Operator-Theoretic Framework for Gradient-Free Federated Learning
🤖 AI Summary
Addressing the tripartite challenges of data heterogeneity, communication/computation constraints, and privacy preservation in federated learning, this paper proposes the first operator-theoretic, gradient-free framework. Our method constructs the global model within a kernel affine subspace via reproducing kernel Hilbert space (RKHS) embedding and forward-inverse operator transformations—eliminating explicit gradient exchange. Leveraging concentration inequalities, we analyze operator norms to establish theoretical guarantees on risk, estimation error, and robustness. The framework enables scalar-level knowledge transfer, end-to-end differential privacy (with kernel smoothing mitigating accuracy degradation under high privacy budgets), and fully homomorphic encryption (FHE) for inference—requiring only $Q imes C$ encryptions of minima and $C$ equality comparisons per test point. Evaluated on four benchmarks with fixed encoders, our approach matches or surpasses strong gradient-based fine-tuning methods, achieving up to a 23.7-point improvement, while maintaining practical inference latency.
Application Category
📝 Abstract
Federated learning must address heterogeneity, strict communication and computation limits, and privacy while ensuring performance. We propose an operator-theoretic framework that maps the $L^2$-optimal solution into a reproducing kernel Hilbert space (RKHS) via a forward operator, approximates it using available data, and maps back with the inverse operator, yielding a gradient-free scheme. Finite-sample bounds are derived using concentration inequalities over operator norms, and the framework identifies a data-dependent hypothesis space with guarantees on risk, error, robustness, and approximation. Within this space we design efficient kernel machines leveraging the space folding property of Kernel Affine Hull Machines. Clients transfer knowledge via a scalar space folding measure, reducing communication and enabling a simple differentially private protocol: summaries are computed from noise-perturbed data matrices in one step, avoiding per-round clipping and privacy accounting. The induced global rule requires only integer minimum and equality-comparison operations per test point, making it compatible with fully homomorphic encryption (FHE). Across four benchmarks, the gradient-free FL method with fixed encoder embeddings matches or outperforms strong gradient-based fine-tuning, with gains up to 23.7 points. In differentially private experiments, kernel smoothing mitigates accuracy loss in high-privacy regimes. The global rule admits an FHE realization using $Q imes C$ encrypted minimum and $C$ equality-comparison operations per test point, with operation-level benchmarks showing practical latencies. Overall, the framework provides provable guarantees with low communication, supports private knowledge transfer via scalar summaries, and yields an FHE-compatible prediction rule offering a mathematically grounded alternative to gradient-based federated learning under heterogeneity.
Problem
Research questions and friction points this paper is trying to address.
Addresses federated learning challenges under heterogeneity, communication limits, and privacy constraints
Develops gradient-free framework using operator theory and RKHS mappings for efficient learning
Enables private knowledge transfer via scalar summaries and FHE-compatible prediction rules
Innovation
Methods, ideas, or system contributions that make the work stand out.
Operator-theoretic framework mapping L2-optimal solution into RKHS
Gradient-free scheme using forward and inverse operators with finite-sample bounds
Scalar space folding measure enabling private communication and FHE compatibility
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