  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 $x^{\top} \theta$, 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.
