PriorBoost: An Adaptive Algorithm for Learning from Aggregate Responses

This work studies algorithms for learning from aggregate responses. We focus on the construction of aggregation sets (called bags in the literature) for event-level loss functions. We prove for linear regression and generalized linear models (GLMs) that the optimal bagging problem reduces to one-dimensional size-constrained $k$-means clustering. Further, we theoretically quantify the advantage of using curated bags over random bags. We then propose the PriorBoost algorithm, which adaptively forms bags of samples that are increasingly homogeneous with respect to (unobserved) individual responses to improve model quality. We study label differential privacy for aggregate learning, and we also provide extensive experiments showing that PriorBoost regularly achieves optimal model quality for event-level predictions, in stark contrast to non-adaptive algorithms.

Learning from Aggregate responses: Instance Level versus Bag Level Loss Functions

Due to the rise of privacy concerns, in many practical applications the training data is aggregated before being shared with the learner, in order to protect privacy of users' sensitive responses. In an aggregate learning framework, the dataset is grouped into bags of samples, where each bag is available only with an aggregate response, providing a summary of individuals' responses in that bag. In this paper, we study two natural loss functions for learning from aggregate responses: bag-level loss and the instance-level loss. In the former, the model is learnt by minimizing a loss between aggregate responses and aggregate model predictions, while in the latter the model aims to fit individual predictions to the aggregate responses. In this work, we show that the instance-level loss can be perceived as a regularized form of the bag-level loss. This observation lets us compare the two approaches with respect to bias and variance of the resulting estimators, and introduce a novel interpolating estimator which combines the two approaches. For linear regression tasks, we provide a precise characterization of the risk of the interpolating estimator in an asymptotic regime where the size of the training set grows in proportion to the features dimension. Our analysis allows us to theoretically understand the effect of different factors, such as bag size on the model prediction risk. In addition, we propose a mechanism for differentially private learning from aggregate responses and derive the optimal bag size in terms of prediction risk-privacy trade-off. We also carry out thorough experiments to corroborate our theory and show the efficacy of the interpolating estimator.

Anonymous Learning via Look-Alike Clustering: A Precise Analysis of Model Generalization

While personalized recommendations systems have become increasingly popular, ensuring user data protection remains a top concern in the development of these learning systems. A common approach to enhancing privacy involves training models using anonymous data rather than individual data. In this paper, we explore a natural technique called \emph{look-alike clustering}, which involves replacing sensitive features of individuals with the cluster's average values. We provide a precise analysis of how training models using anonymous cluster centers affects their generalization capabilities. We focus on an asymptotic regime where the size of the training set grows in proportion to the features dimension. Our analysis is based on the Convex Gaussian Minimax Theorem (CGMT) and allows us to theoretically understand the role of different model components on the generalization error. In addition, we demonstrate that in certain high-dimensional regimes, training over anonymous cluster centers acts as a regularization and improves generalization error of the trained models. Finally, we corroborate our asymptotic theory with finite-sample numerical experiments where we observe a perfect match when the sample size is only of order of a few hundreds.

Learning via Look-Alike Clustering: A Precise Analysis of Model Generalization

While personalized recommendations systems have become increasingly popular, ensuring user data protection remains a paramount concern in the development of these learning systems. A common approach to enhancing privacy involves training models using anonymous data rather than individual data. In this paper, we explore a natural technique called \emph{look-alike clustering}, which involves replacing sensitive features of individuals with the cluster's average values. We provide a precise analysis of how training models using anonymous cluster centers affects their generalization capabilities. We focus on an asymptotic regime where the size of the training set grows in proportion to the features dimension. Our analysis is based on the Convex Gaussian Minimax Theorem (CGMT) and allows us to theoretically understand the role of different model components on the generalization error. In addition, we demonstrate that in certain high-dimensional regimes, training over anonymous cluster centers acts as a regularization and improves generalization error of the trained models. Finally, we corroborate our asymptotic theory with finite-sample numerical experiments where we observe a perfect match when the sample size is only of order of a few hundreds.

Causal Inference with Differentially Private (Clustered) Outcomes

Estimating causal effects from randomized experiments is only feasible if participants agree to reveal their potentially sensitive responses. Of the many ways of ensuring privacy, label differential privacy is a widely used measure of an algorithm's privacy guarantee, which might encourage participants to share responses without running the risk of de-anonymization. Many differentially private mechanisms inject noise into the original data-set to achieve this privacy guarantee, which increases the variance of most statistical estimators and makes the precise measurement of causal effects difficult: there exists a fundamental privacy-variance trade-off to performing causal analyses from differentially private data. With the aim of achieving lower variance for stronger privacy guarantees, we suggest a new differential privacy mechanism, "Cluster-DP", which leverages any given cluster structure of the data while still allowing for the estimation of causal effects. We show that, depending on an intuitive measure of cluster quality, we can improve the variance loss while maintaining our privacy guarantees. We compare its performance, theoretically and empirically, to that of its unclustered version and a more extreme uniform-prior version which does not use any of the original response distribution, both of which are special cases of the "Cluster-DP" algorithm.

Measuring Re-identification Risk

Compact user representations (such as embeddings) form the backbone of personalization services. In this work, we present a new theoretical framework to measure re-identification risk in such user representations. Our framework, based on hypothesis testing, formally bounds the probability that an attacker may be able to obtain the identity of a user from their representation. As an application, we show how our framework is general enough to model important real-world applications such as the Chrome's Topics API for interest-based advertising. We complement our theoretical bounds by showing provably good attack algorithms for re-identification that we use to estimate the re-identification risk in the Topics API. We believe this work provides a rigorous and interpretable notion of re-identification risk and a framework to measure it that can be used to inform real-world applications.

Structured Dynamic Pricing: Optimal Regret in a Global Shrinkage Model

We consider dynamic pricing strategies in a streamed longitudinal data set-up where the objective is to maximize, over time, the cumulative profit across a large number of customer segments. We consider a dynamic probit model with the consumers' preferences as well as price sensitivity varying over time. Building on the well-known finding that consumers sharing similar characteristics act in similar ways, we consider a global shrinkage structure, which assumes that the consumers' preferences across the different segments can be well approximated by a spatial autoregressive (SAR) model. In such a streamed longitudinal set-up, we measure the performance of a dynamic pricing policy via regret, which is the expected revenue loss compared to a clairvoyant that knows the sequence of model parameters in advance. We propose a pricing policy based on penalized stochastic gradient descent (PSGD) and explicitly characterize its regret as functions of time, the temporal variability in the model parameters as well as the strength of the auto-correlation network structure spanning the varied customer segments. Our regret analysis results not only demonstrate asymptotic optimality of the proposed policy but also show that for policy planning it is essential to incorporate available structural information as policies based on unshrunken models are highly sub-optimal in the aforementioned set-up.

Learning Rate Schedules in the Presence of Distribution Shift

We design learning rate schedules that minimize regret for SGD-based online learning in the presence of a changing data distribution. We fully characterize the optimal learning rate schedule for online linear regression via a novel analysis with stochastic differential equations. For general convex loss functions, we propose new learning rate schedules that are robust to distribution shift, and we give upper and lower bounds for the regret that only differ by constants. For non-convex loss functions, we define a notion of regret based on the gradient norm of the estimated models and propose a learning schedule that minimizes an upper bound on the total expected regret. Intuitively, one expects changing loss landscapes to require more exploration, and we confirm that optimal learning rate schedules typically increase in the presence of distribution shift. Finally, we provide experiments for high-dimensional regression models and neural networks to illustrate these learning rate schedules and their cumulative regret.

Prediction Sets for High-Dimensional Mixture of Experts Models

Large datasets make it possible to build predictive models that can capture heterogenous relationships between the response variable and features. The mixture of high-dimensional linear experts model posits that observations come from a mixture of high-dimensional linear regression models, where the mixture weights are themselves feature-dependent. In this paper, we show how to construct valid prediction sets for an $\ell_1$-penalized mixture of experts model in the high-dimensional setting. We make use of a debiasing procedure to account for the bias induced by the penalization and propose a novel strategy for combining intervals to form a prediction set with coverage guarantees in the mixture setting. Synthetic examples and an application to the prediction of critical temperatures of superconducting materials show our method to have reliable practical performance.

GRASP: A Goodness-of-Fit Test for Classification Learning

Performance of classifiers is often measured in terms of average accuracy on test data. Despite being a standard measure, average accuracy fails in characterizing the fit of the model to the underlying conditional law of labels given the features vector ($Y|X$), e.g. due to model misspecification, over fitting, and high-dimensionality. In this paper, we consider the fundamental problem of assessing the goodness-of-fit for a general binary classifier. Our framework does not make any parametric assumption on the conditional law $Y|X$, and treats that as a black box oracle model which can be accessed only through queries. We formulate the goodness-of-fit assessment problem as a tolerance hypothesis testing of the form \[ H_0: \mathbb{E}\Big[D_f\Big({\sf Bern}(\eta(X))\|{\sf Bern}(\hat{\eta}(X))\Big)\Big]\leq \tau\,, \] where $D_f$ represents an $f$-divergence function, and $\eta(x)$, $\hat{\eta}(x)$ respectively denote the true and an estimate likelihood for a feature vector $x$ admitting a positive label. We propose a novel test, called \grasp for testing $H_0$, which works in finite sample settings, no matter the features (distribution-free). We also propose model-X \grasp designed for model-X settings where the joint distribution of the features vector is known. Model-X \grasp uses this distributional information to achieve better power. We evaluate the performance of our tests through extensive numerical experiments.