Analytically Characterized Optimal Power Control for Signal-Level-Integrated Sensing, Computing and Communication in Federated Learning

1. Problem Statement

The paper addresses the challenge of integrating three critical functionalities—Sensing, Computing, and Communication (ISCC)—at the signal level within a Federated Learning (FL) framework. Specifically, it focuses on Signal-level ISCC (Sig-ISCC), where a single waveform simultaneously supports:

• Over-the-Air Computation (AirComp): Aggregating model updates (gradients) from edge devices (EDs) to a base station (BS) for FL.
• Joint Target Detection: Sensing a target using the echoes of the same transmitted signals.

The Core Conflict:

• AirComp Objective: Requires received signals from all EDs to be power-aligned (specifically, $h_k \sqrt{p_k} \approx \text{constant}$) to minimize aggregation distortion (Mean Square Error, MSE). This typically favors devices with poor channels to transmit at higher power.
• Sensing Objective: Requires a specific aggregate signal-to-noise ratio (SNR) at the BS to detect a target. This imposes a constraint on the weighted sum of transmit powers ($\sum p_k b_k \ge \eta_D$), where $b_k$ depends on the radar cross-section and path loss.
• The Challenge: The sensing constraint breaks the monotonicity of standard AirComp power control. A device with a large sensing coefficient ($b_k$) might need high power to satisfy the sensing constraint, even if its channel gain ($h_k$) is poor, which would severely degrade the AirComp aggregation accuracy. The paper seeks to find the joint optimal transmit power ($p_k$) and receive scaling factor ($\alpha$) that minimizes AirComp distortion while strictly satisfying the joint target detection requirement.

2. Methodology

System Model

• FL Framework: Standard FedAvg with $K$ edge devices. The BS aggregates gradients via AirComp.
• Sensing Model: Coordinated joint detection where EDs transmit, receive echoes, and the BS fuses the processed signals to perform a binary hypothesis test (target present vs. absent). The detection probability is linked to a sensing metric $E_I = \sum p_k b_k$.
• Optimization Goal: Minimize the AirComp MSE (which dictates FL convergence speed) subject to:
  1. Joint sensing constraint: $\sum p_k b_k \ge \eta_D$.
  2. Maximum power constraints: $0 \le p_k \le P_{max}$.

Mathematical Formulation & Transformation

The initial problem is non-convex due to the coupling of variables $p_k$ and $\alpha$ in the objective function.

1. Variable Transformation: The authors introduce a bijective change of variables: $x_k = p_k \alpha$. This transforms the original non-convex problem (P1) into an equivalent convex optimization problem (P2).
2. Convexity: The new objective function is convex, and the constraints become affine, allowing for rigorous theoretical analysis.

Analytical Solution & Algorithm

Instead of relying solely on numerical solvers, the authors derive an analytically characterized optimal solution:

• KKT Conditions: They analyze the Karush-Kuhn-Tucker (KKT) conditions for the convex problem.
• Structural Properties: They prove that the optimal solution is characterized by a specific structure where devices are partitioned into two sets:
  • Those transmitting at maximum power ($P_{max}$).
  • Those transmitting at a specific power level determined by a dual variable $\lambda^*$ (related to the sensing constraint).
• Monotonicity: They establish that the relationship between the dual variable $\lambda$ and the system parameters is monotonic.
• Algorithm 1 (Polynomial-Time):
  • The algorithm systematically searches through possible "active sets" (combinations of devices at max power).
  • For each candidate set, it solves a root-finding problem for $\lambda^*$ using the monotonicity property.
  • Complexity: The algorithm runs in $O(|K|^3)$ time, which is polynomial and significantly faster than general non-linear programming (NLP) solvers.
  • Robustness: The method is proven robust to numerical errors in the root-finding step.

3. Key Contributions

1. First Sig-ISCC Framework for FL: Proposes the first uplink power control framework for signal-level ISCC in AirComp-based Federated Learning (AirFEEL), utilizing a single shared waveform for both sensing and computation.
2. Convex Reformulation: Successfully transforms a non-convex joint power-scaling problem into an equivalent convex problem via a novel variable transformation.
3. Analytical Optimal Algorithm: Develops a polynomial-time algorithm with provable global optimality. Unlike black-box solvers, this algorithm exploits the specific mathematical structure of the problem (monotonicity and active sets) to find the exact solution.
4. Robustness Analysis: Provides theoretical guarantees that the algorithm remains stable even with numerical inaccuracies in the root-finding process.

4. Simulation Results

The authors validate their approach using MNIST and CIFAR-10 datasets under various channel conditions and target locations.

• Optimality Verification:
  • Compared against off-the-shelf NLP solvers (e.g., IPOPT), the proposed algorithm achieves numerical precision where the gap is often $<0.01\%$.
  • Standard solvers without the specific variable transformation often fail to reach the global optimum or get stuck in local minima.
• Computational Efficiency:
  • The proposed algorithm is 10x to 100x faster than commercial solvers (average runtime ~0.001s vs. ~0.01s).
• Performance vs. Baselines:
  • Zero-Forcing (ZF): Highly sensitive to deep fades; performance degrades significantly when the target is close to specific ED groups.
  • Greedy Sensing Power: Satisfies sensing constraints but sacrifices FL accuracy, especially in non-i.i.d. data scenarios.
  • Proposed Method: Consistently achieves the lowest aggregation MSE and highest test accuracy, closely matching the "No Sensing" lower bound. This demonstrates that the sensing constraint can be satisfied with negligible impact on FL performance when power is optimized jointly.

5. Significance

• 6G Enabler: This work provides a foundational solution for 6G Integrated Sensing and Communication (ISAC) systems, specifically addressing the "tightest" level of integration (Sig-ISCC) where sensing and communication share the exact same waveform.
• Resource Efficiency: It demonstrates that joint sensing and learning can be performed without significant resource overhead or performance degradation, provided optimal resource allocation is used.
• Practical Deployment: The development of a low-complexity, polynomial-time algorithm makes real-time implementation feasible for edge networks, overcoming the latency and reliability issues associated with iterative numerical solvers.
• Theoretical Insight: The paper bridges the gap between theoretical convex optimization and practical wireless system design, showing how analytical properties can be leveraged to create robust, high-performance control algorithms for distributed learning.