Paper:
Adaptive moment methods have been remarkably successful for optimization under the presence of high dimensional or sparse gradients, in parallel to this, adaptive sampling probabilities for SGD have allowed optimizers to improve convergence rates by prioritizing examples to learn efficiently. Numerous applications in the past have implicitly combined adaptive moment methods with adaptive probabilities yet the theoretical guarantees of such procedures have not been explored. We formalize double adaptive stochastic gradient methods DASGRAD as an optimization technique and analyze its convergence improvements in a stochastic convex optimization setting, we provide empirical validation of our findings with convex and non convex objectives. We observe that the benefits of the method increase with the model complexity and variability of the gradients, and we explore the resulting utility in extensions to transfer learning.

Review:
Summary:
This paper explores the combination of adaptive moment methods and adaptive data sampling procedures for stochastic optimization. The paper presents standard analysis in the stochastic convex setting which raised some question for me (presented below). The empirical results are promising but in my opinion can be further improved through minor changes to experimental procedure.
Overall:
1) There are a few instances of undefined notation. For example, $\beta_1(t)$ in the definition of Adam/AMSGrad or $V_t$ (the latter I assume is the average over all datapoints?).
2) Theorem 1 is stated very informally with some notation left unexplained (e.g. V_t above or E_n[...]). Further, it is not clear to me whether the authors are proving results in the online convex optimization setting or the stochastic convex optimization setting (though, there is some equivalence between them). Finally, I am concerned by the presentation of Theorem 1 and Corollary 1.1 as it has been shown by Reddi et al. that Adam does not have bounded regret in general without algorithmic changes. In particular, bimodal gradient noise distributions can be designed such that the average Regret is unbounded (see Reddi et al. for details). While these issues can be addressed through modifications to the standard Adam algorithm I am unable to find a suitable discussion within the paper.
3) Theorem 2 introduces DASGrad which has not yet been clearly defined (Algorithm 1 depends on averaging functions of the past gradients for flexibility --- which averaging functions are being used here?). Once again, the notation Var_n appears whose definition is not clear to me. Additionally, could you provide a citation for the optimal adaptive probabilities and clearly state under which conditions these adaptive probabilities are optimal?
The crux of Theorem 2 depends on the informal statements present in Section 3.3, in which it is claimed that the variance of the second order moments are sufficiently large to dominate the sum of gradient histories. I am having a hard time seeing how this would be true generally and I believe that further theoretical analysis is necessary to justify this discussion.
4) The introduction of Algorithm 2 felt like a bait-and-switch. This algorithm seems intuitively sensible and uses the same check-pointing ideas that have been introduced in recent variance reduction methods (e.g. [1]). However, this is a different algorithm which is not analyzed in this paper. 
Additionally, the fourth line of Algorithm 2 says to compute p_t lying in the simplex --- I believe this is a typo? Further, could you explain the introduction of epsilon in line 4. This seems unnecessary given the propto symbol but I am worried I am missing something.
5) The empirical results introduce an additional deviation from the proposed algorithm by using batch sizes of 32. Could the authors please explain how the adaptive sampling probabilities are used in this setting? (My apologies if this is obvious, I am not overly familiar with such methods).
Overall I felt that the empirical results were fairly promising but some questions remained for me. The toy experiments in section 6.1 were illustrative but seem to only consider adaptive probabilities without the adaptive moments?
Section 6.3 was also an interesting experimental setup which is well suited to DASGrad but I felt that the baseline comparison was unfair. DASGrad was provided with the ground-truth balance statistics in the importance weights while the baselines were trained without any correction. I would recommend that the authors explore combinations of enabling balance correction in the baseline methods (possibly through reweighting the loss according to the balance statistics) or requiring DASGrad to compute the optimal probabilities through the adaptive moments as suggested throughout the rest of the paper.
Minor:
- Pg 3, typo "we first we extend"
References
[1] Accelerating Stochastic Gradient Descent using Predictive Variance Reduction, Rie Johnson and Tong Zhang