Technical quality: I think that the theory is very complete (bounds are given for pretty much everything relevant to the problem), and the experiments show that this method performs better on large/complicated models (the small/simple models have too little variance for extra servers to help, and the staleness prevents much benefits). I think the biggest limitation of the paper is the lack of comparison against the method in [14] (the paper mostly compares against the non-distributed -- 1 worker -- case, instead of a more standard distributed case). Novelty/originality: My impression is theoretical results are mostly a combination of proof techniques used in other SG-MCMC and asynchronous SGD papers (however, I'm not too sure that this claim is correct). Assuming this is true, I think the results are well-executed, but not too unique. Potential impact or usefulness: I think the theoretical analysis will be useful for people interested in how asynchrony affects SG-MCMC. However, I'm not too clear how much this will help for running SG-MCMC in practice.

I have one key concern, which may be a misunderstanding on my part as I did not check the supplementary section in detail. The update at each time step is computed using gradients of different parameter values (theta), some of which were generated arbitrarily many time steps ago. This dependence on previous samples means that the SAGA-LD chain is not Markov. The proofs seem to be based on a result for SG-MCMC chains, but I am not sure if the result easily applies to SAGA-LD because of the violation of the Markov property. Other than the above point, I think this is a very useful line of work.

In supervised learning, one is interested in minimizing the empirical risk where efficient optimization algorithms become the key. First-order methods such as stochastic gradient descent and its variants are reasonably well understood admitting efficient implementation and parallelization techniques. However there has been a recent interest in making second-order methods such as Newton's method or L-BFGS method efficient for such large-scale problems. This paper is along this direction, presenting a new variant of the stochastic L-BFGS method that is efficient and robust in mainly two settings: The first arises in the presence of node failures in a distributed computing environment, the second occurs when one uses an adaptive batch size that varies over iterations for accelerating learning. The main idea is to form the Hessian estimate based on the overlap between consecutive batches (the intuition why this works is that we have less limitation in choosing the second-order information matrix compared to an estimate of the true gradient).

Comparing with previous work, including generalized phase retrieval problems, this paper has the following differences: 1) Solves a second order function incorporating absolute values of measurements 2) No step size normalization (or variants) 3) No gradient truncation The key intuition of this paper is that for a measurement ai, x with large magnitude, in the local region near global optima, the sign of ai, x and ai, z are likely to be same. In the case, locally the gradient update of reshaped WF (RWF) is as the same as an equivalent least squares problem. For the objective function including all the measurements, if most the components have reasonably large measurements, those with small measurements contribute less, hence using gradient descent good initialization should give good results. Such intuition is formalized in equation (34) and (35). Yet I have a question -- why don't one consider further truncating components with small magnitude when computing gradient? Just like in robust regression, TWF, etc, we can throw out "wrong" directions.

Summary: This paper proposes a stochastic online learning method for the task of structured prediction. In this setting, the learner doest not get the correct structured output during training. Instead, it only gets bandit feedback from the labeler. The paper first proposes an online learning algorithm that learns model parameters via stochastic gradient descent; generalizes the learning method to pair-wise comparison of structured outputs; provides an optimization approach with Cross-Entropy Minimization; and theoretically analyzes the convergence property of the optimization approach. Pros: The paper proposes an online stochastic learning algorithm for minimizing the expected loss of structured predictions; gives a method of learning from pair-wise comparisons; and theoretical analyze the convergence rate.

The extension of SGGMC from previous work (SGRHMC)[1] are in two folds. First, the proposed method use Geodesic flow rather than Riemmannian manifold. Second, the proposed method leverage a symmetric splitting integrator (ABOBA) scheme. However, unfortunately none of extensions have a clear and convincing novelty as far as I can see. The drop-in replacement of D and Q are not surprising.

The initial motivation seems to be the work of Hoffman et al on the use of clustering to speedup stochastic methods for ERM. Their method was not proved to converge to the optimal due to the use of biased stochastic gradients. Also, that work seemed to work only for small clusters due to the approach chosen. This papers goes a long way to develop the basic idea into a satisfying theoretical framework which also gives rise to efficient implementations. This paper is truly a pleasure to read – a very fine example of academic exposition.

However, this paper is not carefully written. For example, the references are missing on page 6, line 192 and page 7, line 206. The legend of red lines are missing for Figure 2c,d. The paper states only the necessary information but not sufficient for the readers to follow easily. I think the clarity of this paper could be greatly improved especially the authors did not use the full 8 pages.

The rebuttal answered all my technical questions, in particular the one about probability and expectations in Theorems 1(ii) and 3. My scores (which were given assuming the technicalities can be fixed) remain the same. The theoretical results do not feel that exciting. The regret bounds seem more or less what one might expect, and there is no comparison to bounds for other algorithms. I understand that there are no directly comparable bounds since you have both budget size and dimensionality of random projections as significant parameters, but still some discussion would be nice. The dependence on budget B in your bounds could perhaps be pointed out more explicitly.

The main contribution in this work is showing that in the context of matrix completion, when equipped with a good initialization, the sequence of the solutions produced by a widely used (but without theoretical guarantee) SGD algorithm converges linearly to the true matrix. The paper reads well and most of theoretical result looks sound. But I have some concerns as follows. Is it possible to access fewer samples, e.g., "O(mu * d * k * log(d))" while still ensuring global convergence? Theorem 3.1 says "for any fixed T 1, with probability at least 1 - T/(d 10), we have linear convergence". That means, if we run the algorithm (i.e., repeatedly reveal the samples) for a longer time, we have a LOWER confidence to obtain the true matrix.