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Linear bandits with polylogarithmic minimax regret (2402.12042v2)

Published 19 Feb 2024 in cs.LG, cs.AI, and stat.ML

Abstract: We study a noise model for linear stochastic bandits for which the subgaussian noise parameter vanishes linearly as we select actions on the unit sphere closer and closer to the unknown vector. We introduce an algorithm for this problem that exhibits a minimax regret scaling as $\log3(T)$ in the time horizon $T$, in stark contrast the square root scaling of this regret for typical bandit algorithms. Our strategy, based on weighted least-squares estimation, achieves the eigenvalue relation $\lambda_{\min} ( V_t ) = \Omega (\sqrt{\lambda_{\max}(V_t ) })$ for the design matrix $V_t$ at each time step $t$ through geometrical arguments that are independent of the noise model and might be of independent interest. This allows us to tightly control the expected regret in each time step to be of the order $O(\frac1{t})$, leading to the logarithmic scaling of the cumulative regret.

Citations (4)

Summary

  • The paper introduces LinUCB-VN, an algorithm achieving polylogarithmic regret by incorporating a vanishing noise model in linear bandits.
  • It utilizes a weighted least squares estimator to finely balance exploration near the unknown parameter, diverging from traditional square-root regret scaling.
  • The work challenges conventional regret norms and paves the way for new theoretical insights and practical applications in adaptive decision-making.

Exploring the Frontiers of Linear Stochastic Bandits with Vanishing Noise

Introduction to Linear Bandits with Vanishing Noise

Linear bandits have been a focal point in the exploration of efficient learning algorithms, providing a fertile ground for understanding the intricate balance between exploration and exploitation. In this context, a novel noise model is studied, presenting a unique challenge and opportunity within the linear stochastic bandit framework. The paper introduces an innovative algorithm capable of achieving a polylogarithmic regret scaling in the time horizon TT for linear bandits, diverging from the typical square root scaling observed in most bandit algorithms.

Overview of the Proposed Model

The paper substantially contributes to the linear bandit literature by introducing a linear stochastic bandit model that incorporates a noise component that decreases as actions are taken closer to an unknown parameter on the unit sphere. Notably, the noise model proposed is not only intuitively appealing in contexts like recommendation systems, where certainty increases as suggestions align more closely with a user’s preferences, but it also sets a new direction for theoretical exploration given its departure from constant noise assumptions prevalent in existing models.

Technical Achievements and Analytical Inspections

The authors propose LinUCB-VN, a variant of the LinUCB algorithm, adjusted to suit the vanishing noise model. This adaptation uses a weighted least squares estimator, factoring in the vanishing nature of the noise, to achieve polylogarithmic scaling of regret. This section of the paper excels in both the novelty of the approach and in the meticulous theoretical analysis that firmly establishes the logarithmic scaling of regret as being attributable to the tailored action selection strategy, which leads to a finer control over exploration in areas close to the estimated unknown parameter.

Theoretical Implications and Practical Considerations

By breaking the traditional square root barrier for regret in linear bandits, this work not only enriches the theoretical understanding of the problem space but also opens new avenues for algorithm design in environments where noise is not stationary. The dependence of noise on the distance to the unknown optimal action introduces a nuanced dimension to the exploration-exploitation trade-off, emphasizing the potential for more refined algorithms that adapt to varying noise levels.

Future Directions and Open Questions

While the paper marks a significant step forward, it also outlines several intriguing open problems and areas for future research. Among these, the exploration of variance estimators in place of sub-Gaussian parameter estimators and the potential adaptation of these techniques to broader classes of reward distributions stand out. Additionally, the authors conjecture about the general necessity of ensuring a relation between the minimum and maximum eigenvalues of the design matrix in minimizing regret for a wide range of noise models, suggesting a fertile ground for future theoretical investigations.

Concluding Thoughts

In conclusion, the exploration of linear stochastic bandits under the novel vanishing noise model presents both a challenging and illuminating foray into the complexities of optimizing action selection in uncertain environments. The introduction of LinUCB-VN and the polylogarithmic regret scaling achieved under this model reflect a considerable advancement in the field, pushing the boundaries of what is understood about the dynamics of learning in the presence of noise and setting a new benchmark for future investigations into linear bandit problems.