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We investigate the contraction properties of locally differentially private mechanisms. More specifically, we derive tight upper bounds on the divergence between $PK$ and $QK$ output distributions of an $\epsilon$-LDP mechanism $K$ in terms of a divergence between the corresponding input distributions $P$ and $Q$, respectively. Our first main technical result presents a sharp upper bound on the $\chi^2$-divergence $\chi^2(PK\|QK)$ in terms of $\chi^2(P\|Q)$ and $\epsilon$. We also show that the same result holds for a large family of divergences, including KL-divergence and squared Hellinger distance. The second main technical result gives an upper bound on $\chi^2(PK\|QK)$ in terms of total variation distance $TV(P, Q)$ and $\epsilon$. We then utilize these bounds to establish locally private versions of the Cram\'er-Rao bound, Le Cam's, Assouad's, and the mutual information methods, which are powerful tools for bounding minimax estimation risks. These results are shown to lead to better privacy analyses than the state-of-the-arts in several statistical problems such as entropy and discrete distribution estimation, non-parametric density estimation, and hypothesis testing.

Differential Privacy (DP) provides tight upper bounds on the capabilities of optimal adversaries, but such adversaries are rarely encountered in practice. Under the hypothesis testing/membership inference interpretation of DP, we examine the Gaussian mechanism and relax the usual assumption of a Neyman-Pearson-Optimal (NPO) adversary to a Generalized Likelihood Test (GLRT) adversary. This mild relaxation leads to improved privacy guarantees, which we express in the spirit of Gaussian DP and $(\varepsilon, \delta)$-DP, including composition and sub-sampling results. We evaluate our results numerically and find them to match the theoretical upper bounds.

Pufferfish privacy (PP) is a generalization of differential privacy (DP), that offers flexibility in specifying sensitive information and integrates domain knowledge into the privacy definition. Inspired by the illuminating equivalent formulation of DP in terms of mutual information due to Cuff and Yu, this work explores PP through the lens of information theory. We provide an information-theoretic formulation of PP, termed mutual information PP (MI-PP), in terms of the conditional mutual information between the mechanism and the secret, given the public information. We show that MI-PP is implied by the regular PP and characterize conditions under which the reverse implication is also true, recovering the DP information-theoretic equivalence result as a special case. We establish convexity, composability, and post-processing properties for MI-PP mechanisms and derive noise levels for the Gaussian and Laplace mechanisms. The obtained mechanisms are applicable under relaxed assumptions and provide improved noise levels in some regimes, compared to classic, sensitivity-based approaches. Lastly, applications of MI-PP to auditing privacy frameworks, statistical inference tasks, and algorithm stability are explored.

Protecting large language models from privacy leakage is becoming increasingly crucial with their wide adoption in real-world products. Yet applying differential privacy (DP), a canonical notion with provable privacy guarantees for machine learning models, to those models remains challenging due to the trade-off between model utility and privacy loss. Utilizing the fact that sensitive information in language data tends to be sparse, Shi et al. (2021) formalized a DP notion extension called Selective Differential Privacy (SDP) to protect only the sensitive tokens defined by a policy function. However, their algorithm only works for RNN-based models. In this paper, we develop a novel framework, Just Fine-tune Twice (JFT), that achieves SDP for state-of-the-art large transformer-based models. Our method is easy to implement: it first fine-tunes the model with redacted in-domain data, and then fine-tunes it again with the original in-domain data using a private training mechanism. Furthermore, we study the scenario of imperfect implementation of policy functions that misses sensitive tokens and develop systematic methods to handle it. Experiments show that our method achieves strong utility compared to previous baselines. We also analyze the SDP privacy guarantee empirically with the canary insertion attack.

One of the most important features of financial time series data is volatility. There are often structural changes in volatility over time, and an accurate estimation of the volatility of financial time series requires careful identification of change-points. A common approach to modeling the volatility of time series data is the well-known GARCH model. Although the problem of change-point estimation of volatility dynamics derived from the GARCH model has been considered in the literature, these approaches rely on parametric assumptions of the conditional error distribution, which are often violated in financial time series. This may lead to inaccuracies in change-point detection resulting in unreliable GARCH volatility estimates. This paper introduces a novel change-point detection algorithm based on a semiparametric GARCH model. The proposed method retains the structural advantages of the GARCH process while incorporating the flexibility of nonparametric conditional error distribution. The approach utilizes a penalized likelihood derived from a semiparametric GARCH model and an efficient binary segmentation algorithm. The results show that in terms of change-point estimation and detection accuracy, the semiparametric method outperforms the commonly used Quasi-MLE (QMLE) and other variations of GARCH models in wide-ranging scenarios.

To investigate the structure of individual differences in performance on behavioral tasks, Haaf and Rouder (2017) developed a class of hierarchical Bayesian mixed models with varying levels of constraint on the individual effects. The models are then compared via Bayes factors, telling us which model best predicts the observed data. One common criticism of their method is that the observed data are assumed to be drawn from a normal distribution. However, for most cognitive tasks, the primary measure of performance is a response time, the distribution of which is well known to not be normal. In this paper, I investigate the assumption of normality for two datasets in numerical cognition. Specifically, I show that using a shifted lognormal model for the response times does not change the overall pattern of inference. Further, since the model-estimated effects are now on a logarithmic scale, the interpretation of the modeling becomes more difficult, particularly because the estimated effect is now multiplicative rather than additive. As a result, I recommend that even though response times are not normally distributed in general, the simplification afforded by the Haaf and Rouder (2017) approach provides a pragmatic approach to modeling individual differences in behavioral tasks.

When applying differential privacy to sensitive data, a common way of getting improved performance is to use external information such as other sensitive data, public data, or human priors. We propose to use the algorithms with predictions framework -- previously applied largely to improve time complexity or competitive ratios -- as a powerful way of designing and analyzing privacy-preserving methods that can take advantage of such external information to improve utility. For four important tasks -- quantile release, its extension to multiple quantiles, covariance estimation, and data release -- we construct prediction-dependent differentially private methods whose utility scales with natural measures of prediction quality. The analyses enjoy several advantages, including minimal assumptions about the data, natural ways of adding robustness to noisy predictions, and novel "meta" algorithms that can learn predictions from other (potentially sensitive) data. Overall, our results demonstrate how to enable differentially private algorithms to make use of and learn noisy predictions, which holds great promise for improving utility while preserving privacy across a variety of tasks.

We propose a sequential, anytime valid method to test the conditional independence of a response $Y$ and a predictor $X$ given a random vector $Z$. The proposed test is based on e-statistics and test martingales, which generalize likelihood ratios and allow valid inference at arbitrary stopping times. In accordance with the recently introduced model-X setting, our test depends on the availability of the conditional distribution of $X$ given $Z$, or at least a sufficiently sharp approximation thereof. Within this setting, we derive a full characterization of e-statistics for testing conditional independence, investigate growth-rate optimal e-statistics and their power properties, and show that our method yields tests with asymptotic power one in the special case of a logistic regression model. A simulation study is done to demonstrate that the approach is robust with respect to violations of the model-X assumption and competitive in terms of power when compared to established sequential and non-sequential testing methods.

Causal discovery aims to recover causal structures generating the observational data. Despite its success in certain problems, in many real-world scenarios the observed variables are not the target variables of interest, but the imperfect measures of the target variables. Causal discovery under measurement error aims to recover the causal graph among unobserved target variables from observations made with measurement error. We consider a specific formulation of the problem, where the unobserved target variables follow a linear non-Gaussian acyclic model, and the measurement process follows the random measurement error model. Existing methods on this formulation rely on non-scalable over-complete independent component analysis (OICA). In this work, we propose the Transformed Independent Noise (TIN) condition, which checks for independence between a specific linear transformation of some measured variables and certain other measured variables. By leveraging the non-Gaussianity and higher-order statistics of data, TIN is informative about the graph structure among the unobserved target variables. By utilizing TIN, the ordered group decomposition of the causal model is identifiable. In other words, we could achieve what once required OICA to achieve by only conducting independence tests. Experimental results on both synthetic and real-world data demonstrate the effectiveness and reliability of our method.

Decentralized optimization is increasingly popular in machine learning for its scalability and efficiency. Intuitively, it should also provide better privacy guarantees, as nodes only observe the messages sent by their neighbors in the network graph. But formalizing and quantifying this gain is challenging: existing results are typically limited to Local Differential Privacy (LDP) guarantees that overlook the advantages of decentralization. In this work, we introduce pairwise network differential privacy, a relaxation of LDP that captures the fact that the privacy leakage from a node $u$ to a node $v$ may depend on their relative position in the graph. We then analyze the combination of local noise injection with (simple or randomized) gossip averaging protocols on fixed and random communication graphs. We also derive a differentially private decentralized optimization algorithm that alternates between local gradient descent steps and gossip averaging. Our results show that our algorithms amplify privacy guarantees as a function of the distance between nodes in the graph, matching the privacy-utility trade-off of the trusted curator, up to factors that explicitly depend on the graph topology. Finally, we illustrate our privacy gains with experiments on synthetic and real-world datasets.