Private Regression via Data-Dependent Sufficient Statistic Perturbation

Sufficient statistic perturbation (SSP) is a widely used method for differentially private linear regression. SSP adopts a data-independent approach where privacy noise from a simple distribution is added to sufficient statistics. However, sufficient statistics can often be expressed as linear queries and better approximated by data-dependent mechanisms. In this paper we introduce data-dependent SSP for linear regression based on post-processing privately released marginals, and find that it outperforms state-of-the-art data-independent SSP. We extend this result to logistic regression by developing an approximate objective that can be expressed in terms of sufficient statistics, resulting in a novel and highly competitive SSP approach for logistic regression. We also make a connection to synthetic data for machine learning: for models with sufficient statistics, training on synthetic data corresponds to data-dependent SSP, with the overall utility determined by how well the mechanism answers these linear queries.

Joint Selection: Adaptively Incorporating Public Information for Private Synthetic Data

Mechanisms for generating differentially private synthetic data based on marginals and graphical models have been successful in a wide range of settings. However, one limitation of these methods is their inability to incorporate public data. Initializing a data generating model by pre-training on public data has shown to improve the quality of synthetic data, but this technique is not applicable when model structure is not determined a priori. We develop the mechanism jam-pgm, which expands the adaptive measurements framework to jointly select between measuring public data and private data. This technique allows for public data to be included in a graphical-model-based mechanism. We show that jam-pgm is able to outperform both publicly assisted and non publicly assisted synthetic data generation mechanisms even when the public data distribution is biased.

Human in-the-Loop Estimation of Cluster Count in Datasets via Similarity-Driven Nested Importance Sampling

Identifying the number of clusters serves as a preliminary goal for many data analysis tasks. A common approach to this problem is to vary the number of clusters in a clustering algorithm (e.g., 'k' in $k$-means) and pick the value that best explains the data. However, the count estimates can be unreliable especially when the image similarity is poor. Human feedback on the pairwise similarity can be used to improve the clustering, but existing approaches do not guarantee accurate count estimates. We propose an approach to produce estimates of the cluster counts in a large dataset given an approximate pairwise similarity. Our framework samples edges guided by the pairwise similarity, and we collect human feedback to construct a statistical estimate of the cluster count. On the technical front we have developed a nested importance sampling approach that yields (asymptotically) unbiased estimates of the cluster count with confidence intervals which can guide human effort. Compared to naive sampling, our similarity-driven sampling produces more accurate estimates of counts and tighter confidence intervals. We evaluate our method on a benchmark of six fine-grained image classification datasets achieving low error rates on the estimated number of clusters with significantly less human labeling effort compared to baselines and alternative active clustering approaches.

DISCount: Counting in Large Image Collections with Detector-Based Importance Sampling

Many modern applications use computer vision to detect and count objects in massive image collections. However, when the detection task is very difficult or in the presence of domain shifts, the counts may be inaccurate even with significant investments in training data and model development. We propose DISCount -- a detector-based importance sampling framework for counting in large image collections that integrates an imperfect detector with human-in-the-loop screening to produce unbiased estimates of counts. We propose techniques for solving counting problems over multiple spatial or temporal regions using a small number of screened samples and estimate confidence intervals. This enables end-users to stop screening when estimates are sufficiently accurate, which is often the goal in a scientific study. On the technical side we develop variance reduction techniques based on control variates and prove the (conditional) unbiasedness of the estimators. DISCount leads to a 9-12x reduction in the labeling costs over naive screening for tasks we consider, such as counting birds in radar imagery or estimating damaged buildings in satellite imagery, and also surpasses alternative covariate-based screening approaches in efficiency.

Kernel Interpolation with Sparse Grids

Structured kernel interpolation (SKI) accelerates Gaussian process (GP) inference by interpolating the kernel covariance function using a dense grid of inducing points, whose corresponding kernel matrix is highly structured and thus amenable to fast linear algebra. Unfortunately, SKI scales poorly in the dimension of the input points, since the dense grid size grows exponentially with the dimension. To mitigate this issue, we propose the use of sparse grids within the SKI framework. These grids enable accurate interpolation, but with a number of points growing more slowly with dimension. We contribute a novel nearly linear time matrix-vector multiplication algorithm for the sparse grid kernel matrix. Next, we describe how sparse grids can be combined with an efficient interpolation scheme based on simplices. With these changes, we demonstrate that SKI can be scaled to higher dimensions while maintaining accuracy.

Sample Average Approximation for Black-Box VI

We present a novel approach for black-box VI that bypasses the difficulties of stochastic gradient ascent, including the task of selecting step-sizes. Our approach involves using a sequence of sample average approximation (SAA) problems. SAA approximates the solution of stochastic optimization problems by transforming them into deterministic ones. We use quasi-Newton methods and line search to solve each deterministic optimization problem and present a heuristic policy to automate hyperparameter selection. Our experiments show that our method simplifies the VI problem and achieves faster performance than existing methods.

U-Statistics for Importance-Weighted Variational Inference

We propose the use of U-statistics to reduce variance for gradient estimation in importance-weighted variational inference. The key observation is that, given a base gradient estimator that requires $m > 1$ samples and a total of $n > m$ samples to be used for estimation, lower variance is achieved by averaging the base estimator on overlapping batches of size $m$ than disjoint batches, as currently done. We use classical U-statistic theory to analyze the variance reduction, and propose novel approximations with theoretical guarantees to ensure computational efficiency. We find empirically that U-statistic variance reduction can lead to modest to significant improvements in inference performance on a range of models, with little computational cost.

Automatically Marginalized MCMC in Probabilistic Programming

Hamiltonian Monte Carlo (HMC) is a powerful algorithm to sample latent variables from Bayesian models. The advent of probabilistic programming languages (PPLs) frees users from writing inference algorithms and lets users focus on modeling. However, many models are difficult for HMC to solve directly, which often require tricks like model reparameterization. We are motivated by the fact that many of those models could be simplified by marginalization. We propose to use automatic marginalization as part of the sampling process using HMC in a graphical model extracted from a PPL, which substantially improves sampling from real-world hierarchical models.

Variational Marginal Particle Filters

Variational inference for state space models (SSMs) is known to be hard in general. Recent works focus on deriving variational objectives for SSMs from unbiased sequential Monte Carlo estimators. We reveal that the marginal particle filter is obtained from sequential Monte Carlo by applying Rao-Blackwellization operations, which sacrifices the trajectory information for reduced variance and differentiability. We propose the variational marginal particle filter (VMPF), which is a differentiable and reparameterizable variational filtering objective for SSMs based on an unbiased estimator. We find that VMPF with biased gradients gives tighter bounds than previous objectives, and the unbiased reparameterization gradients are sometimes beneficial.

Relaxed Marginal Consistency for Differentially Private Query Answering

Many differentially private algorithms for answering database queries involve a step that reconstructs a discrete data distribution from noisy measurements. This provides consistent query answers and reduces error, but often requires space that grows exponentially with dimension. Private-PGM is a recent approach that uses graphical models to represent the data distribution, with complexity proportional to that of exact marginal inference in a graphical model with structure determined by the co-occurrence of variables in the noisy measurements. Private-PGM is highly scalable for sparse measurements, but may fail to run in high dimensions with dense measurements. We overcome the main scalability limitation of Private-PGM through a principled approach that relaxes consistency constraints in the estimation objective. Our new approach works with many existing private query answering algorithms and improves scalability or accuracy with no privacy cost.