Near-Optimal Streaming Heavy-Tailed Statistical Estimation with Clipped SGD (2410.20135v1)

Abstract: We consider the problem of high-dimensional heavy-tailed statistical estimation in the streaming setting, which is much harder than the traditional batch setting due to memory constraints. We cast this problem as stochastic convex optimization with heavy tailed stochastic gradients, and prove that the widely used Clipped-SGD algorithm attains near-optimal sub-Gaussian statistical rates whenever the second moment of the stochastic gradient noise is finite. More precisely, with $T$ samples, we show that Clipped-SGD, for smooth and strongly convex objectives, achieves an error of $\sqrt{\frac{\mathsf{Tr}(\Sigma)+\sqrt{\mathsf{Tr}(\Sigma)|\Sigma|_2}\log(\frac{\log(T)}{\delta})}{T}}$ with probability $1-\delta$, where $\Sigma$ is the covariance of the clipped gradient. Note that the fluctuations (depending on $\frac{1}{\delta}$) are of lower order than the term $\mathsf{Tr}(\Sigma)$. This improves upon the current best rate of $\sqrt{\frac{\mathsf{Tr}(\Sigma)\log(\frac{1}{\delta})}{T}}$ for Clipped-SGD, known only for smooth and strongly convex objectives. Our results also extend to smooth convex and lipschitz convex objectives. Key to our result is a novel iterative refinement strategy for martingale concentration, improving upon the PAC-Bayes approach of Catoni and Giulini.

• The paper introduces a novel Clipped-SGD method for sequential parameter estimation in heavy-tailed data settings, achieving near-optimal sub-Gaussian error rates.
• It rigorously analyzes the algorithm under minimal assumptions, ensuring robust performance with only finite second moment requirements for stochastic gradients.
• The improved error bounds and iterative strategy outperform prior techniques, making the approach practical for high-dimensional, memory-constrained streaming applications.

Near-Optimal Streaming Heavy-Tailed Statistical Estimation with Clipped SGD

The paper tackles the challenging problem of estimating parameters from high-dimensional datasets exhibiting heavy-tailed distributions in a memory-constrained streaming setting. This setting necessitates processing data sequentially, in stark contrast to more traditional batch processing frameworks. The authors recast this scenario as a stochastic convex optimization (SCO) problem characterized by heavy-tailed stochastic gradients and focus on the utilization of Clipped Stochastic Gradient Descent (SGD) as the optimization method of choice.

Key Contributions

1. Problem Formulation and Approach:
   • The authors define the investigation framework as a SCO problem where said gradients exhibit heavy-tailed behavior. This is a natural fit due to the convex nature of many estimation problems and their occurrence across several applications with heavy-tailed data.
2. Algorithm and Theoretical Results:
   • They provide a rigorous analysis of the Clipped-SGD algorithm and demonstrate that it achieves near-optimal sub-Gaussian statistical rates when the second moment of the gradient noise is finite. Notably, they derive the result for smooth and strongly convex objectives.
   • For a given number, $T$, of samples, the authors show that Clipped-SGD achieves an error of the order $\sqrt{\frac{Tr(\Sigma) + \sqrt{Tr(\Sigma)\|\Sigma\|_2}\ln(\nicefrac{\ln(T)}{\delta})}{T}}$, where $\Sigma$ is the covariance matrix of the clipped gradients.
3. Comparison to Prior Rates:
   • The results improve upon previous rates for Clipped-SGD by tightening the bound involving $\ln(\nicefrac{1}{\delta})$. The paper presents an innovative iterative strategy for refining these bounds, surpassing the PAC-Bayes approach of Catoni (2018).
4. Robustness to Gradient Noise:
   • A substantial part of the paper is dedicated to ensuring that the analysis holds under minimal assumptions on the distribution of the gradient noise, specifically assuming only finite second moments, thereby accommodating heavy-tailed distributions robustly.

Implications and Applications

The implications of this work are broad and beneficial for multiple domains where streaming and immediate data usage are practical necessities. The approach is not only theoretically sound but also practical for real-world scenarios requiring robust parameter estimation with sequential data access under memory constraints.

The paper's theoretical advances encourage further exploration of clipped techniques beyond purely convex settings. Moreover, understanding the dynamics in non-convex optimization problems like those encountered in deep learning frameworks would be an intriguing continuation.

Future Outlook

The research opens several channels for extending the existing framework:

• Exploration of Non-Convex Applications:
  • Current work mainly addresses convex problems; however, the framework could potentially be adapted for non-convex landscapes frequently present in neural network training.
• Beyond Strong Convexity:
  • Extensions could investigate alternative structures where strong convexity does not hold, which is more reflective of assorted real-world data optimization scenarios.
• Broader Statistical Estimations:
  • The Clipped-SGD could be employed in diversified statistical estimators, ensuring robust performance in heavy-tailed situations beyond what current empirical means achieve.

Overall, the paper introduces an impactful stride forward in handling complex statistical problems within constrained environments and sets the stage for further refinements and adaptability to broader classes of optimization challenges.