  Poisson likelihood models have been prevalently used in imaging, social
networks, and time series analysis. We propose fast, simple,
theoretically-grounded, and versatile, optimization algorithms for Poisson
likelihood modeling. The Poisson log-likelihood is concave but not
Lipschitz-continuous. Since almost all gradient-based optimization algorithms
rely on Lipschitz-continuity, optimizing Poisson likelihood models with a
guarantee of convergence can be challenging, especially for large-scale
problems.
  We present a new perspective allowing to efficiently optimize a wide range of
penalized Poisson likelihood objectives. We show that an appropriate saddle
point reformulation enjoys a favorable geometry and a smooth structure.
Therefore, we can design a new gradient-based optimization algorithm with
$O(1/t)$ convergence rate, in contrast to the usual $O(1/\sqrt{t})$ rate of
non-smooth minimization alternatives. Furthermore, in order to tackle problems
with large samples, we also develop a randomized block-decomposition variant
that enjoys the same convergence rate yet more efficient iteration cost.
  Experimental results on several point process applications including social
network estimation and temporal recommendation show that the proposed algorithm
and its randomized block variant outperform existing methods both on synthetic
and real-world datasets.
