  In this note, we propose and study the notion of modified Fej\'{e}r
sequences. Within a Hilbert space setting, we show that it provides a unifying
framework to prove convergence rates for objective function values of several
optimization algorithms. In particular, our results apply to forward-backward
splitting algorithm, incremental subgradient proximal algorithm, and the
Douglas-Rachford splitting method including and generalizing known results.
