Over the past decade, a growing body of research has shown that
γ-weak submodularity broadly arises in numerous subset selection tasks, including feature selection, neural network pruning, and video summarization. Despite its prevalence, maximizing a
γ-weakly submodular function subject to a general matroid constraint remains challenging. To date, the only known approximation guarantee is the conservative
(1+1/γ)−2 factor established by \citet{chen2018weakly}. To improve upon this result, this paper proposes a novel algorithm called \MGPE, which repeatedly performs maximum-gain local exchanges through careful control of a non-homogeneous Poisson clock, and proves that this \MGPE\ can attain an approximation ratio arbitrarily close to
ργ=1−(γ/(2−γ))2(1−γ)γ2. In sharp contrast to the previous guarantee, our obtained factor
ργ not only strictly improves upon
(1+1/γ)−2 for every
γ∈(0,1], but also can asymptotically approach the optimal
(1−1/e)-approximation for submodular maximization as
γ→1. Furthermore, we surprisingly find that when the matroid constraint reduces to a cardinality or the objective satisfies the stronger notion of
α-weak DR-submodularity, \MGPE\ can automatically recover the tight approximation ratios of
1−e−γ and
1−e−α, respectively. Here,
α∈(0,1] denotes the DR ratio.