Bi-Criteria Radio Spectrum Sharing With Subspace-Based Pareto Tracing

Abstract

Radio spectrum is a scarce resource. To meet demands, new wireless technologies must operate in shared spectrum over unlicensed bands (coexist). We consider coexistence of Long-Term Evolution (LTE) License-Assisted Access (LAA) with incumbent Wi-Fi systems. Our scenario consists of multiple LAA and Wi-Fi links sharing an unlicensed band; we aim to simultaneously optimize performance of both coexistence systems. To do this, we present a technique to continuously estimate the Pareto frontier of parameter sets (traces) which approximately maximize all convex combinations of network throughputs over network parameters. We use a dimensionality reduction approach known as active subspaces to determine that this near-optimal parameter set is primarily composed of two physically relevant parameters. A choice of two-dimensional subspace enables visualizations augmenting explainability and the reduced-dimension convex problem results in approximations which dominate random grid search.

Keywords: LAA; LTE; Pareto trace; Wi-Fi; active subspace; wireless coexistence.

Fig. 1.

System model consists of multiple LAA and Wi-Fi links sharing an unlicensed band.

Fig. 2.

Stretch sampling a two-dimensional zonotope. The boundary of the zonotope 𝓨 (solid black curve) is depicted along with the boundary of the convex hull of data (dashed-black curve) and N = 10, 000 projected random samples (black dots). Axes correspond to independent active coordinates. The resulting triangulation (gray mesh) is informed by 25 uniformly sampled points per boundary and exterior Voronoi centers (gray dots) constitute the new stretch samples used to improve the ridge approximations.

Fig. 3.

Comparison of the eigenvalue decay for Wi-Fi and LAA throughputs. The eigenvalues resulting from Algorithm 1 are shown as the blue stem plot (solid blue dots) corresponding to N = 10, 000 Monte Carlo samples. This is contrasted with eigenvalues of the convex quadratic Hessian (yellow circles) computed using the N = 10, 000 unperturbed random function evaluations defining the convex problem (11) over the full-dimensional parameter space. Additionally, the eigenvalues of the active subspaces resulting from the full-dimensional convex quadratic fit as a surrogate (blue circles) is contrasted to the unbiased finite difference approach (solid blue dots).

Fig. 4.

Condition number of convex combinations of quadratic Hessians. The convex combination resulting from (11) over the full dimensional space (orange) is contrasted with the significantly improved conditioning over the mixed two-dimensional subspace (blue). Results are depicted at 100 points along the corresponding quadratic traces (10). Black crosses indicate points along the trace that do not pass through the domain.

Fig. 5.

Pareto trace of convex quadratic ridge profiles. The quadratic Pareto trace (red curve and dots) is overlaid on a shadow plot over the mixed coordinates (colored scatter) with the projected bounds and vertices (zonotope) of the domain (black dots and lines), 𝓨. The quadratic approximations (colored contours) are contrasted against the true function evaluations represented by the colors of the scatter. Also depicted is the projection of the non-dominated domain values from the N = 10, 000 random samples (black circles). The active coordinate trace (red dots and curve) begins at the upper left-most boundary with near maximum quadratic Wi-Fi throughput and we move (smoothly) along a trajectory to the lower left-most boundary obtaining near maximum quadratic LAA throughput—maintaining an approximately best trade-off over the entire curve restricted to 𝓨. This is contrasted with a linear interpolation of full dimensional left (τ = 0) and right (τ = 1) approximations to (7) (blue dots and curve) and 15 successive approximations to (7) with uniform discretization of τ ∈ [0, 1] (blue circles) as the “brute force” solution.

Fig. 6.

Approximation of the Pareto front resulting from the quadratic trace. The conditional Pareto front (solid-red curve) is contrasted with the non-dominated random throughput values (black circles) and scatter of N = 10, 000 random responses colored according to the averaged throughput (τ = 0.5) response. The solid-red curve is a Monte Carlo approximation of the conditional Pareto front (22) with corresponding evaluations (overlaid red dots) used to compute conditional means. The dashed-red curve and coincident dots represent a discretization over geodesic (19) between left (τ = 0) and right (τ = 1) inactive maximizing arguments on the manifold of approximately Pareto optimal solutions (18). Also shown is the linear Pareto front (21) through the full parameter space (blue curve and coincident dots) and a set of 15 successive maximizations (7) over a uniform discretization of τ ∈ [0, 1] (blue circles) as the “brute force” solution.