Interpretability in Machine Learning: on the Interplay with Explainability, Predictive Performances and Models

Interpretability has recently gained attention in the field of machine learning, for it is crucial when it comes to high-stakes decisions or troubleshooting. This abstract concept is hard to grasp and has been associated, over time, with many labels and preconceived ideas. In this position paper, in order to clarify some misunderstandings regarding interpretability, we discuss its relationship with significant concepts in machine learning: explainability, predictive performances, and machine learning models. For instance, we challenge the idea that interpretability and explainability are substitutes to one another, or that a fixed degree of interpretability can be associated with a given machine learning model.

Statistical Guarantees for Variational Autoencoders using PAC-Bayesian Theory

Since their inception, Variational Autoencoders (VAEs) have become central in machine learning. Despite their widespread use, numerous questions regarding their theoretical properties remain open. Using PAC-Bayesian theory, this work develops statistical guarantees for VAEs. First, we derive the first PAC-Bayesian bound for posterior distributions conditioned on individual samples from the data-generating distribution. Then, we utilize this result to develop generalization guarantees for the VAE's reconstruction loss, as well as upper bounds on the distance between the input and the regenerated distributions. More importantly, we provide upper bounds on the Wasserstein distance between the input distribution and the distribution defined by the VAE's generative model.

Invariant Causal Set Covering Machines

Rule-based models, such as decision trees, appeal to practitioners due to their interpretable nature. However, the learning algorithms that produce such models are often vulnerable to spurious associations and thus, they are not guaranteed to extract causally-relevant insights. In this work, we build on ideas from the invariant causal prediction literature to propose Invariant Causal Set Covering Machines, an extension of the classical Set Covering Machine algorithm for conjunctions/disjunctions of binary-valued rules that provably avoids spurious associations. We demonstrate both theoretically and empirically that our method can identify the causal parents of a variable of interest in polynomial time.

PAC-Bayesian Generalization Bounds for Adversarial Generative Models

We extend PAC-Bayesian theory to generative models and develop generalization bounds for models based on the Wasserstein distance and the total variation distance. Our first result on the Wasserstein distance assumes the instance space is bounded, while our second result takes advantage of dimensionality reduction. Our results naturally apply to Wasserstein GANs and Energy-Based GANs, and our bounds provide new training objectives for these two. Although our work is mainly theoretical, we perform numerical experiments showing non-vacuous generalization bounds for Wasserstein GANs on synthetic datasets.

A Greedy Algorithm for Building Compact Binary Activated Neural Networks

We study binary activated neural networks in the context of regression tasks, provide guarantees on the expressiveness of these particular networks and propose a greedy algorithm for building such networks. Aiming for predictors having small resources needs, the greedy approach does not need to fix in advance an architecture for the network: this one is built one layer at a time, one neuron at a time, leading to predictors that aren't needlessly wide and deep for a given task. Similarly to boosting algorithms, our approach guarantees a training loss reduction every time a neuron is added to a layer. This greatly differs from most binary activated neural networks training schemes that rely on stochastic gradient descent (circumventing the 0-almost-everywhere derivative problem of the binary activation function by surrogates such as the straight through estimator or continuous binarization). We show that our method provides compact and sparse predictors while obtaining similar performances to state-of-the-art methods for training binary activated networks.

Learning Aggregations of Binary Activated Neural Networks with Probabilities over Representations

Considering a probability distribution over parameters is known as an efficient strategy to learn a neural network with non-differentiable activation functions. We study the expectation of a probabilistic neural network as a predictor by itself, focusing on the aggregation of binary activated neural networks with normal distributions over real-valued weights. Our work leverages a recent analysis derived from the PAC-Bayesian framework that derives tight generalization bounds and learning procedures for the expected output value of such an aggregation, which is given by an analytical expression. While the combinatorial nature of the latter has been circumvented by approximations in previous works, we show that the exact computation remains tractable for deep but narrow neural networks, thanks to a dynamic programming approach. This leads us to a peculiar bound minimization learning algorithm for binary activated neural networks, where the forward pass propagates probabilities over representations instead of activation values. A stochastic counterpart of this new neural networks training scheme that scales to wider architectures is proposed.

Learning Stochastic Majority Votes by Minimizing a PAC-Bayes Generalization Bound

We investigate a stochastic counterpart of majority votes over finite ensembles of classifiers, and study its generalization properties. While our approach holds for arbitrary distributions, we instantiate it with Dirichlet distributions: this allows for a closed-form and differentiable expression for the expected risk, which then turns the generalization bound into a tractable training objective. The resulting stochastic majority vote learning algorithm achieves state-of-the-art accuracy and benefits from (non-vacuous) tight generalization bounds, in a series of numerical experiments when compared to competing algorithms which also minimize PAC-Bayes objectives -- both with uninformed (data-independent) and informed (data-dependent) priors.

Self-Bounding Majority Vote Learning Algorithms by the Direct Minimization of a Tight PAC-Bayesian C-Bound

In the PAC-Bayesian literature, the C-Bound refers to an insightful relation between the risk of a majority vote classifier (under the zero-one loss) and the first two moments of its margin (i.e., the expected margin and the voters' diversity). Until now, learning algorithms developed in this framework minimize the empirical version of the C-Bound, instead of explicit PAC-Bayesian generalization bounds. In this paper, by directly optimizing PAC-Bayesian guarantees on the C-Bound, we derive self-bounding majority vote learning algorithms. Moreover, our algorithms based on gradient descent are scalable and lead to accurate predictors paired with non-vacuous guarantees.

A General Framework for the Derandomization of PAC-Bayesian Bounds

PAC-Bayesian bounds are known to be tight and informative when studying the generalization ability of randomized classifiers. However, when applied to some family of deterministic models such as neural networks, they require a loose and costly derandomization step. As an alternative to this step, we introduce three new PAC-Bayesian generalization bounds that have the originality to be pointwise, meaning that they provide guarantees over one single hypothesis instead of the usual averaged analysis. Our bounds are rather general, potentially parameterizable, and provide novel insights for various machine learning settings that rely on randomized algorithms. We illustrate the interest of our theoretical result for the analysis of neural network training.

Implicit Variational Inference: the Parameter and the Predictor Space

Having access to accurate confidence levels along with the predictions allows to determine whether making a decision is worth the risk. Under the Bayesian paradigm, the posterior distribution over parameters is used to capture model uncertainty, a valuable information that can be translated into predictive uncertainty. However, computing the posterior distribution for high capacity predictors, such as neural networks, is generally intractable, making approximate methods such as variational inference a promising alternative. While most methods perform inference in the space of parameters, we explore the benefits of carrying inference directly in the space of predictors. Relying on a family of distributions given by a deep generative neural network, we present two ways of carrying variational inference: one in \emph{parameter space}, one in \emph{predictor space}. Importantly, the latter requires us to choose a distribution of inputs, therefore allowing us at the same time to explicitly address the question of \emph{out-of-distribution} uncertainty. We explore from various perspectives the implications of working in the predictor space induced by neural networks as opposed to the parameter space, focusing mainly on the quality of uncertainty estimation for data lying outside of the training distribution. We compare posterior approximations obtained with these two methods to several standard methods and present results showing that variational approximations learned in the predictor space distinguish themselves positively from those trained in the parameter space.