Contextual Bandit Algorithms with Supervised Learning Guarantees (1002.4058v3)

Abstract: We address the problem of learning in an online, bandit setting where the learner must repeatedly select among $K$ actions, but only receives partial feedback based on its choices. We establish two new facts: First, using a new algorithm called Exp4.P, we show that it is possible to compete with the best in a set of $N$ experts with probability $1-\delta$ while incurring regret at most $O(\sqrt{KT\ln(N/\delta)})$ over $T$ time steps. The new algorithm is tested empirically in a large-scale, real-world dataset. Second, we give a new algorithm called VE that competes with a possibly infinite set of policies of VC-dimension $d$ while incurring regret at most $O(\sqrt{T(d\ln(T) + \ln (1/\delta))})$ with probability $1-\delta$. These guarantees improve on those of all previous algorithms, whether in a stochastic or adversarial environment, and bring us closer to providing supervised learning type guarantees for the contextual bandit setting.

Contextual Bandit Algorithms with Supervised Learning Guarantees

The paper presents a significant advancement in contextual bandit algorithms by introducing Exp4.P, a modification to the existing Exp4 algorithm. The authors focus on the challenge of achieving a high probability bound on regret in non-stochastic, adversarial settings with a finite but potentially large set of policies. Their results bring closer parity between guarantees provided in contextual bandit settings and those found in supervised learning.

The central contribution is the Exp4.P algorithm, which minimizes regret within the order $O(\sqrt{KT\ln N})$, a notable improvement over the previous Exp4 algorithm, which had high variance in importance-weighted estimates and thus lacked high probability guarantees. For a stochastic version, Exp4.P achieves a regret bound of $O(\sqrt{Td\ln T})$ when competing against an infinite set of policies with finite VC-dimension. Consequently, Exp4.P provides more reliable performance than prior approaches, ensuring robustness under adversarial conditions.

Technical Overview

In the non-stochastic bandit setting, a learner must choose among $K$ actions at each step, observing the reward for only the chosen action. The challenge lies in exploration, where the learner's objective is to optimize reward accumulation compared to a set of $N$ context-informed policies.

Exp4.P builds upon the foundational Exp4 algorithm by Auer et al. The modification involves controlling the estimate variance more effectively to achieve a higher probability bound on regret. This is particularly beneficial in practical applications where having consistently reliable performance is crucial.

Key theoretical results include:

• A proof that Exp4.P achieves regret at most $O(\sqrt{KT\ln N})$ with high probability in adversarial contexts.
• Demonstration of improved performance over existing high probability bounds in purely stochastic settings.

Empirical Evaluation

The authors validate Exp4.P using a large-scale, real-world dataset, highlighting its empirical efficiency. By applying Exp4.P to Yahoo!'s personalized content recommendation system on the front page, they observed significant improvements in click-through rates compared to standard baselines, emphasizing the algorithm's practical feasibility and its potential to outperform traditional strategies.

Implications

Theoretically, Exp4.P's design narrows the gap between bandit algorithms and supervised learning guarantees. Practically, it enables robust decision-making in dynamic environments like online recommendation systems, where failure to explore enough can lead to substantial regret. The algorithm is also adaptable; its structure allows efficient handling of large or infinitely structured expert classes when they are well-organized or polynomially bounded.

Future Directions

The Exp4.P algorithm, while significantly enhanced, inherits some limitations, such as computational inefficiency in cases where $N$ becomes prohibitively large. Future research could explore more efficient implementations for handling these scenarios or further refining the balance between exploration and exploitation in other real-world applications. Alternative strategies for setting probability distributions, as noted by McMahan and Streeter, could also be examined in more varied and complex settings to assess their efficacy and computational viability.

In conclusion, Exp4.P marks a substantive improvement in the analysis and application of contextual bandit algorithms. Its direct comparison with supervised learning regarding regret bounds sets a new standard for assessing and implementing bandit-based decision-making in various technological domains.