The Grammar of Interactive Explanatory Model Analysis

The growing need for in-depth analysis of predictive models leads to a series of new methods for explaining their local and global properties. Which of these methods is the best? It turns out that this is an ill-posed question. One cannot sufficiently explain a black-box machine learning model using a single method that gives only one perspective. Isolated explanations are prone to misunderstanding, which inevitably leads to wrong or simplistic reasoning. This problem is known as the Rashomon effect and refers to diverse, even contradictory interpretations of the same phenomenon. Surprisingly, the majority of methods developed for explainable machine learning focus on a single aspect of the model behavior. In contrast, we showcase the problem of explainability as an interactive and sequential analysis of a model. This paper presents how different Explanatory Model Analysis (EMA) methods complement each other and why it is essential to juxtapose them together. The introduced process of Interactive EMA (IEMA) derives from the algorithmic side of explainable machine learning and aims to embrace ideas developed in cognitive sciences. We formalize the grammar of IEMA to describe potential human-model dialogues. IEMA is implemented in the human-centered framework that adopts interactivity, customizability and automation as its main traits. Combined, these methods enhance the responsible approach to predictive modeling.Comment: 17 pages, 10 figures, 3 table

Boosting insights in insurance tariff plans with tree-based machine learning methods

Pricing actuaries typically operate within the framework of generalized linear models (GLMs). With the upswing of data analytics, our study puts focus on machine learning methods to develop full tariff plans built from both the frequency and severity of claims. We adapt the loss functions used in the algorithms such that the specific characteristics of insurance data are carefully incorporated: highly unbalanced count data with excess zeros and varying exposure on the frequency side combined with scarce, but potentially long-tailed data on the severity side. A key requirement is the need for transparent and interpretable pricing models which are easily explainable to all stakeholders. We therefore focus on machine learning with decision trees: starting from simple regression trees, we work towards more advanced ensembles such as random forests and boosted trees. We show how to choose the optimal tuning parameters for these models in an elaborate cross-validation scheme, we present visualization tools to obtain insights from the resulting models and the economic value of these new modeling approaches is evaluated. Boosted trees outperform the classical GLMs, allowing the insurer to form profitable portfolios and to guard against potential adverse risk selection

The Inverse Shapley Value Problem

For $f$ a weighted voting scheme used by $n$ voters to choose between two candidates, the $n$ \emph{Shapley-Shubik Indices} (or {\em Shapley values}) of $f$ provide a measure of how much control each voter can exert over the overall outcome of the vote. Shapley-Shubik indices were introduced by Lloyd Shapley and Martin Shubik in 1954 \cite{SS54} and are widely studied in social choice theory as a measure of the "influence" of voters. The \emph{Inverse Shapley Value Problem} is the problem of designing a weighted voting scheme which (approximately) achieves a desired input vector of values for the Shapley-Shubik indices. Despite much interest in this problem no provably correct and efficient algorithm was known prior to our work. We give the first efficient algorithm with provable performance guarantees for the Inverse Shapley Value Problem. For any constant \eps > 0 our algorithm runs in fixed poly$(n)$ time (the degree of the polynomial is independent of \eps) and has the following performance guarantee: given as input a vector of desired Shapley values, if any "reasonable" weighted voting scheme (roughly, one in which the threshold is not too skewed) approximately matches the desired vector of values to within some small error, then our algorithm explicitly outputs a weighted voting scheme that achieves this vector of Shapley values to within error \eps. If there is a "reasonable" voting scheme in which all voting weights are integers at most \poly(n) that approximately achieves the desired Shapley values, then our algorithm runs in time \poly(n) and outputs a weighted voting scheme that achieves the target vector of Shapley values to within error $\eps=n^{-1/8}.

CLEAR: Covariant LEAst-square Re-fitting with applications to image restoration

In this paper, we propose a new framework to remove parts of the systematic errors affecting popular restoration algorithms, with a special focus for image processing tasks. Generalizing ideas that emerged for $\ell_1$ regularization, we develop an approach re-fitting the results of standard methods towards the input data. Total variation regularizations and non-local means are special cases of interest. We identify important covariant information that should be preserved by the re-fitting method, and emphasize the importance of preserving the Jacobian (w.r.t. the observed signal) of the original estimator. Then, we provide an approach that has a "twicing" flavor and allows re-fitting the restored signal by adding back a local affine transformation of the residual term. We illustrate the benefits of our method on numerical simulations for image restoration tasks

Multiclass Learning with Simplex Coding

In this paper we discuss a novel framework for multiclass learning, defined by a suitable coding/decoding strategy, namely the simplex coding, that allows to generalize to multiple classes a relaxation approach commonly used in binary classification. In this framework, a relaxation error analysis can be developed avoiding constraints on the considered hypotheses class. Moreover, we show that in this setting it is possible to derive the first provably consistent regularized method with training/tuning complexity which is independent to the number of classes. Tools from convex analysis are introduced that can be used beyond the scope of this paper

A Mathematical Formalization of Hierarchical Temporal Memory's Spatial Pooler

Hierarchical temporal memory (HTM) is an emerging machine learning algorithm, with the potential to provide a means to perform predictions on spatiotemporal data. The algorithm, inspired by the neocortex, currently does not have a comprehensive mathematical framework. This work brings together all aspects of the spatial pooler (SP), a critical learning component in HTM, under a single unifying framework. The primary learning mechanism is explored, where a maximum likelihood estimator for determining the degree of permanence update is proposed. The boosting mechanisms are studied and found to be only relevant during the initial few iterations of the network. Observations are made relating HTM to well-known algorithms such as competitive learning and attribute bagging. Methods are provided for using the SP for classification as well as dimensionality reduction. Empirical evidence verifies that given the proper parameterizations, the SP may be used for feature learning.Comment: This work was submitted for publication and is currently under review. For associated code, see https://github.com/tehtechguy/mHT

Optimization by gradient boosting

Gradient boosting is a state-of-the-art prediction technique that sequentially produces a model in the form of linear combinations of simple predictors---typically decision trees---by solving an infinite-dimensional convex optimization problem. We provide in the present paper a thorough analysis of two widespread versions of gradient boosting, and introduce a general framework for studying these algorithms from the point of view of functional optimization. We prove their convergence as the number of iterations tends to infinity and highlight the importance of having a strongly convex risk functional to minimize. We also present a reasonable statistical context ensuring consistency properties of the boosting predictors as the sample size grows. In our approach, the optimization procedures are run forever (that is, without resorting to an early stopping strategy), and statistical regularization is basically achieved via an appropriate $L^2$ penalization of the loss and strong convexity arguments