  The proximal inertial gradient descent is efficient for the composite
minimization and applicable for broad of machine learning problems. In this
paper, we revisit the computational complexity of this algorithm and present
other novel results, especially on the convergence rates of the objective
function values. The non-ergodic O(1/k) rate is proved for proximal inertial
gradient descent with constant stepzise when the objective function is
coercive. When the objective function fails to promise coercivity, we prove the
sublinear rate with diminishing inertial parameters. In the case that the
objective function satisfies optimal strong convexity condition (which is much
weaker than the strong convexity), the linear convergence is proved with much
larger and general stepsize than previous literature. We also extend our
results to the multi-block version and present the computational complexity.
Both cyclic and stochastic index selection strategies are considered.
