Adaptivity of averaged stochastic gradient descent to local strong convexity for logistic regression (1303.6149v3)

Abstract: In this paper, we consider supervised learning problems such as logistic regression and study the stochastic gradient method with averaging, in the usual stochastic approximation setting where observations are used only once. We show that after $N$ iterations, with a constant step-size proportional to $1/R^2 \sqrt{N}$ where $N$ is the number of observations and $R$ is the maximum norm of the observations, the convergence rate is always of order $O(1/\sqrt{N})$, and improves to $O(R^2 / \mu N)$ where $\mu$ is the lowest eigenvalue of the Hessian at the global optimum (when this eigenvalue is greater than $R^2/\sqrt{N}$). Since $\mu$ does not need to be known in advance, this shows that averaged stochastic gradient is adaptive to \emph{unknown local} strong convexity of the objective function. Our proof relies on the generalized self-concordance properties of the logistic loss and thus extends to all generalized linear models with uniformly bounded features.

• The paper demonstrates that averaged stochastic gradient descent (SGD) adapts its convergence behavior to local strong convexity for logistic regression by leveraging self-concordance without needing prior knowledge of the local convexity parameter.
• The analysis shows that averaged SGD with a constant step-size achieves a general convergence rate of O(1/√N), which improves to O(R²/μN) under conditions of local strong convexity.
• This work extends the applicability of averaged SGD to other generalized linear models with bounded features and self-concordant loss functions, showing that squared gradient norms converge at a rate of O(1/N).

Adaptivity of Averaged Stochastic Gradient Descent

The paper "Adaptivity of Averaged Stochastic Gradient Descent to Local Strong Convexity for Logistic Regression" by Francis Bach investigates the applicability of averaged stochastic gradient descent (SGD) in supervised learning scenarios, specifically focusing on logistic regression. This work builds on the foundational stochastic approximation frameworks while addressing the nuances of local strong convexity through self-concordance properties.

Main Contributions

The paper delineates the following contributions:

1. Convergence Rates with Averaged SGD: It demonstrates that for a constant step-size scaling with the maximum norm of observations, i.e., proportional to $1/R^2 \sqrt{N}$, the averaged SGD achieves a convergence rate of $O(1/\sqrt{N})$. Importantly, this convergence improves under conditions of local strong convexity to $O(R^2/\mu N)$, where $\mu$ is the smallest eigenvalue of the Hessian at the global optimum.
2. Adaptivity Without Prior Knowledge: A salient feature of the derived results is the adaptivity of the algorithm to the local strong convexity without requiring prior knowledge of the local convexity parameter $\mu$. This adaptivity is facilitated by the self-concordant structure of the logistic loss function, which allows the method to leverage local curvature effectively.
3. Analysis Under Self-Concordance: The work extends the applicability of averaged SGD to a class of generalized linear models marked by bounded features and self-concordant loss functions. Self-concordance serves as a pivotal assumption that permits sharper control over deviations and derivation of bounds on moments and tail probabilities of function values and gradients.

Theoretical Insights and Implications

1. Applications Across Convexity Classes: The results provide rigorous insights for both strongly convex and non-strongly convex scenarios. By addressing local strong convexity, the paper bridges gaps where typical global convexity might fail, notably in non-compact and high-dimensional settings such as logistic regression.
2. Convergence of Gradients: The analysis yields a significant result indicating that squared norms of gradients converge at a rate of $O(1/N)$. This behavior underscores the potential for improved function convergence rates in locally strongly convex problems.
3. Probability and High-Order Bounds: By establishing both expected and probabilistic bounds, the work provides a robust statistical framework for understanding the efficacy of SGD in real-world applications. The convergence guarantees extend beyond mere expectation to encompass probability measures and higher-order moments, ensuring stability and reliability of the algorithm's performance.

Future Directions

The paper suggests several avenues for future work:

1. Decaying Step Sizes: Exploration of decaying step sizes instead of constant ones, potentially optimizing convergence rates while simplifying the tuning process.
2. Broader Algorithmic Extensions: Modifications and extensions to SGD that incorporate alternative regularization and acceleration methods could further elucidate the balance between adaptivity, efficiency, and computational overhead.
3. Practical Implementation and Empirical Validation: While theoretically solid, empirical studies validating these theoretical findings in diverse contexts and datasets, particularly involving logistic regression and other non-linear models, remain a valuable direction.

Conclusion

Francis Bach's exploration into averaged stochastic gradient descent extends its adaptability to unknown local strong convexity, particularly for logistic regression. By leveraging the self-concordant nature of the loss functions, the paper provides comprehensive theoretical insights into the convergence behavior. The results not only strengthen the theoretical underpinnings of SGD in machine learning but also widen the scope of its applicability to more complex, non-convex settings with minimal parameter tuning.