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Deep Unfolding with Approximated Computations for Rapid Optimization

Dvir Avrahami, Amit Milstein, Caroline Chaux, Tirza Routtenberg, Nir Shlezinger

TL;DR

This paper introduces a learned optimization framework that jointly tackles iteration count and per-iteration complexity, and demonstrates the effectiveness of the method on two representative problems: hybrid beamforming and robust principal component analysis.

Abstract

Optimization-based solvers play a central role in a wide range of signal processing and communication tasks. However, their applicability in latency-sensitive systems is limited by the sequential nature of iterative methods and the high computational cost per iteration. While deep unfolding has emerged as a powerful paradigm for converting iterative algorithms into learned models that operate with a fixed number of iterations, it does not inherently address the cost of each iteration. In this paper, we introduce a learned optimization framework that jointly tackles iteration count and per-iteration complexity. Our approach is based on unfolding a fixed number of optimization steps, replacing selected iterations with low-complexity approximated computations, and learning extended hyperparameters from data to compensate for the introduced approximations. We demonstrate the effectiveness of our method on two representative problems: (i) hybrid beamforming; and (ii) robust principal component analysis. These fundamental case studies show that our learned approximated optimizers can achieve state-of-the-art performance while reducing computational complexity by over three orders of magnitude. Our results highlight the potential of our approach to enable rapid, interpretable, and efficient decision-making in real-time systems.

Deep Unfolding with Approximated Computations for Rapid Optimization

TL;DR

This paper introduces a learned optimization framework that jointly tackles iteration count and per-iteration complexity, and demonstrates the effectiveness of the method on two representative problems: hybrid beamforming and robust principal component analysis.

Abstract

Optimization-based solvers play a central role in a wide range of signal processing and communication tasks. However, their applicability in latency-sensitive systems is limited by the sequential nature of iterative methods and the high computational cost per iteration. While deep unfolding has emerged as a powerful paradigm for converting iterative algorithms into learned models that operate with a fixed number of iterations, it does not inherently address the cost of each iteration. In this paper, we introduce a learned optimization framework that jointly tackles iteration count and per-iteration complexity. Our approach is based on unfolding a fixed number of optimization steps, replacing selected iterations with low-complexity approximated computations, and learning extended hyperparameters from data to compensate for the introduced approximations. We demonstrate the effectiveness of our method on two representative problems: (i) hybrid beamforming; and (ii) robust principal component analysis. These fundamental case studies show that our learned approximated optimizers can achieve state-of-the-art performance while reducing computational complexity by over three orders of magnitude. Our results highlight the potential of our approach to enable rapid, interpretable, and efficient decision-making in real-time systems.

Paper Structure

This paper contains 34 sections, 2 theorems, 44 equations, 6 figures, 5 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. System Model and Preliminaries
  3. Optimization Model
  4. Challenges of Rapid Optimization
  5. Learned Approximated Optimization
  6. Rationale
  7. Methodology
  8. Increased Parameterization
  9. Constructing the Approximated Iterative Optimizer
  10. Training via Empirical Risk Minimization
  11. Discussion
  12. Case Study 1: Hybrid Beamforming
  13. Hybrid Beamforming Formulation
  14. Channel Model
  15. Hybrid Beamforming
  16. ...and 19 more sections

Key Result

Proposition 1

Let $\mathcal{L}_{\rm o}({\boldsymbol{s}}; {\boldsymbol{x}})$ be a differentiable objective function, and let $\bm{\eta}_k$ be comprised of positive step-sizes. Then the update eqn:ElementWiseGD yields a decrease in the objective value as in eqn:descent to first order, i.e., the method retains the d

Figures (6)

  • Figure 1: Hybrid beamforming case study illustration
  • Figure 2: Rate vs snr, small-scale mimo.
  • Figure 3: Rate vs. snr, large-scale mimo.
  • Figure 4: Low-rank recovery error vs. iterations, synthetic data
  • Figure 5: Convergence for higher-rank ($r=50$) vs lower-rank ($r=5$).
  • ...and 1 more figures

Theorems & Definitions (4)

  • Proposition 1
  • proof
  • Proposition 2
  • proof