Bootstrap Model Aggregation for Distributed Statistical Learning
In distributed, or privacy-preserving learning, we are often given a set of probabilistic models estimated from different local repositories, and asked to combine them into a single model that gives efficient statistical estimation. A simple method is to linearly average the parameters of the local models, which, however, tends to be degenerate or not applicable on non-convex models, or models with different parameter dimensions. One more practical strategy is to generate bootstrap samples from the local models, and then learn a joint model based on the combined bootstrap set. Unfortunately, the bootstrap procedure introduces additional noise and can significantly deteriorate the performance. In this work, we propose two variance reduction methods to correct the bootstrap noise, including a weighted M-estimator that is both statistically efficient and practically powerful. Both theoretical and empirical analysis is provided to demonstrate our methods.
This article proposes two new distributed algorithms to ameliorate the original issue with the bootstrapped KL averaging method. The article shows that a naive bootstrapped KL averaging would result in the same computational burden in combining the estimation even the data is first partitioned for local estimation. The main reason is due to the variance contained in the bootstrap sample. The paper then proposed a variance reduction technique to fix this issue. In particular, the algorithm tries to quantify the bootstrap variance by measuring the difference between the local estimator and local bootstrapped estimator, and correct the global KL estimator based on the quantification. The theory shows that by adding this correction, the convergence rate can be improved to O(N^{-1} + (dn^2)^{-1}).
This is a successive work correcting previous research on using KL averaging combining subset estimators. I think the most appealing point for using KL averaging, despite the computational issue, is its power in dealing with latent variable models. There is another line of work in using geometric median to combine subset estimators the authors might want to compare to, for example, Minsker (2013) and Hsu and Sabato (2013). These algorithms are simple and efficient in most cases, but might not be doing well for latent variable models. The variance reduction technique used in this article is very similar to the de-bias technique used in Javanmard and Montanari (2015) and Lee et a. (2015), so the theoretical contribution is kind of limited.  There are two caveats for the proposed approach. The first lies in the resampling. If the original model is complicated in sampling, then the additional resample requirement from the KL averaging could ruin the whole framework. Second, the variance reduction requires to compute the Fishier information matrix and the inverse, which will adds an O(p^3) computational burden. This will also become huge when p is large.  ========================  after reading feedback ====================  Previously I thought the advantage of this paper over the geometric median or the simple averaging approach is that it can handle latent variable models, such as estimating mixture of Gaussians. It is well known that the latent variable model has this "label switching" problem (the parameters are equivalent up to some permutation), so a direct averaging of the parameters obtained from different subsets might result in bad outcomes due to the permutation. The KL averaging method solves this problem by avoiding directly averaging the sub-estimators. Instead, it resamples data from each subset and pools the new data together to estimate the final result, so as to avoid the "label switching" issue.  However, as now I double checked the paper, I found that the variance reduction technique introduced in this paper actually ruined this nice property of the original KL averaging. If you take a look at (7), which is the main formula in this article, you will realize that (7) also suffers from the "label switching" problem if \theta's are up to some permutation. Then we arrive at some tricky scenario. On the one hand, the KL averaging can handle the latent variable model, but the efficiency is bad (according to this article) compared to the direct averaging. On the other hand, the enhanced KL averaging with variance reduction can improve the efficiency to the same level as direct averaging, but loses the ability to hand latent variable models (essentially the same as direct averaging). So I kind of doubt how much can be gained practically by using this method, which involves much heavier computation compared to averaging.  It is a pity there is only one round of author feedback and it is my fault not to pose this question to the author in the previous round. I would love to know how the authors address this problem. But given this is my problem not to ask, I will raise my score of technical quality from 2 --> 3