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Making Every Bit Count for $A$-Optimal State Estimation

Cameron Khanpour, Daniel Turizo, Samuel Talkington

Abstract

We study the problem of controlling how a limited communication bandwidth budget is allocated across heterogeneously quantized sensor measurements. The performance criterion is the trace of the error covariance matrix of the linear minimum mean square error (LMMSE) state estimator, i.e., an $A$-optimal design criterion. Minimizing this criterion with a bit budget constraint yields a nonconvex optimization problem. We derive a formula that reduces each evaluation of the gradient to a single Cholesky factorization. This enables efficient optimization by both a projection-free Frank-Wolfe method (with a computable convergence certificate) and an interior point method with L-BFGS Hessian approximation over the problem's continuous relaxation. A largest remainder rounding procedure recovers integer bit allocations with a bound on the quality of the rounded solution. Numerical experiments in IEEE power grid test cases with up to 300 buses compare both solvers and demonstrate that the analytic gradient is the key computational enabler for both methods. Additionally, the heterogeneous bit allocation is compared to standard uniform bit allocation on the 500 bus IEEE power grid test case.

Making Every Bit Count for $A$-Optimal State Estimation

Abstract

We study the problem of controlling how a limited communication bandwidth budget is allocated across heterogeneously quantized sensor measurements. The performance criterion is the trace of the error covariance matrix of the linear minimum mean square error (LMMSE) state estimator, i.e., an -optimal design criterion. Minimizing this criterion with a bit budget constraint yields a nonconvex optimization problem. We derive a formula that reduces each evaluation of the gradient to a single Cholesky factorization. This enables efficient optimization by both a projection-free Frank-Wolfe method (with a computable convergence certificate) and an interior point method with L-BFGS Hessian approximation over the problem's continuous relaxation. A largest remainder rounding procedure recovers integer bit allocations with a bound on the quality of the rounded solution. Numerical experiments in IEEE power grid test cases with up to 300 buses compare both solvers and demonstrate that the analytic gradient is the key computational enabler for both methods. Additionally, the heterogeneous bit allocation is compared to standard uniform bit allocation on the 500 bus IEEE power grid test case.

Paper Structure

This paper contains 19 sections, 8 theorems, 58 equations, 4 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Contributions
  4. Problem Formulation
  5. Measurement Model and Estimation
  6. Bandwidth Allocation Problem
  7. Bit Allocation Procedure
  8. Gradient and Linear Minimization Oracle
  9. Convergence Guarantee
  10. Interior Point Method with Analytic Gradient
  11. Rounding Procedure
  12. Numerical Experiments
  13. Comparison of Solver Computation Time
  14. Solver Configuration
  15. Test Cases
  16. ...and 4 more sections

Key Result

Lemma 1

For each measurement $i=1,\dots,m$, Equivalently, In particular, $\partial F/\partial b_i < 0$ whenever $\boldsymbol{h}_i \neq \boldsymbol{0}$. Each gradient evaluation reduces to a single Cholesky factorization of $\boldsymbol{M}(\boldsymbol{b})$: the diagonal entries $\boldsymbol{h}_i^\top \boldsymbol{C}_{\boldsymbol{\varepsilon}}^2 \boldsymbol{h

Figures (4)

  • Figure 3: Bit allocation on a sensor network. Each node $i$ receives $b_i$ bits ( $\textcolor{bitgold}{\blacksquare}$ = 1 bit) subject to budget $\textcolor{bitgold}{\sum_i b_i} \le B$. Inset: 2-bit quantization maps an analog measurement to $2^{b_i}\!=\!4$ discrete levels.
  • Figure 4: Frank--Wolfe iteration on the budget simplex $\mathcal{B}$ for $m=3$. The LMO selects vertex $\boldsymbol{s}^{(t)}$; the next iterate lies on $[\boldsymbol{b}^{(t)}, \boldsymbol{s}^{(t)}]$ with step $\gamma_t$.
  • Figure 5: Percentage improvement of the optimized heterogeneous bit allocation \ref{['eq:bit-alloc-relaxed']} over uniform allocation $\boldsymbol{b} = \boldsymbol{1}\cdot\lfloor B/m \rfloor$ on the case500 test system ($m=499$ sensors). Each boxplot shows 30 randomized instances at the given budget level $c = B/m$. Shaded columns denote integer values of $c$.
  • Figure 6: Median solve time vs. sensor-to-state ratio $m/d$ for $d \in \{10, 20\}$ with $B = 2d$. Error bars show the IQR over 30 instances. Frank--Wolfe scales linearly in $m$ (for $m \gg d$) with $O(d^3 + d^2 m)$ per-iteration cost. Ipopt fails to converge entirely beyond $m/d \approx 200$ ($d{=}10$) and $100$ ($d{=}20$).

Theorems & Definitions (21)

  • Remark : Convexity in precision space
  • Lemma 1: Gradient with respect to bits
  • proof
  • Proposition 1: Budget saturation
  • proof
  • Proposition 2: Lipschitz continuity of the gradient
  • proof
  • Proposition 3: Closed-form LMO
  • proof
  • Corollary 3.1: Explicit Frank--Wolfe gap
  • ...and 11 more