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تسجيل مستخدم جديدCombinatorial Pure Exploration with Full-bandit Feedback and Beyond: Solving Combinatorial Optimization under Uncertainty with Limited Observation
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Combinatorial optimization is one of the fundamental research fields that has been extensively studied in theoretical computer science and operations research. When developing an algorithm for combinatorial optimization, it is commonly assumed that parameters such as edge weights are exactly known as inputs. However, this assumption may not be fulfilled since input parameters are often uncertain or initially unknown in many applications such as recommender systems, crowdsourcing, communication networks, and online advertisement. To resolve such uncertainty, the problem of combinatorial pure exploration of multi-armed bandits (CPE) and its variants have recieved increasing attention. Earlier work on CPE has studied the semi-bandit feedback or assumed that the outcome from each individual edge is always accessible at all rounds. However, due to practical constraints such as a budget ceiling or privacy concern, such strong feedback is not always available in recent applications. In this article, we review recently proposed techniques for combinatorial pure exploration problems with limited feedback.
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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 $x in m
athcal{X}$ and receives a random reward with expectation $x^{top} theta$, with $theta in mathbb{R}^d$ 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 $Delta_{min}$ (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} theta $, 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 $Delta_{min}$ settings while our CPE-PL algorithm is the only one returning correct answers for nonlinear reward functions.
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