# Hammer and Anvil: Toward a Theory of Backdoors in Federated Learning

arXiv: https://arxiv.org/abs/2509.08089 · HTML: https://mlsanity.com/paper/2509.08089/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 28 of its 62 references that are in the map.
- First posted on arXiv 9 Sep 2025.

## Builds on (most cited first, 28 of 28)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Communication-Efficient Learning of Deep Networks from Decentralized Data](https://mlsanity.com/paper/1602.05629/) (2016)
- [Byzantine-Robust Distributed Learning: Towards Optimal Statistical Rates](https://mlsanity.com/paper/1803.01498/) (2018)
- [Analyzing Federated Learning through an Adversarial Lens](https://mlsanity.com/paper/1811.12470/) (2018)
- [Differential Privacy Has Disparate Impact on Model Accuracy](https://mlsanity.com/paper/1905.12101/) (2019)
- [Attack of the Tails: Yes, You Really Can Backdoor Federated Learning](https://mlsanity.com/paper/2007.05084/) (2020)
- [A Little Is Enough: Circumventing Defenses For Distributed Learning](https://mlsanity.com/paper/1902.06156/) (2019)
- [Data Poisoning Attacks Against Federated Learning Systems](https://mlsanity.com/paper/2007.08432/) (2020)
- [Mitigating Sybils in Federated Learning Poisoning](https://mlsanity.com/paper/1808.04866/) (2018)
- [FLDetector: Defending Federated Learning Against Model Poisoning Attacks via Detecting Malicious Clients](https://mlsanity.com/paper/2207.09209/) (2022)
- [CRFL: Certifiably Robust Federated Learning against Backdoor Attacks](https://mlsanity.com/paper/2106.08283/) (2021)
- [D"IoT: A Federated Self-learning Anomaly Detection System for IoT](https://mlsanity.com/paper/1804.07474/) (2018)
- [Learning to Detect Malicious Clients for Robust Federated Learning](https://mlsanity.com/paper/2002.00211/) (2020)
- [Defending against Backdoors in Federated Learning with Robust Learning Rate](https://mlsanity.com/paper/2007.03767/) (2020)
- [DeepSight: Mitigating Backdoor Attacks in Federated Learning Through Deep Model Inspection](https://mlsanity.com/paper/2201.00763/) (2022)
- [Byzantine-Robust Federated Machine Learning through Adaptive Model Averaging](https://mlsanity.com/paper/1909.05125/) (2019)
- [Fine-Tuning Is All You Need to Mitigate Backdoor Attacks](https://mlsanity.com/paper/2212.09067/) (2022)
- [FL-WBC: Enhancing Robustness against Model Poisoning Attacks in Federated Learning from a Client Perspective](https://mlsanity.com/paper/2110.13864/) (2021)
- [Backdoor Attacks and Defenses in Federated Learning: Survey, Challenges and Future Research Directions](https://mlsanity.com/paper/2303.02213/) (2023)
- [On the Vulnerability of Backdoor Defenses for Federated Learning](https://mlsanity.com/paper/2301.08170/) (2023)
- [FreqFed: A Frequency Analysis-Based Approach for Mitigating Poisoning Attacks in Federated Learning](https://mlsanity.com/paper/2312.04432/) (2023)
- [BayBFed: Bayesian Backdoor Defense for Federated Learning](https://mlsanity.com/paper/2301.09508/) (2023)
- [Multi-metrics adaptively identifies backdoors in Federated learning](https://mlsanity.com/paper/2303.06601/) (2023)
- [Data and Model Poisoning Backdoor Attacks on Wireless Federated Learning, and the Defense Mechanisms: A Comprehensive Survey](https://mlsanity.com/paper/2312.08667/) (2023)
- [Semi-Targeted Model Poisoning Attack on Federated Learning via Backward Error Analysis](https://mlsanity.com/paper/2203.11633/) (2022)
- [G2uardFL: Safeguarding Federated Learning Against Backdoor Attacks through Attributed Client Graph Clustering](https://mlsanity.com/paper/2306.04984/) (2023)
- [A Survey on Vulnerability of Federated Learning: A Learning Algorithm Perspective](https://mlsanity.com/paper/2311.16065/) (2023)
- [Identify Backdoored Model in Federated Learning via Individual Unlearning](https://mlsanity.com/paper/2411.01040/) (2024)

## Abstract (arXiv)

Federated Learning (FL) enables distributed model training but is vulnerable to backdoor attacks, where malicious clients embed attacker-controlled behaviors into the global model. Existing defenses fail against adaptive adversaries. In this paper, we present "Hammer and Anvil", a principled theoretical framework that categorizes backdoors by the deviation, δ, of their updates to the mean of the updates. We identify two fundamental defense types: "Type 1 (The Anvil)", comprising outlier detection and robust aggregation effective against large-deviation attacks, and "Type 2 (The Hammer)", consisting of removal-based defenses effective against small-deviation attacks. We demonstrate that defenses of a single type and non-principled combined defenses inherently leave an exploitable gap for adaptive attackers. To bridge this gap, we propose the principled combination of Type 1 and Type 2 defenses. We evaluate our framework against a new, worst-case, full-information adaptive adversary that knows the benign updates, the aggregation algorithm, and its parameters, and yet this adversary fails against our combined defenses. Our empirical evaluation across various datasets and settings shows that single-typed and non-principled combined defenses are easily broken, often by a single malicious client. In contrast, our best combined defense variants, HAFlameCSFT, HAKrumCSFT, and HAMulti-MetricsCSFT, remain undefeated even in the most adversarial settings. Our results provide a principled approach for research on backdoors in federated learning.