# An Effective Adaptive Combination Strategy for Distributed Learning Network
## 3 Adaptive Combination Scheme
### Fixed-Point Iteration Solution
Applying the constraint \(\mathbf{1}_{n_{k}}^{T}\mathbf{b}_{k}(t)=1\) and pre-multiplying both sides of (13) by \(\mathbf{1}_{n_{k}}^{T}\) yields the Lagrange multiplier \(\alpha\)  
\[\alpha=\frac{\mathbf{1}_{n_{k}}^{T}(\mathbf{b}_{k}(t-1)-\eta_{k}(t)\mathbf{\Gamma}_{k}(t- 1)\Re(\mathbf{Q}_{\mathbf{\tau}_{k}})\mathbf{b}_{k}(t-1))-1}{\eta_{k}(t)\mathbf{1}_{n_{k}}^{T} \mathbf{\Gamma}_{k}(t-1)\mathbf{1}_{n_{k}}}. \tag{14}\]  
Substituting \(\alpha\) into (13), and using the constraint \(\mathbf{1}_{n_{k}}^{T}\mathbf{b}_{k}(t-1)=1\) again, we can obtain the update of combiners \(\mathbf{b}_{k}(t)\),  
\[\mathbf{b}_{k}(t)=\mathbf{b}_{k}(t-1)-\eta_{k}(t)\mathbf{G}_{k}(t-1)\mathbf{\Gamma}_{k}(t-1) \Re(\mathbf{Q}_{\mathbf{\tau}_{k}})\mathbf{b}_{k}(t-1). \tag{15}\]  
where \(\mathbf{G}_{k}(t-1)\triangleq\mathbf{I}_{n_{k}}-\frac{\mathbf{D}_{k}(t-1)\mathbf{b}_{k}(t-1) \mathbf{1}_{n_{k}}^{T}}{\mathbf{1}_{n_{k}}^{T}\mathbf{D}_{k}(t-1)\mathbf{b}_{k}(t-1)}\) with \(\mathbf{I}_{n_{k}}\) denoting the \(n_{k}\times n_{k}\) identity matrix. The adaptive combiners (15) can be updated by two incremental steps  
\[\mathbf{g}_{k}(t) =\mathbf{\xi}_{k}(t-1)\Re(\mathbf{Q}_{\mathbf{\tau}_{k}})\mathbf{b}_{k}(t-1) \tag{16}\] \[\mathbf{b}_{k}(t) =\mathbf{b}_{k}(t-1)-\eta_{k}(t)\mathbf{g}_{k}(t) \tag{17}\]  
where  
\[\mathbf{\zeta}_{k}(t-1)=\mathbf{G}_{k}(t-1)\mathbf{\Gamma}_{k}(t-1). \tag{18}\]  
Then, we can obtain the combiner \(\mathbf{a}_{k}(t)\)  
\[\mathbf{a}_{k}(t)=\mathbf{\mathrm{P}}_{k}\mathbf{b}_{k}(t) \tag{19}\]  
which can be used in (3) to update the local weight estimate adaptively. We also define the adaptive combination matrix \(\mathbf{A}(t)\in\mathbb{R}^{N\times N}\) where \(\mathbf{a}_{k}(t)\) is the \(k\)th column vector of it. Please note that \(-\mathbf{g}_{k}(t)\) in (16) can be seen as the product of \(\mathbf{\zeta}_{k}(t-1)\) and the gradient \(\
abla_{\mathbf{b}_{k}}J(\mathbf{b}_{k}(t-1))=\Re(\mathbf{Q}_{\mathbf{\tau}_{k}})\mathbf{b}_{k}(t-1)\), which means \(\mathbf{\zeta}_{k}(t-1)\) can be seen an auxiliary matrix to adjust the update direction \(\
abla_{\mathbf{b}_{k}}J(\mathbf{b}_{k}(t-1))\). On the one hand, note that \(\mathbf{G}_{k}(t-1)\) in (18) is a projection matrix [22], which enables the update direction \(-\mathbf{g}_{k}(t)\) orthogonal to the vector \(\mathbf{1}_{n_{k}}\), i.e., \(\mathbf{1}_{n_{k}}^{T}\mathbf{g}_{k}(t)=0\). Then we have \(\mathbf{1}_{n_{k}}^{T}\mathbf{b}_{k}(t)=\mathbf{1}_{n_{k}}^{T}\mathbf{b}_{k}(t-1)=\cdots=\mathbf{ 1}_{n_{k}}^{T}\mathbf{b}_{k}(0)=1\) if the initial combiners satisfy \(\mathbf{1}_{n_{k}}^{T}\mathbf{b}_{k}(0)=1\). On the other hand, since as a whole, \(\mathbf{\zeta}_{k}(t-1)\) is a positive semi-definite symmetric matrix, the update direction \(-g_{k}(t)\) is still the descent direction of the cost function (7) [21]. Instead of using the positive semi-definite symmetric matrix \(\zeta_{k}(t-1)=G_{k}(t-1)\Gamma_{k}(t-1)\), the classic AC [12] consider \(\zeta_{k}(t-1)\) being replaced by the orthogonal projection matrix characterized by \(\mathbf{1}_{n_{k}}\), i.e., \(\mathbf{1}_{n_{k}}-\frac{\mathbf{1}_{n_{k}}\mathbf{1}_{n_{k}}^{T}}{n_{k}}\), which actually limits the update direction of the adaptive combiners to be lie in the hyperplane spanned by \(\mathbf{1}_{n_{k}}\) and \(\
abla_{\mathbf{b}_{k}}J(\mathbf{b}_{k}(t-1))\). In fact, we find that (16) and (17) could reduce to the classic AC [12] when \(\mathbf{D}_{k}(t)=\text{diag}\{\mathbf{b}_{k}(t)\}^{-1}\). We now consider optimizing the learning factor \(\eta_{k}(t)\). Substituting (16) and (17) into (7) yields a cost function regarding the learning factor \(\eta_{k}(t)\),  
\[h(\eta_{k}(t)) =\mathbf{g}_{k}^{T}(t)\Re(\mathbf{Q}_{\mathbf{\tau}_{k}})\mathbf{g}_{k}(t)\eta_{k }^{2}(t)\] \[-2\mathbf{g}_{k}^{T}(t)\Re(\mathbf{Q}_{\mathbf{\tau}_{k}})\mathbf{b}_{k}(t-1)\eta _{k}(t)\] \[+\mathbf{b}_{k}^{T}(t-1)\Re(\mathbf{Q}_{\mathbf{\tau}_{k}})\mathbf{b}_{k}(t-1). \tag{20}\]  
Obviously, \(h(\eta_{k}(t))\) is a quadratic (convex) function with respect to \(\eta_{k}(t)\).