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

Renato Luis Garrido Cavalcante, Noor Ul Ain, Lorenzo Miretti, Slawomir Stanczak

TL;DR

The paper tackles max-min power control in cellular and cell-less massive MIMO by moving beyond the traditional use-and-then-forget (UatF) rate bound. It develops a monotonic, scalable, and positive (MSP) function framework and a corresponding fixed-point algorithm to optimize tight ergodic-rate bounds that utilize instantaneous CSI at the decoder, enabling direct optimization rather than relying on UatF surrogates. The authors prove the existence and convergence of a fixed-point method to the global solution of the weighted max-min problem under these bounds and show how Monte Carlo sampling can be employed to estimate the required expectations. Numerical results in a cell-less setting demonstrate substantial gains over UatF-based schemes, particularly in scenarios with zero-mean channels and simple beamforming, underscoring the practical impact of directly optimizing under alternative ergodic-rate bounds.

Abstract

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.

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

TL;DR

The paper tackles max-min power control in cellular and cell-less massive MIMO by moving beyond the traditional use-and-then-forget (UatF) rate bound. It develops a monotonic, scalable, and positive (MSP) function framework and a corresponding fixed-point algorithm to optimize tight ergodic-rate bounds that utilize instantaneous CSI at the decoder, enabling direct optimization rather than relying on UatF surrogates. The authors prove the existence and convergence of a fixed-point method to the global solution of the weighted max-min problem under these bounds and show how Monte Carlo sampling can be employed to estimate the required expectations. Numerical results in a cell-less setting demonstrate substantial gains over UatF-based schemes, particularly in scenarios with zero-mean channels and simple beamforming, underscoring the practical impact of directly optimizing under alternative ergodic-rate bounds.

Abstract

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.

Paper Structure

This paper contains 7 sections, 4 theorems, 17 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Mathematical preliminaries
  3. Notation and definitions
  4. Framework for Power Control with Tight Ergodic Rate Bounds
  5. Power control in multi-user MIMO
  6. Simulations
  7. Conclusion

Key Result

Proposition 1

Let $T:{\mathbb R}_{++}^N\to{\mathbb R}_{++}^N$ be an MSP mapping. Given a monotone norm $\|\cdot\|$, there exists exactly one tuple $(\gamma^\star,{\boldsymbol{x}}^\star)\in{\mathbb R}_{++}\times{\mathbb R}_{++}^N$ solving the so-called conditional (nonlinear) eigenvalue problem: Furthermore, the solution $(\gamma^\star,{\boldsymbol{x}}^\star)$ to (eq.condeig) (called the conditional eigenpair)

Figures (1)

  • Figure 1: Convergence of the worst-user rate (OER bound) under centralized MMSE beamforming. For reference, the minimum OER bound obtained with the solution to the UatF max-min problem is included.

Theorems & Definitions (10)

  • Definition 1
  • Proposition 1
  • Proposition 2
  • proof
  • Remark 1
  • Remark 2
  • Remark 3
  • Proposition 3
  • proof
  • Corollary 1