A geometric alternative to Nesterov's accelerated gradient descent (1506.08187v1)

Abstract: We propose a new method for unconstrained optimization of a smooth and strongly convex function, which attains the optimal rate of convergence of Nesterov's accelerated gradient descent. The new algorithm has a simple geometric interpretation, loosely inspired by the ellipsoid method. We provide some numerical evidence that the new method can be superior to Nesterov's accelerated gradient descent.

• The paper proposes a novel geometric method as an alternative to Nesterov's accelerated gradient descent (NAGD) for smooth, strongly convex functions, achieving the same optimal convergence rate.
• This method combines gradient descent and ellipsoid concepts, using gradient information and two line searches per iteration to iteratively shrink enclosing balls and accelerate convergence.
• Experimental results show the proposed geometric descent method (GeoD) is competitive with NAGD, potentially more robust in worst-case scenarios, and offers a more intuitive understanding of acceleration.

A Geometric Alternative to Nesterov's Accelerated Gradient Descent

The paper proposes a novel method for unconstrained optimization of smooth and strongly convex functions that achieves the optimal convergence rate of Nesterov's accelerated gradient descent (NAGD). The method presents a geometric approach inspired by the ellipsoid method, aiming to provide a more intuitive understanding of the acceleration mechanism compared to NAGD. The paper makes a compelling argument for the integration of both zeroth and first order information within the optimization process, positing this method as potentially more effective in practical applications.

Core Contributions

The primary contribution of this paper is the development of a new optimization algorithm that blends elements of gradient descent and the ellipsoid method. The proposed algorithm decreases the squared distance to the optimum at a rate of $1-1/\sqrt{\kappa}$ per iteration, where $\kappa$ is the condition number of the function. This matches the optimal rate of convergence among first-order methods. Unlike NAGD, which has a challenging intuitive understanding, this geometric method offers a clear intuition by maintaining and shrinking enclosing balls centered at iteratively updated points.

Methodology

The algorithm is predicated on defining a strongly convex function $f: \mathbb{R}^n \rightarrow \mathbb{R}$, characterized by smoothness parameter $\beta$ and convexity parameter $\alpha$. It constructs intersections of balls based on gradients and iteratively reduces the radius of these intersections. The algorithm performs two line searches per iteration alongside a gradient evaluation, integrating information from previous iterations to accelerate convergence.

The paper details how one can formulate a suboptimal algorithm using gradient descent mixed with geometric properties and then evolve this to an accelerated method. It leverages the property that gradient information allows further shrinking of the enclosing ball, hence speeding up the convergence.

Experimental Validation

Several experiments were conducted to compare the proposed method against established optimization techniques like Steepest Descent, NAGD, and L-BFGS. The proposed geometric descent method (GeoD) was competitive and in some cases outperformed NAGD. Although L-BFGS remains superior in many scenarios due to its ability to utilize previous gradients effectively, GeoD showed promising results particularly in worst-case scenarios where it converged more rapidly than NAGD, hence indicating its potential robustness.

Implications and Future Directions

This work contributes significantly to the field of optimization, offering a method with simpler geometric interpretations than existing accelerated techniques. The implications for practical machine learning and optimization tasks are substantial, as GeoD integrates smoothness and convexity properties in a naturally interpretable manner. Importantly, this approach could be generalized to other optimization paradigms where convergence rates are critical, providing an alternative pathway to improve efficiency and performance.

Future research may explore optimizations further leveraging geometric insights, potentially extending to non-convex domains or hybrid models that incorporate additional information types. The promising results also suggest possible expansions of GeoD's ball intersection principles to support adaptive or stochastic environments, which are common in real-world data-driven applications.