A Particle-Swarm-Assisted Gradient Meta-Learning Algorithm for Joint Transmit Precoding and STAR-RIS Coefficient Optimization

I. INTRODUCTION

• Prior optimization methods can be computationally complex or locally optimal, while meta-heuristics do not exploit gradients and GML-JO uses random initialization.The paper notes that random initialization can limit achievable WSR on a sharply peaked landscape.Source: §I · pp. 1–2
• An amplitude-split parameterization and collapsed precoder representation automatically satisfy energy conservation and reduce the original problem's search dimension.Source: §I · p. 2
• PSA-GML uses PSO to globally search STAR-RIS coefficients for a warm start, then a coordinate-wise LSTM meta-optimizer jointly refines coefficients and precoder.The LSTM learns per-coordinate adaptive updates through first-order gradient meta-learning.Source: §I · p. 2
• The meta-optimizer is trained offline across channel realizations and applied to unseen channels without further adaptation.Source: §I · p. 2
• 11.06 bits/s/Hz WSR; PSA-GML improves on benchmark AO by 13.1%, multiple-random-restart AO by 6.2%, and random-phase by 35.1%.The introduction also reports 4.73 bits/s/Hz in the interference-limited regime, 83.9% of the 5.64 bits/s/Hz hand-designed Adam result.Source: §I · p. 2

II. SYSTEM MODEL AND PROBLEM FORMULATION

• In energy-splitting mode, the STAR-RIS serves transmission and reflection users on opposite sides, with direct BS-user links assumed blocked.The objective is WSR maximization subject to transmit-power and energy-conservation constraints.Source: §II · pp. 2–3
• The WSR maximization is non-convex because of fractional SINR, the unit-energy constraint, and strong coupling between the precoder and STAR-RIS coefficients.Source: §II · p. 3

III. EQUIVALENT TRANSFORMATIONS OF THE ORIGINAL PROBLEM

• The original non-convex problem is transformed into a sequence of equivalent tractable forms that lays the foundation for PSA-GML.Source: §III · p. 3

A. Amplitude-Split Parameterization

• For each element, the feasible amplitudes are parameterized as βt,n = cos φn and βr,n = sin φn, with φn in [0, π/2].The mapping is unique for every feasible amplitude pair.Source: A. Amplitude-Split Parameterization · pp. 3–4
• This parameterization automatically enforces the energy-conservation constraint on transmission and reflection amplitudes.Source: A. Amplitude-Split Parameterization · p. 3

B. Closed-Form Zero-Forcing Precoder

• For a full-row-rank composite channel, the closed-form ZF precoder nulls inter-user interference and yields an interference-free SINR.The stated rank condition holds when K ≤ Nt.Source: B. Closed-Form Zero-Forcing Precoder · p. 4
• Power-normalizing the ZF precoder scales each useful signal uniformly while preserving its zero-forcing property.Source: B. Closed-Form Zero-Forcing Precoder · p. 4

C. Collapsed Precoder Representation

• With the collapsed representation, the optimization problem is further reformulated for joint optimization of the STAR-RIS coefficients and precoder.Source: C. Collapsed Precoder Representation · p. 4
• The collapsed precoder W = A^H X is without loss of optimality because the null-space component does not affect SINR and only increases transmit power.Source: C. Collapsed Precoder Representation · p. 4
• The collapsed representation reduces the precoder search from NtK to K^2 complex entries, especially benefiting settings with many base-station antennas.The collapsed variable is initialized as the ZF solution X(0) = (AA^H)^−1.Source: C. Collapsed Precoder Representation · p. 4
• For any feasible precoder, the collapsed form achieves the same WSR with no greater transmit power.Source: C. Collapsed Precoder Representation · p. 4

D. Unconstrained Reparameterization

• The sigmoid reparameterization maps an unconstrained real variable to a feasible amplitude-split angle and makes the composite channel differentiable in that variable.The feasible endpoints are approached in the limit; phase shifts can be optimized directly as unconstrained variables because they are 2π-periodic.Source: D. Unconstrained Reparameterization · p. 4

IV. PROPOSED PSA-GML ALGORITHM

• PSA-GML combines global exploration with local refinement because gradient-only methods are initialization-sensitive and heuristic-only methods do not exploit gradient information.Source: §IV · p. 4

A. PSO-Based Global Warm Start

• PSO searches the STAR-RIS coefficient space within the feasible box, using the WSR achieved by the closed-form ZF precoder as particle fitness.This fitness avoids an inner precoder optimization while matching the final objective.Source: A. PSO-Based Global Warm Start · pp. 4–5
• PSO updates particle positions and velocities using inertia, personal and global bests, random coefficients, and projection onto the feasible box.Source: A. PSO-Based Global Warm Start · p. 5
• The global-best fitness never decreases during PSO, and the final global best initializes the refinement stage.Source: A. PSO-Based Global Warm Start · p. 5
• Because PSO explores the feasible region, its warm start is largely insensitive to random initialization, addressing gradient-based methods’ initialization sensitivity.Source: A. PSO-Based Global Warm Start · p. 5

• The collapsed precoder reproduces the Stage 1 ZF solution at initialization, then generalizes it as the variables evolve.Source: B. Gradient Meta-Learning Refinement · p. 5
• At each refinement step, the variables are reconstructed, the precoder is scaled to satisfy the power constraint, and automatic differentiation supplies the WSR gradient.Source: B. Gradient Meta-Learning Refinement · p. 5
• The gradient calculation differentiates the WSR through the SINR and composite-channel coefficients, including explicit derivatives for amplitude and phase variables.Source: B. Gradient Meta-Learning Refinement · pp. 5–6
• The LSTM meta-optimizer jointly refines the STAR-RIS coefficients and precoder by mapping objective gradients to learned updates.Source: B. Gradient Meta-Learning Refinement · pp. 5–6
• Two coordinate-wise LSTMs handle the distinct scales of STAR-RIS coefficients and precoder entries, with each coordinate carrying its own momentum-like recurrent memory.The shared LSTM parameters are reused across coordinates; the per-coordinate memory resides in the recurrent state.Source: B. Gradient Meta-Learning Refinement · p. 6
• Offline training unrolls the update recursion over channel realizations and minimizes negative average WSR across the unrolled steps.Source: B. Gradient Meta-Learning Refinement · p. 6
• Detaching the gradient input avoids numerically unstable second-order backpropagation on the sharply peaked warm-start landscape while preserving per-coordinate adaptive updates.The first-order approximation follows the spirit of FOMAML.Source: B. Gradient Meta-Learning Refinement · p. 6

• Algorithm 1 summarizes the proposed PSA-GML algorithm.Source: C. Proposed PSA-GML Algorithm · p. 6

D. Convergence and Complexity Analysis

• Offline meta-training uses first-order updates of the LSTM parameters over channel minibatches, and the trained optimizer is deployed for S-step refinement.The average loss is described as encouraging fast convergence and high final WSR.Source: Figure 3 · p. 7, D. Convergence and Complexity Analysis · p. 7
• The returned PSA-GML solution has WSR no lower than its PSO warm start because the best refinement iterate is retained.The PSO global-best fitness is also never degraded during Stage 1.Source: D. Convergence and Complexity Analysis · p. 7
• Each Stage 2 step costs O(NNtK + K3 + DH2), with the coordinate-wise LSTM term dominant in the stated example.For N = 32, Nt = 32, K = 4, and H = 32, DH2 is approximately 1.3 × 10^5, versus approximately 4.1 × 10^3 for NNtK + K3.Source: D. Convergence and Complexity Analysis · p. 7
• PSA-GML refinement and AO have the same asymptotic per-iteration complexity order; its practical advantage is parallel PSO search and avoiding AO’s multiple random restarts.The LSTM update is the dominant per-step cost in the stated example.Source: D. Convergence and Complexity Analysis · p. 7

PIPSO(NNtK + K3)

• The LSTM update dimension is D = 3N + 2K2, with H denoting the hidden size.The supplied complexity expression is a fragment; this point reports only the explicitly defined quantities.Source: PIPSO(NNtK + K3) · p. 7

V. SIMULATION RESULTS

• Simulations use Nt = 32 antennas, N = 32 STAR-RIS elements, and K = 4 users by default, with i.i.d. Rayleigh and Saleh-Valenzuela channels.The setup uses S = 300 refinement steps, H = 32, and offline training over 120 channel realizations; results are averaged over independent realizations.Source: §V · p. 7, A. Parameter Setting and Baseline · p. 8

A. Parameter Setting and Baseline

• The baselines include random-phase plus ZF, AO/BCD, PSO without refinement, and refinement initialized from random coefficients.AO/BCD is also evaluated using the best outcome from 20 random restarts to account for initialization sensitivity.Source: A. Parameter Setting and Baseline · p. 8

B. Convergence Behavior

• PSA-GML converges rapidly to high WSR from a higher initial value than refine-only, which ends at a substantially lower WSR.Fig. 4 presents convergence behavior for PSA-GML and the baselines.Source: B. Convergence Behavior · p. 8, Figure 4 · p. 8

C. Performance versus SNR

• 11.06 bits/s/Hz at 10 dB exceeds single-run AO’s 9.78 bits/s/Hz by 13.1% and 20-restart AO’s 10.42 bits/s/Hz by 6.2%.The reported advantage over 20-restart AO is stated to rule out an unfavorable AO initialization as the explanation.Source: C. Performance versus SNR · p. 8, Figure 5 · p. 8

D. Learned Optimizer in the Interference-Limited Regime

• In the interference-limited regime, the learned optimizer reaches 83.9% of Adam’s WSR with the same 300-step budget, while SGD is nearly equal to PSO-only.The main-regime ablation reports 11.06 bits/s/Hz for the learned optimizer, 99.4% of Adam’s 11.12 bits/s/Hz; the refinement gain is smaller there because the PSO warm start is higher quality.Source: D. Learned Optimizer in the… · p. 9, Table I · p. 9
• A model trained in the main regime transfers zero-shot to the interference-limited setting, achieving an average WSR of 4.85 bits/s/Hz, approximately 86% of Adam’s reference.This transfer uses no fine-tuning.Source: D. Learned Optimizer in the… · p. 9

E. Impact of the Number of Elements and Users

• WSR grows with N for all methods, while PSA-GML achieves the highest WSR across the considered N and K values, with its relative advantage strongest at small-to-moderate N.Source: E. Impact of the Number of Elements… · p. 9, Figure 8 · p. 10

F. Warm-Start Gain Analysis

• SV warm-start gain is at least 20% across N, peaking at 37.4% at N = 16; i.i.d. gain is around 10% for N ≤64 and 6.9% at N = 128.Source: F. Warm-Start Gain Analysis · pp. 9–10, Figure 9 · p. 10

G. Runtime and Complexity

• At N = 32, K = 4, PSA-GML takes about 0.9 s versus AO’s 0.4 s, despite equal-order per-iteration complexity; offline training adds a one-time cost.Source: G. Runtime and Complexity · p. 10, Figure 10 · p. 10

H. Optimized Coefficients

• For one channel realization, transmission and reflection powers are adaptively balanced across elements, while phase shifts span [0, 2π).Source: H. Optimized Coefficients · p. 10, Figure 11 · p. 10

VI. CONCLUSION

• 11.06 bits/s/Hz WSR exceeds AO by 13.1%, multiple-random-restart AO by 6.2%, and the random-phase scheme by 35.1%.Source: §VI · p. 11
• In the interference-limited regime, the learned optimizer reaches 83.9% of the hand-designed Adam refinement.Source: §VI · p. 11