[Paper Review] Exploiting the Surrogate Gap in Online Multiclass Classification

Dirk van der Hoeven|arXiv (Cornell University)|Jul 24, 2020

Advanced Bandit Algorithms Research32 references4 citations

TL;DR

This paper introduces Gaptron, a randomized first-order algorithm for online multiclass classification that exploits the gap between the non-convex zero-one loss and convex surrogate losses (logistic, hinge, smooth hinge). It achieves $O(K)$ regret in the full information setting and $O(K\sqrt{T})$ regret in the bandit setting, with no dependence on feature dimension $d$, making it the first linear-time algorithm with this regret bound in the bandit setting.

ABSTRACT

We present Gaptron, a randomized first-order algorithm for online multiclass classification. In the full information setting we show expected mistake bounds with respect to the logistic loss, hinge loss, and the smooth hinge loss with constant regret, where the expectation is with respect to the learner's randomness. In the bandit classification setting we show that Gaptron is the first linear time algorithm with $O(K\sqrt{T})$ expected regret, where $K$ is the number of classes. Additionally, the expected mistake bound of Gaptron does not depend on the dimension of the feature vector, contrary to previous algorithms with $O(K\sqrt{T})$ regret in the bandit classification setting. We present a new proof technique that exploits the gap between the zero-one loss and surrogate losses rather than exploiting properties such as exp-concavity or mixability, which are traditionally used to prove logarithmic or constant regret bounds.

Motivation & Objective

• To develop a first-order online multiclass classification algorithm with improved regret bounds in both full information and bandit settings.
• To eliminate dependence on feature dimension $d$ in regret bounds, unlike prior $O(K\sqrt{T})$ bandit algorithms.
• To introduce a new proof technique based on the gap between zero-one and surrogate losses, rather than exp-concavity or mixability.
• To achieve linear-time $O(K\sqrt{T})$ regret in the bandit multiclass setting, a first for such bounds.

Proposed method

• Gaptron uses online gradient descent (OGD) to update the weight matrix $\W_t$ based on surrogate loss gradients.
• It introduces a randomized prediction strategy where the learner samples from a distribution over classes to balance exploration and exploitation.
• The algorithm exploits the gap between the zero-one loss (performance measure) and surrogate losses (optimization target), using this to derive regret bounds.
• A novel proof technique bounds the surrogate gap by $O(K)$ in the full information case and $O(K\sqrt{T})$ in the bandit case, independent of $d$.
• The analysis leverages AdaHedge for probability vector updates and Bhatia-Davis inequality to control variance terms in the loss gap.
• The method is generalized to handle abstention in classification, with a cost $c_t < 1/2$ ensuring bounded regret.

Experimental results

Research questions

• RQ1 Can a first-order online multiclass classifier achieve $O(K)$ regret in the full information setting without dependence on feature dimension?
• RQ2 Is it possible to design a linear-time algorithm with $O(K\sqrt{T})$ regret in the bandit multiclass classification setting?
• RQ3 Can the gap between zero-one and surrogate losses be systematically exploited to derive regret bounds, instead of relying on exp-concavity or mixability?
• RQ4 How can the dependence on feature dimension $d$ be eliminated in bandit multiclass regret bounds?

Key findings

• Gaptron achieves $O(K)$ expected regret in the full information setting with respect to logistic, hinge, and smooth hinge losses.
• In the bandit setting, Gaptron achieves $O(K\sqrt{T})$ expected regret with no dependence on the feature dimension $d$.
• Gaptron is the first linear-time algorithm with $O(K\sqrt{T})$ regret in the bandit multiclass classification setting.
• The regret bounds are derived via a novel proof technique that exploits the gap between zero-one and surrogate losses, not traditional properties like exp-concavity.
• The algorithm maintains a randomized prediction strategy that ensures bounded surrogate and abstention gaps through adaptive probability updates.
• The analysis shows that the regret remains $O(K\sqrt{T})$ even when the abstention cost $c_t < 1/2$, with improved constants compared to prior work.