Paper:
Stochastic AUC maximization has garnered an increasing interest due to better fit to imbalanced data classification. However, existing works are limited to stochastic AUC maximization with a linear predictive model, which restricts its predictive power when dealing with extremely complex data. In this paper, we consider stochastic AUC maximization problem with a deep neural network as the predictive model. Building on the saddle point reformulation of a surrogated loss of AUC, the problem can be cast into a non-convex concave min-max problem. The main contribution made in this paper is to make stochastic AUC maximization more practical for deep neural networks and big data with theoretical insights as well. In particular, we propose to explore Polyak-Łojasiewicz (PL) condition that has been proved and observed in deep learning, which enables us to develop new stochastic algorithms with even faster convergence rate and more practical step size scheme. An AdaGrad-style algorithm is also analyzed under the PL condition with adaptive convergence rate. Our experimental results demonstrate the effectiveness of the proposed algorithms.

Review:
This paper proposes two algorithms for the non-convex concave AUC maximization problem, along with theoretical analysis. Experiments show the proposed methods are effective, especially in data imbalanced scenarios.
Strengths:
This paper might be useful and interesting to related research, which overcomes some limitations in previous works such as: 1. the convex assumptions; 2. only considering simple models like linear models; 3. the need of extra memory to store/maintain samples. The proposed method extends existing works to a non-convex setting, which can be applied to deep neural networks, and is applicable for batch-learning and online learning.
The proposed methods achieve better experimental results, especially in the data imbalanced scenarios, which is a real problem that may arise in many scenarios. The paper provides theoretical analysis on the proposed methods, based on Assumption 1, and inspired by the PL condition.
Weaknesses:
I think some comparisons with AdaGrad and related methods should be performed in experiments. Since PPD-AdaGrad is “AdaGrad style”.
The assumptions seem a bit unclear. What does the first assumption in Assumption 1 imply?
Minor Comments:
1. Since the experiments label the first 5/50 classes as negative, and the last 5/50 classes as positive for CIFAR10/CIFAR100, is it possible to provide results on experiments that label in the opposite way (or randomly label 5/50 classes) and add these results in the paper/appendix? Just to make results more convincing and reduce some potential dataset influences.
2. Is it possible to provide some results on more imbalanced positive-negative ratio like 20:1?
3. Is it possible to provide some comparison in terms of actual time, like learning curves with time as x-axis?
4. In the multi-class problems, why are the lower layers shared while last layer separated? 
5. Since the extension to multi-classes problems are mentioned in the paper. I like to see some experimental results on this setting.
6. How do the proposed methods perform on models other than NN?
7. I think there is a typo on Page 4, the definition of AUC definition: the latter y should be -1.