Directional Smoothness and Gradient Methods: Convergence and Adaptivity (2403.04081v2)

Abstract: We develop new sub-optimality bounds for gradient descent (GD) that depend on the conditioning of the objective along the path of optimization rather than on global, worst-case constants. Key to our proofs is directional smoothness, a measure of gradient variation that we use to develop upper-bounds on the objective. Minimizing these upper-bounds requires solving implicit equations to obtain a sequence of strongly adapted step-sizes; we show that these equations are straightforward to solve for convex quadratics and lead to new guarantees for two classical step-sizes. For general functions, we prove that the Polyak step-size and normalized GD obtain fast, path-dependent rates despite using no knowledge of the directional smoothness. Experiments on logistic regression show our convergence guarantees are tighter than the classical theory based on $L$-smoothness.

• The paper introduces directional smoothness to capture local gradient variations, yielding tighter convergence bounds.
• It distinguishes between point-wise and path-wise smoothness, offering practical alternatives to global Lipschitz conditions.
• Empirical studies validate that adaptive step-size schemes based on directional smoothness significantly enhance gradient descent performance.

Directional Smoothness and Gradient Descent: Convergence and Adaptivity

Introduction to Directional Smoothness

In the optimization community, the recent focus has been on understanding how the local geometry of the loss landscape influences the convergence properties of gradient descent (GD) methods. Traditional analyses, rooted in global Lipschitz smoothness assumptions, often yield pessimistic or unrealistic convergence rates when applied outside their standard theoretical settings. This discrepancy has led to a new wave of research seeking finer measures of smoothness that can better capture the local geometry encountered along the optimization path.

This paper introduces the concept of directional smoothness as a measure of gradient variation that depends only on the path of optimization, rather than on global, worst-case constants. Specifically, directional smoothness is defined along the linear segment (or 'chord') between consecutive GD iterates. This approach allows for a more nuanced analysis, providing convergence bounds that adapt to the actual path taken by the gradient descent algorithm.

Developments in Directional Smoothness

Point-wise and Path-wise Smoothness

The paper distinguishes between point-wise directional smoothness and path-wise directional smoothness. Point-wise directional smoothness, denoted as $D(y, x)$, measures the gradient variation at the endpoints of the optimization path. It offers an easily computable upper bound on function values along the path but does so at a potentially less tight bound compared to global Lipschitz smoothness. Path-wise directional smoothness, denoted as $A(x,y)$, captures the gradient variation along the entire chord between $x$ and $y$. Although providing a tighter bound, it is more challenging to compute due to its dependence on the properties of the function along the entire linear segment.

The paper robustly defines these forms of smoothness, proving that they satisfactorily bound the objective function's upper-bound for any differentiable and convex function, without necessarily presuming global smoothness.

Theoretical Implications and Experimental Validation

The theoretical framework developed for directional smoothness allows the derivation of new convergence bounds for gradient descent on convex objectives. These bounds not only reflect the algorithm's locality but are also more precise than classical bounds when adapted step-size sequences are used.

Experimental evaluations further demonstrate the practical relevance of these contributions. The application to gradient descent with adapted step-sizes, rooted in directional smoothness, displays significant improvement over methods grounded in global smoothness assumptions. Notably, such adapted approaches often operate at the 'edge of stability,' pushing the boundaries of convergence rates achievable under traditional analyses.

Future Prospects and Practical Considerations

Directional smoothness paves the way toward gradient methods that are genuinely adaptive to the landscape's local geometry, promising efficient optimization procedures without the baggage of global assumptions. Future investigations could explore the integration of directional smoothness concepts with more sophisticated optimization frameworks or explore the implications of adaptive step-size schemes for a broader range of optimization problems.

The paper sets a solid foundation for future exploration in this area, offering a compelling alternative to traditional global smoothness metrics. As the field progresses, the tools and theories surrounding directional smoothness are expected to refine further, enhancing our understanding and application of gradient-based optimization methods in complex settings.