The last post in the Machine Learning algorithm frenzy that I'm currently on was based on Support Vector Machines. This post is going to look at another classification algorithm called a Decision Tree and again see if this can improve on our classification problem of predicting whether lines from Simpson's episodes are said by Bart or Homer. Again, a Decision Tree is a supervised learning algorithm that can be used to classify data based on a model it has built on training data. Like the other models, it tries to split data into two or more sets but the most significant variable that creates the best split is calculated by the algorithm. To distinguish between Homer and Bart from a set of images the decision tree would split the data on each of them and choose which performs best.

This post originally appeared on the Yhat blog. Yhat is a Brooklyn based company whose goal is to make data science applicable for developers, data scientists, and businesses alike. Yhat provides a software platform for deploying and managing predictive algorithms as REST APIs, while eliminating the painful engineering obstacles associated with production environments like testing, versioning, scaling and security. SVM is a supervised machine learning algorithm which can be used for classification or regression problems. It uses a technique called the kernel trick to transform your data and then based on these transformations it finds an optimal boundary between the possible outputs.

We consider the problem of estimating a regression function in the common situation where the number of features is small, where interpretability of the model is a high priority, and where simple linear or additive models fail to provide adequate performance. To address this problem, we present GapTV, an approach that is conceptually related both to CART and to the more recent CRISP algorithm, a state-of-the-art alternative method for interpretable nonlinear regression. GapTV divides the feature space into blocks of constant value and fits the value of all blocks jointly via a convex optimization routine. Our method is fully data-adaptive, in that it incorporates highly robust routines for tuning all hyperparameters automatically. We compare our approach against CART and CRISP and demonstrate that GapTV finds a much better trade-off between accuracy and interpretability.

We describe stochastic Newton and stochastic quasi-Newton approaches to efficiently solve large linear least-squares problems where the very large data sets present a significant computational burden (e.g., the size may exceed computer memory or data are collected in real-time). In our proposed framework, stochasticity is introduced in two different frameworks as a means to overcome these computational limitations, and probability distributions that can exploit structure and/or sparsity are considered. Theoretical results on consistency of the approximations for both the stochastic Newton and the stochastic quasi-Newton methods are provided. The results show, in particular, that stochastic Newton iterates, in contrast to stochastic quasi-Newton iterates, may not converge to the desired least-squares solution. Numerical examples, including an example from extreme learning machines, demonstrate the potential applications of these methods.

In this paper, we propose PCKID, a novel, robust, kernel function for spectral clustering, specifically designed to handle incomplete data. By combining posterior distributions of Gaussian Mixture Models for incomplete data on different scales, we are able to learn a kernel for incomplete data that does not depend on any critical hyperparameters, unlike the commonly used RBF kernel. To evaluate our method, we perform experiments on two real datasets. PCKID outperforms the baseline methods for all fractions of missing values and in some cases outperforms the baseline methods with up to 25 percentage points.

In application domains such as healthcare, we want accurate predictive models that are also causally interpretable. In pursuit of such models, we propose a causal regularizer to steer predictive models towards causally-interpretable solutions and theoretically study its properties. In a large-scale analysis of Electronic Health Records (EHR), our causally-regularized model outperforms its L1-regularized counterpart in causal accuracy and is competitive in predictive performance. We perform non-linear causality analysis by causally regularizing a special neural network architecture. We also show that the proposed causal regularizer can be used together with neural representation learning algorithms to yield up to 20% improvement over multilayer perceptron in detecting multivariate causation, a situation common in healthcare, where many causal factors should occur simultaneously to have an effect on the target variable.

We introduce a simple permutation equivariant layer for deep learning with set structure.This type of layer, obtained by parameter-sharing, has a simple implementation and linear-time complexity in the size of each set. We use deep permutation-invariant networks to perform point-could classification and MNIST-digit summation, where in both cases the output is invariant to permutations of the input. In a semi-supervised setting, where the goal is make predictions for each instance within a set, we demonstrate the usefulness of this type of layer in set-outlier detection as well as semi-supervised learning with clustering side-information.

By restricting the iterate on a nonlinear manifold, the recently proposed Riemannian optimization methods prove to be both efficient and effective in low rank tensor completion problems. However, existing methods fail to exploit the easily accessible side information, due to their format mismatch. Consequently, there is still room for improvement in such methods. To fill the gap, in this paper, a novel Riemannian model is proposed to organically integrate the original model and the side information by overcoming their inconsistency. For this particular model, an efficient Riemannian conjugate gradient descent solver is devised based on a new metric that captures the curvature of the objective. Numerical experiments suggest that our solver is more accurate than the state-of-the-art without compromising the efficiency.

Modern convolutional networks, incorporating rectifiers and max-pooling, are neither smooth nor convex. Standard guarantees therefore do not apply. Nevertheless, methods from convex optimization such as gradient descent and Adam are widely used as building blocks for deep learning algorithms. This paper provides the first convergence guarantee applicable to modern convnets. The guarantee matches a lower bound for convex nonsmooth functions. The key technical tool is the neural Taylor approximation -- a straightforward application of Taylor expansions to neural networks -- and the associated Taylor loss. Experiments on a range of optimizers, layers, and tasks provide evidence that the analysis accurately captures the dynamics of neural optimization. The second half of the paper applies the Taylor approximation to isolate the main difficulty in training rectifier nets: that gradients are shattered. We investigate the hypothesis that, by exploring the space of activation configurations more thoroughly, adaptive optimizers such as RMSProp and Adam are able to converge to better solutions.