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# Identifying a minimal class of models for high–dimensional data

tl;dr: a technique for feature selection in regression which might be useful for exploratory analysis and which can provide guidelines for designing subsequent costly experiments by hinting at which features need not be collected. The main weaknesses are multiple non-discoverable hyperparameters, a blind random search for optimization, and a not so easily actionable output of the algorithm.

Consider sparse regression with a number of features/predictors $p$ greater than the number of datapoints $n$. In this setting it is vital to extract features relevant to the regression and compute using the “effective” $p$ of the dataset. The authors define a model of order $k$ as a particular choice of $k$ predictors $S$ among the $p$ available.

The goal is then to explore as many possible models of order $k$ using some non-greedy search strategy (to avoid similar models in the results), then use the trajectory of the search to make statements about the relevance of different predictors. The idea is to be able to say things like “predictor #xxx was present in all models of size $6>k>1$” and extract useful information from there for posterior analysis.

A key question is that of the consistency of the estimated models.1 Recall that the Lasso (regression with $l^1$ penalty) and the ElasticNet (which uses a convex combination of $l^1$ and $l^2$ penalties) provide no guarantees (under conditions applicable in practice) about the consistency of the estimation of the subset of predictors S with non-zero weights.2

The key idea to this respect is to change the problem and not to try to find the “true” $S_0$ (which is unlikely to exist, and even more so if $p>n$) but a set of models. The authors define then a minimal class of models of size $k$ and efficiency $\eta$ as the set of all models with risk within $\eta$ margin of the optimal risk among all models of order $k$. This is clearly an untractable class, since it is defined in terms of unknown population quantities, but it can be approximated with sample quantities. There seem to be no bounds on the quality of approximation.

The method is as follows: Start with a reduced range of orders $\mathcal{M}$, then apply simulated annealing to explore the space of models. The transition from one model to another is done by changing the state (in/out) of just one predictor at a time. Which one is taken is determined by probabilities $p^{\operatorname{in}}, p^{\operatorname{out}}$ depending on scores $\gamma$ for each predictor. These are used in computing a threshold $q$ for the Metropolis-Hastings criterion. See p.7. in the paper for the details

Examples:

1. Expression of $p=4088$ genes in $n=71$ samples of Bacilus subtilis, the target is the production of Riboflavin. The method picked 112 models providing some (inconclusive) insights into which genes might be worth exploring by experimentalists (e.g. gene #xxx appears so and so many times, genes this and that almost never do…).
2. Air pollution dataset, more of the same idea: try to guess what predictors are relevant by exploring the search path of the model estimators.

1. A model estimator $\hat{S}_n$ is consistent if $\underset{n \rightarrow \infty}{\lim} \mathbb{P} (\hat{S}_n = S_0) = 0$, where $S_0$ is the true set of relevant predictors (i.e. with non-zero coefficients in the true regression function, $\mathbb{E}(Y|X)$).
2. See e.g. Larry Wasserman’s lecture notes at Carnegie Mellon.