  Classical global convergence results for first-order methods rely on uniform
smoothness and the \L{}ojasiewicz inequality. Motivated by properties of
objective functions that arise in machine learning, we propose a non-uniform
refinement of these notions, leading to \emph{Non-uniform Smoothness} (NS) and
\emph{Non-uniform \L{}ojasiewicz inequality} (N\L{}). The new definitions
inspire new geometry-aware first-order methods that are able to converge to
global optimality faster than the classical $\Omega(1/t^2)$ lower bounds. To
illustrate the power of these geometry-aware methods and their corresponding
non-uniform analysis, we consider two important problems in machine learning:
policy gradient optimization in reinforcement learning (PG), and generalized
linear model training in supervised learning (GLM). For PG, we find that
normalizing the gradient ascent method can accelerate convergence to
$O(e^{-t})$ while incurring less overhead than existing algorithms. For GLM, we
show that geometry-aware normalized gradient descent can also achieve a linear
convergence rate, which significantly improves the best known results. We
additionally show that the proposed geometry-aware descent methods escape
landscape plateaus faster than standard gradient descent. Experimental results
are used to illustrate and complement the theoretical findings.
