Statistical inference methods are fundamentally important in machine learning. Most state-of-the-art inference algorithms are variants of Markov chain Monte Carlo (MCMC) or variational inference (VI). However, both methods struggle with limitations in practice: MCMC methods can be computationally demanding; VI methods may have large bias. In this work, we aim to improve upon MCMC and VI by a novel hybrid method based on the idea of reducing simulation bias of finite-length MCMC chains using gradient-based optimisation. The proposed method can generate low-biased samples by increasing the length of MCMC simulation and optimising the MCMC hyper-parameters, which offers attractive balance between approximation bias and computational efficiency. We show that our method produces promising results on popular benchmarks when compared to recent hybrid methods of MCMC and VI.
This paper proposes a new combination of Markov chain Monte Carlo (MCMC) and variational inference (VI) for improving approximate inference. The main contribution is the optimization objective that allows improving the quality of samples obtained from the combination of VI and MCMC. Specifically, the authors minimize the "approximate" version of the Kullback-Leibler (KL) divergence between the distribution of MCMC + VI and the true distribution. The authors validate the effectiveness of their formulation through experiments on 6 synthetic benchmarks and generative modeling of MNIST (experiments on Bayesian neural networks are also provided in the appendix). 
Overall, I think the paper provides a solid contribution towards combining MCMC and VI by proposing a way to optimize the MCMC part. The experiments validate the method by showing consistent improvement over existing methods. However, I believe the justification behind the proposed formulation, i.e., Equation (4) and (5), needs to be improved before being published at the conference.
First, for Equation (4), the explanation behind "replacing" H(P_{T}) with ELBO w.r.t. P_{0} is confusing. Specifically, it is reasoned that ELBO w.r.t. P_{t} only increase after MCMC steps. This statement is misleading since the replacement was done for H(P_{T}), not the ELBO w.r.t. P_{T}. 
I also think the Equation (5) is not properly justified. it is stated that the constraint is needed for preventing P_{T} to be closer to P_{0}. However, nothing is stated about the reason on why P_{T} gets closer to \pi when Equation (5) is satisfied.  Note that even if the expected log-likelihood of the distribution is high, it does not necessarily mean that the distribution is more similar. 
Minor comments:
- I was unable to understand why the algorithm is named "ergodic" inference. Both HVI and the proposed EI rely on the ergodic property of Markov chain for improving the variational distribution. I hope the authors could better illustrate on this point. I also think the term "ergodic approximation" in page 3. is hard to understand.
- I (weakly) suggest changing y-axis of Figure 5. to log-scale for better readability. It almost seems that the brown plot does not converge in Fig 5-(a).
- The paper could have been strengthened by performing experiments on more challenging datasets, e.g., CIFAR-10 or CIFAR-100.