Paper:
We propose a novel method to estimate the parameters of a collection of Hidden Markov Models (HMM), each of which corresponds to a set of known features. The observation sequence of an individual HMM is noisy and/or insufficient, making parameter estimation solely based on its corresponding observation sequence a challenging problem. The key idea is to combine the classical ExpectationMaximization (EM) algorithm with a neural network, while these two are jointly trained in an end-to-end fashion, mapping the HMM features to its parameters and effectively fusing the information across different HMMs. In order to address the numerical difficulty in computing the gradient of the EM iteration, simultaneous perturbation stochastic approximation (SPSA) is employed to approximate the gradient. We also provide a rigorous proof that the approximated gradient due to SPSA converges to the true gradient almost surely. The efficacy of the proposed method is demonstrated on synthetic data as well as a real-world e-Commerce dataset.

Review:
The authors propose to use numerical differentiation (using random perturbation) to approximate the Jacobian of a particular update (essentially equations 5~7) which plays an important role in the estimation of HMMs.  To do so, the authors provide first a concise intro to HMM models (well known stuff in S2), presenting the iteration in detail, jump into their model (cryptically presented in my opinion in S3) and then propose a numerical approximation scheme using SPSA (building upon literature from the 90's, with Theo 1 being the main contribution), before moving onto experiments.
I have found the paper poorly presented. Its general motivation stands on a shaky ground (as illustrated by the choice of words by the authors, see below). In terms of presentation, reminders on HMM are welcome, but unfortunately the authors have not kept the same standard for clarity of notation in Section 3, which makes reading and understanding what the authors are doing quite difficult. Not being a specialist in this field, I have struggled a bit to understand the model itself, and the practical motivation of adding a DNN in the middle of what is otherwise an unrolled back-and-forth between k steps of EM estimation of transition parameters and the addition of a DNN layer. Despite the complexity in the story, what the authors propose is essentially to apply a numerical approximation scheme for Jacobians of these EM updates instead of backprop. Since this is the crux of the contribution,  I feel some more numerical evidence that their approach works compared to baselines (e.g. Hinton et al 2018) is needed. For these reasons my assessment is a bit on the lower side. 
- parenthesis bug in b_j(... in Eq.4
- in Eq. 5, index i appears both in numerator (as regular index) and denominator (as sum index)
- what is \Psi in Eq.8 ?
- "While HMM is arguably less prevalent in the era of deep learning": odd way to start an intro. All papers cited date back to more than 2011, 2 in 2006, all the rest in 20th century. This is particularly strange given the few citations to papers >2015 in Section 5.
- the observation sequence o_{t,1:T(u)} is "weakly" indexed by u (since T(u) is just a length)
- What is the \forall u notation below Eq. 9?
- "the number of nodes required to build the forward and backward
probability in the computation graph of an automatic differentiation engine is on the order of O(T^2). Empirically we found this leads to intractable computation cost." since this is critical, where is this empirical evidence? this seems to be a storage problem and cannot be a complexity issue. There are ways to mitigate this problem by only storing partially information, I feel this comparison would add a lot of value to the authors' claim.
- Where is J^{(k)} defined (as opposed to \hat{J}^{(k)}) defined in Eq.14?