--- title: 'Localized Cost Functions: Theory & Applications' url: https://www.emergentmind.com/topics/localized-cost-functions type: topic --- # Localized Cost Functions: Theory & Applications Searching arXiv for the cited papers to ground the article in current records. Localized cost functions are objective constructions in which cost evaluation, optimization, or consistency enforcement is restricted to a subset of variables, coordinates, orbitals, or samples rather than being expressed only as a single globally coupled term. In the literature considered here, the notion appears in at least four technically distinct forms: distributed convex optimization with local domains, global cost functions in cost function networks under local transformations, selectively localized spread functionals for Wannier orbitals, and sample-wise local regression objectives coupled by graph regularization [1305.1885] [1502.02414] [1407.5124] [1603.06743]. The common theme is not a single formalism but a recurring design principle: exploit structure that is intrinsically local while still solving a global optimization or inference problem. ## 1. Conceptual scope and formal uses In distributed optimization, locality is defined by a family of index sets \(S_p \subseteq \{1,\dots,n\}\), one per node \(p\), so that each local function \(f_p\) depends only on \(x_{S_p}\). The resulting problem is \[ \min_x \sum_{p=1}^P f_p(x_{S_p}), \] with \(x \in \mathbb{R}^n\), and is explicitly described as a “localized” or “partial-variable” optimization problem [1305.1885]. In cost function networks, the relevant object is a global cost function: a mapping assigning an integer cost to every assignment of an unbounded set of variables. Locality enters through scope-restricted cost functions \(W_S\), unary costs, and Equivalence Preserving Transformations (EPTs), which shift cost between a scope \(S_1\) and a sub-scope \(S_2 \subset S_1\) without changing the cost of any full assignment [1502.02414]. In electronic-structure theory, localization refers to the spread functional of Wannier functions. The standard Marzari–Vanderbilt spread \(\Omega_{MV}\) is replaced by a selectively localized functional \(\Omega_{SL}\) that minimizes spread only for a chosen subset \(J' < J\) of “objective Wannier functions,” while the remaining functions are left as spectators [1407.5124]. In high-dimensional regression, the “localized Lasso” assigns a distinct coefficient vector \(w_i \in \mathbb{R}^d\) to each sample \(i\). Its objective combines per-sample squared loss, sample-wise network regularization, and sample-wise exclusive group sparsity: \[ \min_W \sum_{i=1}^n (y_i-w_i^\top x_i)^2 +\lambda_1 \sum_{ic} x_\ell^{(j),k}, \] solves \[ x_{S_p}^{(p),k+1} = \arg\min_{z\in\mathbb{R}^{|S_p|}} f_p(z) +\sum_{\ell\in S_p} \left[ (v_\ell^{(p),k})^\top z_\ell +\frac{\rho}{2}D_{p,\ell}\|z_\ell\|^2 \right], \] and then exchanges the updated coordinate copies with neighbors. After all colors update, the dual step is \[ \gamma_\ell^{(p),k+1} \leftarrow \gamma_\ell^{(p),k} +\rho \sum_{j\in \mathcal{N}_p\cap \mathcal{V}_\ell} \bigl(x_\ell^{(p),k+1}-x_\ell^{(j),k+1}\bigr). \] For non-connected variables, the method introduces Steiner paths by solving a centralized minimum-Steiner-tree problem with terminals \(\mathcal{V}_\ell\), producing an augmented induced subgraph \(\mathcal{G}_\ell'\). The same color-ADMM scheme is then applied on enlarged local domains \(S_p' = S_p \cup \{\ell \mid p \text{ is Steiner for } \ell\}\). For Steiner nodes, the corresponding coordinate update reduces to a simple quadratic and has closed form because \(f_p\) does not depend on that coordinate. The convergence theorem states that, under the standing convexity assumptions and assuming either \(C=2\) or strong convexity of each group-cost \(\sum_{p\in \mathcal{C}_c} f_p\), the Extended ADMM scheme converges to a primal–dual solution. Communication complexity is described in terms of locality: per iteration each node \(p\) sends \(|S_p|\) real numbers to each neighbor in \(\mathcal{N}_p\), only for shared coordinates, so one ADMM iteration costs \(\sum_p |S_p|\) scalar transmissions. In the reported experiments, the algorithm required fewer communications to converge than prior distributed methods. ## 3. Local transformations and tractability in cost function networks The theory developed in "Tractability and Decompositions of Global Cost Functions" places locality at the level of cost scope and cost movement rather than distributed communication [1502.02414]. A Cost Function Network is \((X,W,\top)\), where \(X=\{x_1,\dots,x_n\}\) is a set of discrete variables with domains \(D(x_i)\), and \(W\) contains a zero-arity constant \(W_\emptyset\), unary costs \(W_i(x_i)\), and higher-arity cost functions \(W_S:D^S\to[0,\dots,\top]\). The cost of a full assignment \(\ell\in D^X\) is \[ \mathrm{cost}(\ell)=\bigoplus_{W_S\in W} W_S(\ell[S]), \] with \(([0,\dots,\top],\oplus,\le)\) defined by \(a\oplus b=\min(\top,a+b)\). Locality is operationalized through EPTs. An \(r\)-EPT shifts cost between a larger scope \(S_1\) and a sub-scope \(S_2\subset S_1\) with \(|S_2|=r\), while preserving the cost of every complete assignment. In the Project operation, for a tuple \(\ell\in D^{S_2}\) and shift amount \(\alpha\) satisfying \[ -W_{S_2}(\ell) \le \alpha \le \min_{\ell' \in D^{S_1}:\,\ell'[S_2]=\ell} W_{S_1}(\ell'), \] one updates \[ W_{S_2}(\ell)\leftarrow W_{S_2}(\ell)\oplus \alpha, \] and \[ W_{S_1}(\ell')\leftarrow W_{S_1}(\ell')\ominus \alpha \] for every \(\ell'\) extending \(\ell\), where \(a\ominus b = a-b\) if \(a\neq \top\), else \(\top\). The central question is tractable projection-safety. A global cost function \(F\) is tractable if minimizing \(F\) can be done in polynomial time, and it is tractable \(r\)-projection-safe if it remains tractable after any sequence of \(r\)-EPTs. The paper gives a sharp dependence on \(r\). Any tractable global cost function is \(0\)-projection-safe. If \(F\) is tractable and returns only finite costs, then for any \(r\ge 2\), \(F\) is not \(r\)-projection-safe unless \(P=NP\). The case \(r=1\) is explicitly identified as borderline: some tractable global cost functions remain tractable after unary EPTs, others do not. The positive side of the \(r=1\) boundary is represented by flow-based global cost functions such as soft AllDifferent variable-based and soft GCC, which remain flow-based after unary EPTs and therefore are tractable \(1\)-projection-safe. The negative side is illustrated by a cost function \(W_{2SAT}\) encoding Max-2-SAT: minimizing \(F\) is polynomial, but \(1\)-extensions can encode Weighted-SAT with weight limit \(k\), which is NP-complete. Local cost movement is therefore not innocuous; its tractability depends decisively on the arity of the scope onto which cost is projected. Two structural mechanisms preserve tractability. The first is polynomial DAG-filterability. A safe DAG-filter represents a cost function \(W_S\) as a directed acyclic graph whose internal nodes aggregate sub-costs through associative–commutative functions and whose leaves are unary costs. If the graph is polynomial in \(|S|+|D|\) and projections and extensions distribute safely through the DAG, then minimization can be done by bottom-up dynamic programming and tractability is preserved under unary EPTs. The paper provides examples based on soft grammar cost, soft Among cost, soft Regular cost, and \(W_{\mathrm{Max}}(S,f)\), with explicit polynomial complexities. The second mechanism is decomposition into a Berge-acyclic bounded-arity network. A \(p\)-network-decomposition of \(W_S\) is a polynomial-size CFN over \(X\cup E\) with cost functions of arity at most \(p\) such that \[ W_S(\ell)= \min_{\ell' \in D^{X\cup E}:\,\ell'[X]=\ell} \bigoplus_{W_T\in F} W_T(\ell'[T]). \] Under this form, soft local consistencies such as Terminal Directional Arc Consistency (T-DAC*) and Virtual Arc Consistency (VAC) can emulate dynamic programming. The paper proves that, with a suitable variable order, T-DAC* on \(W_S\) yields the same updated unary cost on the last variable as T-DAC* on the full decomposition, and that VAC yields the same lower bound \(W_\emptyset\) on either representation. A plausible implication is that locality in cost function networks is ambivalent: it can be the source of tractability when confined to unary projections or safe decompositions, but it can also destroy tractability when cost is pushed onto larger scopes. ## 4. Selective localization of spread functionals in Wannier theory In "Selectively Localized Wannier Functions," localization is not attached to variable subsets in an optimization network, but to the spatial spread of a chosen orbital subspace [1407.5124]. The standard Marzari–Vanderbilt spread functional for \(J\) Wannier functions \(|R\,n\rangle\) is \[ \Omega_{MV}\equiv \sum_{n=1}^J \left[\langle r^2\rangle_n-\bar r_n^2\right], \] with \(\bar r_n = \langle 0n|r|0n\rangle\) and \(\langle r^2\rangle_n=\langle 0n|r^2|0n\rangle\). On a discrete \(k\)-mesh, \[ \Omega_{MV}=\Omega_{IOD}+\Omega_D, \] where \[ \Omega_{IOD} = \frac1N \sum_{n=1}^J \sum_{k,b} w_b\bigl[1-|M_{nn}^{k,b}|^2\bigr], \] \[ \Omega_D = \frac1N \sum_{n=1}^J \sum_{k,b} w_b \bigl[\Im\ln M_{nn}^{k,b}+b\cdot \bar r_n\bigr]^2, \] and \(M_{mn}^{k,b}=\langle u_{m,k}|u_{n,k+b}\rangle\). The selectively localized functional restricts the outer summation to a subset \(J'