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Data-Driven Learning of Two-Stage Beamformers in Passive IRS-Assisted Systems with Inexact Oracles

Spyridon Pougkakiotis, Hassaan Hashmi, Dionysis Kalogerias

TL;DR

Overall it is demonstrated that, by leveraging the developed theory and its insights on the propagation of oracle errors, this work can create highly efficient and scalable algorithms for the solution of general IRS-assisted optimal beamforming problems, under realistic assumptions.

Abstract

We develop an efficient data-driven and model-free unsupervised learning algorithm for achieving fully passive intelligent reflective surface (IRS)-assisted optimal short/long-term beamforming in wireless communication networks. The proposed algorithm is based on a zeroth-order stochastic gradient ascent methodology, suitable for tackling two-stage stochastic nonconvex optimization problems with continuous uncertainty and unknown (or "black-box") terms present in the objective function, via the utilization of inexact evaluation oracles. We showcase that the algorithm can operate under realistic and general assumptions, and establish its convergence rate close to some stationary point of the associated two-stage (i.e., short/long-term) problem, particularly in cases where the second-stage (i.e., short-term) beamforming problem (e.g., transmit precoding) is solved inexactly using an arbitrary (inexact) oracle. The proposed algorithm is applicable on a wide variety of IRS-assisted optimal beamforming settings, while also being able to operate without (cascaded) channel model assumptions or knowledge of channel statistics, and over arbitrary IRS physical configurations; thus, no active sensing capability at the IRS(s) is needed. Our algorithm is numerically demonstrated to be very effective in a range of experiments pertaining to a well-studied MISO downlink model, including scenarios demanding physical IRS tuning (e.g., directly through varactor capacitances), even in large-scale regimes.

Data-Driven Learning of Two-Stage Beamformers in Passive IRS-Assisted Systems with Inexact Oracles

TL;DR

Overall it is demonstrated that, by leveraging the developed theory and its insights on the propagation of oracle errors, this work can create highly efficient and scalable algorithms for the solution of general IRS-assisted optimal beamforming problems, under realistic assumptions.

Abstract

We develop an efficient data-driven and model-free unsupervised learning algorithm for achieving fully passive intelligent reflective surface (IRS)-assisted optimal short/long-term beamforming in wireless communication networks. The proposed algorithm is based on a zeroth-order stochastic gradient ascent methodology, suitable for tackling two-stage stochastic nonconvex optimization problems with continuous uncertainty and unknown (or "black-box") terms present in the objective function, via the utilization of inexact evaluation oracles. We showcase that the algorithm can operate under realistic and general assumptions, and establish its convergence rate close to some stationary point of the associated two-stage (i.e., short/long-term) problem, particularly in cases where the second-stage (i.e., short-term) beamforming problem (e.g., transmit precoding) is solved inexactly using an arbitrary (inexact) oracle. The proposed algorithm is applicable on a wide variety of IRS-assisted optimal beamforming settings, while also being able to operate without (cascaded) channel model assumptions or knowledge of channel statistics, and over arbitrary IRS physical configurations; thus, no active sensing capability at the IRS(s) is needed. Our algorithm is numerically demonstrated to be very effective in a range of experiments pertaining to a well-studied MISO downlink model, including scenarios demanding physical IRS tuning (e.g., directly through varactor capacitances), even in large-scale regimes.

Paper Structure

This paper contains 16 sections, 5 theorems, 45 equations, 6 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Formulation and Preliminary Results
  3. Problem Formulation
  4. Assumptions
  5. Technical Results
  6. Compound Channel Zeroth-order Gradient Approximation
  7. Algorithm and Convergence Analysis
  8. A Zeroth-order Projected Stochastic Gradient Ascent
  9. Convergence Analysis
  10. Controlling the inexact oracle error
  11. Numerical Results
  12. iZoSGA with fixed WMMSE iterations
  13. iZoSGA with varying WMMSE iterations
  14. Tuned IRS tested with inexact WMMSE precodings
  15. iZoSGA with Physical IRS
  16. ...and 1 more sections

Key Result

Lemma 1

For every $\bm{\theta} \in \Theta$, $\bm{W} \in \mathcal{W}$ and a.e. $\omega\in \Omega$, the gradient of ${F}\left(\bm{W},\bm{H}(\bm{\theta},\omega)\right)$ with respect to $\bm{\theta}$ reads where $\frac{\partial^{\circ}}{\partial \bm{z}}(\cdot)$ is the Wirtinger cogradient operator. Moreover, there exists a constant $B_F > 0$ such that

Figures (6)

  • Figure 1: Flowchart of the proposed algorithm, termed iZoSGA, which is the inexact extension of the so-called ZoSGA algorithm developed in IEEE_tran_Hashmietal.
  • Figure 2: 1000 element IRS-aided network configuration.
  • Figure 3: Average sumrates achieved by WMMSE wmmseShi2011 (random IRS phase-shifts) and iZoSGA, with a 1000 phase-shifter IRS using 1,2,3,5,10,20 and 50 WMMSE iterations.
  • Figure 4: Average sumrates achieved by WMMSE wmmseShi2011 (random IRS phase-shifts) and two separate iZoSGA experiments with WMMSE iterations decreasing after every $8000$ iterations ($20 {\to} 10 {\to} 7 {\to} 6 {\to} 5$ and $20 {\to} 5 {\to} 4 {\to} 3 {\to} 2$).
  • Figure 5: Average sumrates achieved by WMMSE wmmseShi2011 (at 1,2,3,4 and 5 iterations) with iZoSGA fine-tuned 1000 IRS phase-shifters.
  • ...and 1 more figures

Theorems & Definitions (12)

  • Lemma 1
  • proof
  • Theorem 1
  • proof
  • Remark 1
  • Definition 1
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • ...and 2 more