pepr-hal-data / bio2flow_json /AssessingPerSampleMembershipInferenceVulnerabilitywithoutRetraining.grobid.tei.xml.json
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| "text": "Recent work in the privacy literature shows that sample-targeted membership inference attacks (MIAs) significantly outperform untargeted approaches by a wide margin. Motivated by this observation, we address the following question: can the privacy vulnerability of individual training points be assessed without training shadow models? We show that per-sample exposure to MIA is governed not only by a point's loss, but also by a data-dependent geometric measure. In the linear setting, we derive a closed-form decomposition of individual black-box MIA vulnerability into a population leverage score and a residual loss term, making explicit how sample-dependent geometry translates into privacy exposure. Since the final layer of most modern architectures is linear, we extend this framework to deep networks and propose a surrogate score operating on last-layer representations that requires only a single trained model and no shadow models. Empirical evaluations across diverse datasets and architectures show that our score outperforms loss and gradient-norm baselines at identifying the highest-risk points under state-of-the-art attacks, providing a computationally efficient and theoretically grounded tool for per-sample privacy risk assessment.", | |
| "entities": [] | |
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| "index": 1, | |
| "text": "Modern machine learning models, and deep neural networks in particular, are known to memorize aspects of their training data [Zhang et al., 2017, Carlini et al., 2019]. This memorization induces privacy vulnerabilities that can be exploited by Membership Inference Attacks (MIAs), which aim to determine whether a specific data point was included in the training set or not [Shokri et al., 2017, Carlini et al., 2022a]. A principled defense against membership inference is provided by Differential Privacy (DP) [Dwork, 2006], implemented in deep learning through noise injection in stochastic gradient methods [Abadi et al., 2016]. However, controlling the trade-off between privacy protection and model utility remains challenging. Noise calibration typically relies on worst-case theoretical accounting, paired with empirical privacy auditing via MIAs, and often leads to either overly conservative noise levels or insufficient privacy protection. In this context, designing MIAs for empirical auditing is essential to quantify privacy leakage in non-privately trained models [Yeom et al., 2018] or to validate the practical tightness of DP guarantees in privately trained models [Nasr et al., 2021, Jagielski et al., 2020]. 1 While such auditing is now standard practice [Carlini et al., 2022a, Nasr et al., 2021, Zarifzadeh et al., 2024], relying on aggregate metrics such as average accuracy or AUC is insufficient. These global measures can obscure critical risk heterogeneity, as outliers and rare subgroups are significantly more prone to memorization than typical samples [Carlini et al., 2022a, Feldman andZhang, 2020]. Consequently, a model certified as private on average may still expose specific points to high privacy risks. To address this heterogeneity, recent work is increasingly focusing on per-sample privacy risk assessment, aiming to quantify membership leakage at the level of each data point, separately, rather than in aggregate. State-of-the-art methods for per-sample auditing [Carlini et al., 2022a, Zarifzadeh et al., 2024] predominantly rely on shadow models. These techniques train multiple shadow models on random data splits to characterize each data point's influence on the model's behavior. This enables identification of which points exhibit increased sensitivity to their presence in the training dataset. However, these techniques are computationally prohibitive, especially for large-scale models, as they require retraining the model many times. This leads to our central question stated in the abstract: can the privacy vulnerability of individual training points be assessed without training shadow models?", | |
| "entities": [] | |
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| "index": 2, | |
| "text": "To address this question in a principled way, we first consider the baseline case of Gaussian linear regression. Even in this simple model, the task poses interesting challenges: the fixed-design analysis from classical statistics leads to the leverage score (see below), which quantifies a data point's geometric influence solely through its feature vector, ignoring the response. The fully randomized approach, on the other hand, provides metrics that depend on general problem characteristics (dimensions, covariance operator of the features) but do not account for specific data points (see, e.g., Tan et al. [2022]). To obtain relevant per-sample risk exposure metrics, we propose to consider non-asymptotic results in the randomized Gaussian setting by conditioning on a single data point (i.e., its feature vector and its response).", | |
| "entities": [] | |
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| "index": 3, | |
| "text": "For Gaussian linear regression, we derive a closed-form expression for the expected leaveone-out generalization gap, the signal exploited by state-of-the-art black-box attacks such as LiRA [Carlini et al., 2022a], and show that it decomposes, at first order, into a response-free geometric term, the population leverage score hii , and a response-dependent structural error ε i . Membership inference vulnerability is therefore governed jointly by a point's geometric measure and its structural error. Samples that are both geometrically influential in shaping the model's parameters and poorly fit by the model are at the highest risk of privacy leakage.", | |
| "entities": [ | |
| { | |
| "text": "LiRA", | |
| "type": "Tool", | |
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| "begin": 184, | |
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| "confidence": 1.0 | |
| } | |
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| "index": 4, | |
| "text": "To extend this analysis beyond the linear setting, we approximate the leave-one-out loss gap using two classical estimators: an influence-function estimate and a Newton-step estimate.", | |
| "entities": [] | |
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| "index": 5, | |
| "text": "Restricted to the last layer of a trained network, both admit closed-form expressions in terms of the leverage score of the learned representations, and can be computed from a single trained model at negligible cost. This yields a scalable, theoretically grounded proxy for per-sample privacy risk that tracks the rankings produced by state-of-the-art shadow-model attacks, without any retraining.", | |
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| "index": 6, | |
| "text": "The paper is organized as follows. In Section 3 we derive a closed-form expression for the expected leave-one-out generalization gap of a target point in Gaussian linear regression, showing that the signal driving LiRA-style attacks decomposes, at first order, into a label-free geometric term, the population leverage hii , and a label-dependent structural error ε i . In Section 4, we propose two retraining-free per-sample vulnerability scores for general differentiable models, based, respectively, on influence-function and Newton-step approximations of the leave-one-out loss. Restricted to the last layer, both admit closed-form expressions in terms of the leverage score of the last-layer representation, which can be efficiently computed for deep networks. Finally, Section 5 reports the performance of the proposed vulnerability scores in terms of recovery of the results obtained by state-of-the-art retraining-based MIA methods. In particular, we observe that the proposed metrics consistently outperform commonly used baselines, such as training loss or terminal gradient-norm, across various configurations of dataset/architecture pairs. We start with a quick review of the relevant literature.", | |
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| "index": 7, | |
| "text": "Membership inference aims to determine if a specific sample was used to train a model. Early approaches relied on simple metric-based classifiers, exploiting overfitting signals such as prediction confidence, entropy, or the magnitude of gradients [Shokri et al., 2017, Yeom et al., 2018]. While computationally inexpensive, these methods often struggle to distinguish between a \"vulnerable\" member and a \"hard\" non-member.", | |
| "entities": [] | |
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| "index": 8, | |
| "text": "To address this, current state-of-the-art methods adopt the shadow model paradigm [Shokri et al., 2017]. By training multiple models on different data splits, attacks like LiRA (Likelihood Ratio Attack) [Carlini et al., 2022a] or RMIA (Robust Membership Inference Attack) [Zarifzadeh et al., 2024] essentially perform a hypothesis test on the loss distribution of a target point [Zarifzadeh et al., 2024]. While highly effective, these approaches are computationally prohibitive, often requiring hundreds of training runs to estimate the risk for a single point accurately. Our work seeks to achieve the precision of these hypothesis-test based methods without their computational burden, by substituting retraining with geometric analysis.", | |
| "entities": [ | |
| { | |
| "text": "LiRA", | |
| "type": "Tool", | |
| "attributes": [], | |
| "begin": 172, | |
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| "confidence": 1.0 | |
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| { | |
| "text": "RMIA", | |
| "type": "Tool", | |
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| "begin": 230, | |
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| } | |
| ] | |
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| "index": 9, | |
| "text": "Recent work has begun to explore the connection between influence and privacy. Notably, Feldman and Zhang [2020] and Feldman [2020] link the memorization of a sample to its influence on the learning process, arguing that memorization is necessary for generalization in long-tailed distributions. Other work has used influence functions to craft new attack methods [Cohen andGiryes, 2024, Suri et al., 2024], but these approaches inherit two well-known limitations of influence functions in deep learning: a prohibitive computational cost, and a fragility that often renders the resulting estimates unreliable [Basu et al., 2021].", | |
| "entities": [] | |
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| { | |
| "index": 10, | |
| "text": "Our work addresses these limitations by confining influence estimation to the last layer, where closed-form expressions are available. This reduces the computational overhead and avoids the fragility of full-network influence, while remaining a faithful proxy of per-sample influence as shown in our experiments.", | |
| "entities": [] | |
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| "index": 11, | |
| "text": "Privacy Auditing and Heterogeneity. Differential Privacy (DP) [Dwork, 2006] provides worst-case guarantees for membership privacy. Using algorithms such as DP-SGD [Abadi et al., 2016], one can train Deep Learning models to be private. However, these bounds are often loose and do not capture the empirical reality that privacy risk is non-uniform: some outlier samples are far more exposed than others [Zarifzadeh et al., 2024].", | |
| "entities": [ | |
| { | |
| "text": "DP-SGD", | |
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| "begin": 156, | |
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| "confidence": 1.0 | |
| } | |
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| "text": "Auditing per-sample privacy risk, i.e., estimating the specific (ε, δ) DP parameters for a given sample, remains an open challenge. Existing auditing tools rely heavily on the aforementioned shadow model techniques or randomized smoothing [Lecuyer et al., 2019], which scale poorly. Our work contributes to this domain by proposing an efficient individual privacy-risk metric for identifying vulnerable samples. This approach complements existing auditing frameworks by providing a first-pass, zero-retraining filter for high-risk data points. Removing these vulnerable points from the training set is not an option since, in practice, such deletions often expose new points, an effect described as the privacy onion effect by Carlini et al. [2022b]. Instead, our method can serve as a tool for canary selection [Carlini et al., 2019] or per-sample privacy training [Wang, 2018, Alaggan et al., 2017].", | |
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| "index": 13, | |
| "text": "To rigorously understand the privacy leakage exploited by black-box attacks, we first analyze their behavior within the context of linear models. State-of-the-art MIAs, such as shadow model classifiers [Shokri et al., 2017], LiRA [Carlini et al., 2022a], and RMIA [Zarifzadeh et al., 2024], compute membership scores via classifiers or likelihood ratio tests fitted to per-sample losses. In practice, they estimate the loss distribution of each sample, conditioned on whether it was included in the training set, by training shadow models on multiple data splits.", | |
| "entities": [ | |
| { | |
| "text": "LiRA", | |
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| "begin": 225, | |
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| }, | |
| { | |
| "text": "RMIA", | |
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| "index": 14, | |
| "text": "To model the stochasticity of training on random subsamples, we isolate a fixed target point i and treat the remainder of the dataset as being drawn from an underlying probability distribution. We then analyze the marginal distribution of the target's loss under two scenarios: the IN setting, where the model is trained on the fixed point alongside the samples drawn from the distribution, and the OUT setting, where the model is trained exclusively on the sampled distribution without the target point.", | |
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| "text": "Let Z ⊂ R p × R be the data space, and let z i = (x i , y i ) ∈ Z be a fixed target observation. Throughout our analysis, we condition on this target point, treating it as a deterministic constant. We consider a full training dataset S of size n that contains this fixed target z i . The other n -1 points in S, denoted collectively as S -i = {(X j , Y j )} j̸ =i , are random variables sampled from the underlying data distribution.", | |
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| "text": "Because z i is fixed, the randomness in our estimators arises entirely from these n -1 background observations. We assume their covariates are drawn i.i.d. from a multivariate Gaussian distribution, X j ∼ N (0, Σ), and their responses follow the linear generative model Y j | X j ∼ N (X ⊤ j θ, σ 2 ) where θ ∈ R p is the unknown true parameter and σ 2 the structural noise variance. We consider the scalar response case for simplicity, and the results are extended to multivariate responses in Appendix A.", | |
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| "index": 17, | |
| "text": "Let X ∈ R n×p denote the full design matrix, where the i-th row is the deterministic x ⊤ i and the remaining rows form the random submatrix X -i ∈ R (n-1)×p . Similarly, let Y ∈ R n be the response vector containing y i and the random subvector", | |
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| "index": 18, | |
| "text": "be the ordinary least squares (OLS) estimator trained on the full dataset S, and θ-i be the estimator trained on S -i . We denote e i = y i -x ⊤ i θ the residual for the target point under the full model and e -i = y i -x ⊤ i θ-i the residual for the target point under the leave-one-out model. The corresponding squared residuals are the training loss ℓ i = (y i -x ⊤ i θ) 2 and the leave-one-out loss ℓ -i = (y i -", | |
| "entities": [ | |
| { | |
| "text": "θ", | |
| "type": "Tool", | |
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| "index": 19, | |
| "text": "the empirical leverage score of the target point, which is the i-th diagonal element of H. This score is a random variable due to its dependence on X -i , and quantifies the geometric influence of the point on its own fitted value [Belsley et al., 1980] illustrated by the leave-one-out identity e i = (1 -h ii ) e -i [Allen, 1974].", | |
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| "index": 20, | |
| "text": "We define two deterministic quantities characterizing the target point: its population leverage score hii = x ⊤ i Σ -1 x i , which corresponds to the squared Mahalanobis distance to the feature distribution, and its structural error ε i = y i -x ⊤ i θ. Our goal is to characterize the distribution of the losses ℓ i and ℓ -i and see how these distributions relate to the target point's MIA vulnerability.", | |
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| { | |
| "index": 21, | |
| "text": "We begin by analyzing the leave-one-out Gram matrix G -i = X ⊤ -i X -i , which follows a Wishart distribution W p (Σ, n-1) [Muirhead, 1982]. Applying the Sherman-Morrison-Woodbury identity allows us to express both the empirical leverage score h ii and the leave-one-out residual e -i as functions of G -i , thereby isolating the geometric randomness of the design from the structural noise. The following lemma makes this decomposition explicit. Lemma 3.1 (Joint Law of Leverage Scores and Residuals). Let G -i = X ⊤ -i X -i denote the leave-one-out Gram matrix, which follows a Wishart distribution W p (Σ, n -1) [Muirhead, 1982]. We define", | |
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| { | |
| "index": 22, | |
| "text": "Then, the empirical leverage score h ii and the leave-one-out residual e -i satisfy the identities:", | |
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| "index": 23, | |
| "text": "Moreover, conditioned on (x i , y i ), V ∼ χ 2 n-p and Z ∼ N (0, 1), with Z ⊥ ⊥ V . The independence of Z and V in Lemma 3.1 is an important property: given X -i , the standardized prediction error Z is N (0, 1) regardless of the design, hence independent of any function of it, including V . Combining the leave-one-out identity e i = (1 -h ii )e -i [Allen, 1974], which gives ℓ i = (1 -h ii ) 2 ℓ -i , with Lemma 3.1, we obtain a full characterization of the joint distribution of the losses at point i in Proposition 3.2. Proposition 3.2 (Per-sample Loss Distribution). Under the Gaussian linear regression model, the leave-one-out loss ℓ -i and the training loss ℓ i at a point (x i , y i ) satisfy", | |
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| "index": 24, | |
| "text": "where V and Z are defined in Lemma 3.1, satisfying V ∼ χ 2 n-p and Z ∼ N (0, 1), with Z ⊥ ⊥ V .", | |
| "entities": [] | |
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| "index": 25, | |
| "text": "Using Proposition 3.2 and the Neyman-Pearson Lemma [Neyman and Pearson, 1933], we derive the optimal black-box attack in closed form. In the large-sample regime n -p ≫ 1, the variable V ∼ χ 2 n-p concentrates sharply around its mean, so we replace it by the constant n -p and Proposition 3.2 reduces to a deterministic rescaling in distribution:", | |
| "entities": [] | |
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| { | |
| "index": 26, | |
| "text": "The likelihood ratio is then monotone in the observed loss, so the Neyman-Pearson optimal test thresholds ℓ i,obs directly. Corollary 3.3 (Optimal MIA Trade-off). Let F i,OUT denote the CDF of ℓ -i under H OUT . For any target false positive rate α i ∈ (0, 1), the false negative rate of the Neyman-Pearson optimal attack at point i is", | |
| "entities": [] | |
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| "index": 27, | |
| "text": "The closed-form expression itself follows from a standard Neyman-Pearson argument; the corollary's value lies in clarifying which features of the problem dictate vulnerability. The full trade-off function α i → β i (α i ) depends on the data only through ( hii , ε i ) and the global parameters (n, p, σ 2 ). In particular, the entire design matrix X -i enters only through the scalar population leverage hii . The pair ( hii , ε i ) thus acts as a per-point sufficient statistic for membership-inference vulnerability.", | |
| "entities": [] | |
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| "index": 28, | |
| "text": "0 50 100 Population leverage hii 0 50 100 150 200 Squared error ε 2 i 1 0 5 0 1 0 0 0.0 0.5 1.0 FPR (α i ) 0.0 0.2 0.4 0.6 0.8 1.0", | |
| "entities": [] | |
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| "index": 29, | |
| "text": "Alongside the trade-off curve, the expected generalization gap E[∆ℓ i ] has been identified as the fundamental theoretical bound on the power of loss-based MIAs [Yeom et al., 2018]. In our model, it admits a theoretical closed-form and a tractable asymptotic expression that make the dependence on ( hii , ε i ) explicit. Proposition 3.4 (Expected Generalization Gap). For n -p > 2, the expected difference between the leave-one-out and training loss for a fixed point (x i , y i ) is exactly:", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 30, | |
| "text": "The asymptotic expansion of this gap in powers of (n -p) -1 is:", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 31, | |
| "text": "Proposition 3.4 establishes that, to first order, the generalization gap is governed by the interaction between a point's structural error ε 2 i and its leverage hii . Intuitively, the points exhibiting the largest generalization gap are those that are simultaneously outliers in feature space (high hii ) and difficult to predict (high ε 2 i ). Moreover, the gap vanishes at a rate of 1/n, which confirms that the privacy risk associated with any individual point diminishes as the size of the dataset grows. i and the leverage score hii . The middle and right panels present the trade-off curves of the optimal MIA for varying hii and ε 2 i , respectively, confirming that the attack's statistical power increases monotonically in both parameters.", | |
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| "index": 32, | |
| "text": "Proofs for all the results presented in this section are deferred to Appendix A.", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 33, | |
| "text": "In the linear regression case, we could derive the exact value of the loss gap ℓ -i -ℓ i without retraining. This is because the OLS estimator has a closed-form solution and the loss is quadratic. For more general models, such as generalized linear models or deep neural networks, we do not have closed-form solutions. In the following, we discuss how common techniques from influence functions and Newton steps can be used to estimate the loss gap without retraining, and how they relate to the previous analysis in both the linear regression and the logistic regression case.", | |
| "entities": [] | |
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| "index": 34, | |
| "text": "Influence Function. Influence functions [Hampel, 1974, Koh andLiang, 2017] provide a way to estimate the effect of removing a training point z i on the model parameters θ without retraining. Formally, the influence of z i is defined as:", | |
| "entities": [] | |
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| { | |
| "index": 35, | |
| "text": "Using a first-order Taylor expansion, this defines our first estimator for the loss gap, which we call the Influence Function estimator :", | |
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| }, | |
| { | |
| "index": 36, | |
| "text": "Newton Step. Alternatively, one can approximate the loss gap by taking a single Newton step on the leave-one-out objective starting from θ. Using a first-order Taylor expansion on the loss, this defines our second estimator, which we call the Newton Step estimator :", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 37, | |
| "text": "Notably, when the model has fully converged and minimizes the empirical risk (i.e.,", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 38, | |
| "text": "Under this convergence assumption, the Newton Step estimator simplifies to:", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 39, | |
| "text": "a form also known as the Approximate Leave-One-Out (ALO) estimator in the statistics literature and used in approximate unlearning methods [Guo et al., 2020, Sekhari et al., 2021].", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 40, | |
| "text": "Remarkably, ∆ℓ IF i and ∆ℓ", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 41, | |
| "text": "coincide up to a rank-one correction of the Hessian, despite being derived from distinct principles. In the following section, we derive closed-form expressions for these two estimators in the context of linear and logistic regression, and show how they can serve as scalable proxies for the loss gap to evaluate vulnerability to membership inference attacks.", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 42, | |
| "text": "We now derive the exact expressions of ∆ℓ IF i and ∆ℓ NS i for linear and logistic regression. These are not further approximations: they are the direct, closed-form instantiations of Equations ( 7) and ( 8) for the respective model classes.", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 43, | |
| "text": "In linear regression, the OLS estimator θ = (X ⊤ X) -1 X ⊤ Y is available in closed form and the loss is quadratic, so both estimators admit exact closed-form expressions. Letting e i denote the residual and h ii the leverage score, the two estimators yield:", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 44, | |
| "text": "The Newton Step estimator introduces a leverage correction (1 -h ii ) -1 , bringing it closer to the exact leave-one-out formula:", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 45, | |
| "text": "Logistic Regression. In binary logistic regression, with pi = σ( θ⊤ x i ), the two estimators yield:", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 46, | |
| "text": "where", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 47, | |
| "text": "is the weighted leverage score, with W = diag(w j ) j=1,...,n . This is the logistic regression analogue of the OLS leverage score [Pregibon, 1981]: in both cases, h ii captures the local sensitivity of the fitted response to the target, ∂ ŷi /∂y i in linear regression and ∂ pi /∂y i in logistic regression. As in Equation ( 9), the Newton Step estimator applies a leverage correction relative to the Influence Function estimator.", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 48, | |
| "text": "We generalize both of these results for the multivariate case in Appendix B.6. Motivated by these findings, as well as our theoretical analysis in Proposition 3.4, we propose to use these estimates as proxies for vulnerability assessment in deep neural networks. In practice, we consider only the linear last layer of the network, and apply the estimators defined in equations ( 9) and (10) (or their multivariate counterparts) depending on the loss used for the training (cross-entropy for classification, quadratic for regression).", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 49, | |
| "text": "In both linear and logistic regression, the two estimators differ by a single multiplicative factor:", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 50, | |
| "text": "At low leverage they therefore agree to leading order. The correction matters only as h ii → 1, a regime reached, for instance, when x i is orthogonal to every other row of X, in which case h ii = 1 exactly. We have: Proofs for all the results presented in this section are deferred to Appendix B.", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 51, | |
| "text": "We evaluate our two surrogates ∆ℓ IF i and ∆ℓ", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 52, | |
| "text": "for per-sample membership-inference vulnerability across a diverse set of modalities, architectures and learning regimes. Datasets and architectures. We cover both classification and regression on tabular, vision and text data: (i) CIFAR-10 with a small CNN; (ii) CIFAR-100 with ResNet-20; (iii)", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 53, | |
| "text": "BloodMNIST with a DINOv2 ViT-small classifier [Oquab et al., 2023], trained with a linear probe on top of frozen features; (iv) SST-2 with a BERT [Devlin et al., 2019] classifier in two regimes: full fine-tuning of bert_uncased_L-2_H-128_A-2 and a linear probe on top of a frozen bert-base-uncased; (v) California Housing with both a closed-form ridge regression and a deep MLP; (vi) UCI Energy with a tabular MLP. For each task we use the standard train split, randomly partitioned in half between members and non-members (N train = N/2) with a per-model seed, following the LiRA shadow-model protocol [Carlini et al., 2022a].", | |
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| { | |
| "index": 54, | |
| "text": "Shadow and target models. For every dataset/architecture pair we train 200 shadow (reference) models with i.i.d. random member/non-member splits, plus 16 independent target models used to compute the per-sample vulnerability surrogates. Optimizers, learning rates, batch sizes, weight decay and number of epochs follow standard recipes for each architecture (AdamW for vision and text, SGD-momentum for ResNet-20, Adam for the regression MLPs); the exact hyper-parameters are listed in Appendix C.1 and in the release scripts.", | |
| "entities": [ | |
| { | |
| "text": "AdamW", | |
| "type": "Tool", | |
| "attributes": [], | |
| "begin": 359, | |
| "end": 364, | |
| "confidence": 1.0 | |
| }, | |
| { | |
| "text": "SGD-momentum", | |
| "type": "Tool", | |
| "attributes": [], | |
| "begin": 386, | |
| "end": 398, | |
| "confidence": 1.0 | |
| }, | |
| { | |
| "text": "Adam", | |
| "type": "Tool", | |
| "attributes": [], | |
| "begin": 414, | |
| "end": 418, | |
| "confidence": 1.0 | |
| } | |
| ] | |
| }, | |
| { | |
| "index": 55, | |
| "text": "Attacks and surrogates. We use LiRA [Carlini et al., 2022a] as the reference per-sample attack, since it achieved the highest Attack Success Rate (ASR) in our experiments; we compute its per-sample ASR by averaging over the 200 reference models. We compare four retraining-free surrogates of vulnerability, all computed from a single target model and without shadow models: the training loss ℓ i , the gradient-norm ∥∇ℓ i ∥, the influence-function estimate ∆ℓ IF i and the Newton-step estimate ∆ℓ NS i . For deep networks, both variants are restricted to the last layer, which keeps their cost essentially negligible. In Appendix D, we extend this analysis to include comparisons with both the RMIA attack [Zarifzadeh et al., 2024] and the original shadow-model attack [Shokri", | |
| "entities": [ | |
| { | |
| "text": "LiRA", | |
| "type": "Tool", | |
| "attributes": [], | |
| "begin": 31, | |
| "end": 35, | |
| "confidence": 1.0 | |
| } | |
| ] | |
| }, | |
| { | |
| "index": 56, | |
| "text": "We compute it using the inverse Hessian Products [Pearlmutter, 1994] and the Conjugate Gradient algorithm [Hestenes and Stiefel, 1952]. More details can be found in Appendix D.1.", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 57, | |
| "text": "Metrics. Following standard per-sample auditing practice [Shokri et al., 2017, Carlini et al., 2022a, Zarifzadeh et al., 2024], we evaluate surrogates based on top-k% recall: the fraction of the most vulnerable points (highest ASR) correctly identified within the surrogate's top-k% rankings. The Spearman rank correlation is provided in Appendix D.2 for reference, but we find it less informative in our context as it is dominated by the high density of points near the 50% ASR (random guess), where surrogates provide little discriminative signal.", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 58, | |
| "text": "All experiments were run on an Nvidia-V100 GPU cluster; the full pipeline, training, statistics, attacks, scores and figures, is reproducible from the released code through the per-dataset launch scripts, and will be released publicly upon acceptance of the paper.", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 59, | |
| "text": "We present the recall @5% of the true LiRA top-1% (measured by Average Success Rate) in Table 1. Figure 2 illustrates the performance of these surrogates for multiple recall percentages. We also present more extensive results for different percentages of recall in Appendix D, which show similar trends.", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 60, | |
| "text": "Across all eight settings, our two surrogates ∆ℓ IF i and ∆ℓ NS i , consistently recover the samples LiRA flags as most vulnerable, well above the random baseline, often by one to two orders of magnitude.", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 61, | |
| "text": "They also surpass the loss baseline on seven out of eight settings, with recall@1% improving by more than 20% on some datasets. The sole exception is the BERT linear probe, where the loss baseline marginally outperforms our method; we hypothesize that this setting, characterized by simple training dynamics and few vulnerable points, is one in which the raw loss already provides a strong signal. We further observe high variance in recall@1% across all surrogates on the UCI Energy dataset, which we attribute to the mismatch between the small dataset size (N = 768) and the high-dimensional tabular MLP (p = 512). At this scale, recall@1% is evaluated over only 7 points, making the metric inherently noisy.", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 62, | |
| "text": "Interestingly, our leverage-based surrogates have greater difficulty recovering the most vulnerable points for RMIA and Shokri classifier-based attacks (Appendix D.3). For RMIA, our method still outperforms the loss baseline in all cases, although by a narrower margin. For the classifier-based approach, this is likely because the attack learns a non-parametric decision boundary that is not well captured by the leverage-based scores, which are derived from a local quadratic approximation of the loss landscape. Finally, the full-network influence function proved worse in both speed and recall compared to our last-layer approach (see Appendix D.1).", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 63, | |
| "text": "Overall, these scores yield a shadow-free, single-model proxy that approaches LiRA's perpoint ASR ranking at a fraction of its computational cost. Scoring all points with our leverage-based surrogates on a single target model is 1,000× to 10,000× faster than training the 200 shadow models required by LiRA, excluding the target-model training cost, which is shared by both approaches. Detailed timings for each surrogate and attack are reported in the Appendix C.2.", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 64, | |
| "text": "We proposed a retraining-free framework for per-sample membership inference vulnerability assessment. In Gaussian linear regression, we derived a closed-form decomposition of the expected leave-one-out gap into a response-free population leverage term and a responsedependent structural error. We extended this view to deep networks via influence-function and Newton-step surrogates restricted to the last layer, both computable from a single trained model. Across eight dataset/architecture pairs, our surrogates outperform loss and gradient-norm baselines at recovering LiRA's most vulnerable points, at a fraction of the cost.", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 65, | |
| "text": "Limitations remain: the theory is exact only in the linear case, the last-layer restriction overlooks earlier feature learning, and the surrogates track LiRA more closely than RMIA or classifier-based attacks. Natural extensions include layer-wise or kernel-based leverage scores, integration into DP-SGD training as a canary-selection or per-sample weighting tool, and a finer characterization of which memorization signals escape local quadratic approximations.", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 66, | |
| "text": "Table of Contents A Omitted Proofs from Section 3 15 A.1 Proof of Lemma 3.1 (Joint Law of Leverage Scores and Residuals) . . . . . . 15 A.2 Proof of Proposition 3.2 (Per-sample Loss Distribution) . . . . . . . . . . . . 16 A.3 Proof of Corollary 3.3 (Optimal MIA Trade-off) . . . . . . . . . . . . . . . . . 17 A.4 Proof of Proposition 3.4 (Expected Generalization Gap) . . . . . . . . . . . . 17 A.5 Generalization to the Multivariate Case . . . . . . . . . . . . . . . . . . . . . 18 B Omitted Proofs from Section 4 19 B.1 Proof of the Influence Function Estimate (Eq. (7)) . . . . . . . . . . . . . . . 19 B.2 Proof of the Newton Step Estimate (Eq. (8)) . . . . . . . . . . . . . . . . . . 20 B.3 Results in the Linear Regression (Eq. (9)) . . . . . . . . . . . . . . . . . . . . 20 B.4 Results in the Logistic Regression (Eq. (10)) . . . . . . . . . . . . . . . . . . . 21 B.5 Proof that ∂ pi /∂y i = h ii . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 B.6 Generalization to the Multivariate Case . . . . . . . . . . . . . . . . . . . . . 22 C Details on the Experimental Setup 25 C.1 Model Architectures and Training Hyperparameters . . . . . . . . . . . . . . 25 C.2 Time and Compute Resources . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 D Additional Experimental Results 26 D.1 Comparison with full-network Influence functions . . . . . . . . . . . . . . . . 26 D.2 Top-k recall and Spearman correlation for LiRA attack . . . . . . . . . . . . . 27 D.3 Top-k recall and Spearman correlation for RMIA attack and Classifier Shokri's attack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 D.4 Additional visualizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 67, | |
| "text": "This appendix provides the rigorous proofs for the theoretical results characterizing the finite-sample generalization gap. Throughout these proofs, we maintain the conditioning on the fixed target observation (x i , y i ), treating x i and y i as deterministic constants. The randomness originates entirely from X -i and Y -i .", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 68, | |
| "text": "Proof. We first derive the distribution of the empirical leverage score", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 69, | |
| "text": "Applying the Sherman-Morrison formula to invert the rank-one update:", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 70, | |
| "text": "Multiplying both sides by x ⊤ i on the left and x i on the right yields:", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 71, | |
| "text": "Because the rows of X -i are i.i.d. N (0, Σ), G -i follows a Wishart distribution W p (Σ, n -1).", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 72, | |
| "text": "A standard property of the Inverse-Wishart distribution states that for any fixed vector x i , the quadratic form scales as an inverse chi-squared random variable. Specifically:", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 73, | |
| "text": "Let this random variable be denoted as V . Recalling that hii = x ⊤ i Σ -1 x i , we have V = hii / hii , which implies hii = hii /V . Substituting this into our expression for h ii gives:", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 74, | |
| "text": "which concludes the first part of the lemma.", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 75, | |
| "text": "We now derive the distribution of the leave-one-out residual e -i . The leave-one-out OLS estimator is", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 76, | |
| "text": "Crucially, because the conditional distribution of Z given X -i is standard normal and does not depend on X -i , Z is unconditionally independent of X -i . Since V is entirely a function of X -i , it strictly follows that Z ⊥ ⊥ V . Thus, we can write the residual as e", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 77, | |
| "text": "Finally, by standard regression update rules, the training residual is e i = e -i /(1 + hii ). Squaring this yields the training loss:", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 78, | |
| "text": "Proof. The results follow directly from Lemma 3.1 by writing that the training loss is", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 79, | |
| "text": "and the leave-one-out loss is ℓ -i = e 2 -i .", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 80, | |
| "text": "Proof. Under the non-member hypothesis H OUT , the optimal decision rule rejects the null hypothesis if the observed loss is unusually small: ℓ obs,i < γ α . For a fixed False Positive Rate α, the threshold is strictly determined by the non-member cumulative distribution function: α = P(ℓ obs,i < γ α | H OUT ), yielding γ α = F -1 i,OUT (α). Under the member hypothesis H IN , the loss is scaled by c 2 i . The True Positive Rate is the probability that this scaled loss falls below the same threshold:", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 81, | |
| "text": "We finish the proof by writing", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 82, | |
| "text": "Proof. The generalization gap is", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 83, | |
| "text": ", we can factor out e 2 -i :", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 84, | |
| "text": "To compute the expectation E[∆ℓ i | x i , y i ], we leverage the independence of Z and V established in Lemma 3.1. We first take the inner expectation with respect to Z:", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 85, | |
| "text": "Substituting this inner expectation into the gap yields the exact finite-sample formula:", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 86, | |
| "text": "To derive the asymptotic expansion, let k = n -p and define x = 1/V . We expand the rational term as x → 0 using the Maclaurin series for (1 + hii x) -2 :", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 87, | |
| "text": "We now apply the exact inverse moments of the chi-squared distribution V ∼ χ 2 k . For k > 4:", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 88, | |
| "text": "Substituting these moments into our expansion:", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 89, | |
| "text": "Finally, we convert this falling-factorial series into standard powers of 1/k using geometric series expansions: (k -2) -1 = k -1 + 2k -2 + O(k -3 ) and (k -2) -1 (k -4) -1 = k -2 + O(k -3 ).", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 90, | |
| "text": "Gathering like terms gives:", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 91, | |
| "text": "Substituting k = n -p completes the proof.", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 92, | |
| "text": "We now extend Proposition 3.4 to a q-dimensional response. Let Y j ∈ R q be generated as", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 93, | |
| "text": ", and define the training and leave-one-out losses", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 94, | |
| "text": "The structural error is now a vector", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 95, | |
| "text": "with the asymptotic expansion in", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 96, | |
| "text": "Proof. The leverage analysis of Lemma 3.1 depends only on X and is unchanged:", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 97, | |
| "text": "Writing E = Y -XΘ ∈ R n×q for the noise matrix (rows i.i.d. N (0, Σ y )), we have", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 98, | |
| "text": "Since this conditional law does not depend on X -i , Z is unconditionally independent of X -i and therefore of V . We thus write", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 99, | |
| "text": "The univariate Sherman-Morrison contraction e i = (1 -h ii ) e -i applies column-wise to Θ, so it carries over to vector residuals and yields", | |
| "entities": [ | |
| { | |
| "text": "Θ", | |
| "type": "Tool", | |
| "attributes": [], | |
| "begin": 89, | |
| "end": 90, | |
| "confidence": 1.0 | |
| } | |
| ] | |
| }, | |
| { | |
| "index": 100, | |
| "text": "Combining the two displays,", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 101, | |
| "text": "By independence of Z and V ,", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 102, | |
| "text": "which yields the exact finite-sample expression in the proposition. The asymptotic expansion follows the same algebra as the proof of Theorem 3.4, replacing ε 2 i by ∥ε i ∥ 2 and σ 2 by tr(Σ y ).", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 103, | |
| "text": "The univariate result of Proposition 3.4 is recovered when q = 1 and Σ y = σ 2 . Crucially, the leverage hii remains label-free and so retains the same first-order role; only the structural-noise contribution is replaced by the sum of marginal output variances tr(Σ y ).", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 104, | |
| "text": "B.1 Proof of the Influence Function Estimate (Eq. ( 7))", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 105, | |
| "text": "We recall here the derivation of the influence function estimate, following the perturbationbased approach introduced by Koh and Liang [2017].", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 106, | |
| "text": "Proof. Let the empirical risk over the full dataset be R(θ) = 1 n n j=1 ℓ(θ, z j ), and let θ = arg min θ R(θ) be the empirical risk minimizer.", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 107, | |
| "text": "Rather than strictly removing the i-th point, we consider the effect of infinitesimally upweighting the loss of z i by some small continuous value ϵ. We define the perturbed empirical risk minimizer as:", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 108, | |
| "text": "By the first-order optimality condition, the gradient of the perturbed objective at θϵ,zi must be zero:", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 109, | |
| "text": "We then differentiate this entire condition with respect to ϵ and evaluate it at ϵ = 0 (where θϵ,zi = θ). Applying the chain rule yields:", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 110, | |
| "text": "be the Hessian of the empirical risk. Solving for the derivative gives us the influence of up-weighting z i on the parameters:", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 111, | |
| "text": "Removing the point z i from the dataset corresponds to reducing its weight from 1 n to 0. We can approximate this by setting ϵ = -1 n . Using a first-order Taylor expansion around ϵ = 0, the change in the parameters is:", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 112, | |
| "text": "Finally, to estimate the loss gap for the i-th point, we apply a first-order Taylor expansion to the loss function itself:", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 113, | |
| "text": "Expanding the Hessian H and pulling the factor n out of the inverse gives the final influence function estimate:", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 114, | |
| "text": "This gives us our influence function estimate for the loss gap ∆ℓ IF i when removing the i-th point from the training set.", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 115, | |
| "text": "Proof. The Newton step estimate is derived from the second-order Taylor expansion of the loss function around the empirical risk minimizer θ. The change in parameters when removing the i-th point can be approximated by the Newton update:", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 116, | |
| "text": "This is equivalent to performing a single Newton step on the loss of the i-th point, treating it as a separate optimization problem. The estimated loss gap is then:", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 117, | |
| "text": "which gives our Newton step estimate for the loss gap ∆ℓ", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 118, | |
| "text": "This estimate captures the curvature of the loss landscape around θ for the specific point z i , and can provide a more accurate approximation of the loss gap compared to the influence function, especially when the loss is highly non-linear or when the Hessian is not wellconditioned.", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 119, | |
| "text": "We specialize Eq. ( 7) and Eq. ( 8) to the squared loss ℓ(θ, z j ) = (y j -", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 120, | |
| "text": "Proof. For each j, 7):", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 121, | |
| "text": "Substituting in Eq. ( 8):", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 122, | |
| "text": "The Newton estimate is exact up to the first-order Taylor expansion of the loss; for the quadratic OLS loss, the second-order remainder is itself", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 123, | |
| "text": "We specialize Eq. ( 7) and Eq. ( 8) to the binary cross-entropy loss with pj = σ(x ⊤ j θ) and", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 124, | |
| "text": "j=1 and define the leverage score for the logistic model [Pregibon, 1981] as:", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 125, | |
| "text": "Proof. The per-sample loss ℓ(θ, z j ) = -y j log pj -(1-y j ) log(1-pj ) has gradient and Hessian", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 126, | |
| "text": "Influence. Substituting in Eq. ( 7):", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 127, | |
| "text": "Substituting in Eq. ( 8):", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 128, | |
| "text": "Proof. The MLE θ satisfies the score equation", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 129, | |
| "text": "where p = σ(X θ). Differentiating with respect to y i and applying the chain rule, with ∂ p/∂ η = W and η = X θ, yields", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 130, | |
| "text": "We extend the closed-form expressions to multivariate-output settings: multivariate linear regression and multinomial logistic regression. The arguments rely only on the per-sample gradient and Hessian, so Eq. ( 7) and Eq. ( 8) apply. We derive the exact formulas for the influence and Newton estimates in these two cases.", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 131, | |
| "text": "For multivariate linear regression over q outputs with quadratic loss ℓ(Θ) = n j=1 ∥y j -Θ ⊤ x j ∥ 2 , the problem decouples into q independent OLS regressions. For a single dimension k, the residual is e i,k ∈ R, the gradient is g i,k = -2e i,k x i , and the scalar leverage is h", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 132, | |
| "text": "The influence approximation (using the full-data Hessian) for the leave-one-out change in loss is:", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 133, | |
| "text": "where e i = y i -Θ⊤ x i ∈ R q is the full residual vector.", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 134, | |
| "text": "Proof. The full Hessian for dimension k is H Θ = 2X ⊤ X and is independent of k. The loss gap for dimension k using the influence approximation presented in equation* (7) is:", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 135, | |
| "text": "Summing the independent changes over all q dimensions yields the total loss gap:", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 136, | |
| "text": "The Newton approximation, using the leaveone-out Hessian for the leave-one-out change in loss is:", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 137, | |
| "text": "Applying the Sherman-Morrison formula to invert this rank-1 downdate yields", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 138, | |
| "text": "The loss gap per dimension using this LOO Hessian is:", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 139, | |
| "text": "Summing over all q dimensions yields the total loss gap:", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 140, | |
| "text": "We denote y j ∈ {0, 1} m the one-hot encoded label, Θ ∈ R p×m the learned parameter, and ℓ j ( Θ) = -m k=1 y jk log pjk the total loss, with pj = softmax( Θ⊤ x j ) the predicted class probabilities. The results for the influence and Newton estimates are given in the following proposition. Proposition B.3. In the multivariate case for multinomial logistic regression over m classes, we have:", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 141, | |
| "text": "Where", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 142, | |
| "text": "•", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 143, | |
| "text": "is the block-diagonal leverage matrix.", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 144, | |
| "text": "• Xi = x i ⊗ I m ∈ R pm×m is the expanded feature matrix.", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 145, | |
| "text": "• H Θ = n j=1 x j x ⊤ j ⊗ W j ∈ R pm×pm is the full Hessian of the loss.", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 146, | |
| "text": "Proof. Let L( Θ) = n j=1 ℓ j ( Θ) be the total cross-entropy loss. For convenience, we parameterize the model by the vectorized weights Θ ∈ R pm instead of a matrix ∈ R p×m . Using the provided definitions, the per-sample gradient and Hessian evaluated at the optimal full-data parameters Θ can be written compactly as:", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 147, | |
| "text": "Summing the per-sample Hessians gives the full Hessian H Θ = n j=1 Xj W j X⊤ j . When we leave out the i-th sample, the leave-one-out Hessian H -i is defined as:", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 148, | |
| "text": "Using a single Newton step approximation, the change in the estimated parameters upon removing the i-th sample is:", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 149, | |
| "text": "The approximate change in the loss function for the left-out sample, ∆ℓ i , evaluates the gradient against this parameter shift:", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 150, | |
| "text": "To evaluate the inner term without explicitly inverting the pm × pm LOO Hessian, we apply the Woodbury matrix identity to H -1 -i = (H Θ -Xi W i X⊤ i ) -1 . Because the weight matrix W i is not strictly invertible (its columns sum to zero), we use the form of the Woodbury identity that does not require W -1 i :", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 151, | |
| "text": "We substitute this expanded inverse into our target expression X⊤ i H -1 -i Xi . Leveraging the definition H ii = X⊤ i H -1 Θ Xi , we obtain:", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 152, | |
| "text": "Next, we factor out H ii on the left side:", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 153, | |
| "text": "Applying the push-through matrix identity W i (I m -H ii W i ) -1 = (I m -W i H ii ) -1 W i , the term inside the brackets becomes:", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 154, | |
| "text": "To combine these terms, we multiply the identity matrix I m by (I m -", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 155, | |
| "text": "Factoring out (I m -W i H ii ) -1 from both terms in the bracket leaves:", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 156, | |
| "text": "The -W i H ii and +W i H ii cleanly cancel out, yielding the final simplified matrix expression:", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 157, | |
| "text": "Substituting this matrix back into our earlier equation for the scalar change in loss yields the desired formulation:", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 158, | |
| "text": "In the multivariate case for multinomial logistic regression over m classes, we have ∆ℓ", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 159, | |
| "text": "Where g i , H ii are defined as in the previous proposition.", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 160, | |
| "text": "Proof. The influence function approximation is a special case of the Newton step approximation where we take only the first-order term in the Taylor expansion of the loss. This corresponds to replacing the matrix inverse (I m -W i H ii ) -1 with the identity matrix I m .", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 161, | |
| "text": "For m = 2 the per-sample weight collapses to the scalar w j , H Θ = X ⊤ W X factorizes as a single Kronecker product, and we recover the binary formulas of the previous subsection.", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 162, | |
| "text": "In general H does not factorize as a Kronecker product because W j depends on j, so the multinomial leverage cannot be separated into a sample term and a class term as in the binary case.", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 163, | |
| "text": "Table 4: Compute budget for our experimental setting. Values are observed run-times in GPU-hours. Surrogates aggregates the GPU time spent computing the influence / Newton scores surrogates over all targets.", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 164, | |
| "text": "surrogate ∇ℓ ⊤ i H -1 θ ∇ℓ i . The values are reported in the CIFAR-10 and California Housing (MLP) cells in Table 5.", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 165, | |
| "text": "We report full results for the LiRA MIA in Table 5. Notably, we report the Spearman rank correlation between the surrogate scores and the per-point ASR for reference, but we do not find a strong correlation. We attribute this to the fact that points near the center of the vulnerability distribution yield an Attack Success Rate (ASR) close to 0.5, offering little informative value. In contrast, top-k recall effectively isolates the highly vulnerable points, which are the primary focus for practical security auditing and defense.", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 166, | |
| "text": "For classification models we also report the self-entropy defined as H i = -m k=1 pi,k log pi,k , where pi,k is the model's predicted probability of class k on input x i . Self-entropy is a standard confidence-based MIA baseline [Salem et al., 2018, Song andMittal, 2021]: training samples typically receive more peaked predictions and hence lower entropy than non-members.", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 167, | |
| "text": "We provide the top-k recall curves for RMIA and Shokri's attack in Figure 4 and Figure 5 respectively.", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 168, | |
| "text": "We provide additional visualizations of the samples with the highest and lowest influence scores and Newton scores for the BloodMNIST, CIFAR-10 and CIFAR-100 datasets in Figure 6. We also provide the top-10 highest and lowest influence scores and Newton scores for the BERT model trained on the SST-2 dataset with linear probing in Table 6. The joint distribution of local loss ℓ i and influence score for every training sample of a single target model, colored by per-sample LiRA ASR, is presented in Figure 7.", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 169, | |
| "text": "10 -2 10 -1 10 0 10 1 10 2 SST-2 (BERT full ft.) SST-2 (BERT linear probe) CalHousing (Ridge) CalHousing (MLP) 10 -2 10 -1 10 0 10 1 10 -2 10 -1 10 0 10 1 10 2 CIFAR-10 (CNN) 10 -2 10 -1 10 0 10 1 CIFAR-100 (ResNet-20) 10 -2 10 -1 10 0 10 1 BloodMNIST (DINOv2) 10 -2 10 -1 10 0 10 1 UCI Energy (MLP) top-k percentile (%) recall (%) Loss ℓ i ∇ℓ i ∆ℓ NS i ∆ℓ IF i random y = x 16-model mean single model ±1 std band", | |
| "entities": [ | |
| { | |
| "text": "BloodMNIST", | |
| "type": "Tool", | |
| "attributes": [], | |
| "begin": 241, | |
| "end": 251, | |
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| } | |
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| { | |
| "index": 170, | |
| "text": "0.00 ± 0.00 10.90 ± 19.74 28.25 ± 7.50 0.541 ± 0.032 ∆ℓ IF i 6.25 ± 25.00 14.72 ± 23.99 30.32 ± 7.89 0.540 ± 0.032", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 171, | |
| "text": "We evaluate our surrogates on a deliberately diverse benchmark spanning tabular regression, image classification, and text classification, with model sizes ranging from a single linear layer to a fully fine-tuned transformer. For each (dataset, architecture) pair, target and reference (shadow) models are trained with the same hyperparameters; only the random seed, which controls both weight initialization and the training split differs. We train 200 reference models and 16 target models per setting. Table 2 summarizes the architectures and Table 3 the training hyperparameters. The detailed implementation can be found in the codebase.", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 172, | |
| "text": "All experiments were run using V100 (32 GB) GPU. The values presented in Table 4 are the wall-clock times reported using dedicated packages. They split into four phases: (i) training the reference ensemble, (ii) training the target models, (iii) running the LiRA, RMIA and Shokri attacks plus the per-point statistics, and (iv) computing the leverage-score / influence / Newton surrogates on each target. We report the total time spent in each phase, as well as the total time for the entire benchmark.", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 173, | |
| "text": "To evaluate cost-effectiveness, surrogate performance is reported as the time required to compute scores for a single target, allowing a direct comparison with the training of the 200 reference models. While actual implementation involves computing scores across all targets, resulting in a total runtime approximately 16 times the per-target figure, the single-target metric highlights the efficiency gains. For example, using the DiNO linear model, computing", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 174, | |
| "text": "Training Attacks+Stats Surrogates IF (full network) Total Dataset / Model per-model 200 refs 16 targets 200 refs per-target per-target GPU-h UCI Energy / MLP 30 s 1.7 h 8 min 30 s 3 s -1.7 h California / Linear reg. 1 min 3.3 h 0.3 h 7 min 10 s -3.6 h California / MLP 1 min 3.3 h 0.3 h 7 min 10 s 20 min 8.9 h BloodMNIST / DINO (linear) 15 min 60.0 h 4.0 h 5 min 15 s -64.0 h CIFAR-10 / CNN 4 min 13.0 h 1.0 h 20 min 30 s 2.2 h 49.2 h CIFAR-100 / ResNet-20 13 min 44.4 h 3.5 h 45 min 1 min -48.5 h SST-2 / BERT-tiny (full FT) 6 min 20 h 1.6 h 17 min 45 s -20.6 h SST-2 / BERT-base (linear) 19 min 64.6 h 5.0 h 18 min 3 min -70 h Total 266 GPU-h", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 175, | |
| "text": "surrogates for one target takes only 15 seconds; this represents a 50, 000× speedup compared to the 60 hours required to train all the reference (shadow) models.", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 176, | |
| "text": "The training phase dominates the total cost. Once all models are trained, surrogates and attacks reuse the same models.", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 177, | |
| "text": "Detailed results for the LiRA ASR top-k% recall are reported in Table 5. We observe that the Newton and Influence surrogates outperform the loss surrogate across most datasets and architectures. Although the Spearman correlation is often higher for our surrogates, the absolute values remain quite low (except for the CIFAR datasets).", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 178, | |
| "text": "This suggests that the ranking of samples by our surrogates is not perfectly aligned with the ranking by the LiRA score, mostly for the points that are in the middle of the distribution. This is not surprising as the LiRA ASR score is very noisy in this part of the distribution. This intuition is confirmed by Figure 7: points in the middle of the distribution are scattered, whereas in the tails the ranking induced by the surrogates is much more consistent with the LiRA score.", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 179, | |
| "text": "To evaluate the impact of the last-layer approximation, we computed full-network influence functions for CIFAR10 (CNN) and California Housing (MLP). These estimates were calculated using inverse Hessian-vector products [Pearlmutter, 1994] via the Conjugate Gradient (CG) algorithm [Hestenes and Stiefel, 1952]. We ensured CG convergence by setting a tolerance of 10 -2 and applying a notably high damping value of 10; while strictly necessary for the algorithm to converge in our case, such heavy damping inherently alters the resulting influence estimates.", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 180, | |
| "text": "As presented in Table 4, the computational overhead for the full-network approach is substantial: for CIFAR10, processing a single model took more than 2 hours, compared to only 30 seconds for the last-layer surrogate. Despite this significantly higher cost, the full-network influence function performed worse in identifying vulnerable samples. This suggests that the last-layer approximation not only offers a massive gain in efficiency but may also act as a useful regularizer by focusing on the most discriminative features. Furthermore, as noted by Basu et al. [2021], this underperformance may come from the inherent fragility and poor scaling of influence functions when applied to the non-convex landscapes of deep networks.", | |
| "entities": [] | |
| }, | |
| { | |
| "index": 181, | |
| "text": "Figure 3 illustrates the top-k% recall with the additional full network influence function", | |
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