Machine Learning (stat.ML)

Abstract: Deep neural networks (NNs) are known to lack uncertainty estimates and struggle to incorporate new data. We present a method that mitigates these issues by converting NNs from weight space to function space, via a dual parameterization. Importantly, the dual parameterization enables us to formulate a sparse representation that captures information from the entire data set. This offers a compact and principled way of capturing uncertainty and enables us to incorporate new data without retraining whilst retaining predictive performance. We provide proof-of-concept demonstrations with the proposed approach for quantifying uncertainty in supervised learning on UCI benchmark tasks.

Abstract: Best Arm Identification (BAI) problems are progressively used for data-sensitive applications, such as designing adaptive clinical trials, tuning hyper-parameters, and conducting user studies to name a few. Motivated by the data privacy concerns invoked by these applications, we study the problem of BAI with fixed confidence under $\epsilon$-global Differential Privacy (DP). First, to quantify the cost of privacy, we derive a lower bound on the sample complexity of any $\delta$-correct BAI algorithm satisfying $\epsilon$-global DP. Our lower bound suggests the existence of two privacy regimes depending on the privacy budget $\epsilon$. In the high-privacy regime (small $\epsilon$), the hardness depends on a coupled effect of privacy and a novel information-theoretic quantity, called the Total Variation Characteristic Time. In the low-privacy regime (large $\epsilon$), the sample complexity lower bound reduces to the classical non-private lower bound. Second, we propose AdaP-TT, an $\epsilon$-global DP variant of the Top Two algorithm. AdaP-TT runs in arm-dependent adaptive episodes and adds Laplace noise to ensure a good privacy-utility trade-off. We derive an asymptotic upper bound on the sample complexity of AdaP-TT that matches with the lower bound up to multiplicative constants in the high-privacy regime. Finally, we provide an experimental analysis of AdaP-TT that validates our theoretical results.

Abstract: Classical machine learning methods may lead to poor prediction performance when the target distribution differs from the source populations. This paper utilizes data from multiple sources and introduces a group distributionally robust prediction model defined to optimize an adversarial reward about explained variance with respect to a class of target distributions. Compared to classical empirical risk minimization, the proposed robust prediction model improves the prediction accuracy for target populations with distribution shifts. We show that our group distributionally robust prediction model is a weighted average of the source populations' conditional outcome models. We leverage this key identification result to robustify arbitrary machine learning algorithms, including, for example, random forests and neural networks. We devise a novel bias-corrected estimator to estimate the optimal aggregation weight for general machine-learning algorithms and demonstrate its improvement in the convergence rate. Our proposal can be seen as a distributionally robust federated learning approach that is computationally efficient and easy to implement using arbitrary machine learning base algorithms, satisfies some privacy constraints, and has a nice interpretation of different sources' importance for predicting a given target covariate distribution. We demonstrate the performance of our proposed group distributionally robust method on simulated and real data with random forests and neural networks as base-learning algorithms.

Abstract: SHAP (SHapley Additive exPlanations) has become a popular method to attribute the prediction of a machine learning model on an input to its features. One main challenge of SHAP is the computation time. An exact computation of Shapley values requires exponential time complexity. Therefore, many approximation methods are proposed in the literature. In this paper, we propose methods that can compute SHAP exactly in polynomial time or even faster for SHAP definitions that satisfy our additivity and dummy assumptions (eg, kernal SHAP and baseline SHAP). We develop different strategies for models with different levels of model structure information: known functional decomposition, known order of model (defined as highest order of interaction in the model), or unknown order. For the first case, we demonstrate an additive property and a way to compute SHAP from the lower-order functional components. For the second case, we derive formulas that can compute SHAP in polynomial time. Both methods yield exact SHAP results. Finally, if even the order of model is unknown, we propose an iterative way to approximate Shapley values. The three methods we propose are computationally efficient when the order of model is not high which is typically the case in practice. We compare with sampling approach proposed in Castor & Gomez (2008) using simulation studies to demonstrate the efficacy of our proposed methods.

Abstract: Maximum mean discrepancy (MMD) refers to a general class of nonparametric two-sample tests that are based on maximizing the mean difference over samples from one distribution $P$ versus another $Q$, over all choices of data transformations $f$ living in some function space $\mathcal{F}$. Inspired by recent work that connects what are known as functions of $\textit{Radon bounded variation}$ (RBV) and neural networks (Parhi and Nowak, 2021, 2023), we study the MMD defined by taking $\mathcal{F}$ to be the unit ball in the RBV space of a given smoothness order $k \geq 0$. This test, which we refer to as the $\textit{Radon-Kolmogorov-Smirnov}$ (RKS) test, can be viewed as a generalization of the well-known and classical Kolmogorov-Smirnov (KS) test to multiple dimensions and higher orders of smoothness. It is also intimately connected to neural networks: we prove that the witness in the RKS test -- the function $f$ achieving the maximum mean difference -- is always a ridge spline of degree $k$, i.e., a single neuron in a neural network. This allows us to leverage the power of modern deep learning toolkits to (approximately) optimize the criterion that underlies the RKS test. We prove that the RKS test has asymptotically full power at distinguishing any distinct pair $P \not= Q$ of distributions, derive its asymptotic null distribution, and carry out extensive experiments to elucidate the strengths and weakenesses of the RKS test versus the more traditional kernel MMD test.

Abstract: Recent papers have used machine learning architecture to fit low-order functional ANOVA models with main effects and second-order interactions. These GAMI (GAM + Interaction) models are directly interpretable as the functional main effects and interactions can be easily plotted and visualized. Unfortunately, it is not easy to incorporate the monotonicity requirement into the existing GAMI models based on boosted trees, such as EBM (Lou et al. 2013) and GAMI-Lin-T (Hu et al. 2022). This paper considers models of the form $f(x)=\sum_{j,k}f_{j,k}(x_j, x_k)$ and develops monotone tree-based GAMI models, called monotone GAMI-Tree, by adapting the XGBoost algorithm. It is straightforward to fit a monotone model to $f(x)$ using the options in XGBoost. However, the fitted model is still a black box. We take a different approach: i) use a filtering technique to determine the important interactions, ii) fit a monotone XGBoost algorithm with the selected interactions, and finally iii) parse and purify the results to get a monotone GAMI model. Simulated datasets are used to demonstrate the behaviors of mono-GAMI-Tree and EBM, both of which use piecewise constant fits. Note that the monotonicity requirement is for the full model. Under certain situations, the main effects will also be monotone. But, as seen in the examples, the interactions will not be monotone.