Paper:
We consider composition problems of the form
Composition optimization arises in many important machine learning applications: reinforcement learning, variance-aware learning, nonlinear embedding, and many others. Both gradient descent and stochastic gradient descent are straightforward solution, but both require to compute 1 n n j=1 G j (x) in each single iteration, which is inefficient-especially when n is large. Therefore, with the aim of significantly reducing the query complexity of such problems, we designed a stochastically controlled compositional gradient algorithm that incorporates two kinds of variance reduction techniques, and works in both strongly convex and non-convex settings. The strategy is also accompanied by a mini-batch version of the proposed method that improves query complexity with respect to the size of the mini-batch. Comprehensive experiments demonstrate the superiority of the proposed method over existing methods.

Review:
This paper proposes a new method for empirical composition problems to which the vanilla SGD is not applicable because it has a finite-sum structure inside non-linear loss functions. A proposed method (named SCCG) is a combination of stochastic compositional gradient descent (SCGD) and stochastically controlled stochastic gradient (SCSG). In a theoretical analysis part, a linear convergence rate and a sub-linear convergence rate are derived under the strong convex and non-convex settings, respectively. In experiments, the superior performance of the method to competitors is verified on both strongly convex and non-convex problems.
Clarity:
The paper is clear and well written.
Quality:
The work is of good quality and is technically sound.
Significance:
The problem treated in this paper is important and contains several applications as mentioned in the paper. Hence, developing an efficient method for this problem is important and interesting. Although derived convergence rates are better than existing primal methods, this paper lacks a comparison with the recently proposed primal-dual method by [A.Devraj & J.Chen (2019)].
[A.Devraj & J.Chen (2019)] Stochastic Variance Reduced Primal Dual Algorithms for Empirical Composition Optimization. NeurIPS, 2019.
A convergence rate obtained in [A.Devraj & J.Chen (2019)] seems faster than that of SCCG for ill-conditioned strongly convex problems. However, there exists a certain setting (large-scale setting) where SCCG outperforms their method. Thus, the contribution of the paper is not lost, but it is better to compare SCCG with the method in [A.Devraj & J.Chen (2019)], empirically and theoretically.
If the authors can show an empirical advantage over their method, it will make the paper stronger.
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Update:
I thank the authors for the response and hard work. I am convinced of the advantage of the proposed method. I would like to keep my score.