adaptive filters: Performance and convergence properties[END_REF], [START_REF] Nascimento | A low-complexity strategy for speeding up the convergence of convex combinations of adaptive filters[END_REF], [START_REF] Chamon | Combination of adaptive filters with coefficients feedback[END_REF] to combinations of diffusion strategies.

1) Weight transfer: At each time instant n and each node k, parameter k,n indicates which component strategy locally performs better. So, at each time instant n and each node k, the weight vector of the best component strategy can be transferred to the other component strategies in order to improve the overall performance of the combination layer. Depending on how the selected weight vector is shared, weight transfer strategies can be further divided into two categories, the copying one and the leakage one. Note that the sharing procedures described below have to be run after the combination step has been completed. The results obtained with these procedures will then be used by each node at the adaptation step of the next iteration.

Copying weights: The component strategy with the weakest performance copies the received weight vector for itself when the following conditions are satisfied simultaneously:

• k,n > 1 or k,n < 1
1 , where 0 ⌧ 1 < 1 is a pre-defined threshold value with typical value 0.95. This condition means that the weight transfer can occur when one component greatly outperforms another.

• mod(n, N 0 ) = 0, where the mod(•, •) function returns the remainder after division, and N 0 2. This condition implies that weight transfers can occur periodically with period N 0 2. By combining the above two conditions, we finally have:

w (1) k,n = ( w (2) k,n , if k,n < 1 1 and mod(n, N 0 ) = 0 w (1) k,n , otherwise (44) and w
(2) k,n = ( w (1) k,n , if k,n > 1 and mod(n, N 0 ) = 0 w (2) k,n , otherwise. (45)
Leakage transfer: The component strategy with the weakest performance partially absorbs the received weight vector when

k,n > 2 or k,n < 1 2 , that is, w (1)
k,n = ( ⇢w (1)
k,n + (1 ⇢)w (2) k,n , if k,n < 1 2 w (1) k,n , otherwise (46)
and

w (2) k,n = ( ⇢w (2) k,n + (1 ⇢)w (1) k,n , if k,n > 2 w (2) k,n m otherwise ( 47
)
where 2 and ⇢ are two non-negative parameters satisfying 0 ⌧ 2 < 1 and 0 ⌧ ⇢ < 1, with typical value of 0.95.

2) Weight feedback: Since the combined estimate at each node cannot be worse than the estimate of each component strategy, we can feedback the combined estimate to each component strategy periodically to improve the performance, that is,

w (1) k,n = ( w k,n , if mod(n, N 0 0 ) = 0 w (1) k,n , otherwise (48)
w (2) k,n = ( w k,n , if mod(n, N 0 0 ) = 0 w (2) k,n , otherwise, (49)
with period N 0 0 a large positive integer.

B. Convex combination of multiple strategies

We shall now extend the convex combination scheme to multiple component strategies. The general scheme follows the description in Section III except that we have M component diffusion strategies. We introduce M convex combination coefficients

(1) k,n , (2)
k,n , • • • , (M )
k,n at each node k and time instant n, satisfying the non-negativity and sum-to-one constraints. By combining the M local estimates at each agent k, we obtain the overall system coefficients w k,n and estimation error e k,n at the combination layer, defined as follows:

w k,n = M X i=1 (i) k,n w (i) k,n (50)
e k,n = M X i=1 (i) k,n e (i) k,n (51)
We adapt (i) k,n by minimizing the MSE of the combination layer. To satisfy the non-negativity and sum-to-one constraints, we introduce a nonlinear modified softmax function to calculate

(i) k,n as: (i) k,n = exp(↵ (i) k,n ) + P M j=1 exp(↵ (j) k,n ) + M , , i = 1, • • • , M, (52)
where 0, ↵

k,n are newly introduced auxiliary variables, and exp(•) denotes the exponential function. Parameterization of (i) k,n via (52) satisfies the non-negativity and the sum-to-one constraints.

We shall now directly update

↵ (i) k,n instead of (i)
k,n by considering the following adaptation scheme. Using stochastic gradient descent to minimize [START_REF] Chen | Multitask diffusion adaptation over networks with common latent representations[END_REF], we obtain the multiple strategies LMS scheme as:

↵ (i) k,n+1 = ↵ (i) k,n v ↵ k @J MSE n @↵ (i) k,n ⇡ ↵ (i) k,n + v ↵ k exp(↵ (i) k,n )(e k,n e (i) k,n ) P M j=1 exp(↵ (j) k,n ) + M e k,n . (53)
Specifically, when setting = 0, (53) reduces to:

↵ (i) k,n+1 ⇡ ↵ (i) k,n + v ↵ k (i) k,n e k,n (e k,n e (i) k,n ) (54)
To bound the dynamic of

↵ (i) k,n+1 , parameters ↵ (i) k,n are further required to be in interval [ ↵ + 0 , ↵ + 0 ] with ↵ + 0 > 0.

VI. DISCUSSION

A. Computation complexity and communication overhead

The computation complexity and communication overhead of the convex combination schemes result from those of the diffusion strategy layer and the combination layer. We shall consider the case of M component strategies S (i) .

• Regarding the computation overhead, since convex combination schemes can be applied to any component strategies, we cannot describe all scenarios in a unified manner since the computation overhead of each component strategy is unknown. To alleviate this, we shall adopt the following description. We denote the computation cost of each component strategy by q (i) , and we adopt the notation p k to denote the computation overhead of node k at the combination layer. Then the total computational complexity q can be evaluated as:

q = M X i=1 q (i) + N X k=1 p k . (55)
Given any candidate diffusion strategy, q (i) is related to the filter length L and the total number of nodes N , as well as the complexity in evaluating the stochastic matrices A 1 , A 2 and C. Quantity p k in convex combination schemes is related to the total number M of candidate diffusion strategies and the complexity in evaluating the exponential function in [START_REF] Simões | FADE: Fast and asymptotically efficient distributed estimator for dynamic networks[END_REF], or the sigmoid function [START_REF] Chen | Multitask diffusion adaptation over networks[END_REF].

If the exponential function can be evaluated in advance and its values stored in a table, the computation overhead can be reduced greatly. As a conclusion, the total computation complexity q of convex combination schemes is larger than the sum of the computation complexities of all component strategies. • We denote the communication cost of each component strategy by h (i) . Since the combination schemes are conducted in a distributed manner at each node, there is no communication overhead at the combination layer. Therefore, the total communication cost h of the convex combination scheme can be evaluated as:

h = M X i=1 h (i) , (56)
which means that h is equal to the sum of the communication costs of all component strategies. Given any candidate diffusion strategy, h (i) is related to the filter length L and the network topology. The total communication complexity h can be reduced, for instance, by using compression coding methods before communicating with neighboring nodes [START_REF] Shlezinger | Federated learning with quantization constraints[END_REF], [START_REF] Harrane | On reducing the communication cost of the diffusion LMS algorithm[END_REF]. B. Comparison with the affine combination scheme Without loss of generality, we focus on the case of two component strategies. For ease of comparison, we recall the update equation of the affine combination schemes in [START_REF] Jin | Affine combination of diffusion strategies over networks[END_REF]:

k,n+1 = k,n + v 0 ↵ k e k,n u > k,n (w (1) k,n w (2) k,n ) (57)
Compared with the affine combination scheme (57), the update equation of the convex combination scheme ( 17) is more complex, in particular because of the truncation operation and the nonlinear function used to evaluate coefficients k,n+1 . However, as derived in the current work, the convex combination scheme is more stable than the affine one.

VII. SIMULATION RESULTS

In this section, we present simulation results to illustrate the proposed convex combination schemes and theoretical results. All simulated curves were obtained by averaging over 100 Monte Carlo runs.

A. Validation of convex combination schemes

We considered a non-stationary system identification scenario where w ?

k varies over time. The network consisted of 10 nodes with connection topology depicted in Fig. 2(a). The regressors were generated from a zero-mean multivariate Gaussian distribution with covariance matrix R u,k = 2 u,k I 60 . The noise signals was generated from Gaussian distribution N (0, 2 z,k ). Variances 2 u,k and 2 z,k were generated randomly as depicted in Fig. 2(b). given by [START_REF] Chen | Diffusion LMS over multitask networks[END_REF] and [START_REF] Zhao | Clustering via diffusion adaptation over networks[END_REF], respectively. The evolution of w ?

k was divided into five stationary stages and four transient episodes. During the stationary stages, we set w ? k of each agent so that, from n = 1 to 1000, and from n = 4500 to 6000, the entire network pursued the same target. While from instant n = 1500 to 2500, from n = 3000 to 4000 and from n = 6500 to 8000, the network split to pursue 2, 4, and 6 targets, respectively. The transient episodes between two adjacent stationary stages were designed by using linear interpolation over 500 time instants. The results are plotted in Figs. 3 and4.

In Fig. 3(a), as expected, both the power-normalized scheme and the sign-regressor scheme with static fusion matrices tend to the best component strategy at each stage. Their behavior is similar to that of the non-cooperative diffusion LMS when estimating multiple targets, and similar to that of the diffusion LMS with averaging rule when pursuing the same target. The evolution of the convex combination coefficients in Fig. 3(b) validates the effectiveness of the combination scheme.

The results with adaptive fusion matrices are illustrated in Fig. 4. A similar learning behavior can be observed and a similar conclusion can be drawn.

2) Combination of two distinct diffusion strategies:

We considered the multitask diffusion strategies for clustered networks proposed in [START_REF] Chen | Multitask diffusion adaptation over networks[END_REF] and [START_REF] Nassif | Multitask diffusion LMS with sparsity-based regularization[END_REF], where the former uses squared `2-norm co-regularizer to promote cooperation within clusters, and the latter uses `1-norm co-regularizer. The simulation setting was the same to that in Section VII-A1, except that the 10 nodes were divided into three clusters to pursue three groups of different but related targets, and the targets for nodes within the same cluster were identical. For four stationary stages, the weight vectors w ?

Ci were generated according to w ? Ci = w o + Ci w Ci . When Ci for i = 1, 2, 3 are the same, the component strategy with `1-norm co-regularizer should perform better. Otherwise, the component strategy with squared `2-norm co-regularizer should have a better performance. We set the regularization strengths of both coregularizers to 0.1, and a uniform A (i) 2 was used such that:

a (i) 2,`k = |N k \ C(k)| 1
The results are plotted in Figs. 5 and6. Both the powernormalized scheme and the sign-regressor scheme have quite Fig. 6. Simulation results of the sign-regressor scheme with the two diffusion strategies in [START_REF] Chen | Multitask diffusion adaptation over networks[END_REF] and [START_REF] Nassif | Multitask diffusion LMS with sparsity-based regularization[END_REF].

the same behavior as the optimum component strategy at the different stages.

3) Combination of multiple diffusion strategies:

We shall now examine the performance of the combination scheme (53) via simulation. The simulation setting was similar to that in Section VII-A1, except that we changed the duration time of the stationary stages. We considered three component diffusion strategies for illustration purposes: A k control the trade-off between convergence rate and steady-state performance, it should be helpful to adopt two different step-sizes and combine them, such as from time instant n = 1000 to 5000 when the network pursues multiple targets, to obtain a faster initial convergence speed and lower misadjustment at steady state simultaneously.

The results are illustrated in Fig. 7. As expected, at the different stages the proposed combination scheme tracks the best component strategy, which is further validated from the evolution of affine combination coefficients of node 9 in Fig. 7(b). These results illustrate the effectiveness of combination framework in combining multiple component strategies.

4) Influence of the step-size v ↵,k :

We shall now examine the influence of the parameters in the power-normalized scheme, sign-regressor scheme and the multiple strategies LMS scheme. Based on various experiments, we found that the performance of the combination schemes is not sensitive to the temporal smoothing factor ⌘, to parameters ↵ + or ↵ + 0 , and to small-valued ". We therefore set ⌘ to a typical value of 0.95 and ↵ + to 4, ↵ + 0 to 2.5 and " to 0.05. We shall also examine the influence of the step-size v ↵,k . The simulation settings were identical to those used in the first experiment, and we only considered static combination matrices.

The results are plotted in Figs. 8 to 10. For all these three schemes, a small-valued v ↵,k results in a weak ability of tracking the best component, while a large-valued v ↵,k leads to bias. Thus the value of v ↵,k needs to be fine-tuned to ensure tracking performance and estimation accuracy.

B. Adjustments to improve performance

We shall now check the effectiveness of the three adjustment strategies proposed to improve the performance of the combination schemes. The simulation setting was similar to that in Section VII-A, except that we merely considered the first four stationary stages and the duration of stationary stages were set to different values. We considered the diffusion LMS with two static fusion matrices: non-cooperative A (1) 2 = I with network step-size being set to 0.01, and the averaging rule for

A (2)
2 with network step-size being set to 0.0015. Besides, for the three adjustment strategies, we set 1 , 2 and ⇢ to 0.95, N 0 to 50 and N 0 0 to 170. The results are illustrated in Fig. 11. As depicted by grey dashed lines in sub-figures (a), (b) and (c), the results with the three proposed adjustments are not worse than the original power-normalized scheme, and sometimes even offer a faster convergence rate and a lower misadjustment, such as from time instant n = 1 to n = 2500 and from n = 9000 to n = 12000. Besides, though obtaining similar steady-state MSDs, the convergence rate of the weight feedback adjustment is faster than those of the leakage adjustment and copying adjustment.

C. Validation of improvement with power-normalized scheme

We shall now validate the improvement in the EMSE obtained with the power-normalized scheme. As stated in Theorem 1, the improvement occurs when conditions

J (1,2) ex,k,1 < J (i) ex,k,1 and ¯ k,1 2 1 ✓ + k , ✓ + k are satisfied.
We considered the same setting as that used in Fig. 4 of Section VII-A1, except that we only considered the second stationary stage. The simulation results are plotted in Fig. 12 and given in Table I. The EMSE learning curve of Fig. 12(a) validates the improvement in convex power-normalized scheme at steady state. The estimated steady-state EMSEs J (i) ex,k,1 with i = 1, 2 and steady-state cross-EMSE J (1,2) ex,k,1 , averaged over the last 1500 iterations, are plotted in Fig. 12(b).As can be observed in Fig. 12(b), for several nodes the condition

J (1,2) ex,k,1 < J (i) ex,k,1 is satisfied. It can be further checked in Table I that ¯ k,1 2 1 ✓ + k , ✓ + k
for nodes 1 and 6. All these results in Fig. 12 and Table I validate the improvement in the EMSE at steady state with the power-normalized scheme. Node Index k J (1,2) ex,k,1

J (2)
ex,k,1

J (1) ex,k,1 (a) (b)
Fig. 12. Validation of the improvement obtained via the convex powernormalized scheme.

D. Theoretical model validation

As has been done in [START_REF] Jin | Affine combination of diffusion strategies over networks[END_REF], to illustrate the theoretical results as well as to challenge the assumptions and approximations adopted in the theoretical analysis, we considered two networks with different connectivity parameters as described in Table II. Net1 consisted of 10 nodes with the network topology given in Fig. 2(a). Net2 was generated by splitting 20 nodes into seven fully connected clusters, with 3 nodes in each of the first six clusters and 2 nodes in the last cluster. These seven clusters were connected in chain, with a single edge connecting adjacent clusters: agent 3 (in cluster 1) was connected with agent 4 (in cluster 2), and agent 6 (in cluster 2) was connected with agent 7 (in cluster 3), and so on until agent 18 (in cluster 6) was connected to agent 19 (in cluster 7).

The unknown system coefficients to be estimated were of length L = 2. The regressors were generated from a zero-mean Gaussian distribution with covariance matrix R u,k = 2 u,k I L for white inputs and with for colored inputs. Variances 2 u,k and 2 z,k at each agent were generated randomly. For white inputs, by varying 2 z,k , we changed the signal-to-noise ratio (SNR) [START_REF] Das | Distributed state estimation in multi-agent networks[END_REF] to two levels as described in Table III. For illustration purpose, we plot 2 u,k and 2 z,k of each agent with SNR1 in Fig. 2(b). The powernormalized scheme was run with network step-sizes being set to 0.01 and 0.004, respectively. We first validate the theoretical results for the mean and mean-square behaviors of k,n , the transient and steady-state MSDs of each component strategies, and the cross-MSD of two component strategies over the entire network defined by MSD cross , 1

R u,k = 2 u,k ✓ 1 0.5 0.5 1 ◆
N E v (1)> n v (2) n
. All these quantities are necessary in evaluating the MSD behavior of the convex power-normalized scheme. Then by using these results, we evaluate the theoretical MSD behaviors of the convex powernormalized scheme. All results are plotted in Figs. 13 to 19.

We observe in Figs. 13 to 15 that the simulated and theoretical transient values, and theoretical steady-state values of network cross-MSD and MSDs of two component diffusion strategies are superimposed, respectively, which illustrates the accuracy of the theoretical analysis for MSD at each component. Since the analysis of E{ k,n } and E{ 2 k,n } are based on the Taylor series expansion, there are biases between simulated and theoretical transient values of E{ k,n } and E{ 2 k,n }. However the theoretical results for the powernormalized scheme are still acceptable and satisfying.

The results of the power-normalized diffusion for white Gaussian inputs with Net2 and SNR1 are plotted in Fig. 16. - - The results with Net1 and SNR2 are plotted in Fig. 17. Together with Figs. 14 and 15, all these results validate the accuracy of theoretical analyses under different SNR conditions and network connectivity parameters.

The results for moderately colored inputs with Net1 and SNR1 are provided in Fig. 18-Fig. 19. Though assumption A2 is violated, the superimposition of simulated and theoretical curves validates the accuracy of the theoretical results for - sufficiently small step-sizes.

In addition, observe in Figs 14 to 19 that the combination scheme performs worse than the best component strategy. On the one hand, the performance of the power-normalized scheme is closely related to the system parameters, and when L is large, it is easier for the power-normalized scheme to track the best component strategy. Unfortunately, in the simulation setting of Figs. 14 to 19, for the purpose of saving computations, we set parameter L = 2. On the other hand, we have proved in Theorem 1 that the power-normalized scheme is universal at steady state. The Monte-Carlo curves coincide with the theoretical results.

VIII. CONCLUSIONS

Combining diffusion strategies enables a network to reach a better performance. In this paper, we proposed several schemes for the convex combination of multiple component diffusion strategies, as well as several adjustments to further improve the performance. We conducted theoretical analysis for the convex power-normalized scheme. Based on the theoretical results, we conclude that the convex power-normalized scheme is universal at steady state, meanwhile the mean and meansquare stabilities of power-normalized scheme require only the stability of its two component diffusion strategies. Thus the convex power-normalized schemes are more stable than affine combination schemes. Several open problems still have to be addressed. For instance, it would be interesting to conduct a theoretical analysis for the convex sign-regressor scheme. It would also be interesting to explore other combination frameworks and schemes.

APPENDIX A PROOF OF UNIVERSALITY ANALYSIS RESULT

We first consider two extreme situations of ↵ k,n : ex,k,1 . Based on the above results, to evaluate the EMSE after combination at steady state, it is necessary to examine the limiting behavior of E{↵ k,n }. To do so, taking the expectation of (17), we have:

• Situation 1: If lim n!1 E{↵ k,n } = ↵ + ,
E{↵ k,n+1 } ⇡ ⇥ E{↵ k,n + µ 0 ↵ k k,n (1 k,n ) e k,n u > k,n (w (1) k,n w (2) k,n )} ⇤ ↵ + ↵ + , (58)
where we have exchanged the order of expectation and truncation operations to simplify the derivation. This approximation is reasonable, since the likelihood of ↵ k,n+1 to be greater than ↵ + or less than ↵ + before truncation is small due to the existence of factor k,n (1 k,n ) in the update equation. From ( 9) and ( 13), we have:

u > k,n (w (1) k,n w (2) k,n ) = ẽ(2) k,n ẽ(1) k,n (59)
e k,n = k,n ẽ(1) k,n + (1 k,n )ẽ (2) k,n + z k,n . (60)
Under assumptions A1 and A2, substituting (59), (60) into (58), and taking the limit with n ! 1, we obtain:

E{↵ k,n+1 } = ⇥ E{↵ k,n } v↵ k,n E 2 k,n (1 k,n ) 4J (1) k,1 +v ↵ k,n E k,n (1 k,n ) 2 4J (2) k,1 ⇤ ↵ + ↵ + with n !1, (61)
where v↵ k,n , E v↵ k "+p k,n , and

4J (i) k,1 , J (i) ex,k,1 J (1,2) ex,k,1
measures the difference between the steady-state EMSE and steady-state cross-EMSE, with the latter being defined by

J (1,2) ex,k,1 , lim n!1 E{ẽ (1) k,n ẽ(2) k,n }.
From Cauchy-Schwartz inequality and according to the relations between J (1,2) ex,k,1 and J (i) ex,k,1 , we further divide the problem of evaluating the limiting behavior of E{↵ k,n } into three cases:

• Case 1: J (1) ex,k,1  J (1,2) ex,k,1  J (2)
ex,k,1 . We then have 4J

✓ + k , ✓ + k ], both E 2 k,n (1 k,n ) and E k,n (1 k,n ) 2 are lower bounded by bk , ✓ + k (1 ✓ + k ) 2 . Define bk , lim n!1 v↵ k,n . Then (61)
writes to:

E{↵ k,n+1 } ⇥ E{↵ k,n }+b k ⇤ ↵ + ↵ + with n ! 1 (62)
where b k , bk bk (4J

(2) k,1 4J (1)
k,1 ) > 0. It follows from (62) that the unique stationary point of E{↵ k,n+1 } with n ! 1 is ↵ + . According to the previous conclusion drawn in Situation 1, we conclude that J ex,k,1 ⇡ J

(1) ex,k,1 . • Case 2: J (1) ex,k,1 J (1,2) ex,k,1 J (2)
ex,k,1 . We then have 4J

k,1  0. Then (61) writes to:

E{↵ k,n+1 }  ⇥ E{↵ k,n } b 0 k ⇤ ↵ + ↵ + with n ! 1 (63)
where b 0 k , bk bk (4J

k,1 4J

k,1 ) > 0. Thus the unique stationary point of E{↵ k,n+1 } with n ! 1 is ↵ + . According to the conclusion drawn in Situation 2, we have J ex,k,1 ⇡ J (2) ex,k,1 .

• Case 3:

J (i) ex,k,1 > J (1,2)
ex,k,1 for i = 1, 2. We then have 4J

E{ 2 k,n (1 k,n )}4J (1)
k,1 = E{ k,n (1 k,n ) 2 }4J (2)
k,1 with n ! 1. (64) To make equation (64) tractable, by using Ap7 which assumes that the variance of k,n is small with n ! 1, we obtain:

¯ k,1 , lim n!1 E{ k,n } =  4J (2) k,1 4J (1)
k,1 +4J

(2) k,1

✓ + k 1 ✓ + k . (65)
Since the steady-state EMSE of node k after combination writes to:

J ex,k,1 = ¯ 2 k,1 J (1) ex,k,1 + (1 ¯ k,1 ) 2 J (2) ex,k,1 + 2¯ k,1 (1 ¯ k,1 )J (1,2) ex,k,1 , (66)
by substituting ¯ k,1 of (65) without truncation operation into (66) and after some algebraic manipulations, we arrive at:

J ex,k,1 = J (1,2) ex,k,1 + 4J (1) k,1 4J (2) k,1 4J (1) k,1 + 4J (2) k,1 . (67)
Since

J (i) ex,k,1 > J (1,2) ex,k,1 for i = 1, 2, we have 4J (1) k,1 4J (2) k,1 4J (1)
k,1 +4J

(

2) k,1 < 4J (i)
k,1 , and we conclude:

J ex,k,1 < J (i) ex,k,1 , (68)
which means that there is an improvement of EMSE after combination. Since the cross-EMSE is lower than EMSE of each component, we may extract extra information through combination, which brings the gain in EMSE.

As for ¯ k,1 = ✓ + k or 1 ✓ + k obtained with truncation, we have derived in Situation 1 and Situation 2 that

J ex,k,1 ⇡ min{J (1) ex,k,1 , J (2) ex,k,1 }. ( 69
)
For the steady-state EMSE of the entire network defined by:

J ex,net,1 , N X k=1 J ex,k,1 , (70)
we have:

J ex,net,1  min J (1)
ex,net,1 , J

ex,net,1 ,

which means that the EMSE of diffusion network after combination is no worse than that of the best component strategy, leading to the universality of convex power-normalized scheme at steady state.

APPENDIX B ITERATION OF E v (i) n+1

Under assumption A1 and along the lines developed in [START_REF] Chen | Diffusion LMS over multitask networks[END_REF], we have:

v (i) n+1 = B (i) n v (i) n + g (i) n r (i) n , (72)
E v (i) n+1 = B (i) E v (i) n r (i) , (73)
with quantities defined by:

B (i) n = A A A (i)> 2 I NL U (i) H (i) n A A A (i)> 1 , (74)
B (i) = A A A (i)> 2 I NL U (i) H (i) A A A (i)> 1 , (75)
A A A (i) j = A (i) j ⌦ I L , 8j = 1, 2, (76)
U (i) = diag µ (i) 1 , • • • , µ (i) N ⌦ I L , (77)
H (i) n = diag n X `2N k c (i) `k x `,n x > `,n o N k=1 , (78)
H (i) = diag R (i) 1 , • • • , R (i) N , (79)
R (i) k , X `2N k c (i) `k R u,`, (80)
g (i) n = A A A (i)> 2 U (i) p (i) zx,n , (81)
p (i) zx,n = col n X `2N k c (i) `k x `,n z `,n o N k=1 , (82)
h (i) u,n = col n X `2N k c (i) `k x `,n x > `,n w ? k w ? ` o N k=1 , (83) h
(i) u = col n X `2N k c (i) `k R u,k w ? k w ? ` o N k=1 , (84)
r (i) n , A A A (i)> 2 U (i) h (i) u,n | {z } r (i) u,n h A A A (i)> 2 I NL U (i) H (i) n A A A (i)> 1 I NL +(A A A (i)> 2 I NL ) i w ? | {z } r (i) w,n (85)
r (i) , E{r (i) n } = r (i) u r (i) w . (86)
Expression (73) helps to evaluate the iteration of the mean behavior E{v

(i) n+1 }. APPENDIX C EVOLUTION OF E kv n+1 k 2 ⌃
Using (40), we have:

E kv n+1 k 2 ⌃ = 2E v (1)> n+1 n+1 ⌃(I NL n+1 )v (2) n+1 +E (I NL n+1 )v (2) n+1 2 ⌃ + E n+1 v (1) n+1 2 ⌃ . (87) Define: ⌃ (1) n+1 , E > n+1 ⌃ n+1 (88) ⌃ (2) n+1 , E (I NL n+1 ) > ⌃ (I NL n+1 ) (89)
(i) n+1 = vec ⌃ (i) n+1 . (90)
Under approximation Ap3, the last two terms on the RHS of (87) can be written in compact form as

E n v (i)> n+1 ⌃ (i) n+1 v (i) n+1
o for i = 1, 2, which are evaluated in [START_REF] Jin | Affine combination of diffusion strategies over networks[END_REF] under Ap4 as:

E n v (i) n+1
2 (i) n+1 o = E n v (i) n 2 K (i) (i) n+1 o + ⇥ vec{G (i)> } ⇤ > (i) n+1 + f r (i) , ⌃ (i) n+1 , E v (i) n , (91)
where k

• k 2 ⌃ (i) n+1 and k • k 2 (i) n+1
are used interchangeably, with:

f r (i) ,⌃ (i) n+1 , E v (i) n , r (i) 2 ⌃ (i) n+1 2r (i)> ⌃ (i) n+1 B (i) E v (i) n (92) K (i) ⇡ B (i)> ⌦ B (i)> (93)
G (i) , E g (i) n g (i)> n . (94)
Expression (91) was derived in Appendix D of [START_REF] Jin | Affine combination of diffusion strategies over networks[END_REF]. For the first term on RHS of (87), we have: 1) , B 2) , (95) where f x r (1) , r (2) , ⌃ x,n+1 , E v (1) n 1) , B

E v (1)> n+1 ⌃ x,n+1 v (2) n+1 = E v (1)> n ⌃ xc,n+1 v (2) n + ⇥ vec{G > x } ⇤ > x,n+1 + f x r (1) , r (2) , ⌃ x,n+1 , E v (1) n , E v (2) n , B (
, E v (2) n , B (
, r

> ⌃ x,n+1 r (2) E v (1)> n B (1)> ⌃ x,n+1 r (2) r (1)> ⌃ x,n+1 B (2) E v (2) n (96) with ⌃ x,n+1 , E n+1 ⌃(I NL n+1 ) , (1)
⌃ xc,n+1 , vec 1 K x x,n+1 , (97)
x,n+1 , vec ⌃ x,n+1 , (99)

K x ⇡ B (2)> ⌦ B (1)> , (100)
G x , E g (2) n g (1)> n , (101)
and vec

(i) n 2 K (i) (i) n+1
, that is the last two terms of (87) compactly, is given in [START_REF] Chen | Diffusion LMS over multitask networks[END_REF] as:

⇠ (i) n+1 = ⇠ (i) n + ⇥ vec{G (i)> } > (K (i) ) n (i) + kr (i) k 2 (K (i) ) n (i) kv (i) 0 k 2 (I K (i) )(K (i) ) n (i) 2 ⇤ (i) n + (B (i) E{v (i) n }) > ⌦ r (i)> (i) ⇤ , (102) and ⇤
(i) n+1 = ⇤ (i) n K (i) + (B (i) E{v (i) n }) > ⌦r (i)> K (i) I (103) with ⇤
(i) 0 = 0 1⇥(NL) 2 , ⇠ (i) n+1 = E v (i) n+1 2 K (i) (i) n+1 , ⇠ (i) 0
= v (i) 0 2 K (i) (i) n+1 , (i) = K (i) (i) n+1 for i = 1, 2. Following the same routine, E v (1)> n ⌃ xc,n+1 v (2)
n can be evaluated as follows:

⇠ x,n+1 = ⇠ x,n + vec{G > x } > (K x ) n x + (⇧ (1) n + ⇧ (2) n ) x ⇥ B (2) E{v (2) n } > ⌦ r (1)> + r (2)> ⌦ B (1) E{v (1) n } > ⇤ x v (1)> 0 vec 1 (I K x ) (K x ) n x v (2) 0 + r (1)> vec 1 (K x ) n x r (2) , (104) and ⇧
(1)

n+1 = ⇧ (1) n K x + ⇥ r (2)> ⌦(B (1) E{v (1) n }) > ⇤ I K x , (105)
⇧ (2) n+1 = ⇧ (2) n K x + ⇥ (B (2) E{v (2) n }) > ⌦r (1)> ⇤ I K x (106) with ⇧ (i) 0 = 0 1⇥(NL) 2 , x = K x x,n+1 , ⇠ x,n+1 = E v (1)> n+1 ⌃ xc,n+1 v (2) n+1 , ⇠ x,0 = E v (1)> 0 ⌃ xc,n+1 v (2) 0
. By substituting (102) and (104) into (87), we evaluate the transient MSD of the convex power-normalized scheme.

APPENDIX E STEADY-STATE MSD OF DIFFUSION NETWORK WITH POWER-NORMALIZED SCHEME

MSD steady = ⇥ vec{G (1)> } ⇤ > (1) 1 +f r (1) , ⌃ (1) 1 , E v (1)

1 + ⇥ vec{G (2)> } ⇤ > (2)
1 +f r (2) , ⌃ (2) 1 , E v (2)

1 + ⇥ vec{G > x } ⇤ > x,1
+ f x r (1) ,r (2) ,⌃ x,1 , E v (1) 1 , E v (2) 1 ,B (1) ,B

(107) with

⌃ (i) 1 = vec 1 { (i) 1 }, 8 i = 1, 2 (108)
⌃ x,1 = vec 1 { x,1 }. (109)
Expression (107) characterizes the steady-state MSD of the convex power-normalized scheme.

APPENDIX F PROOF OF THEOREM 4

Since k,n is related to ↵ k,n via the mapping [START_REF] Chen | Multitask diffusion adaptation over networks[END_REF], we evaluate the behaviors of ↵ k,n first, then k,n .

Substituting (59) and ( 60) into ( 17), we have:

↵ k,n+1 ⇡ ⇥ ↵ k,n +v 0 ↵ k k,n (1 k,n ) (2 k,n 1)ẽ (1) k,n ẽ(2) k,n k,n (ẽ (1)
k,n ) 2 +(1 k,n )(ẽ (2)
k,n ) 2 +(ẽ (2) k,n ẽ(1) k,n )z k,n ⇤ ↵ + ↵ + . (110)
Define:

⇣ 1 , (ẽ (1)
k,n ) 2 (111) ⇣ 2 , (ẽ (2)
k,n ) 2 (112)

⇣ 3 , ẽ (1) k,n ẽ(2)
k,n (113)
⇣ 4 , (ẽ

(2) k,n ẽ(1) k,n )z k,n (114)
f 1 (↵ k,n ) , 2 k,n (1 k,n ) (115)
f 2 (↵ k,n ) , k,n (1 k,n ) 2 (116)
f 3 (↵ k,n ) , k,n (1 k,n )(2 k,n 1) (117)
f 4 (↵ k,n ) , k,n (1 k,n ) (118)
Using definitions (111)-(118), expression (110) becomes:

↵ k,n+1 ⇡ h ↵ k,n + v 0 ↵ k 4 X `=1 f `(↵ k,n )⇣ `i↵ + ↵ + . ( 119
)
Observing ( 110) and (119), the iterations for ↵ k,n are coupled with k,n . It is difficult, if not impossible, to evaluate the behavior of ↵ k,n directly since the explicit probability distribution of ↵ k,n is unknown. We alleviate this problem by using the first-order Taylor series expansion. Though not accurate, the first-order Taylor series expansion is widely adopted in the analysis of adaptive filters to simplify the derivations [START_REF] Nascimento | A transient analysis for the convex combination of adaptive filters[END_REF], [START_REF] Silva | A transient analysis for the convex combination of two adaptive filters with transfer of coefficients[END_REF]. Define ↵k,n , E{↵ k,n }. Expanding f `(↵ k,n ) around ↵k,n to its first-order, we have:

f `(↵ k,n ) ⇡ f `( ↵k,n ) + f 0 `( ↵k,n )(↵ k,n ↵k,n ) (120)
for `= 1, • • • , 4, where f 0 `( ↵k,n ) = df `( ↵k,n ) d↵ k,n is the first-order derivative of f `(↵ k,n ) over ↵ k,n and evaluated at ↵ k,n = ↵k,n .

Under approximations Ap1, Ap5, substituting (120) into (119) and taking the expectation, we get the mean behavior of ↵ k,n :

E{↵ k,n+1 } ⇡ ⇥ ↵k,n + v↵ k,n f 1 (↵ k,n )J (1)
ex,k,n + v↵ k,n f 2 (↵ k,n )J a,k,n is the cross-EMSE at node k and time instant n. Besides, the following approximation in (121) is adopted to simplify the derivation:

v↵ k,n = E n v ↵ k " + p k,n o ⇡ v ↵ k " + pk,n with pk,n = ⌘ pk,n 1 +(1 ⌘)(J (1)
ex,k,n +J

ex,k,n 2J

(1,2) ex,k,n ). (122) Then, for the mean behavior of k,n , we evaluate it via the approximation:

E{ k,n } ⇡ 1 1 + e E{↵ k,n } . ( 123
)
For the steady-state value ¯ k,1 , E{ k,1 }, taking the limit of (121) with n ! 1 and solving for ¯ k,1 , we obtain the same steady-state value as (65).

Transient value E{ k,n } of (123) and steady-state value E{ k,1 } of (65) are used in evaluating the transient and steady-state mean behavior of the convex power-normalized scheme, respectively.

APPENDIX G PROOF OF THEOREM 5

Similarly, we evaluate the mean-square behavior of k,n based on the first-order Taylor series expansion. Define f 5 (↵ k,n ) , 1 1+e ↵ k,n . Using the first-order Taylor series expansion, we have:

f 5 (↵ k,n ) ⇡ f 5 (↵ k,n ) + f 0 5 (↵ k,n )(↵ k,n ↵k,n ), (124)
where f 0 5 (↵ k,n ) ,

df5( ↵k,n ) d↵ k,n
is the first-order derivative of f 5 (↵ k,n ) and evaluated at ↵ k,n = ↵k,n . Substituting the explicit expression of f 5 (↵ k,n ), we have that f 0 5 (↵ k,n ) = f 5 (↵ k,n ) ⇥ 1 f 5 (↵ k,n ) ⇤ . Since k,n = f 5 (↵ k,n ) and using (124), we have:

E{ 2 k,n } ⇡ ⇥ f 5 (↵ k,n ) ⇤ 2 + ⇥ f 0 5 (↵ k,n ) ⇤ 2 2 ↵ k,n , (125)
where 2 ↵ k,n is the variance of ↵ k,n , and we have:

2 ↵ k,n = E{↵ 2 k,n } ⇥ E{↵ k,n } ⇤ 2 . (126)
In order to use (126) in calculating E{ 2 k,n } of (125), we need to evaluate E{↵ 2 k,n }. By first discarding the truncation operation, squaring both sides of (119) and taking the expectation, we have:

E{↵ 2 k,n+1 } ⇡ E{↵ 2 k,n } + 2v ↵ k,n 4 X `=1 E ↵ k,n ⇣ `f`( ↵ k,n ) + v2 ↵ k,n 4 X `=1 4 X m=1 E f `(↵ k,n )f m (↵ k,n )⇣ `⇣m , (127)
where in the derivation of (127), we used Ap1, (122) and approximation E (

v k "+p k,n ) 2 ⇡ v2 ↵ k,n .
According to the firstorder Taylor series expansion, we have:

↵ k,n f `(↵ k,n ) ⇡ ↵k,n f `( ↵k,n )+ ⇥ f `( ↵k,n ) + ↵k,n f 0 `( ↵k,n ) ⇤ (↵ k,n ↵k,n ) (128)
and

f `(↵ k,n )f m (↵ k,n ) ⇡ f `( ↵k,n )f m (↵ k,n )+ ⇥ f 0 `( ↵k,n )f m (↵ k,n )+f `( ↵k,n )f 0 m (↵ k,n ) ⇤ (↵ k,n ↵k,n ). (129)
Using ( 128), (129) and approximation Ap5 yields: 132) into (127) leads to:

E ↵ k,n ⇣ `f`( ↵ k,n ) ⇡ ↵k,n f `( ↵k,n )E ⇣ ` , 8`= 1, 2, 3 (130) E ↵ k,n ⇣ 4 f 4 (↵ k,n ) = 0, (131) E f `(↵ k,n )f m (↵ k,n )⇣ `⇣m ⇡ f `( ↵k,n )f m (↵ k,n )E ⇣ `⇣m
E{↵ 2 k,n+1 }⇡ E{↵ 2 k,n }+2v ↵ k,n 3 X `=1
↵k,n f `( ↵k,n )E{⇣ `}+ v2

↵ k,n  3 X `=1 3 X m=1 f `(↵ k,n )f m (↵ k,n )E{⇣ `⇣m }+f 4 (↵ k,n )f 4 (↵ k,n )E{⇣ 2 4 } (133)
Under approximation Ap6 and substituting (115)-( 118) into (133), we obtain the explicit expression of E{↵ 2 k,n+1 }. Now, by taking the truncation operation of ↵ k,n into consideration and using the non-negative property of 2 ↵ k,n in (126), we obtain that E{↵ 2 k,n } is constrained to be in the interval ⇥ (E{↵ k,n }) 2 , (↵ + ) 2 ⇤ . Then by using (125), ( 126) and (133) together, we evaluate the mean-square value E{ 2 k,n }. Besides, the steady-state value E{ 2 k,1 } is approximated by:

E{ 2 k,1 } ⇡ ⇥ E{ k,1 } ⇤ 2 . ( 134
)
Transient value E{ 2 k,n } of (125) and steady-state value E{ k,1 } 2 of (134) are used in evaluating the transient and steady-state mean-square behavior of the convex powernormalized scheme, respectively.

+ 1 Fig. 1 .

 11 Fig. 1. Illustration of the combination framework with M component diffusion strategies. Section III is a special case by setting M = 2.

Fig. 2 .

 2 Fig. 2. (a) Network topology; (b) Agent input and noise variances.

1 )

 1 Combination of two diffusion LMS strategies: We considered two ATC diffusion LMS strategies with C = I N and A (i) 1 = I N as component strategies. For matrices A

Fig. 3 .Fig. 4 .

 34 Fig. 3. Simulation results with static fusion matrices. (a) Network MSD learning curves; (b) Evolution of convex combination coefficient k,n for power-normalized scheme.

Fig. 5 .

 5 Fig.5. Simulation results of the power-normalized scheme with the two component strategies in[START_REF] Chen | Multitask diffusion adaptation over networks[END_REF] and[START_REF] Nassif | Multitask diffusion LMS with sparsity-based regularization[END_REF]. (a) Network MSD learning curves; (b) Evolution of the convex combination coefficients k,n .

  the identity matrix corresponding to non-cooperative strategy, with the network step-sizes being set to µ respectively. When the entire network cooperates to pursue the same target, i.e., from time instant n = 1 to 800 and from instant n = 5000 to 6300 in Fig.7(a), the diffusion strategy with averaging rule A (2) 2 should perform better. At other time instants when the network is split to pursue 2, 4 and 6 targets, the non-cooperative diffusion strategies with fusion matrices A

Fig. 7 .

 7 Fig. 7. Simulation results of combination scheme for three diffusion strategies. (a) Network MSD learning curves; (b) Evolution of combination coefficients (i) k,n at agent 9.

1 Fig. 8 . 1 Fig. 9 .

 1819 Fig. 8. Simulation results of the convex power-normalized scheme with different step-sizes v ↵,k .

5 Fig. 10 .

 510 Fig. 10. Simulation results of the multiple strategies LMS scheme with different step-sizes v ↵,k .

Fig. 11 .

 11 Fig. 11. Simulation results of convex power-normalized scheme with three adjustments. (a) Network MSD learning curves with copying weights; (b) Network MSD learning curves with leakage transfer; (c) Network MSD learning curves with weight feedback.

(

  Cross-)EMSE (dB)

Fig. 13 .

 13 Fig. 13. Illustration of simulation results (model vs. Monte Carlo) for the convex power-normalized scheme. Transient and steady-state values of E{ k,n } derived in (123) and (65) (top), as well as these of E{ 2 k,n } derived in (125) and (134) (bottom) for network step-size 0.01 (a) and 0.004 (c); (b) Transient and steady-state cross-MSDs derived in (104) and (107).

2 Diffusion

 2 Fig. 14. Network MSD performance of convex power-normalized scheme (model vs. Monte Carlo) with network step-size 0.01 for white input in Net1 and SNR1.

Fig. 15 .

 15 Fig.[START_REF] Chen | Diffusion LMS over multitask networks[END_REF]. Network MSD performance of convex power-normalized scheme (model vs. Monte Carlo) with network step-size 0.004 for white input in Net1 and SNR1.

Fig. 16 .

 16 Fig.[START_REF] Chen | Multitask diffusion adaptation over networks with common latent representations[END_REF]. Network performance with the power-normalized scheme (model vs. Monte Carlo) with network step-size 0.01 for white input in Net2 and SNR1.

Fig. 17 .

 17 Fig. 17. Network performance with the power-normalized scheme (model vs. Monte Carlo) with network step-size 0.01 for white input in Net1 and SNR2.

Fig. 18 .

 18 Fig.[START_REF] Gogineni | Diffusion affine projection algorithm for multitask networks[END_REF]. Network MSD performance of convex power-normalized scheme (model vs. Monte Carlo) with network step-size 0.01 for colored input in Net1 and SNR1.

Fig. 19 .

 19 Fig. 19. Network MSD performance of convex power-normalized scheme (model vs. Monte Carlo) with network step-size 0.004 for colored input in Net1 and SNR1.

0.

  Since k,n and 1 k,n lie in the interval [1

  > 0. Iteration (61) converges to the stationary point if and only if:

( 2 )

 2 ex,k,n + v↵ k,n f 3 (↵ k,n )J (1,2) ex,k,n ⇤ ↵ + ↵ + ,(121) where the order of the expectation and truncation operations are changed to facilitate the analysis, and J

  8`, m = 1, 2, 3, 4. (132) Substituting (130)-(

  , g

	(i) n and r	(i) n are defined
	in (81) and (85) of Appendix B, respectively.
	Approximation 5: Parameter ↵ k,n varies slowly enough so
	that E{↵ k,n Approximation 6: The a priori estimation errors ẽ(m) k,n ẽ(n) k,n } ⇡ E{↵ k,n }E{ẽ (m) k,n ẽ(n) k,n }, m, n = 1, 2. ẽ(1) k,n and ẽ(2) k,n

  Theorem 4 (Mean behavior of k,n ): Assume data model (1) and approximations Ap1, Ap5 hold. Then for any initial conditions, the convex combination coefficients k,n are stable in the mean sense. The mean behavior of k,n is described by (123) in Appendix F. The value of E{ k,n } at steady-state is given by (65).

	Proof: See Appendix G.	⌅
	V. EXTENSIONS OF THE SCHEME

Proof: See Appendix F. ⌅ Theorem 5 (Mean-square behavior of k,n ): Assume data model (1) and approximations Ap1, Ap5, Ap6 hold. Then for any initial conditions, the convex combination coefficients k,n are stable in the mean-square sense. The mean-square behavior of k,n is described by (125) in Appendix G. The value of E{ 2 k,n } at steady-state is given by (134).

TABLE I ESTIMATED

 I STEADY-STATE VALUES ¯ k,1 AT EACH NODE. SINCE WE SET ↵ + TO 4, ✓ + k ,

		1 1+e ↵ + = 0.982 AND 1 ✓ + k = 0.018.
	Node k	1	2-5	6	7-10
	¯ k,1	0.0541 0.018 0.0376 0.982

TABLE II NETWORK

 II STATISTICS FOR THEORETICAL MODELS VALIDATION. L IS THE LAPLACIAN MATRIX ASSOCIATED WITH THE GRAPH (NETWORK), 2 (L) IS THE ALGEBRAIC CONNECTIVITY [51] OF GRAPH, S I Z E IS THE NUMBER OF NODES, D E N S I T Y IS THE NUMBER OF NON-ZERO ENTRIES OF THE ADJACENCY MATRIX OF GRAPH, AND D I A M E T E R IS THE MAXIMUM DISTANCE BETWEEN ANY TWO NODES[START_REF] Simões | FADE: Fast and asymptotically efficient distributed estimator for dynamic networks[END_REF].

	Network Size Density	2 (L)	Diameter
	Net1	10	44%	0.8576	3
	Net2	20	17.25%	0.0439	13

  we conclude that ↵ k,n ! ↵ + with n ! 1 is almost sure. Then we have If lim n!1 E{↵ k,n } = ↵ + , we deduce that ↵ k,n ! ↵ + as n ! 1 in a high probability. Then we have k,n !

	k,n ! ✓ + k ,	1 1+e ↵ + ⇡ 1, and J ex,k,1 ⇡ J	(1) ex,k,1 .
	J	(2)	1 1+e ↵ + = 1 ✓ + k ⇡ 0, and J ex,k,1 ⇡

• Situation 2:


Member, IEEE Wei Gao
email: weigao@ujs.edu.cn

Senior Member, IEEE Jie Chen
email: dr.jie.chen@ieee.org

Senior Member, IEEE Cédric Richard
email: cedric.richard@unice.fr

Transient Theoretical Analysis of Diffusion RLS Algorithm for Cyclostationary Colored Inputs

Keywords: Adaptive network, diffusion RLS algorithm, transient theoretical analysis, cyclostationary colored inputs. I

Convergence of the diffusion RLS (DRLS) algorithm to steady-state has been extensively studied in the literature, whereas no analysis of its transient convergence behavior has been reported yet. In this letter, we conduct a theoretical analysis of the transient behavior of the DRLS algorithm for cyclostationary colored inputs, in the mean and mean-square error sense. The resulting analytical models allows us to thoroughly investigate the convergence behavior of the algorithm over adaptive networks in such complex scenarios. Simulation results support the accuracy and correctness of the theoretical findings.

introduced the robust DRLS algorithm to improve convergence performance in the presence of impulsive noise. The reducedcommunication DRLS algorithm and its steady-state analytical models were provided in [START_REF] Rastegarnia | Reduced-communication diffusion RLS for distributed estimation over multi-agent networks[END_REF].

Almost all existing works that report analyses of DRLS-type algorithms using the energy conservation principle only focus on the steady-state mean and mean-square error performances under the assumption of stationary inputs. Nevertheless, cyclostationary inputs with periodical variations are ubiquitous in real-world applications [START_REF] Gardner | Exploitation of spectral redundancy in cyclostationary signals[END_REF], [START_REF] Gardner | Cyclostationarity: Half a century of research[END_REF]. In that way, the theoretical performance of LMS-type algorithms were studied further based on this assumption [START_REF] Bershad | Stochastic analysis of the LMS and NLMS algorithms for cyclostationary white Gaussian inputs[END_REF]- [START_REF] Shlezinger | Performance analysis of LMS filters with non-Gaussian cyclostationary signals[END_REF]. Then, the convergence behavior of the diffusion LMS (DLMS) was also analyzed for cyclostationary inputs in [START_REF] Wang | Performance analysis of diffusion LMS algorithm for cyclostationary inputs[END_REF]- [START_REF] Gao | Stochastic analysis of diffusion dual sign algorithm for cyclostationary white Gaussian inputs and nonstationary system[END_REF]. Inspired by a recent work analyzing the non-cooperative RLS algorithm with new tools [START_REF] Eweda | Stochastic analysis of the recursive least squares algorithm for cyclostationary colored inputs[END_REF], in this letter we propose a theoretical analysis of the transient behavior of the DRLS algorithm over adaptive networks for cyclostationary colored inputs. To the best of our knowledge, this is the first work which provides insight into the transient convergence behavior of the DRLS in such scenarios. More importantly, our work proposes a standard framework that can be readily applied to the transient analysis of other DRLS-type algorithms. Simulation results show the accuracy and correctness of the theoretical findings.

II. PRELIMINARIES AND DRLS ALGORITHM

A. Distributed system model

Consider an adaptive network consisting of K nodes with some communication links between these nodes. Every node k has access to a sequence of the collected input-output data pairs {x k,n , d k,n } N n=1 , where x k,n = [x k,n , . . . , x k,n-L+1 ] is the input regression vector and d k,n is the scalar desired response assumed to be zero-mean. At time instant n for each node k, the desired output d k,n is assumed to be generated from the input vector x k,n via the linear regression model

d k,n = x k,n w + z k,n (1)
where w ∈ R L is the optimal weight vector to be estimated, and z k,n denotes the zero-mean Gaussian observation noise with variance σ 2 z,k . Specifically, z k,n is assumed to be independent of any other signals. In addition, the cyclostationary input vector x k,n at node k and time instant n is modeled as a colored random process with the periodically time-varying variance [START_REF] Eweda | Stochastic analysis of the recursive least squares algorithm for cyclostationary colored inputs[END_REF]:

x k,n = σ x,k (n) u k,n (2)
where σ x,k (n) is a deterministic periodic sequence with repetition period T , and u k,n is a zero-mean colored Gaussian random sequence with variance σ 2 u,k = 1. If not the most general one, model ( 2) is a typical form of cyclostationary signal used in many practical applications. The time-varying autocorrelation matrix of input vector x k,n in (2) is given by:

R x,k (n) = E{x k,n x k,n } = Σ x,k (n)R u,k Σ x,k (n) (3) with Σ x,k (n) = diag σ x,k (
n), . . . , σ x,k (n-L+1) a diagonal matrix, and R u,k = E{u k,n u k,n } the autocorrelation matrix. The sinusoidal and pulsed variation models [START_REF] Bershad | Stochastic analysis of the LMS and NLMS algorithms for cyclostationary white Gaussian inputs[END_REF]- [START_REF] Shlezinger | Performance analysis of LMS filters with non-Gaussian cyclostationary signals[END_REF] are often adopted for the periodic sequences σ 2

x,k (n) to study the impact of cyclostationary colored inputs on transient behavior.

B. Diffusion RLS algorithm

Distributed adaptive filtering algorithms are usually implemented using two common diffusion strategies, the adapt-thencombine (ATC) and the combine-then-adapt (CTA) [START_REF] Sayed | Adaptive networks[END_REF], [START_REF]Diffusion adaptation over networks[END_REF]. Subsequently, we shall focus on the DRLS with ATC diffusion strategy by sequentially solving two local least-squares problems based on local measurements, which correspond to the adaptation and combination steps, respectively.

Let us first define the local estimate w k,n-1 and intermediate estimate ψ k,n of w at node k and time n, respectively. Since w k,n-1 is a good guess for ψ k,n as a prior information, we thus solve the following local cost function:

ψ k,n = arg min ψ∈R L ψ -w k,n-1 2 Λ k,n +(d k,n -x k,n ψ) 2 (4) with Λ k,n = Φ k,n -x k,n
x k,n a positive-definite weighting matrix, and Φ k,n the time-averaged correlation matrix of input data for node k defined by [START_REF] Yu | Robust distributed diffusion recursive least squares algorithms with side information for adaptive networks[END_REF]:

Φ k,n = λΦ k,n-1 + x k,n x k,n (5)
with forgetting factor 0 λ < 1 and initial condition Φ k,0 = δI L with a small positive value δ. By substitution of variables,

i.e., v = ψ -w k,n-1 and b = d k,n -x k,n w k,n-1 , we get: v = arg min v∈R L v Λ k,n v + (b -x k,n v) 2 . ( 6
)
Setting the derivative of ( 6) with respect to v to zero, we then arrive at the solution:

ψ k,n = w k,n-1 + P k,n x k,n e k,n (7)
with P k,n = Φ -1 k,n and e k,n the instantaneous estimation error at node k and time instant n, i.e.,

e k,n = d k,n -x k,n w k,n-1 . (8)
Applying the matrix inversion lemma to the r.h.s. of ( 5), the update equation of the inverse of autocorrelation matrix at node k is given by [START_REF] Haykin | Adaptive Filter Theory[END_REF], [START_REF] Sayed | Fundamentals of Adaptive Filtering[END_REF]

P k,n = λ -1 P k,n-1 - λ -1 P k,n-1 x k,n x k,n P k,n-1 1 + λ -1 x k,n P k,n-1 x k,n (9)
with the initial condition P k,0 = δ -1 I L . We collect the intermediate estimates ψ k,n for all nodes into a column vector of length KL, and stack K identity matrices on top of each other into a KL × L matrix, as follows:

ψ n = col{ψ 1,n , . . . , ψ K,n }, I = col{I L , . . . , I L }. (10)
We then attempt to further improve the estimation precision of ψ k,n for each node k by sharing local data available within its neighborhood. Given the entire intermediate estimate ψ n , we solve the following weighted least squares problem [START_REF] Cattivelli | Diffusion recursive least-squares for distributed estimation over adaptive networks[END_REF]:

w k,n = arg min w∈R L { ψ n -Iw 2 Π k } (11)
with Π k = diag a 1k I L , . . . , a Kk I L the node-dependent weighting block diagonal matrix, and {a k } K k=1 the set of nonnegative combination coefficients [START_REF] Sayed | Adaptive networks[END_REF], [START_REF]Diffusion adaptation over networks[END_REF]. Each coefficient a k is the ( , k)-th entry of a left-stochastic matrix A, i.e., A 1 K = 1 K . Likewise, setting the derivative of [START_REF] Arablouei | Adaptive distributed estimation based on recursive least-squares and partial diffusion[END_REF] with respect to w to zero, we finally find the solution

w k,n = ∈N k a k ψ ,n . (12)
Even though the local intermediate estimates ψ k,n may suffer from random estimation error fluctuations, the combined solution ( 12) is able to achieve better estimation accuracy. Note that the weight update equation ( 7) and recursion [START_REF] Liu | Distributed sparse recursive least-squares over networks[END_REF] with combination (12) still offer good estimation performance instead of implementing an incremental-type scheme as in [START_REF] Cattivelli | Diffusion recursive least-squares for distributed estimation over adaptive networks[END_REF]. This will be illustrated by simulation results later.

III. TRANSIENT PERFORMANCE ANALYSIS

We now study the transient performance analysis of DRLS with ATC diffusion strategy for cyclostationary colored inputs in the mean and mean-square sense. The error vectors for node k at time instant n are defined respectively as:

ψ k,n ψ k,n -w , w k,n w k,n -w . (13)
Then, let ψ n and w n denote the block weight error vectors by collecting the error vectors for all nodes as follows:

ψ n col{ ψ 1,n , . . . , ψ K,n }, w n col{ w 1,n , . . . , w K,n }. (14)
We introduce the following K × K block diagonal matrices with individual entries of size L × L:

R x,n diag{x 1,n x 1,n , . . . , x K,n x K,n } (15)
P n diag{P 1,n , . . . , P K,n } (16)
Φ n diag{Φ 1,n , . . . , Φ K,n } (17)
and the K × 1 block column vector with vectors of length L:

s xz,n col{z 1,n x 1,n , . . . , z K,n x K,n }. ( 18
)
Before proceeding, we introduce the following independence assumption.

A1. The regression input random vector x k,n is modeled by a cyclostationary random process that is temporally and spatially independent with covariance matrix R x,k (n).

A. Mean weight error analysis

Collecting both sides of (5) for each nodes, using [START_REF] Gardner | Cyclostationarity: Half a century of research[END_REF] and [START_REF] Eweda | Stochastic analysis of the signed LMS algorithms for cyclostationary white Gaussian inputs[END_REF], we obtain:

Φ n = λΦ n-1 + R x,n . (19)
Then taking expectations of both sides leads to:

E{Φ n } = λE{Φ n-1 } + R x,n (20)
with the initial condition

E{Φ 0 } = δ -1 diag{I L , . . . , I L } and R x,n = E{R x,n } = diag R x,1 (n), . . . , R x,K (n) a block diagonal matrix. Note that Φ n only depends on R x,n .
The recursive relation ( 20) is very useful and crucial in the following theoretical analysis.

Replacing (1) into (8) and using definition w k,n , the instantaneous estimation error can be rewritten as:

e k,n = z k,n -x k,n w k,n-1 . (21)
Subtracting w from both sides of the recursive update equation [START_REF] Cattivelli | Diffusion recursive least-squares for distributed estimation over adaptive networks[END_REF], then using ( 13) and ( 21), leads to the intermediate weight error vector update equation:

ψ k,n = w k,n-1 + P k,n x k,n (z k,n -x k,n w k,n-1 ). ( 22
)
Subtracting w from both sides of combination relation [START_REF] Yu | Robust distributed diffusion recursive least squares algorithms with side information for adaptive networks[END_REF], and using (13), we have the weight error vector:

w k,n = ∈N k a k ψ ,n . (23)
Substituting ( 22) into ( 23), and using the previously introduced expressions ( 14)-( 16) and ( 18), then the recursive update equation of global weight error vector can be formulated as:

w n = A w n-1 -P n R x,n w n-1 + P n s xz,n (24)
with the matrix A = A ⊗ I L . Pre-multiplying both sides of (24) by P -1 n A -1 , and using [START_REF] Bermudez | Stochastic analysis of the LMS algorithm for cyclostationary colored Gaussian inputs[END_REF], it follows that:

Φ n A -1 w n = λΦ n-1 w n-1 + s xz,n . (25)
The purpose of these matrix manipulations was to remove matrix P n that multiplies vector s xz,n on the r.h.s. of [START_REF] Gao | Stochastic analysis of diffusion dual sign algorithm for cyclostationary white Gaussian inputs and nonstationary system[END_REF], which results in the mathematical intractability of the mean weight error analysis. Taking expectation of both sides of [START_REF] Eweda | Stochastic analysis of the recursive least squares algorithm for cyclostationary colored inputs[END_REF] and applying the statistical properties of noise z k,n , we obtain:

E{Φ n A -1 w n } = λE{Φ n-1 w n-1 }. (26)
For the sake of mathematical tractability, we follow [START_REF] Eweda | Stochastic analysis of the recursive least squares algorithm for cyclostationary colored inputs[END_REF] where the authors show that the following two approximations hold for cyclostationary inputs:

E{Φ n A -1 w n } ≈ E{Φ n }A -1 E{ w n } (27)
E{Φ n w n } ≈ E{Φ n }E{ w n }. (28)
To understand these two approximations, we further assume that ∆ n is the random fluctuation of Φ n around E Φ n , which can be expressed as [START_REF] Eweda | Stochastic analysis of the recursive least squares algorithm for cyclostationary colored inputs[END_REF]:

Φ n = E Φ n + ∆ n . (29)
Correspondingly, ( 27) and ( 28) can be written as:

E Φ n A -1 w n = E ∆ n A -1 w n +E Φ n A -1 E w n (30) E Φ n w n = E ∆ n w n + E Φ n E w n . (31)
As shown in [START_REF] Eweda | Stochastic analysis of the recursive least squares algorithm for cyclostationary colored inputs[END_REF], since random matrix ∆ n is small with respect to E Φ n for cyclostationary inputs, the first terms on the r.h.s. of ( 30) and ( 31) are small enough to be elimi-nated [START_REF] Eweda | Stochastic analysis of the recursive least squares algorithm for cyclostationary colored inputs[END_REF], respectively. Substituting the approximations ( 27) and ( 28) into ( 26), yields:

E{Φ n }A -1 E{ w n } = λ E{Φ n-1 }E{ w n-1 }. (32)
Pre-multiplying both sides of (32) by A E{Φ n } -1 leads to the mean weight error behavior of DRLS algorithm, given by:

E{ w n } = λ A E{Φ n } -1 E{Φ n-1 }E{ w n-1 }. ( 33
)
Theorem 1 (Convergence in the mean) Given a left-stochastic matrix A, the weight error vector of the DRLS algorithm with ATC strategy converges to a zero vector as n → ∞, that is,

lim n→∞ E{ w n } = 0 KL ( 34
)
which ensures that the estimate of DRLS algorithm is asymptotically unbiased and converge in the mean sense, i.e., lim n→∞ E{w k,n } = w for all nodes k. Proof: Since E{Φ n } -1 E{Φ n-1 } ≈ I KL as n → ∞ and 0 λ < 1, we can conclude from (33).

B. Mean-square weight error analysis

We shall now proceed with the mean-square transient analysis of the DRLS algorithm by considering the network meansquare deviation (MSD) defined as MSD n = tr K n /K, where K n = E w n w n is the second-order moment matrix of the global weight error vector across all nodes [START_REF] Sayed | Adaptive networks[END_REF], [START_REF]Diffusion adaptation over networks[END_REF]. Hence, our aim is to determine the recursive update equation of K n to evaluate the network transient MSD.

Post-multiplying (25) by its transpose, and taking the expectation of its both sides, we obtain:

E Φ n A -1 w n w n (A -1 ) Φ n = E s xz,n s xz,n + λ 2 E Φ n-1 w n-1 w n-1 Φ n-1 (35)
where the zero-valued cross terms have been eliminated. To make the analysis tractable, we introduce two approximations that hold for cyclostationary inputs [START_REF] Eweda | Stochastic analysis of the recursive least squares algorithm for cyclostationary colored inputs[END_REF]:

E Φ n A -1 w n w n (A -1 ) Φ n ≈ E{Φ n }A -1 K n (A -1 ) E{Φ n } (36) E Φ n w n w n Φ n ≈ E Φ n K n E Φ n . (37)
To construct these two approximations, we use (29) with the l.h.s. terms of (36) and ( 37):

E Φ n A -1 w n w n (A -1 ) Φ n = E{Φ n }A -1 K n (A -1 ) E{Φ n } + E{Φ n }E A -1 w n w n (A -1 ) ∆ n + E ∆ n A -1 w n w n (A -1 ) E{Φ n } (38)
+ E ∆ n A -1 w n w n (A -1 ) ∆ n E Φ n w n w n Φ n = E Φ n K n E Φ n + E ∆ n w n w n ∆ n + E ∆ n w n w n E Φ n + E Φ n E w n w n ∆ n . (39)
Similarly to the justifications for approximations ( 27) and ( 28), observe that (38) and (39) reasonably give rise to approximations (36) and (37). See [START_REF] Eweda | Stochastic analysis of the recursive least squares algorithm for cyclostationary colored inputs[END_REF] for more details. Their rationality and effectiveness will be validated by simulation results later. Substituting approximations (36) and ( 37) into (35), yields:

E{Φ n }A -1 K n (A -1 ) E{Φ n } = λ 2 E{Φ n-1 }K n-1 E{Φ n-1 } + S xz,n . (40)
Considering assumption A1 and the statistical independence of observation noise z k,n , we obtain:

S xz,n = E diag{z 2 1,n x 1,n x 1,n , . . . , z 2 K,n x K,n x K,n } (41) = diag σ 2 z,1 R x,1 (n), . . . , σ 2 z,K R x,K (n) = Σ z R x,n with the observation noise variance block diagonal matrix Σ z = diag{σ 2 z,1 I L , . . . , σ 2 z,K I L }.
Pre-multiplying (40) by A E{Φ n } -1 and post-multiplying (40) by E{Φ n } -1 A simultaneously, we finally obtain:

K n = A λ 2 E{Φ n } -1 E{Φ n-1 }K n-1 E{Φ n-1 }E{Φ n } -1 + E{Φ n } -1 Σ z R x,n E{Φ n } -1 A (42)
where the recursive relation [START_REF] Bershad | Stochastic analysis of the LMS algorithm for cyclostationary colored Gaussian and non-Gaussian inputs[END_REF] is invoked for the update of matrix K n . As a consequence, (42) enables us to investigate the network transient convergence behavior of the DRLS with ATC diffusion strategy in the mean-square sense. In contrast, almost all existing analyses of DRLS-type algorithms are limited to the network steady-state MSD for stationary inputs. More importantly, this work proposes a generic enough framework that can be readily applied to the transient analysis of other DRLS-type algorithms in different scenarios.

IV. SIMULATION RESULTS

Consider the network consisting of 20 nodes with topology illustrated in Fig. 1(a). Figure 1(b) shows the noise variance σ 2 z,k at each node. The optimal weight vector w ∈ R 32 arises from a standard normal distribution scaled by an exponential decaying factor 0.5, and normalized so that w 2 2 = 1. We considered the cyclostationary input model in [START_REF] Sayed | Adaptive networks[END_REF]. Each colored Gaussian sequence u k,n was generated independently across all nodes k according to the autoregressive model:

u k,n = ρu k,n-1 + σ u,k 1 -ρ 2 w k,n (43)
with ρ = 0.8 the normalized correlation factor, and w k,n a zero-mean white Gaussian noise with unit variance. In this way, the (i, j)-th entry of autocorrelation matrix of u k,n was given by

[R u,k ] ij = E u k,n-i+1 u k,n-j+1 = σ 2 u,k ρ |i-j| .
A pulsed power time variation model was used for σ x,k (n); see [START_REF] Gao | Stochastic analysis of diffusion dual sign algorithm for cyclostationary white Gaussian inputs and nonstationary system[END_REF]. The duty cycle α, the lowest amplitude V l , and the highest amplitude V h were set to 0.5, 2 × 10 -3 , and 2, respectively. Three different periods of cyclostationary variations were considered: T = 4, 32 and 512. We set the forgetting factor λ to 0.995 for RLS and DRLS algorithms. Initialization parameter δ used in [START_REF] Bershad | Stochastic analysis of the LMS algorithm for cyclostationary colored Gaussian and non-Gaussian inputs[END_REF] was set to different values depending on the scenario; see Fig. 2. Figure 2 shows that the DRLS algorithm with ATC diffusion outperformed the non-cooperative RLS algorithm. We observe that the transient and steady-state MSD periodically fluctuated with the same slow repetition period (T = 512) as the pulsed input data. On the contrary, the convergence of the DRLS was not affected by the moderate (T = 32) and fast (T = 4) variations in the variance of input data. It can be also seen in Fig. 2 that the theoretical transient MSD curves of DRLS algorithm predicted by (42) coincide with the empirical MSD curves, which demonstrates the effectiveness of approximations (36) and (37). As a conclusion, the perfect match between theoretical predictions and simulated results demonstrate the accuracy of the analytical models derived in this paper.

V. CONCLUSION

In this letter, we theoretically analyzed the transient behavior of the DRLS algorithm in the mean and mean-square sense. The derived analytical models allow to thoroughly investigate the variations in the transient convergence behavior. Simulation results illustrated the accuracy and effectiveness of the theoretical findings.

  Pulsed fast variation (T = 4).

Fig. 2 .

 2 Fig. 2. Comparisons of empirical and theoretical transient MSD of RLS and DRLS algorithms for ρ = 0.8 and pulsed variations of input variance.

Fig. 1 .

 1 Fig. 1. Experimental setup.

Manuscript received MM DD, 2021; revised MM DD, 2021. This work was supported in part by the National NSFC under Grant 61701200 and Grant 61671382.