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Finite Horizon Optimization for Large-Scale MIMO

Yi Feng, Kaiming Shen

TL;DR

This paper addresses the computational bottleneck of WMMSE beamforming in large-scale MIMO by reframing the problem through fractional programming and a finite-horizon optimization lens. It converts the WSR beamforming problem into an unconstrained quadratic form using FP, then solves it with a gradient-descent scheme that runs only a finite number of iterations, with step sizes optimized via Chebyshev polynomial theory. The main contributions are (i) a finite-horizon gradient-descent method that avoids large $M\times M$ inversions, (ii) a principled step-size design using Chebyshev polynomials that accelerates convergence, and (iii) an extension to multi-cell networks with per-cell optimization. Simulations show substantial CPU-time gains over WMMSE while delivering competitive sum-rate performance, highlighting practical benefits for real-time precoding in massive MIMO systems.

Abstract

Large-scale multiple-input multiple-output (MIMO) is an emerging wireless technology that deploys thousands of transmit antennas at the base-station to boost spectral efficiency. The classic weighted minimum mean-square-error (WMMSE) algorithm for beamforming is no suited for the large-scale MIMO because each iteration of the algorithm then requires inverting a matrix whose size equals the number of transmit antennas. While the existing methods such as the reduced WMMSE algorithm seek to decrease the size of matrix to invert, this work proposes to eliminate this large matrix inversion completely by applying gradient descent method in conjunction with fractional programming. Furthermore, we optimize the step sizes for gradient descent from a finite horizon optimization perspective, aiming to maximize the performance after a limited number of iterations of gradient descent. Simulations show that the proposed algorithm is much more efficient than the WMMSE algorithm in optimizing the large-scale MIMO precoders.

Finite Horizon Optimization for Large-Scale MIMO

TL;DR

This paper addresses the computational bottleneck of WMMSE beamforming in large-scale MIMO by reframing the problem through fractional programming and a finite-horizon optimization lens. It converts the WSR beamforming problem into an unconstrained quadratic form using FP, then solves it with a gradient-descent scheme that runs only a finite number of iterations, with step sizes optimized via Chebyshev polynomial theory. The main contributions are (i) a finite-horizon gradient-descent method that avoids large inversions, (ii) a principled step-size design using Chebyshev polynomials that accelerates convergence, and (iii) an extension to multi-cell networks with per-cell optimization. Simulations show substantial CPU-time gains over WMMSE while delivering competitive sum-rate performance, highlighting practical benefits for real-time precoding in massive MIMO systems.

Abstract

Large-scale multiple-input multiple-output (MIMO) is an emerging wireless technology that deploys thousands of transmit antennas at the base-station to boost spectral efficiency. The classic weighted minimum mean-square-error (WMMSE) algorithm for beamforming is no suited for the large-scale MIMO because each iteration of the algorithm then requires inverting a matrix whose size equals the number of transmit antennas. While the existing methods such as the reduced WMMSE algorithm seek to decrease the size of matrix to invert, this work proposes to eliminate this large matrix inversion completely by applying gradient descent method in conjunction with fractional programming. Furthermore, we optimize the step sizes for gradient descent from a finite horizon optimization perspective, aiming to maximize the performance after a limited number of iterations of gradient descent. Simulations show that the proposed algorithm is much more efficient than the WMMSE algorithm in optimizing the large-scale MIMO precoders.

Paper Structure

This paper contains 7 sections, 4 theorems, 61 equations, 7 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. System Model
  3. Conversion to Quadratic Program
  4. Finite Horizon Optimization
  5. Multi-Cell Case
  6. Simulation Results
  7. Conclusion

Key Result

Lemma 1

Denote by $g(\bm V)$ the objective function in prob:MIMO:unconstrained. For any nonzero $\bm V$, we have $g(\bm V)=g(c\bm V)$ given any nonzero factor $c$, and thus for any $\bm V$.

Figures (7)

  • Figure 1: According to the conventional optimization theory, algorithm A is better because its asymptotic convergence rate is higher when the number of iterations is sufficiently large. However, in the context of finite horizon optimization, algorithm B can outperform algorithm A when the attention is restricted to a finite range up to $T$ iterations.
  • Figure 2: Sum rates vs. number of iterations in the single-cell case.
  • Figure 3: Sum rates vs. CPU time in the single-cell case.
  • Figure 4: Objective value in \ref{['prob:QP']} vs. number of iterations in the single-cell case.
  • Figure 5: Sum rates vs. number of transmit antennas in the single-cell case.
  • ...and 2 more figures

Theorems & Definitions (6)

  • Lemma 1
  • Theorem 1
  • Proposition 1
  • Proposition 2
  • Remark 1
  • Remark 2