Privacy-preserving data release leveraging optimal transport and particle gradient descent

We present a novel approach for differentially private data synthesis of protected tabular datasets, a relevant task in highly sensitive domains such as healthcare and government. Current state-of-the-art methods predominantly use marginal-based approaches, where a dataset is generated from private estimates of the marginals. In this paper, we introduce PrivPGD, a new generation method for marginal-based private data synthesis, leveraging tools from optimal transport and particle gradient descent. Our algorithm outperforms existing methods on a large range of datasets while being highly scalable and offering the flexibility to incorporate additional domain-specific constraints.

Hidden yet quantifiable: A lower bound for confounding strength using randomized trials

In the era of fast-paced precision medicine, observational studies play a major role in properly evaluating new treatments in clinical practice. Yet, unobserved confounding can significantly compromise causal conclusions drawn from non-randomized data. We propose a novel strategy that leverages randomized trials to quantify unobserved confounding. First, we design a statistical test to detect unobserved confounding with strength above a given threshold. Then, we use the test to estimate an asymptotically valid lower bound on the unobserved confounding strength. We evaluate the power and validity of our statistical test on several synthetic and semi-synthetic datasets. Further, we show how our lower bound can correctly identify the absence and presence of unobserved confounding in a real-world setting.

Strong inductive biases provably prevent harmless interpolation

Classical wisdom suggests that estimators should avoid fitting noise to achieve good generalization. In contrast, modern overparameterized models can yield small test error despite interpolating noise -- a phenomenon often called "benign overfitting" or "harmless interpolation". This paper argues that the degree to which interpolation is harmless hinges upon the strength of an estimator's inductive bias, i.e., how heavily the estimator favors solutions with a certain structure: while strong inductive biases prevent harmless interpolation, weak inductive biases can even require fitting noise to generalize well. Our main theoretical result establishes tight non-asymptotic bounds for high-dimensional kernel regression that reflect this phenomenon for convolutional kernels, where the filter size regulates the strength of the inductive bias. We further provide empirical evidence of the same behavior for deep neural networks with varying filter sizes and rotational invariance.

Tight bounds for maximum $\ell_1$-margin classifiers

Popular iterative algorithms such as boosting methods and coordinate descent on linear models converge to the maximum $\ell_1$-margin classifier, a.k.a. sparse hard-margin SVM, in high dimensional regimes where the data is linearly separable. Previous works consistently show that many estimators relying on the $\ell_1$-norm achieve improved statistical rates for hard sparse ground truths. We show that surprisingly, this adaptivity does not apply to the maximum $\ell_1$-margin classifier for a standard discriminative setting. In particular, for the noiseless setting, we prove tight upper and lower bounds for the prediction error that match existing rates of order $\frac{\|\wgt\|_1^{2/3}}{n^{1/3}}$ for general ground truths. To complete the picture, we show that when interpolating noisy observations, the error vanishes at a rate of order $\frac{1}{\sqrt{\log(d/n)}}$. We are therefore first to show benign overfitting for the maximum $\ell_1$-margin classifier.

Fast rates for noisy interpolation require rethinking the effects of inductive bias

Good generalization performance on high-dimensional data crucially hinges on a simple structure of the ground truth and a corresponding strong inductive bias of the estimator. Even though this intuition is valid for regularized models, in this paper we caution against a strong inductive bias for interpolation in the presence of noise: Our results suggest that, while a stronger inductive bias encourages a simpler structure that is more aligned with the ground truth, it also increases the detrimental effect of noise. Specifically, for both linear regression and classification with a sparse ground truth, we prove that minimum $\ell_p$-norm and maximum $\ell_p$-margin interpolators achieve fast polynomial rates up to order $1/n$ for $p > 1$ compared to a logarithmic rate for $p = 1$. Finally, we provide experimental evidence that this trade-off may also play a crucial role in understanding non-linear interpolating models used in practice.

Tight bounds for minimum l1-norm interpolation of noisy data

We provide matching upper and lower bounds of order $\sigma^2/\log(d/n)$ for the prediction error of the minimum $\ell_1$-norm interpolator, a.k.a. basis pursuit. Our result is tight up to negligible terms when $d \gg n$, and is the first to imply asymptotic consistency of noisy minimum-norm interpolation for isotropic features and sparse ground truths. Our work complements the literature on "benign overfitting" for minimum $\ell_2$-norm interpolation, where asymptotic consistency can be achieved only when the features are effectively low-dimensional.

Interpolation can hurt robust generalization even when there is no noise

Numerous recent works show that overparameterization implicitly reduces variance for min-norm interpolators and max-margin classifiers. These findings suggest that ridge regularization has vanishing benefits in high dimensions. We challenge this narrative by showing that, even in the absence of noise, avoiding interpolation through ridge regularization can significantly improve generalization. We prove this phenomenon for the robust risk of both linear regression and classification and hence provide the first theoretical result on robust overfitting.

How rotational invariance of common kernels prevents generalization in high dimensions

Kernel ridge regression is well-known to achieve minimax optimal rates in low-dimensional settings. However, its behavior in high dimensions is much less understood. Recent work establishes consistency for kernel regression under certain assumptions on the ground truth function and the distribution of the input data. In this paper, we show that the rotational invariance property of commonly studied kernels (such as RBF, inner product kernels and fully-connected NTK of any depth) induces a bias towards low-degree polynomials in high dimensions. Our result implies a lower bound on the generalization error for a wide range of distributions and various choices of the scaling for kernels with different eigenvalue decays. This lower bound suggests that general consistency results for kernel ridge regression in high dimensions require a more refined analysis that depends on the structure of the kernel beyond its eigenvalue decay.

Efficient Smoothing of Dilated Convolutions for Image Segmentation

Dilated Convolutions have been shown to be highly useful for the task of image segmentation. By introducing gaps into convolutional filters, they enable the use of larger receptive fields without increasing the original kernel size. Even though this allows for the inexpensive capturing of features at different scales, the structure of the dilated convolutional filter leads to a loss of information. We hypothesise that inexpensive modifications to Dilated Convolutional Neural Networks, such as additional averaging layers, could overcome this limitation. In this project we test this hypothesis by evaluating the effect of these modifications for a state-of-the art image segmentation system and compare them to existing approaches with the same objective. Our experiments show that our proposed methods improve the performance of dilated convolutions for image segmentation. Crucially, our modifications achieve these results at a much lower computational cost than previous smoothing approaches.