Any-stepsize Gradient Descent for Separable Data under Fenchel--Young Losses (2502.04889v1)

Abstract: The gradient descent (GD) has been one of the most common optimizer in machine learning. In particular, the loss landscape of a neural network is typically sharpened during the initial phase of training, making the training dynamics hover on the edge of stability. This is beyond our standard understanding of GD convergence in the stable regime where arbitrarily chosen stepsize is sufficiently smaller than the edge of stability. Recently, Wu et al. (COLT2024) have showed that GD converges with arbitrary stepsize under linearly separable logistic regression. Although their analysis hinges on the self-bounding property of the logistic loss, which seems to be a cornerstone to establish a modified descent lemma, our pilot study shows that other loss functions without the self-bounding property can make GD converge with arbitrary stepsize. To further understand what property of a loss function matters in GD, we aim to show arbitrary-stepsize GD convergence for a general loss function based on the framework of \emph{Fenchel--Young losses}. We essentially leverage the classical perceptron argument to derive the convergence rate for achieving $\epsilon$-optimal loss, which is possible for a majority of Fenchel--Young losses. Among typical loss functions, the Tsallis entropy achieves the GD convergence rate $T=\Omega(\epsilon^{-1/2})$, and the R{\'e}nyi entropy achieves the far better rate $T=\Omega(\epsilon^{-1/3})$. We argue that these better rate is possible because of \emph{separation margin} of loss functions, instead of the self-bounding property.

• The paper investigates the convergence of Gradient Descent with arbitrary stepsizes for separable data under Fenchel--Young losses, extending analysis beyond loss functions with self-bounding properties.
• It uses the Fenchel--Young framework to show arbitrary stepsize convergence is possible without self-bounding properties, highlighting the importance of the separation margin.
• These findings offer practical insights for optimizing models by choosing loss functions based on their separation margin and provide theoretical understanding of GD behavior beyond traditional stepsize constraints.

Summary of "Any-stepsize Gradient Descent for Separable Data under Fenchel--Young Losses"

The paper investigates the convergence properties of Gradient Descent (GD) with arbitrary stepsizes, specifically focusing on its application to separable data using a class of loss functions known as Fenchel--Young losses. Gradient Descent is a widely used optimization algorithm in machine learning. Traditionally, its convergence is ensured under a stable regime where the stepsize is set properly in relation to the problem's smoothness parameters. However, this paper extends beyond these conventional constraints.

Key Contributions

1. Generalization Beyond Self-bounding Losses: While previous studies, such as Wu et al., showed arbitrary stepsize convergence for logistic regression and described the importance of self-bounding properties, this paper examines loss functions without the self-bounding property. It finds that convergence can still occur, introducing the concept of separation margin as a critical factor for this behavior.
2. Fenchel--Young Losses Framework: The research leverages the framework of Fenchel--Young losses, which encompasses a wide range of convex loss functions. This allows for a broader application beyond models restricted to specific loss functions like logistic.
3. Convergence Rates: Strong numerical results are presented, indicating that different loss functions in the Fenchel--Young framework induce distinct convergence rates. For example, the Tsallis entropy achieves a convergence rate of $T = \Omega(\varepsilon^{-1/2})$, whereas the Rényi entropy reaches $T = \Omega(\varepsilon^{-1/3})$.
4. Importance of Separation Margin: The paper highlights that convergence rates can be notably improved when the loss function has a separation margin, aligning the GD trajectory toward a specific direction more effectively.

Implications and Future Directions

• Practical Advances: Understanding these properties offers practical insights for training models more efficiently by selecting appropriate loss functions that either possess a separation margin or can benefit from large stepsizes.
• Theoretical Contributions: Theoretical implications include expanding the understanding of GD behavior under non-standard conditions. This paves the way for exploring optimization beyond traditional settings and could influence how models are trained in complex, realistic scenarios.
• Further Exploration: The paper opens avenues for further investigation into non-separable data cases and the role of different entropy measures. Expanding this framework could lead to more effective algorithms that exploit the benefits of large stepsizes across various machine learning tasks.

Overall, this paper pushes the boundaries of traditional gradient descent analysis by integrating the Fenchel--Young losses framework, thereby offering new perspectives on how loss function properties can influence optimization dynamics. The implications span both practical applications and theoretical advancements, setting the stage for future research in optimization and machine learning.