  Based on SGD, previous works have proposed many algorithms that have improved
convergence speed and generalization in stochastic optimization, such as SGDm,
AdaGrad, Adam, etc. However, their convergence analysis under non-convex
conditions is challenging. In this work, we propose a unified framework to
address this issue. For any first-order methods, we interpret the updated
direction $g_t$ as the sum of the stochastic subgradient $\nabla f_t(x_t)$ and
an additional acceleration term $\frac{2|\langle v_t, \nabla f_t(x_t)
\rangle|}{\|v_t\|_2^2} v_t$, thus we can discuss the convergence by analyzing
$\langle v_t, \nabla f_t(x_t) \rangle$. Through our framework, we have
discovered two plug-and-play acceleration methods: \textbf{Reject Accelerating}
and \textbf{Random Vector Accelerating}, we theoretically demonstrate that
these two methods can directly lead to an improvement in convergence rate.
