  We propose \textit{Meta-Regularization}, a novel approach for the adaptive
choice of the learning rate in first-order gradient descent methods. Our
approach modifies the objective function by adding a regularization term on the
learning rate, and casts the joint updating process of parameters and learning
rates into a maxmin problem. Given any regularization term, our approach
facilitates the generation of practical algorithms. When
\textit{Meta-Regularization} takes the $\varphi$-divergence as a regularizer,
the resulting algorithms exhibit comparable theoretical convergence performance
with other first-order gradient-based algorithms. Furthermore, we theoretically
prove that some well-designed regularizers can improve the convergence
performance under the strong-convexity condition of the objective function.
Numerical experiments on benchmark problems demonstrate the effectiveness of
algorithms derived from some common $\varphi$-divergence in full batch as well
as online learning settings.
