MHACL for Secure SIM-MIMO Systems

• The paper introduces a framework integrating geometric product-manifold optimization with multi-agent continual learning to boost secrecy rates in secure MIMO systems.
• MHACL utilizes Riemannian gradients and dual-scale policy updates to efficiently solve high-dimensional, non-convex joint optimization problems under practical hardware constraints.
• SIMHACL, a reduced-complexity variant, achieves millisecond-level training and near-optimal performance, making it viable for dynamic and secure wireless communications.

Manifold-Enhanced Heterogeneous Multi-Agent Continual Learning (MHACL) is a framework for solving high-dimensional, non-convex joint optimization problems that arise in secure, stacked intelligent metasurface (SIM)-assisted multi-user multiple-input multiple-output (MIMO) wireless systems. MHACL integrates geometric product-manifold optimization, multi-agent continual policy learning, and dual-scale adaptive policy updates to efficiently maximize weighted sum secrecy rate (WSSR) under practical physical and computational constraints. A reduced-complexity template termed SIMHACL further enables millisecond-level training and near-optimal communication secrecy in dynamic environments (Shi et al., 2 Feb 2026).

1. System Model and Optimization Formulation

MHACL is derived for a MIMO downlink system where a base station, referred to as Alice, is equipped with $L$ antennas and a $M$-layered SIM. Each SIM layer comprises $N$ nearly-passive meta-atoms. $K$ single-antenna legitimate users (Bobs) and a single-antenna eavesdropper (Eve) are located in the far field of the SIM.

Each antenna transmits an independent Gaussian data stream. Wave-based beamforming is realized exclusively in the electromagnetic domain through phase shift manipulation:

• Phase-Shift Matrices: $$\Phi_m = \mathrm{diag}(e^{j\phi_m^1}, ..., e^{j\phi_m^N})$$; each phase element $\phi_m^n \in [0,2\pi)$.
• Inter-Layer Coupling: Encoded by fixed complex-valued matrices $W_m \in \mathbb{C}^{N \times N}$ for $m$th layer ($W_1 \in \mathbb{C}^{N \times L}$ connects antennas to SIM).
• Overall SIM Beamformer: $G = \Phi_M W_M \Phi_{M-1} W_{M-1} ... \Phi_1$.

The composite downlink channel for user $k$ is $h_k = W_1^H G^H h_{\text{SIM},k}$, with $$h_{\text{SIM}, k} \sim \mathcal{CN}(0, R_{\text{SIM}, k})$$.

The k-th Bob's and Eve's stream-$k$ SINRs are given by:

$$\gamma_k = \frac{p_k |h_k^H w_1^k|^2}{\sum_{j\ne k} p_j |h_k^H w_1^j|^2 + \sigma_k^2},\quad \gamma_k^e = \frac{p_k |h_e^H w_1^k|^2}{\sum_{j\ne k} p_j |h_e^H w_1^j|^2 + \sigma_e^2}.$$

The secrecy rate is $$R_k^s = [\log_2(1 + \gamma_k) - \log_2(1 + \gamma_k^e)]^+$$.

The main objective—joint precoding optimization—is:

$$\max_{\mathbf{p}, \boldsymbol{\theta}} \sum_{k=1}^K w_k R_k^s(\mathbf{p}, \boldsymbol{\theta}),$$

subject to $\sum_{k=1}^K p_k \leq P_{\max}$, $p_k \geq 0$, and discretized phase constraints $$\phi_m^n \in \{0, 2\pi/2^b, \ldots, 2\pi(2^b-1)/2^b\}$$. This formulation is highly non-convex due to variable coupling, discrete unit-modulus phases, and the large solution space (Shi et al., 2 Feb 2026).

2. Product-Manifold Geometry and Riemannian Gradients

Phase coordination in MHACL is handled via geometric optimization over the product manifold $\mathcal{M} = (S^1)^{MN}$, with each $S^1$ corresponding to a phase element on the unit circle. This approach provides inherent enforcement of the unit-modulus constraint for each phase shift, reducing extraneous parameterization. The manifold representation reduces the search to $MN$ real dimensions.

• Riemannian Gradient: For WSSR objective $f(\mathbf{p}, \boldsymbol{\theta})$, the Euclidean gradient $\nabla_{\boldsymbol{\theta}} f$ is backpropagated through the beamformer. The Riemannian gradient at each phase scalar $\phi_n$ is

$$\operatorname{grad}_{S^1} f = \mathrm{Im}\{e^{-j\phi_n} \frac{\partial f}{\partial \phi_n}\},$$

which is projected onto the tangent space $T_{\phi_n} S^1$ for geometric consistency.

This manifold-based optimization preserves physical feasibility and supports efficient phase updates. It eliminates the need for auxiliary constraints and enables hardware-compatible phase mask updates (Shi et al., 2 Feb 2026).

3. Multi-Agent Continual and Dual-Scale Policy Learning

MHACL leverages a heterogeneous multi-agent formulation: each agent is associated with a decision variable—either BS power allocation or SIM phase shifts per layer. Continual learning is realized by separating adaptation into two timescales:

• Local (Fast) Updates: At each timeslot, agents perform a fixed number of Riemannian gradient steps on phase and power variables, using inner-loop step sizes $\alpha_p, \alpha_\theta$.
• Global (Slow) Meta-Updates: After several slots or iterations, network-level parameters—masks, preconditioners, Transformer weights—are updated via Adam using step sizes $\eta_p, \eta_\theta$ accumulated from recent gradient activity.

This dual-scale architecture enables rapid response to fast channel variations while ensuring long-term stability and policy consolidation via meta-learning. Continual learning is enforced through a regularizer penalizing deviation from previous solutions, stored in a prioritized memory buffer (Shi et al., 2 Feb 2026).

4. Algorithm Structure and SIMHACL Low-Complexity Template

MHACL Algorithm Steps

1. Initialization: Uniform power allocation, phase from prior memory.
2. CSI Observation: Gradient tensors for power and phase are computed from the observed channel state.
3. Inner Loop: Iterative Riemannian descent on power and phase, respecting system and manifold constraints.
4. Instantaneous Loss Evaluation: Includes task loss and regularization.
5. Meta-Update: Periodically aggregate inner-loop gradients to update higher-level network parameters.
6. Memory Update: Update buffer and proceed to the next time slot.

SIMHACL Variant

SIMHACL reduces complexity by:

• Embedding all MN phases in a single compact coordinate and enforcing unit modulus via Riemannian flows.
• Replacing cubic-cost inversions with diagonal preconditioners, allowing phase updates at $O(MN)$ cost, compared to $O((MN)^3)$.
• Utilizing Proposition 1 for power: per-iteration normalization is sufficient, so power updates cost only $O(K)$.

The combined effect yields per-iteration complexity $O(\max\{KL, M\} N )$ for SIMHACL, compared to $O(L N K + M N)$ for base MHACL and $O( K [L M^3 + L^2 N^2] )$ for classical alternating optimization (Shi et al., 2 Feb 2026).

5. Convergence, Complexity, and Theoretical Guarantees

MHACL's updates on the product manifold ensure that, under Lipschitz-continuous Riemannian gradients and sufficiently small step sizes, the iterates $(\mathbf{p}^i, \boldsymbol{\theta}^i)$ converge to a first-order stationary point. The addition of a continual-learning regularizer is shown to maintain bounded deviation from previously learned solutions, so the overall process converges to an $\varepsilon$-stationary regime when the meta-update step size is much smaller than the inner loop step size.

The low-complexity SIMHACL variant further attains near-optimal solutions with provably linear per-iteration cost, reducing hardware overhead and learning latency (Shi et al., 2 Feb 2026).

6. Simulation Setup and Performance Outcomes

MHACL and SIMHACL were validated in a setting where Alice had $L=4$ antennas, $K=4$ users, SIM layers $M=2\ldots8$, and $N=64$ meta-atoms per layer. Environmental conditions included a $28$ GHz carrier, $10$ MHz bandwidth, quasi-static correlated Rayleigh fading, and the eavesdropper situated at the user cluster center.

Key performance metrics and results are summarized below:

Metric	MHACL	SIMHACL
Convergence (iterations)	$\sim2000$ (to within 1% of final WSSR)	$\sim500$
Training time per iteration	$1.4$ ms	$1.0$ ms (30% reduction)
WSSR gain (M up to 6 layers)	$+86\%$	Comparable
Phase quantization penalty (1 bit, $M=6$)	$\sim$10\% WSSR loss	$<2$\% gap at 4 bits
User scaling (WSSR)	Peaks at $K=4$ for $L=4$	Similar trend
Power allocation at $P=10$ dBm	$\sim70\%$ of MHACL	Gap closes at $P=30$ dBm

Beyond $M=6$ layers, inter-layer loss saturates WSSR improvement. 1-bit phase quantization causes a ∼10% WSSR loss; this drops below 2% at 4-bit phase resolution. WSSR is maximized when the number of users matches the number of transmit antennas, with degradation beyond this point due to power dilution and increased inter-user interference. SIMHACL approaches MHACL performance as transmit power increases (Shi et al., 2 Feb 2026).

7. Context and Implications

The MHACL family enables efficient, scalable, and resource-conscious learning and adaptation in secure MIMO systems with SIMs, directly leveraging the physical geometry and system constraints. A plausible implication is that the product manifold and continual multi-agent learning principles embedded in MHACL may generalize to other high-dimensional, non-convex wireless optimization scenarios, particularly those involving hardware-constrained programmable metasurfaces. The linear per-iteration cost, millisecond response time, and near-optimal secrecy metrics position the approach as a candidate for future 6G secure communication deployments (Shi et al., 2 Feb 2026).