Paper:
This paper proposes a new approach for step size adaptation in gradient methods. The proposed method called step size optimization (SSO) formulates the step size adaptation as an optimization problem which minimizes the loss function with respect to the step size for the given model parameters and gradients. Then, the step size is optimized based on alternating direction method of multipliers (ADMM). SSO does not require the second-order information or any probabilistic models for adapting the step size, so it is efficient and easy to implement. Furthermore, we also introduce stochastic SSO for stochastic learning environments. In the experiments, we integrated SSO to vanilla SGD and Adam, and they outperformed state-of-the-art adaptive gradient methods including RMSProp, Adam, L 4 -Adam, and AdaBound on extensive benchmark datasets.

Review:
This paper proposes a new step size adaptation in first-order gradient methods. The proposed method establishes a new optimization problem with the first-order expansion of loss function and the regularization, where the step size is treated as a variable.  ADMM is adopted to solve the optimization problem.
This paper should be rejected because (1) the proposed method does not show the convergence rate improvement of the gradient method with other step sizes adaptation methods. (2) the linearization of the objective function leads the step size to be small ($0<\eta<\epsilon$), which could slow down the convergence in some cases. (3) the experiments generally do not support a significant contribution. In table 1, the results of the competitor are not with the optimal step sizes. The limit grid search range could not verify the empirical superiority of the proposed method.
Minor comments:
The y-axis label of (a) panel in each figure is wrong. I guess it should be "Training loss ".