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Title: Ergodic Inference: Accelerate Convergence by Optimisation

Paper content:
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.
Statistical inference methods in machine learning are dominated by two approaches: simulation and optimisation.
Markov chain Monte Carlo (MCMC) is a well-known simulation-based method, which promises asymptotically unbiased samples from arbitrary distributions at the cost of expensive Markov simulations.
Variational inference (VI) is a well-known method using optimisation, which fits a parametric approximation to the target distribution.
VI is biased but offers a computationally efficient generation of approximate samples.
There is a recent trend of hybrid methods of MCMC and VI to achieve a better balance between computational efficiency and bias.
Hybrid methods often use MCMC or VI as an algorithmic component of the other.
In particular, Salimans et al. (2015) proposed a promising modified VI method that reduces approximation bias by using MCMC transition kernels.
Another technique reduces the computational complexity of MCMC by initialising the Markov simulation from a pretrained variational approximation (Hoffman, 2017; Han et al., 2017) .
Levy et al. (2018) proposed to improve MCMC using flexible non-linear transformations given by neural networks and gradientbased auto-tuning strategies.
In this work, we propose a novel hybrid method, called ergodic inference (EI).
EI improves over both MCMC and VI by tuning the hyper-parameters of a flexible finite-step MCMC chain so that its last state sampling distribution converges fast to a target distribution.
EI optimises a tractable objective function which only requires to evaluate the logarithm of the unnormalized target density.
Furthermore, unlike in traditional MCMC methods, the samples generated by EI from the last state of the MCMC chain are independent and have no correlations.
EI offers an appealing option to balance computational complexity vs. bias on popular benchmarks in machine learning.
Compared with previous hybrid methods, EI has following advantages:
• EI's hyperparameter tuning produces sampling distributions with lower approximation bias.
• The bias is guaranteed to decrease as the length of the MCMC chain increases.
• By stopping gradient computations, EI has less computational cost than related baselines.
We also state some disadvantages of our method:
• The initial state distribution in EI's MCMC chain has to have higher entropy than the target.
• The computational complexity per simulated sample of EI is in general higher than in VI.

assistant: text: 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.