Byzantine-Robust Distributed Learning: Towards Optimal Statistical Rates (1803.01498v2)

Abstract: In large-scale distributed learning, security issues have become increasingly important. Particularly in a decentralized environment, some computing units may behave abnormally, or even exhibit Byzantine failures -- arbitrary and potentially adversarial behavior. In this paper, we develop distributed learning algorithms that are provably robust against such failures, with a focus on achieving optimal statistical performance. A main result of this work is a sharp analysis of two robust distributed gradient descent algorithms based on median and trimmed mean operations, respectively. We prove statistical error rates for three kinds of population loss functions: strongly convex, non-strongly convex, and smooth non-convex. In particular, these algorithms are shown to achieve order-optimal statistical error rates for strongly convex losses. To achieve better communication efficiency, we further propose a median-based distributed algorithm that is provably robust, and uses only one communication round. For strongly convex quadratic loss, we show that this algorithm achieves the same optimal error rate as the robust distributed gradient descent algorithms.

• The paper introduces robust median-based and trimmed-mean-based gradient descent algorithms that achieve provable optimal error rates despite Byzantine failures.
• It rigorously analyzes performance across strongly convex, non-strongly convex, and smooth non-convex loss functions, establishing theoretical lower bounds.
• The study proposes a one-round, communication-efficient median-based algorithm for quadratic losses, balancing robustness and reduced communication overhead.

Byzantine-Robust Distributed Learning: Towards Optimal Statistical Rates

The paper under review addresses the critical issue of robustness in decentralized machine learning systems, particularly focusing on achieving optimal statistical error rates when worker machines may exhibit Byzantine failures. These failures constitute arbitrary and potentially adversarial behavior, posing significant challenges for distributed learning algorithms.

Key Contributions

The paper makes several key contributions:

• Robust Gradient Descent Algorithms: Two robust gradient descent (GD) algorithms are proposed:
  • Median-based GD: This algorithm leverages the coordinate-wise median operation.
  • Trimmed-mean-based GD: This algorithm uses coordinate-wise trimmed mean operations.
• Statistical Error Rates: The algorithms achieve provable statistical error rates for different types of population loss functions—strongly convex, non-strongly convex, and smooth non-convex.
• Communication-Efficient Algorithm: A median-based distributed algorithm is introduced that is provably robust and requires only one communication round, maintaining the same optimal error rate for strongly convex quadratic loss.

Analytical Insights

Robust Gradient Descent Algorithms

The robustness of the proposed GD algorithms relies on two key mechanisms: the coordinate-wise median and the coordinate-wise trimmed mean. The paper derives the following results for these algorithms:

• Median-based GD:
  • Achieves a statistical error rate of $$\widetilde{O}(\frac{\alpha}{\sqrt{n}} + \frac{1}{\sqrt{nm}} + \frac{1}{n})$$ for strongly convex loss functions, provided $n \gtrsim m$.
  • This algorithm does not require prior knowledge of the fraction $\alpha$ of Byzantine machines and handles skewness in the gradient distributions.
• Trimmed-mean-based GD:
  • Obtains an optimal statistical error rate of $$\widetilde{O}(\frac{\alpha}{\sqrt{n}} + \frac{1}{\sqrt{nm}})$$ for strongly convex loss functions under the assumption of sub-exponential gradients.
  • It achieves better performance than the median-based GD when $n$ is small, even in the presence of Byzantine failures.

Communication-Efficient Algorithm

The paper also proposes a robust one-round algorithm:

• Median-based One-round Algorithm: For strongly convex quadratic losses, this algorithm achieves a $$\widetilde{O}(\frac{\alpha}{\sqrt{n}} + \frac{1}{\sqrt{nm}} + \frac{1}{n})$$ error rate, matching the performance of the robust GD algorithms with significantly reduced communication overhead.

Lower Bound on Error Rates

The authors establish a lower bound on the achievable error rate in the presence of Byzantine failures, demonstrating that the dependence on $\alpha$, $n$, and $m$ is unimprovable up to logarithmic factors. Specifically, the bound shows that no algorithm can achieve an error rate lower than $$\widetilde{\Omega}(\frac{\alpha}{\sqrt{n}} + \frac{1}{\sqrt{nm}})$$.

Implications and Future Directions

Practical Implications:

• Robustness: These algorithms enhance reliability in distributed learning settings, such as federated learning, where worker machines may be unreliable or compromised.
• Communication Efficiency: The one-round algorithm is particularly beneficial in scenarios with high communication costs, reducing the necessity for multiple rounds of data transmission without compromising robustness.

Theoretical Implications:

• Optimality: The results affirm the theoretical limits of robust distributed learning in the presence of adversarial behavior, effectively bridging a gap in the current literature.

Future Research:

• High-Dimensional Data: Exploring robust distributed learning in high-dimensional settings remains an open challenge.
• Advanced Robust Algorithms: Developing robust variants of more sophisticated distributed algorithms (e.g., SVRG, DANE) is a promising direction.
• Hybrid Robustness and Efficiency: Balancing robustness, communication efficiency, and computational efficiency in new algorithms offers fruitful avenues for exploration.

Conclusion

In summary, the authors provide robust distributed learning algorithms that achieve optimal statistical error rates against Byzantine failures. Their work presents both systems-oriented and theoretical advancements, making significant strides in the domain of reliable distributed machine learning. Future research can build on these foundations to tackle more complex and large-scale learning environments.