The gradient-push algorithm is a fundamental algorithm for the distributed optimization problem \(\begin{aligned} \min _{x \in {\mathbb{R}}^d} f(x) = \sum _{i=1}^n f_i (x), \end{aligned}\) where each local cost \(f_i\) is only known to agent \(a_i\) for \(1 \le i \le n\) and the agents are connected by a directed graph. In this paper, we obtain convergence results for the gradient-push algorithm with constant stepsize whose range is sharp in terms the order of the smoothness constant \(L>0\) . Precisely, under the two settings: (1) Each local cost \(f_i\) is strongly convex and L-smooth, (2) each local cost \(f_i\) is convex quadratic and L-smooth while the aggregate cost f is strongly convex, we show that the gradient-push algorithm with stepsize \(\alpha>0\) converges to an \(O(\alpha )\) -neighborhood of the minimizer of f for a range \(\alpha \in (0, c/L]\) with a value \(c>0\) independent of \(L>0\) . As a benefit of the result, we suggest a hybrid algorithm that performs the gradient-push algorithm with a relatively large stepsize \(\alpha>0\) for a number of iterations and then go over to perform the Push-DIGing algorithm. It is verified by a numerical test that the hybrid algorithm enhances the performance of the Push-DIGing algorithm significantly. The convergence results of the gradient-push algorithm are also supported by numerical tests.
