SGD with Clipping is Secretly Estimating the Median Gradient (2402.12828v1)

Abstract: There are several applications of stochastic optimization where one can benefit from a robust estimate of the gradient. For example, domains such as distributed learning with corrupted nodes, the presence of large outliers in the training data, learning under privacy constraints, or even heavy-tailed noise due to the dynamics of the algorithm itself. Here we study SGD with robust gradient estimators based on estimating the median. We first consider computing the median gradient across samples, and show that the resulting method can converge even under heavy-tailed, state-dependent noise. We then derive iterative methods based on the stochastic proximal point method for computing the geometric median and generalizations thereof. Finally we propose an algorithm estimating the median gradient across iterations, and find that several well known methods - in particular different forms of clipping - are particular cases of this framework.

• The paper shows that using median gradient estimators in SGD improves resilience against heavy-tailed noise and outliers, achieving a convergence rate of 1/√T.
• The paper introduces an efficient stochastic proximal point algorithm to practically compute median gradients in noisy optimization scenarios.
• The paper unifies several robust optimization methods by revealing that many existing techniques implicitly estimate the median gradient.

Exploring the Depths of SGD with Median Gradient Estimation

Introduction to Gradient-Based Optimization

Gradient-based optimization algorithms, specifically Stochastic Gradient Descent (SGD), have been the cornerstone of numerous advances in machine learning and data science. These algorithms have facilitated the training of complex models in various domains by providing a mechanism to minimize loss functions effectively. At each step, SGD uses a noisy estimate of the gradient to iteratively update the model parameters, aiming for the model that best fits the data according to the specified loss function.

Challenges in Gradient Estimation

One of the critical challenges that arise in stochastic optimization is the quality of the gradient estimation. When the noise in the gradient estimation exhibits certain properties, such as having significant outliers, being heavy-tailed, or being manipulated adversarially, the performance of SGD can deteriorate dramatically. It becomes crucial to devise methods that can robustly estimate the gradient amidst such adverse conditions to ensure the convergence and robustness of the optimization procedure.

SGD with Median Gradient Estimation

The paper investigates SGD's robustness by utilizing median gradient estimates rather than the traditional mean. This approach is particularly beneficial in domains characterized by high levels of noise, including distributed learning with corrupted nodes, training data with large outliers, learning under privacy constraints, and scenarios where the algorithm's dynamics introduce heavy-tailed noise.

Median Gradient-Based Optimization

The paper introduces a novel perspective on robust stochastic optimization by leveraging median-based estimators for gradient computation. This method exhibits resilience against heavy-tailed noise and outliers, thereby providing a powerful tool for optimizing under challenging conditions.

Contribution: Theoretical Insights and Algorithmic Framework

The paper makes several significant contributions:

• Theoretical Analysis: It establishes that SGD with median gradient estimates converges at a rate of $1/\sqrt{T}$, even under heavy-tailed noise with potentially infinite variance. This result underscores the robustness of median-based estimation in stochastic optimization.
• Practical Median Computation: Addressing the computational challenges associated with median estimation, the paper introduces an efficient algorithm based on the stochastic proximal point method, facilitating the practical use of median-based gradient estimates in optimization.
• Unification of Robust Optimization Techniques: The research highlights how several established robust optimization algorithms essentially estimate the median gradient, implicitly or explicitly. This insight offers a unified theoretical foundation for understanding and further developing robust optimization techniques.

Implications and Future Perspectives

The findings of this paper hold substantial implications, both theoretical and practical, for the field of machine learning and optimization. By elucidating the robustness properties of median-based gradient estimation and providing a viable algorithm for its implementation, this research paves the way for the development of more resilient optimization methods capable of handling the complexities of modern datasets and learning environments.

Conclusion

In summary, this paper contributes to the ongoing efforts to enhance the robustness and reliability of gradient-based optimization methods. By shifting the focus towards median gradient estimation, it opens up new avenues for research and practical applications in scenarios plagued by noise and outliers. The direction laid out in this paper holds promise for advancing the state of the art in robust stochastic optimization, with potential impacts across a wide array of applications in machine learning and beyond.