Stochastic convex optimization with bandit feedback (1107.1744v2)

Abstract: This paper addresses the problem of minimizing a convex, Lipschitz function $f$ over a convex, compact set $\xset$ under a stochastic bandit feedback model. In this model, the algorithm is allowed to observe noisy realizations of the function value $f(x)$ at any query point $x \in \xset$. The quantity of interest is the regret of the algorithm, which is the sum of the function values at algorithm's query points minus the optimal function value. We demonstrate a generalization of the ellipsoid algorithm that incurs $\otil(\poly(d)\sqrt{T})$ regret. Since any algorithm has regret at least $\Omega(\sqrt{T})$ on this problem, our algorithm is optimal in terms of the scaling with $T$.

• The paper presents an innovative extension of the ellipsoid algorithm that achieves a regret bound scaling as poly(d)√T.
• It employs a center-point device and pyramidal sampling to effectively process noisy function evaluations.
• The approach bridges stochastic optimization and bandit problems, offering scalable performance in high-dimensional settings.

Stochastic Convex Optimization with Bandit Feedback

The paper under discussion presents a comprehensive investigation into a central challenge in the field of convex optimization when subject to stochastic bandit feedback, a scenario commonplace in sequential decision-making environments. The focus of this paper is the minimization of a convex, Lipschitz continuous function over a convex and compact domain where only noisy function evaluations at queried points are available—a setting akin to the multi-armed bandit problem but extended into a continuous space.

Model and Objectives

The authors model the problem by allowing the optimization algorithm to receive stochastic feedback solely in the form of noisy realizations of the function values at points of interest. The performance is evaluated in terms of regret, i.e., the cumulative difference between the function values at queried points and the optimal function value. A paramount contribution of the paper is the development of a generalization of the ellipsoid algorithm that achieves a regret scale of $(\poly(d)\sqrt{T})$, where $T$ represents the number of queries. This solution is shown to be optimal in the way regret scales with $T$, given the constraint that any algorithm will incur a regret of at least $\Omega(\sqrt{T})$ in this problem setting.

Methodology

In developing their algorithm, the authors build upon the theoretical foundations laid by the classic ellipsoid method, modifying it to gracefully handle the lack of precise gradient information typical of the noisy oracle setting. Key methodological innovations include the utilization of a geometrically inspired "center-point device" and the strategic deployment of pyramidal sampling structures within the domain. These elements ingeniously intertwine to blend the ellipsoid's ability to hone in on the most promising regions of the search space with a robust mechanism for separating optimal from suboptimal regions despite noise.

The paper delineates its approach through a multi-phase iterative framework, each phase further deconstructed into epochs and rounds. Within a single epoch, the algorithm adaptively queries and processes stochastic feedback, refining an encompassing feasible region and intelligently discarding large suboptimal portions. This results in a progressive tightening of focus around potential optima.

Results and Implications

The results indicate that the proposed algorithm performs significantly well, with a regret bound scaling polynomially with the dimension $d$, which is advantageous considering the curse of dimensionality commonly encountered in high-dimensional bandit problems. The work also furnishes a complete regret analysis, ensuring that the algorithm’s adaptability maintains efficiency across varying phase lengths and noise characteristics.

The demonstrated regret bounds suggest that this approach effectively bridges the substantial gap between stochastic bandit feedback settings and more conventional stochastic optimization scenarios. Theoretically, this serves as a crucial advancement in non-parametric bandit problems, which previously faced daunting challenges in scalability with respect to dimension.

Future Prospects

From a practical perspective, the implications of this research are vast. Potential applications span a range of domains where decision-making is sequential and feedback is noisy, such as in online learning environments or adaptive control systems. The theoretical contributions pave pathways for further research, with natural extensions encompassing alternative noise models, different convexity assumptions, or more diversified function classes.

Replication of these techniques or theoretical expansions could yield novel algorithms that further stanch the negative impacts of dimensionality. Moreover, the challenge of reducing the polynomial dependence on $d$ remains an open question, foreshadowing potential breakthroughs should alternative methods (such as random walk-based strategies) be successfully integrated.

In conclusion, this paper presents an essential contribution to stochastic convex optimization under bandit feedback, offering a rigorous yet practical approach that respects both the theoretical complexity and the pragmatic demands of real-world applications. It invites further explorations into enhancing algorithmic efficiency in similarly constrained environments.