We consider general Gaussian latent tree models in which the observed variables are not restricted to be leaves of the tree. Extending related recent work, we give a full semi-algebraic description of the set of covariance matrices of any such model. In other words, we find polynomial constraints that characterize when a matrix is the covariance matrix of a distribution in a given latent tree model. However, leveraging these constraints to test a given such model is often complicated by the number of constraints being large and by singularities of individual polynomials, which may invalidate standard approximations to relevant probability distributions. Illustrating with the star tree, we propose a new testing methodology that circumvents singularity issues by trading off some statistical estimation efficiency and handles cases with many constraints through recent advances on Gaussian approximation for maxima of sums of high-dimensional random vectors. Our test avoids the need to maximize the possibly multimodal likelihood function of such models and is applicable to models with larger number of variables. These points are illustrated in numerical experiments.

Random forests are learning algorithms that build large collections of random trees and make predictions by averaging the individual tree predictions. In this paper, we consider various tree constructions and examine how the choice of parameters affects the generalization error of the resulting random forests as the sample size goes to infinity. We show that subsampling of data points during the tree construction phase is important: Forests can become inconsistent with either no subsampling or too severe subsampling. As a consequence, even highly randomized trees can lead to inconsistent forests if no subsampling is used, which implies that some of the commonly used setups for random forests can be inconsistent. As a second consequence we can show that trees that have good performance in nearest-neighbor search can be a poor choice for random forests.

We consider the optimization of an uncertain objective over continuous and multi-dimensional decision spaces in problems in which we are only provided with observational data. We propose a novel algorithmic framework that is tractable, asymptotically consistent, and superior to comparable methods on example problems. Our approach leverages predictive machine learning methods and incorporates information on the uncertainty of the predicted outcomes for the purpose of prescribing decisions. We demonstrate the efficacy of our method on examples involving both synthetic and real data sets.

While deep reinforcement learning has successfully solved many challenging control tasks, its real-world applicability has been limited by the inability to ensure the safety of learned policies. We propose an approach to verifiable reinforcement learning by training decision tree policies, which can represent complex policies (since they are nonparametric), yet can be efficiently verified using existing techniques (since they are highly structured). The challenge is that decision tree policies are difficult to train. We propose VIPER, an algorithm that combines ideas from model compression and imitation learning to learn decision tree policies guided by a DNN policy (called the oracle) and its Q-function, and show that it substantially outperforms two baselines. We use VIPER to (i) learn a provably robust decision tree policy for a variant of Atari Pong with a symbolic state space, (ii) learn a decision tree policy for a toy game based on Pong that provably never loses, and (iii) learn a provably stable decision tree policy for cart-pole. In each case, the decision tree policy achieves performance equal to that of the original DNN policy.

Learning a decision tree from data is a difficult optimization problem. The most widespread algorithm in practice, dating to the 1980s, is based on a greedy growth of the tree structure by recursively splitting nodes, and possibly pruning back the final tree. The parameters (decision function) of an internal node are approximately estimated by minimizing an impurity measure. We give an algorithm that, given an input tree (its structure and the parameter values at its nodes), produces a new tree with the same or smaller structure but new parameter values that provably lower or leave unchanged the misclassification error. This can be applied to both axis-aligned and oblique trees and our experiments show it consistently outperforms various other algorithms while being highly scalable to large datasets and trees. Further, the same algorithm can handle a sparsity penalty, so it can learn sparse oblique trees, having a structure that is a subset of the original tree and few nonzero parameters. This combines the best of axis-aligned and oblique trees: flexibility to model correlated data, low generalization error, fast inference and interpretable nodes that involve only a few features in their decision.

This article was written by Will Koehrsen. Here's the complete code: just copy and paste into a Jupyter Notebook or Python script, replace with your data and run: The final result is a complete decision tree as an image. To read the rest of this article with code and illustrations, click here.

Here is our list of featured articles and resources posted since Monday.

Due to their structure, decision trees are easy to understand, interpret and visualize. In doing so, a variable check or feature selection is implicitly performed. Both numerical and non-numerical data can be processed simultaneously relatively little effort on the part of the user for the data preparation requires. On the other hand, too complex trees can be created that do not generalize the data well. Small variations in the data can also make the trees unstable, creating a tree that does not solve the problem.

This study proposes a logic architecture for the high-speed and power efficiently training of a gradient boosting decision tree model of binary classification. We implemented the proposed logic architecture on an FPGA and compared training time and power efficiency with three general GBDT software libraries using CPU and GPU. The training speed of the logic architecture on the FPGA was 26-259 times faster than the software libraries. The power efficiency of the logic architecture was 90-1,104 times higher than the software libraries. The results show that the logic architecture suits for high-performance and edge computing.

Optimal stopping is the problem of deciding when to stop a stochastic system to obtain the greatest reward, arising in numerous application areas such as finance, healthcare and marketing. State-of-the-art methods for high-dimensional optimal stopping involve approximating the value function or the continuation value, and then using that approximation within a greedy policy. Although such policies can perform very well, they are generally not guaranteed to be interpretable; that is, a decision maker may not be able to easily see the link between the current system state and the policy's action. In this paper, we propose a new approach to optimal stopping, wherein the policy is represented as a binary tree, in the spirit of naturally interpretable tree models commonly used in machine learning. We formulate the problem of learning such policies from observed trajectories of the stochastic system as a sample average approximation (SAA) problem. We prove that the SAA problem converges under mild conditions as the sample size increases, but that computationally even immediate simplifications of the SAA problem are theoretically intractable. We thus propose a tractable heuristic for approximately solving the SAA problem, by greedily constructing the tree from the top down. We demonstrate the value of our approach by applying it to the canonical problem of option pricing, using both synthetic instances and instances calibrated with real S&P 500 data. Our method obtains policies that (1) outperform state-of-the-art non-interpretable methods, based on simulation-regression and martingale duality, and (2) possess a remarkably simple and intuitive structure.