Faster Rates for the Frank-Wolfe Method over Strongly-Convex Sets (1406.1305v2)

Abstract: The Frank-Wolfe method (a.k.a. conditional gradient algorithm) for smooth optimization has regained much interest in recent years in the context of large scale optimization and machine learning. A key advantage of the method is that it avoids projections - the computational bottleneck in many applications - replacing it by a linear optimization step. Despite this advantage, the known convergence rates of the FW method fall behind standard first order methods for most settings of interest. It is an active line of research to derive faster linear optimization-based algorithms for various settings of convex optimization. In this paper we consider the special case of optimization over strongly convex sets, for which we prove that the vanila FW method converges at a rate of $\frac{1}{t^2}$. This gives a quadratic improvement in convergence rate compared to the general case, in which convergence is of the order $\frac{1}{t}$, and known to be tight. We show that various balls induced by $\ell_p$ norms, Schatten norms and group norms are strongly convex on one hand and on the other hand, linear optimization over these sets is straightforward and admits a closed-form solution. We further show how several previous fast-rate results for the FW method follow easily from our analysis.

• The paper demonstrates that the Frank-Wolfe method achieves an O(1/t²) convergence rate on strongly convex sets, markedly improving the traditional O(1/t) performance.
• It establishes that various norm-induced balls are strongly convex, enabling simple, closed-form linear optimization steps.
• By extending previous accelerated results, the research bridges theory and practice, enhancing optimization in large-scale machine learning tasks.

Faster Rates for the Frank-Wolfe Method over Strongly-Convex Sets

The paper discusses advancements in the Frank-Wolfe (FW) method, a first-order optimization technique that eschews projections in favor of linear optimization within the feasible domain. This projection-free characteristic makes it particularly suitable for large-scale machine learning applications, such as matrix completion and structural SVMs. However, the traditional FW method, despite its utility, suffers from suboptimal convergence rates of $\mathcal{O}(1/t)$, as opposed to Nesterov's accelerated gradient methods, which achieve convergence rates of $\mathcal{O}(1/t^2)$ for smooth problems. The authors confront this disparity by exploring the convergence properties of FW under specific conditions.

Key Contributions

1. Accelerated Convergence over Strongly Convex Sets: The pivotal contribution of this paper lies in demonstrating that when both the objective function and the feasible set are strongly convex, the FW method achieves a convergence rate of $\mathcal{O}(1/t^2)$. This represents a significant improvement over the traditional $\mathcal{O}(1/t)$ rate, aligning more closely with the faster rates seen in projected gradient methods. The attainable acceleration does not rely on dimensionality, thus retaining computational efficiency.
2. Characterization of Strongly Convex Sets: The authors establish that various norm-induced balls—including $\ell_p$ norms for $p \in (1, 2]$, Schatten norms, and group norms—are strongly convex. This property is crucial since linear optimization over these sets is straightforward and can be resolved in closed-form, allowing the FW method to remain computationally feasible.
3. Extension to Prior Results: By employing their analytical framework, the authors show that previous accelerated results—such as those requiring strong convexity of the objective—can be elegantly derived. This positions their work as a unifying theory that bridges isolated instances of faster rates in existing literature.

Implications and Future Directions

The research has both theoretical and practical implications. Theoretically, it advances understanding of the conditions under which projection-free methods can achieve superior convergence speeds, thus enriching the literature on optimization algorithms. Practically, it highlights scenarios where the Frank-Wolfe method can be optimally employed, providing guidance for its implementation in real-world machine learning tasks.

Potential future directions include:

• Logarithmic Rate Exploration: Investigating whether even faster convergence, such as a $\mathcal{O}(\log(1/\epsilon))$ rate, is achievable when both the objective function and feasible domain are strongly convex.
• Non-Smooth Extensions: Extending the techniques to non-smooth optimization problems, which would broaden the method's applicability significantly.
• Complexity of Linear Optimization: Further exploration into optimizing the complexity of the linear optimization step, particularly when the feasible domain is not naturally strongly convex but could be transformed into an equivalent form.

In summary, this paper effectively elevates the Frank-Wolfe method's performance on strongly convex domains, offering pathways for its enhanced application in optimization-challenging fields like machine learning.