Projection-free Online Learning (1206.4657v1)

Abstract: The computational bottleneck in applying online learning to massive data sets is usually the projection step. We present efficient online learning algorithms that eschew projections in favor of much more efficient linear optimization steps using the Frank-Wolfe technique. We obtain a range of regret bounds for online convex optimization, with better bounds for specific cases such as stochastic online smooth convex optimization. Besides the computational advantage, other desirable features of our algorithms are that they are parameter-free in the stochastic case and produce sparse decisions. We apply our algorithms to computationally intensive applications of collaborative filtering, and show the theoretical improvements to be clearly visible on standard datasets.

• The paper introduces projection-free algorithms using the Frank-Wolfe method that achieve sublinear regret while eliminating costly projection steps.
• The approach replaces traditional projection with linear optimization per iteration, drastically reducing computational time especially in high-dimensional problems like collaborative filtering.
• Experimental results confirm faster per-iteration computation and overall convergence, highlighting the practical scalability of these methods in online convex optimization.

Projection-free Online Learning: A Summary

The paper by Elad Hazan and Satyen Kale introduces novel algorithms for online convex optimization that forgo the computationally expensive projection steps prevalent in traditional methods. Instead, these algorithms employ linear optimization, based on the Frank-Wolfe technique, enabling greater efficiency. The work provides a theoretical framework and proves regret bounds, identifying computational and representation advantages.

Online learning, especially within the convex optimization paradigm, often relies on projections to maintain feasibility within a convex domain. The authors argue that, while the projection steps facilitate staying within the domain, they impose significant computational overhead, especially with large datasets or complex domains. Instead, the proposed algorithms solve a linear optimization problem per iteration, simplifying the procedure while maintaining comparable regret bounds.

Theoretical Contributions

The central theorem demonstrates sublinear regret bounds for these projection-free algorithms over various cost structures—specifically, showing that regret is sublinear in both stochastic and adversarial scenarios, albeit with different efficiencies:

• Stochastic, smooth convex costs: The bounds are optimal and show better convergence relative to traditional methods.
• Non-smooth costs: The regret bounds are not always optimal, indicating slower convergence compared to the optimal decision point but providing overall computational benefits due to less complexity per iteration.

Remarkably, the algorithms maintain a distribution over boundary points, ensuring sparse end solutions and obviating the need for explicit decompositions as required in some projected gradient descent approaches.

Practical Implications

One key illustrative application is collaborative filtering, common in recommendation systems, where the domain involves positive semidefinite matrices with bounded trace. The task of projection in such settings often requires computing the entire SVD of the matrix, a costly operation. The paper's approach only necessitates the computation of top singular vectors, thus significantly reducing computational time.

Experimental Validation

The authors validate their approach with experiments on standard collaborative filtering datasets, demonstrating that projection-free algorithms achieve faster computation per iteration and overall faster convergence in practice compared to Online Gradient Descent (OGD). This is particularly notable given the non-stochastic nature of certain cost functions in the experiments. The results suggest a performance increase by several folds in computational time, varying across datasets.

Implications for Future Research

The research opens several avenues for future investigation. The main open question lies in achieving optimal regret bounds while maintaining the computational simplicity afforded by linear optimization. Further work could explore extending these methods to broader problem domains and refining the techniques to handle a wider array of constraints without compromising computational efficiency.

This contribution lies at the intersection of theoretical advancements and practical applicability, offering a clear improvement in the efficiency of online learning algorithms across domains where traditional projection methods pose significant challenges. The exploration and integration of the Frank-Wolfe technique into online learning algorithms represent a significant step forward in addressing the computational bottlenecks faced by existing methods.