Optimal Distributed Learning Under Data Poisoning and Byzantine Failures

Core Concepts

The best learning guarantees that a first-order distributed algorithm can achieve under the Byzantine failure threat model are optimal even in the weaker data poisoning threat model. Furthermore, fully-poisonous local data is a stronger adversarial setting than partially-poisonous local data in distributed ML with heterogeneous datasets.

Abstract

The paper analyzes the problem of distributed machine learning (ML) in the presence of data poisoning and Byzantine failures. It makes the following key contributions:

Lower Bound with Data Poisoning:

Characterizes the suboptimality gap and convergence rate limitations of any stochastic first-order distributed algorithm under the data poisoning threat model.
Shows that the error is in Ω(f/n * ζ^2/μ) and the convergence rate is in Ω((1 + f/n) * σ^2/(με) + L/μ * log(Q0/ε)).

Matching Upper Bound with Byzantine Failure:

Presents a Byzantine-robust adaptation of Distributed Stochastic Gradient Descent (DSGD) that incorporates distributed Polyak's momentum and coordinate-wise trimmed mean aggregation.
Proves that this algorithm achieves an error in O(f/n * ζ^2/μ + ε) with a convergence rate in O((1 + f/n) * Kσ^2/(με) + L/μ * log(Q0/ε)), where K is the condition number.
Shows that the Byzantine-robust scheme yields optimal solutions even against the weaker data poisoning threat model.

Partially-Poisonous vs Fully-Poisonous Local Data:

Considers a scenario where in addition to having f out of n workers with fully-poisonous local datasets, each worker can have partially-poisonous local data.
Proves that the optimization error is in Θ(f/n * ζ^2/μ + b/m * σ^2/μ), which can be achieved using a Byzantine-robust first-order method with an exponential convergence rate.
Shows that fully-poisonous local data alone is a stronger adversarial setting than partially-poisonous local data alone when considering the same fraction of corrupted data points in the system.

Overall, the paper demonstrates the tightness of Byzantine-robust schemes even against the weaker data poisoning threat model, and provides a comprehensive analysis of the impact of different forms of data corruption in distributed ML.

Stats

The global loss function is Q(θ) = (1/n) * Σ_i Q^(i)(θ), where Q^(i)(θ) = E_x~D^(i)[q(θ, x)].
The number of fully corrupted workers is bounded by f < n/2.
The global gradient dissimilarity is bounded by ζ^2.
The local gradients of honest workers have bounded covariance trace of σ^2.
The condition number of the average loss function for honest workers is K = L/μ.

Quotes

"We prove that, while tolerating a wider range of faulty behaviors, Byzantine ML yields solutions that are, in a precise sense, optimal even under the weaker data poisoning threat model."
"We show that in real-world applications when workers' datasets are heterogeneous, fully-poisonous local data is a stronger adversarial setting."

Key Insights Distilled From

On the Relevance of Byzantine Robust Optimization Against Data Poisoning

by Sadegh Farha... at arxiv.org 05-02-2024

https://arxiv.org/pdf/2405.00491.pdf

Deeper Inquiries

How can the analysis be extended to handle non-convex loss functions or more general notions of gradient diversity beyond the bounded dissimilarity assumption

To extend the analysis to handle non-convex loss functions or more general notions of gradient diversity beyond the bounded dissimilarity assumption, several adjustments and considerations need to be made.

Non-Convex Loss Functions:

For non-convex loss functions, the analysis would need to incorporate more sophisticated optimization techniques that can handle the non-convexity of the objective function. This may involve exploring methods like stochastic gradient descent with restarts, evolutionary algorithms, or other optimization strategies tailored for non-convex functions.
The Lyapunov function used in the analysis may need to be modified to account for the non-convexity of the loss function. This could involve adapting the convergence proofs to accommodate the challenges posed by non-convex optimization landscapes.

General Notions of Gradient Diversity:

Beyond the bounded dissimilarity assumption, a more nuanced understanding of gradient diversity could be explored. This could involve considering more complex measures of dissimilarity between gradients, such as higher-order statistics or divergence metrics.
Techniques from information theory or statistical learning theory could be leveraged to quantify and manage gradient diversity in a more comprehensive manner.

By incorporating these adjustments and exploring more advanced optimization and analysis techniques, the framework can be extended to handle non-convex loss functions and more general notions of gradient diversity.

What are the implications of the results on the design of practical distributed ML systems, especially in terms of the trade-off between robustness and efficiency

The results of the analysis have significant implications for the design of practical distributed ML systems, particularly in balancing robustness and efficiency. Here are some key implications:

Robustness vs. Efficiency Trade-off:

The analysis provides insights into the trade-off between robustness against data poisoning and Byzantine failures and the efficiency of distributed ML systems. By understanding the optimal strategies for handling different types of faults, system designers can make informed decisions on the level of robustness required without compromising efficiency.

Algorithm Design:

The techniques developed in the analysis can guide the design of algorithms for distributed ML systems. By incorporating robust aggregation methods, trimmed mean operations, and other Byzantine-robust strategies, system designers can enhance the resilience of their algorithms to various types of faults.

Heterogeneous Data Handling:

The consideration of partially-poisonous local data in the analysis highlights the importance of handling heterogeneous data in distributed systems. Designing algorithms that can effectively manage data diversity among workers can lead to more robust and accurate models.

Real-World Applications:

Practical implications include the application of these robust optimization techniques in critical domains such as healthcare, finance, and cybersecurity, where data integrity and model accuracy are paramount. Implementing robust distributed ML systems based on these findings can enhance the reliability and security of AI applications.

By leveraging the insights from this analysis, system designers can strike a balance between robustness and efficiency in their distributed ML systems, leading to more reliable and effective machine learning models.

Can the techniques developed in this work be applied to other distributed optimization problems beyond machine learning, such as federated optimization in decentralized networks

The techniques developed in this work can be applied to a wide range of distributed optimization problems beyond machine learning, including federated optimization in decentralized networks. Here's how these techniques can be extended to other domains:

Federated Learning:

The robust optimization strategies developed in this work can be directly applied to federated learning settings, where multiple edge devices collaborate to train a shared model without sharing raw data. By incorporating Byzantine-robust techniques and trimmed mean aggregation, federated learning systems can be made more resilient to data poisoning and faulty participants.

Decentralized Networks:

In decentralized networks where nodes collaborate to solve optimization problems, the analysis can be adapted to ensure the integrity of the optimization process. By implementing robust aggregation methods and considering partial data corruption, decentralized networks can achieve more reliable and accurate optimization outcomes.

Cyber-Physical Systems:

The techniques can also be extended to cyber-physical systems where distributed optimization is used for control and coordination. By applying the principles of Byzantine-robust optimization and handling heterogeneous data, these systems can improve fault tolerance and system reliability.

By leveraging the robust optimization techniques developed in this work, various distributed optimization problems in different domains can benefit from enhanced resilience and efficiency.