The elastic net combines lasso and ridge regression to fuse the sparsity property of lasso with the grouping property of ridge regression. The connections between ridge regression and gradient descent and between lasso and forward stagewise regression have previously been shown. Similar to how the elastic net generalizes lasso and ridge regression, we introduce elastic gradient descent, a generalization of gradient descent and forward stagewise regression. We theoretically analyze elastic gradient descent and compare it to the elastic net and forward stagewise regression. Parts of the analysis are based on elastic gradient flow, a piecewise analytical construction, obtained for elastic gradient descent with infinitesimal step size. We also compare elastic gradient descent to the elastic net on real and simulated data and show that it provides similar solution paths, but is several orders of magnitude faster. Compared to forward stagewise regression, elastic gradient descent selects a model that, although still sparse, provides considerably lower prediction and estimation errors.

A common belief in unbiased active learning is that, in order to capture the most informative instances, the sampling probabilities should be proportional to the uncertainty of the class labels. We argue that this produces suboptimal predictions and present sampling schemes for unbiased pool-based active learning that minimise the actual prediction error, and demonstrate a better predictive performance than competing methods on a number of benchmark datasets. In contrast, both probabilistic and deterministic uncertainty sampling performed worse than simple random sampling on some of the datasets.

Recently, new methods for model assessment, based on subsampling and posterior approximations, have been proposed for scaling leave-one-out cross-validation (LOO-CV) to large datasets. Although these methods work well for estimating predictive performance for individual models, they are less powerful in model comparison. We propose an efficient method for estimating differences in predictive performance by combining fast approximate LOO surrogates with exact LOO sub-sampling using the difference estimator and supply proofs with regards to scaling characteristics. The resulting approach can be orders of magnitude more efficient than previous approaches, as well as being better suited to model comparison.

Model inference, such as model comparison, model checking, and model selection, is an important part of model development. Leave-one-out cross-validation (LOO) is a general approach for assessing the generalizability of a model, but unfortunately, LOO does not scale well to large datasets. We propose a combination of using approximate inference techniques and probability-proportional-to-size-sampling (PPS) for fast LOO model evaluation for large datasets. We provide both theoretical and empirical results showing good properties for large data.