Beyond the Use-and-then-Forget (UatF) Bound: Fixed Point Algorithms for Statistical Max-Min Power Control

We introduce mathematical tools and fixed point algorithms for optimal statistical max-min power control in cellular and cell-less massive MIMO systems. Unlike previous studies that rely on the use-and-then-forget (UatF) lower bound on Shannon achievable (ergodic) rates, our proposed framework can deal with alternative bounds that explicitly consider perfect or imperfect channel state information (CSI) at the decoder. In doing so, we address limitations of UatF-based power control algorithms, which inherit the shortcomings of the UatF bound. For example, the UatF bound can be overly conservative: in extreme cases, under fully statistical (nonadaptive) beamforming in zero-mean channels, the UatF bound produces trivial (zero) rate bounds. It also lacks scale invariance: merely scaling the beamformers can change the bound drastically. In contrast, our framework is compatible with information-theoretic bounds that do not suffer from the above drawbacks. We illustrate the framework by solving a max-min power control problem considering a standard bound that exploits instantaneous CSI at the decoder.

💡 Research Summary

This paper addresses a fundamental limitation in the design of power control algorithms for massive MIMO and cell‑less networks: the widespread reliance on the Use‑and‑then‑Forget (UatF) lower bound on ergodic Shannon rates. While the UatF bound often yields tractable optimization problems, it replaces the instantaneous effective channel by its mean and treats the deviation as uncorrelated noise. Consequently, in scenarios with zero‑mean channels and non‑adaptive (statistical) beamforming, the signal‑to‑interference‑plus‑noise ratio (SINR) numerator collapses, leading to a trivial zero‑rate bound. Moreover, the bound is not scale‑invariant; scaling the beamforming vectors can dramatically alter the bound, even though a decoder that knows the scaling should be unaffected. These shortcomings become especially problematic in cell‑less architectures and in cooperative multi‑cell deployments.

To overcome these issues, the authors develop a new mathematical framework based on a class of functions they call “MSP” (Monotonic, Scalable, Positive). An MSP function f : ℝ⁺ⁿ → ℝ⁺ satisfies three properties on the strictly positive orthant: (i) monotonicity (x ≤ y ⇒ f(x) ≤ f(y)), (ii) strict scalability (α > 1 ⇒ f(αx) < αf(x)), and (iii) a strict positive lower bound away from zero. The authors show that MSP functions are a proper superset of the classic standard interference functions, but they retain key analytical features: they are closed under addition, positive scaling, and finite maxima/minima; and any MSP mapping T : ℝ⁺ⁿ → ℝ⁺ⁿ is a contraction under Thompson’s metric, guaranteeing at most one fixed point.

Two central propositions are proved. Proposition 1 establishes that for any MSP mapping T, there exists a unique pair (γ*, x*) solving the conditional eigenvalue problem T(x) = γx with ‖x‖ = 1. Moreover, the normalized mapping G(x) = T(x)/‖T(x)‖ has x* as its unique fixed point, and the iterative scheme x^{(k+1)} = G(x^{(k)}) converges globally to x*. Proposition 2 shows that if a random function g(x, ω) and a deterministic function h(x) are such that for almost every ω the ratio z(x, ω) = h(x)/g(x, ω) is an MSP function, then the deterministic function f(x) = h(x)·E