The Variational Walkback Algorithm
A recognized obstacle to training undirected graphical models with latent variables such as Boltzmann machines is that the maximum likelihood training procedure requires sampling from Monte-Carlo Markov chains which may not mix well, in the inner loop of training, for each example. We first propose the idea that it is sufficient to locally carve the energy function everywhere so that its gradient points in the *right* direction (i.e., towards generating the data). Following on previous work on contrastive divergence, denoising autoencoders, generative stochastic networks and unsupervised learning using non-equilibrium dynamics, we propose a variational bound on the marginal log-likelihood of the data which corresponds to a new learning procedure that first walks away from data points by following the model transition operator and then trains that operator to walk backwards for each of these steps, back towards the training example. The tightness of the variational bound relies on gradually increasing temperature as we walk away from the data, at each step providing a gradient on the parameters to maximize the probability that the transition operator returns to its previous state. Interestingly, this algorithm admits a variant where there is no explicit energy function, i.e., the parameters are used to directly define the transition operator. This also eliminates the explicit need for symmetric weights which previous Boltzmann machine or Hopfield net models require, and which makes these models less biologically plausible.
This paper proposes a new kind of generative model based on an annealing process, where the transition probabilities are learned directly to maximize a variational lower bound on the log-likelihood. Overall, the idea is clever and appealing, but I think the paper needs more quantitative validation and better discussion of the relationship with prior work. In terms of prior work, AIS and RAISE are both closely related algorithms, and share much of the mathematical structure with the proposed method. For this reason, it’s not sufficient to mention them in passing in the related work section; those methods and their relationship to variational walkback need to be discussed in detail. If I understand correctly, the proposed method is essentially an extension of RAISE where the transition probabilities are learned rather than fixed based on an existing MRF. I think this is an interesting and worthwhile extension, but the relationship to existing work needs to be clarified. The analysis of Appendix D seems incorrect. It derives a formula for the ratios of prior and posterior probabilities, but this formula only holds under the assumption of constant temperature (in which case the ratio is very large). When the temperature is varied, the analysis of Neal (2001) applies, and the answer is different. One of the main selling points of the method is that it optimizes a variational lower bound on the log-likelihood; even more accurate estimates can be obtained using importance sampling. It ought to be easy to report log-likelihood estimates for this method, so I wonder why such estimates aren’t reported. There are lots of prior results to compare against on MNIST. (In addition, a natural baseline would be RAISE, so that one can check if the ability to learn the transitions actually helps.) I think the basic idea here is a sound one, so I would be willing to raise my score if the above issues are addressed in a revised version. Minor comments: “A recognized obstacle to training undirected graphical models… is that ML training requires sampling from MCMC chains in the inner loop of training, for each example.” This seems like an unfair characterization, since the standard algorithm is PCD, which usually takes only a single step per mini-batch. Some of the methods discussed in the related work are missing citations. The method is justified in terms of “carving the energy function in the right direction at each point”, but I’m not sure this is actually what’s happening. Isn’t the point of the method that it can optimize a lower bound on the log-likelihood, and therefore learn a globally correct allocation of probability mass?
You are a member of the scientific community tasked with peer review. Review the following paper title and abstract.
### Title + Abstract
The Variational Walkback Algorithm
A recognized obstacle to training undirected graphical models with latent variables such as Boltzmann machines is that the maximum likelihood training procedure requires sampling from Monte-Carlo Markov chains which may not mix well, in the inner loop of training, for each example. We first propose the idea that it is sufficient to locally carve the energy function everywhere so that its gradient points in the *right* direction (i.e., towards generating the data). Following on previous work on contrastive divergence, denoising autoencoders, generative stochastic networks and unsupervised learning using non-equilibrium dynamics, we propose a variational bound on the marginal log-likelihood of the data which corresponds to a new learning procedure that first walks away from data points by following the model transition operator and then trains that operator to walk backwards for each of these steps, back towards the training example. The tightness of the variational bound relies on gradually increasing temperature as we walk away from the data, at each step providing a gradient on the parameters to maximize the probability that the transition operator returns to its previous state. Interestingly, this algorithm admits a variant where there is no explicit energy function, i.e., the parameters are used to directly define the transition operator. This also eliminates the explicit need for symmetric weights which previous Boltzmann machine or Hopfield net models require, and which makes these models less biologically plausible.
user
<think>Okay, to do a decent peer review, I should start with summarizing what the work is about. This includes novelty, contributions, clarity, and impact. I might include suggestions to issues that I find critical to the usefulness of the work. Starting with novelty, there are different types of novelty. The first aspect of novelty is scope and scale. Is the work covering a broader perspective than previous works? The second aspect is methodology. Is the proposed methodology novel compared to previous methods? Do they provide any significant advantage over previous methodologies? The third aspect is results. Are the results different compared previous work? Does this lead to any novel theories or an extension of current theories? Next is clarity, I will be reading the work in depth and analyze if any sentences can be potentially ambiguous or misleading. If any sentences are ambiguous, I will state them and give a suggestion on how to rephrase such sentence or how to make more clear to the audience. Lastly, impact is important. I will attempt to predict how such work will impact the scientific community. This might include the applications of such work and how the work might transform the way we study a certain topic. Finally, I will might only mention issues that are critical. This is to keep the review as concise and helpful as possible.</think>



This paper proposes a new kind of generative model based on an annealing process, where the transition probabilities are learned directly to maximize a variational lower bound on the log-likelihood. Overall, the idea is clever and appealing, but I think the paper needs more quantitative validation and better discussion of the relationship with prior work. In terms of prior work, AIS and RAISE are both closely related algorithms, and share much of the mathematical structure with the proposed method. For this reason, it’s not sufficient to mention them in passing in the related work section; those methods and their relationship to variational walkback need to be discussed in detail. If I understand correctly, the proposed method is essentially an extension of RAISE where the transition probabilities are learned rather than fixed based on an existing MRF. I think this is an interesting and worthwhile extension, but the relationship to existing work needs to be clarified. The analysis of Appendix D seems incorrect. It derives a formula for the ratios of prior and posterior probabilities, but this formula only holds under the assumption of constant temperature (in which case the ratio is very large). When the temperature is varied, the analysis of Neal (2001) applies, and the answer is different. One of the main selling points of the method is that it optimizes a variational lower bound on the log-likelihood; even more accurate estimates can be obtained using importance sampling. It ought to be easy to report log-likelihood estimates for this method, so I wonder why such estimates aren’t reported. There are lots of prior results to compare against on MNIST. (In addition, a natural baseline would be RAISE, so that one can check if the ability to learn the transitions actually helps.) I think the basic idea here is a sound one, so I would be willing to raise my score if the above issues are addressed in a revised version. Minor comments: “A recognized obstacle to training undirected graphical models… is that ML training requires sampling from MCMC chains in the inner loop of training, for each example.” This seems like an unfair characterization, since the standard algorithm is PCD, which usually takes only a single step per mini-batch. Some of the methods discussed in the related work are missing citations. The method is justified in terms of “carving the energy function in the right direction at each point”, but I’m not sure this is actually what’s happening. Isn’t the point of the method that it can optimize a lower bound on the log-likelihood, and therefore learn a globally correct allocation of probability mass?
assistant
This paper proposes a new kind of generative model based on an annealing process, where the transition probabilities are learned directly to maximize a variational lower bound on the log-likelihood. Overall, the idea is clever and appealing, but I think the paper needs more quantitative validation and better discussion of the relationship with prior work. In terms of prior work, AIS and RAISE are both closely related algorithms, and share much of the mathematical structure with the proposed method. For this reason, it’s not sufficient to mention them in passing in the related work section; those methods and their relationship to variational walkback need to be discussed in detail. If I understand correctly, the proposed method is essentially an extension of RAISE where the transition probabilities are learned rather than fixed based on an existing MRF. I think this is an interesting and worthwhile extension, but the relationship to existing work needs to be clarified. The analysis of Appendix D seems incorrect. It derives a formula for the ratios of prior and posterior probabilities, but this formula only holds under the assumption of constant temperature (in which case the ratio is very large). When the temperature is varied, the analysis of Neal (2001) applies, and the answer is different. One of the main selling points of the method is that it optimizes a variational lower bound on the log-likelihood; even more accurate estimates can be obtained using importance sampling. It ought to be easy to report log-likelihood estimates for this method, so I wonder why such estimates aren’t reported. There are lots of prior results to compare against on MNIST. (In addition, a natural baseline would be RAISE, so that one can check if the ability to learn the transitions actually helps.) I think the basic idea here is a sound one, so I would be willing to raise my score if the above issues are addressed in a revised version. Minor comments: “A recognized obstacle to training undirected graphical models… is that ML training requires sampling from MCMC chains in the inner loop of training, for each example.” This seems like an unfair characterization, since the standard algorithm is PCD, which usually takes only a single step per mini-batch. Some of the methods discussed in the related work are missing citations. The method is justified in terms of “carving the energy function in the right direction at each point”, but I’m not sure this is actually what’s happening. Isn’t the point of the method that it can optimize a lower bound on the log-likelihood, and therefore learn a globally correct allocation of probability mass?