Accelerated Objective Gap and Gradient Norm Convergence for Gradient Descent via Long Steps (2403.14045v4)

Abstract: This work considers gradient descent for L-smooth convex optimization with stepsizes larger than the classic regime where descent can be ensured. The stepsize schedules considered are similar to but differ slightly from the recent silver stepsizes of Altschuler and Parrilo. For one of our stepsize sequences, we prove a $O\left(N^{- 1.2716\dots}\right)$ convergence rate in terms of objective gap decrease and for the other, we show the same rate of decrease for squared-gradient-norm decrease. This first result improves on the recent result of Altschuler and Parrilo by a constant factor, while the second results improve on the exponent of the prior best squared-gradient-norm convergence guarantee of $O\left(N^{-1}\right)$.

• The paper introduces long-step techniques that achieve O(1/N^(1.2716)) convergence for both objective gap and gradient norm in L-smooth convex optimization.
• It utilizes left-heavy and right-heavy step-size sequences with recursive gluing to rigorously prove accelerated convergence rates.
• The enhanced convergence reduces computational cost and paves the way for further research in adaptive and non-convex optimization methods.

Accelerated Objective Gap and Gradient Norm Convergence for Gradient Descent via Long Steps

The paper under consideration, "Accelerated Objective Gap and Gradient Norm Convergence for Gradient Descent via Long Steps," introduces innovative techniques to enhance gradient descent's convergence rates for $L$-smooth convex optimization problems. The research builds upon existing studies by exploring step-size strategies beyond the conventional "short step-size" regime. The authors present two important findings: leveraging sequences of "long steps," they achieve $O(1/N^{1.2716})$ convergence rates for both objective gap and squared-gradient-norm descent, considerably improving upon previously established rates by a constant factor.

Gradient descent is an optimization technique that minimizes a function iteratively by moving in the direction of steepest descent. Classically, for $L$-smooth convex functions, gradient descent with appropriately set short step sizes provides an objective gap convergence rate of $O(1/N)$. Recent theoretical advancements reveal the potential to further optimize convergence by modifying step-size sequences. However, these typically compromise the monotonic reduction in objective function value, making analysis more challenging.

This paper addresses these challenges by considering two categories of step-size sequences termed as "left-heavy" ($h^{(k)}$) and "right-heavy" ($h^{(k)}_\text{right}$). Both sequences capitalize on specific patterns to achieve faster convergence. The paper proves an accelerated convergence rate for the objective gap and gradient norm, $O(1/N^{\log_2(\phi - 1)})$, where $\phi$ denotes the silver ratio $1 + \sqrt{2}$. This outcome surpasses the recent silver stepsizes proposed by Altschuler and Parrilo, which achieved a similar rate by a constant factor; it is noteworthy for the objective gap.

The methodology involves meticulous derivation of sufficient conditions and a sequence of recursive constructions. The proof framework heavily utilizes the concept of coercivity and performance estimation problems in assessing smooth convex optimization landscapes, capitalizing on established mathematics around interpolation theory. The authors deploy "recursive gluing" techniques in tandem with specifically crafted certificates to substantiate their convergence claims. This method extends the intricate induction structure used in sequential steps to yield compact and demonstrative proofs for their accelerated methodology.

These findings have significant implications both practically and theoretically. Practically, the enhanced convergence rates can substantially reduce computational costs in large-scale optimization settings. Theoretically, the paper suggests the promise of further exploiting step-size flexibility in routine optimization algorithms beyond current standards. The results present new avenues for exploring optimization landscapes using non-traditional approaches. Future research could focus on expanding these techniques to more general non-convex or stochastic settings, potentially incorporating adaptive strategies for real-time optimization procedures in dynamic environments.

In conclusion, the paper "Accelerated Objective Gap and Gradient Norm Convergence for Gradient Descent via Long Steps" provides profound insight into leveraging structural properties of step-size sequences to enhance convergence rates in gradient descent. The well-founded analysis and significant numerical improvements offered by this paper potentially encourage a rethinking of standard approaches within gradient-based optimization algorithms. As the field of optimization continues to evolve alongside increasing computational demands, such innovative theoretical contributions are invaluable in aligning academic pursuits with practical applications.