Combinatorial Pure Exploration with Full-Bandit or Partial Linear Feedback
This work addresses sample-efficient decision-making in complex action spaces for machine learning and optimization, though it is incremental as it builds on prior combinatorial pure exploration frameworks.
The paper tackles the combinatorial pure exploration problem with full-bandit feedback (CPE-BL) by designing a polynomial-time adaptive algorithm that matches sample complexity lower bounds and has a light dependence on the smallest gap, and extends it to a novel generalization with partial linear feedback (CPE-PL) that handles limited feedback, general reward functions, and combinatorial action spaces. The results include empirical evaluations showing algorithms run orders of magnitude faster than existing ones, with robustness across gap settings and correctness for nonlinear rewards.
In this paper, we first study the problem of combinatorial pure exploration with full-bandit feedback (CPE-BL), where a learner is given a combinatorial action space $\mathcal{X} \subseteq \{0,1\}^d$, and in each round the learner pulls an action $x \in \mathcal{X}$ and receives a random reward with expectation $x^{\top} θ$, with $θ\in \mathbb{R}^d$ a latent and unknown environment vector. The objective is to identify the optimal action with the highest expected reward, using as few samples as possible. For CPE-BL, we design the first {\em polynomial-time adaptive} algorithm, whose sample complexity matches the lower bound (within a logarithmic factor) for a family of instances and has a light dependence of $Δ_{\min}$ (the smallest gap between the optimal action and sub-optimal actions). Furthermore, we propose a novel generalization of CPE-BL with flexible feedback structures, called combinatorial pure exploration with partial linear feedback (CPE-PL), which encompasses several families of sub-problems including full-bandit feedback, semi-bandit feedback, partial feedback and nonlinear reward functions. In CPE-PL, each pull of action $x$ reports a random feedback vector with expectation of $M_{x} θ$, where $M_x \in \mathbb{R}^{m_x \times d}$ is a transformation matrix for $x$, and gains a random (possibly nonlinear) reward related to $x$. For CPE-PL, we develop the first {\em polynomial-time} algorithm, which simultaneously addresses limited feedback, general reward function and combinatorial action space, and provide its sample complexity analysis. Our empirical evaluation demonstrates that our algorithms run orders of magnitude faster than the existing ones, and our CPE-BL algorithm is robust across different $Δ_{\min}$ settings while our CPE-PL algorithm is the only one returning correct answers for nonlinear reward functions.