# Operator-Theoretic Framework for Gradient-Free Federated Learning

arXiv: https://arxiv.org/abs/2512.01025 · HTML: https://mlsanity.com/paper/2512.01025/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 37 of its 74 references that are in the map.
- First posted on arXiv 30 Nov 2025.

## Builds on (most cited first, 30 of 37)

- [Communication-Efficient Learning of Deep Networks from Decentralized Data](https://mlsanity.com/paper/1602.05629/) (2016)
- [Deep Learning with Differential Privacy](https://mlsanity.com/paper/1607.00133/) (2016)
- [Learning Imbalanced Datasets with Label-Distribution-Aware Margin Loss](https://mlsanity.com/paper/1906.07413/) (2019)
- [Personalized Federated Learning with Moreau Envelopes](https://mlsanity.com/paper/2006.08848/) (2020)
- [Exploiting Shared Representations for Personalized Federated Learning](https://mlsanity.com/paper/2102.07078/) (2021)
- [Clustered Federated Learning: Model-Agnostic Distributed Multi-Task Optimization under Privacy Constraints](https://mlsanity.com/paper/1910.01991/) (2019)
- [Data-Free Knowledge Distillation for Heterogeneous Federated Learning](https://mlsanity.com/paper/2105.10056/) (2021)
- [FedProto: Federated Prototype Learning across Heterogeneous Clients](https://mlsanity.com/paper/2105.00243/) (2021)
- [Semi-supervised Knowledge Transfer for Deep Learning from Private Training Data](https://mlsanity.com/paper/1610.05755/) (2016)
- [Why gradient clipping accelerates training: A theoretical justification for adaptivity](https://mlsanity.com/paper/1905.11881/) (2019)
- [Personalized Cross-Silo Federated Learning on Non-IID Data](https://mlsanity.com/paper/2007.03797/) (2020)
- [XGLUE: A New Benchmark Dataset for Cross-lingual Pre-training, Understanding and Generation](https://mlsanity.com/paper/2004.01401/) (2020)
- [FjORD: Fair and Accurate Federated Learning under heterogeneous targets with Ordered Dropout](https://mlsanity.com/paper/2102.13451/) (2021)
- [Practical and Private (Deep) Learning without Sampling or Shuffling](https://mlsanity.com/paper/2103.00039/) (2021)
- [How to DP-fy ML: A Practical Guide to Machine Learning with Differential Privacy](https://mlsanity.com/paper/2303.00654/) (2023)
- [Understanding Gradient Clipping in Private SGD: A Geometric Perspective](https://mlsanity.com/paper/2006.15429/) (2020)
- [Privacy- and Utility-Preserving Textual Analysis via Calibrated Multivariate Perturbations](https://mlsanity.com/paper/1910.08902/) (2019)
- [Robust Federated Learning: The Case of Affine Distribution Shifts](https://mlsanity.com/paper/2006.08907/) (2020)
- [Automatic Clipping: Differentially Private Deep Learning Made Easier and Stronger](https://mlsanity.com/paper/2206.07136/) (2022)
- [Local and Central Differential Privacy for Robustness and Privacy in Federated Learning](https://mlsanity.com/paper/2009.03561/) (2020)
- [Personalized Federated Learning with Gaussian Processes](https://mlsanity.com/paper/2106.15482/) (2021)
- [Federated Learning on Heterogeneous and Long-Tailed Data via Classifier Re-Training with Federated Features](https://mlsanity.com/paper/2204.13399/) (2022)
- [Personalized Federated Learning for Intelligent IoT Applications: A Cloud-Edge based Framework](https://mlsanity.com/paper/2002.10671/) (2020)
- [Iterative Methods for Private Synthetic Data: Unifying Framework and New Methods](https://mlsanity.com/paper/2106.07153/) (2021)
- [Improved Differential Privacy for SGD via Optimal Private Linear Operators on Adaptive Streams](https://mlsanity.com/paper/2202.08312/) (2022)
- [Natural Language Understanding with Privacy-Preserving BERT](https://mlsanity.com/paper/2104.07504/) (2021)
- [Federated Dynamic Sparse Training: Computing Less, Communicating Less, Yet Learning Better](https://mlsanity.com/paper/2112.09824/) (2021)
- [Multi-Epoch Matrix Factorization Mechanisms for Private Machine Learning](https://mlsanity.com/paper/2211.06530/) (2022)
- [FedSoft: Soft Clustered Federated Learning with Proximal Local Updating](https://mlsanity.com/paper/2112.06053/) (2021)
- [FedML-HE: An Efficient Homomorphic-Encryption-Based Privacy-Preserving Federated Learning System](https://mlsanity.com/paper/2303.10837/) (2023)

## Built on by (most cited first, 2 of 2)

- [Kernel Affine Hull Machines as Compute-Efficient Encoders for Frozen Semantic Spaces](https://mlsanity.com/paper/2605.02950/) (2026)
- [Prediction Limits and Koopman Closure of Geometry-Induced Soft State Abstractions](https://mlsanity.com/paper/2609.32652/) (2026)

## Abstract (arXiv)

Federated learning must address heterogeneity, strict communication and computation limits, and privacy while ensuring performance. We propose an operator-theoretic framework that maps the L2-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 × 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.