Title: Anonymous Learning via Look-Alike Clustering: A Precise Analysis of Model Generalization
Venue: NeurIPS 2023
Accepted: yes
Keywords: high-dimensional regression

generalization error

asymptotic analysis

Convex Gaussian Minimax Theorem

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

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

Abstract

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

1 Introduction

Look-alike modeling in machine learning encompasses a range of techniques that focus on identifying users who possess similar characteristics, behaviors, or preferences to a specific target individual. This approach primarily relies on the principle that individuals with shared attributes are likely to exhibit comparable interests and behaviors. By analyzing the behavior of these look-alike users, look-alike modeling enables accurate predictions for the target user. This technique has been widely used in various domains, including targeted marketing and personalized recommendations, where it plays a crucial role in enhancing user experiences and driving tailored outcomes [26, 19, 18, 21].

In this paper, we use look-alike clustering for a different purpose, namely to anonymize sensitive information of users. Consider a supervised regression setup where the training set contains n pairs ( x i , y i ) , for i ∈ [ n ] , with y i ∈ R denoting the response and x i ∈ R d representing a high-dimensional vector of features. We consider two groups of features: sensitive features, which contain some personal information about users and should be protected from the leaner, and the non-sensitive features. We assume that that the learner has access to a clustering structure on users, which is non-private information (e.g. based on non-sensitive features or other non-sensitive data set on users).

We propose a look-alike clustering approach, where we anonymize the individuals' sensitive features by replacing them with the cluster's average values. Only the anonymized dataset will be shared with the learner who then uses it to train a model. We refer to Figure 1 for an illustration of this approach. Note that the learner never gets access to the individuals' sensitive features and so this approach is safe from re-identification attacks where the learner is given access to the pool of individuals' sensitive information (up to permutation) and may use the non-sensitive features to re-identify the

look-alike

representations

non-sensitive

features

sensitive

features

non-sensitive

features

users in

cluster 1

users in

cluster 2

users in

cluster 3

!

1

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users. Also note that since a common representation (average sensitive features) is used for all the users in a cluster, this approach offers m -anonymity provided that each cluster is of size at least m (minimum size clustering).

Minimum size clustering has received an increased attention mainly as a tool for anonymization and when privacy considerations are in place [7, 2, 3]. A particular application is for providing anonymity for user targeting in online advertising with the goal of replacing the use of third-party cookies with a more privacy-respecting entity [10]. There are a variety of approximation algorithms for clustering with minimum size constraint [23, 9, 1, 24], as well as parallel and dynamic implementation [10].

In this paper, we focus on linear regression and derive a precise characterization of model generalization 1 using the look-alike clustering approach, in the so-called proportional regime where the size of training set grows in proportion to the number of parameters (which for the linear regression is equal to the number of features). The proportional regime has attracted a significant attention as overparametrized models have become greatly prevalent. It allows to understand the effect under/overparametrization in feature-rich models, providing insights to several intriguing phenomena, including double-descent behavior in the generalization error [22, 8, 14].

Our precise asymptotic theory allows us to demystify the effect of different factors on the model generalization under look-alike clustering, such as the role of cluster size, number of clusters, signalto-noise ratio of the model as well as the strength of sensitive and non-sensitive features. A key tool in our analysis is a powerful extension of Gordon's Gaussian process inequality [13] known as the Convex Gaussian Minimax Theorem (CGMT), which was developed in [30] and has been used for studying different learning problems; see e.g, [29, 8, 15, 14, 16].

Initially, it might be presumed that look-alike clustering would hinder model generalization by suppressing sensitive features of individuals, suggesting a possible tradeoff between anonymity (privacy) and model performance. However, our analysis uncovers scenarios in which look-alike clustering actually enhances model generalization! We will develop further insights on these results by arguing that the proposed look-alike clustering can serve as a form of regularization, mitigating model overfitting and consequently improving the model generalization.

Before summarizing our key contributions in this paper, we conclude this section by discussing some of the recent work on the tradeoff between privacy and model generalization at large. An approach to study such potential tradeoff is via the lens of memorization. Modern deep neural networks, with remarkable generalization property, operate in the overparametrized regime where there are more tunable parameters than the number of training samples. Such overparametrized models tend to interpolate the training data and are known to fit well even random labels [34, 33]. Similar phenomenon has been observed in other models, such as random forest [4], Adaboost [25, 32], and kernel methods [5, 17]. Beyond label memorization, [6] studies setting where learning algorithms with near-optimal generalization must encode most of the information about the entire high-dimensional (and high-entropy) covariates of the training examples. Clearly, memorization of training data imposes significant privacy risks when this data contains sensitive personal information, and therefore these results hint to a potential trade-off between privacy protection and model generalization [27, 12, 20]. Lastly, [11] studies settings where data is sampled from a mixture of subpopulations, and shows that label memorization is necessary for achieving near-optimal generalization error, whenever the

distribution of subpopulation frequencies is long-tailed. Intuitively, this corresponds to datasets with many small distinct subpopulations. In order to predict more accurately on a subpopulation from which only a very few examples are observed, the algorithm needs to memorize their labels.

1.1 Summary of contributions

We consider a linear regression setting for response variable y given feature x , and posit a Gaussian Mixture Model on the features to model the clustering structure on the samples. We focus on the high-dimensional asymptotic regime where the number of training samples n , the dimension of sensitive features ( p ), and the dimension of non-sensitive features ( d -p ) grow in proportion ( p / n → ψ p and d / n → ψ d , for some constants 0 < ψ p ≤ ψ d ). Asymptotic analysis in this particular regime, characterized by a fixed sample size to feature size ratio, has recently garnered significant attention due to its relevance to the regime where modern neural networks operate. This analysis allows for the study of various intriguing phenomena related to both statistical properties (such as double-descent) and the tractability of optimizing the learning process in such networks [22, 8, 14], where the population analysis n / d → ∞ fails to capture. Let T n = {( x i , y i ) , i ∈ [ n ]} denote the (unanonymized) training set and T n L be the set obtained after replacing the sensitive features with the look-alike representations of clusters. We denote by ̂ θ and ̂ θ L the min-norm estimators fit to T n and T n L , respectively. Under this asymptotic setting:

We provide a precise characterization of the generalization error of ̂ θ and ̂ θ L . Despite the randomness in data generating model, we show that in the high-dimensional asymptotic, the generalization errors of these estimators converge in probability to deterministic limits for which we provide explicit expressions.

Our characterizations reveal several interesting facts about the generalization of the estimators:

( i ) For the min-norm estimator ̂ θ we observe significantly different behavior in the underparametrized regime ( ψ d ≤ 1 ) than in the overparametrized regime ( ψ d > 1 ) . Note that in the underparametrized regime, the min-norm estimator coincides with the standard least squares estimator. For the look-alike estimator ̂ θ L our analysis identifies the underparametrized regime as ψ d -ψ p ≤ 1 and the overparametrized regime as ψ d -ψ p > 1 .

( ii ) In the underparametrized regime, our analysis shows that, somewhat surprisingly, the generalization error (for both estimators) does not depend on the number or size of the clusters, nor the scaling of the cluster centers.

( iii ) In the overparametrized regime, our analysis provides a precise understanding of the role of different factors, including the number of clusters, energy of cluster centers, and the alignment of the model with the constellation of cluster centers, on the generalization error.

Using our characterizations, we discuss settings where the look-alike estimator ̂ θ L has better generalization than its non-private counterpart ̂ θ . A relevant quantity that shows up in our analysis is the ratio of the norm of the model component on the sensitive features over the noise in the response, which we refer to as signal-to-noise ratio (SNR). Using our theory, we show that if SNR is below a certain threshold, then look-alike estimator ̂ θ L has lower generalization error than ̂ θ . This demonstrates scenarios where anonymizing sensitive features via look-alike clustering does 'not' hinder model generalization. We give an interpretation for this result, after Theorem 5.1, by arguing that at low-SNR, look-alike clustering acts as a regularization and mitigates overfitting, which consequently improves model generalization.

In our analysis in the previous parts, we assume that the learner has access to the exact underlying clustering structure on the users, to disentangle the clustering estimation error from look-alike modeling. However, in practice the learner needs to estimate the clustering structure from data. In Section 3.2, we combine our analysis with a perturbation analysis to extend our results to the case of imperfect clustering estimation.

Due to space constraint, we refer to the supplementary material for an overview of our proof techniques as well as proof of theorems and technical lemmas.

2 Model

We consider a linear regression setting, where we are given n i.i.d pairs ( x i , y i ) , where the response y i is given by

We assume that there is a clustering structure on features x i , i ∈ [ n ] , independent from the responses. We model this structure via Gaussian-Mixture model.

Gaussian-Mixture Model (GMM) on features. Each example x belong to cluster glyph[lscript] ∈ [ k ] , with probability π glyph[lscript] . We let π = [ π, π 2 , . . . , π k ] ∈ R k with π ≥ 0 and 1 T π = 1 . The cluster conditional distribution of an example x in cluster glyph[lscript] follows an isotropic Gaussian with mean µ glyph[lscript] ∈ R d , namely

By scaling the model (2.1), without loss of generality we assume τ = 1 . Writing in the matrix form, we let

It is also convenient to encode the cluster membership as one-hot encoded vectors λ i ∈ R k , where λ i is one at entry glyph[lscript] (with glyph[lscript] being the cluster of example x i ) and zero everywhere else. The GMM can then be written as

with Z ∈ R d × n is a Gaussian matrix with i.i.d N ( 0 , 1 ) entries, and Λ ∈ R k × n is the matrix obtained by stacking vectors λ i as its column.

Sensitive and non-sensitive features. We assume that some of the features are sensitive for which we have some reservation to share with the learner and some non-sensitive features. Without loss of generality, we write it as x = ( x s , x ns ) , where x s ∈ R p representing the sensitive features and x ns ∈ R d -p representing the non-sensitive features. We also decompose the model θ 0 (2.1) as θ 0 = ( θ 0 , s , θ 0 , ns ) with θ 0 , s ∈ R p and θ 0 , ns ∈ R d -p . Likewise, the cluster mean vector µ is decomposed as µ = ( µ s , µ ns ) . The idea of look-alike clustering is to replace the sensitive features of an example x s with the center of its cluster µ s . This way, if each cluster is of size at least m , then look-alike clustering offers m -anonymity.

Our goal in this paper is to precisely characterize the effect of look-alike clustering on model generalization. We focus on the high-dimensional asymptotic regime, where the number of training data n , and features sizes d, p grow in proportion.

We formalize the high-dimensional asymptotic setting in the assumption below:

Assumption 1 We assume that the number of clusters k is fixed and focus on the asymptotic regime where n, d, p →∞ at a fixed ratio d / n → ψ d and p / n → ψ p .

To study the generalization of a model θ (performance on unseen data) via the out-of-sample prediction risk defined as Risk ( θ ) ∶= E [( y -x T θ ) 2 ] , where ( y, x ) is generated according to (2.1). Our next lemma characterizes the risk when the feature x is drawn from GMM.

Lemma 2.1 Under the linear response model (2.1) and a GMM for features x , the out-of-sample prediction risk of a model θ is given by

The proof of Lemma 2.1 is deferred to the supplementary.

3 Main results

Consider the minimum glyph[lscript] 2 norm (min-norm) least squares regression estimator of y on X defined by

where ( XX T ) † denotes the Moore-Penrose pseudoinverse of XX T . This estimator can also be formulated as

We also define the 'look-alike estimator' denoted by ̂ θ L , where the sensitive features are first anonymized via look-like modeling, and then the min-norm estimator is computed based on the resulting features. Specifically the sensitive feature x s of each sample is replaced by the center of its cluster. In our notation, writing X T = [ X T s , X T ns ] , we define X T L = [( M s Λ ) T , X T ns ] the features matrix obtained after look-alike modeling on the sensitive features. The look-alike estimator is then given by

Our main result is to provide a precise characterization of the risk of look-alike estimator ̂ θ L as well as ̂ θ (non-look-alike) in the asymptotic regime, as described in Assumption 1. We then discuss regimes where look-alike clustering offers better generalization.

As our analysis shows there are two majorly different setting in the behavior of the look-alike estimator: ( i ) ψ d -ψ p ≤ 1 , i.e., the sample size n is asymptotically larger than d -p , the number of non-sensitive features (referred to as underparametrized asymptotics); ( ii ) ψ d -ψ p ≥ 1 , which is referred to as overparametrized asymptotics.

Our first theorem is on the risk of look-alike estimator in the underparametrized setting. To present our result, we consider the following singular value decomposition for M s , the matrix of cluster centers restricted to sensitive features:

Theorem 3.1 ( Look-alike estimator, underparametrized regime ) Consider the linear response model (2.1) , where the features are coming from the GMM (2.4) . Also assume that ∥ θ 0 , s ∥ = r s and ∥ U T s θ 0 , s ∥ = √ ρr s , for all n, p . Under Assumption 1 with ψ d -ψ p ≤ 1 , the out-of-sample prediction risk of look-alike estimator ̂ θ L , defined by (3.2) , converges in probability,

There are several intriguing observations about this result. In the underparametrized regime:

The risk depends on θ 0 , s (model component on the sensitive features), only through the norms ∥ θ 0 , s ∥ = r s and ∥ U T s θ 0 , s ∥ = √ ρr s . Note that ∥ U T s θ 0 , s ∥ measures the alignment of the model with the left singular vectors of the cluster centers.

The cluster structure on the non-sensitive features plays no role in the risk, nor does θ 0 , ns the model component corresponding to the non-sensitive features.

The cluster prior probabilities π does not impact the risk.

We next proceed to the overparametrized setting. For technical convenience, we make some simplifying assumption, however, we believe a similar derivation can be obtained for the general case, albeit with a more involved analysis.

Assumption 2 Suppose that there is no cluster structure on the non-sensitive features ( M ns = 0 ). Also, assume orthogonal, equal energy centers for the clusters on the sensitive features ( M s = µ U s with U T s U s = I k ).

Our next theorem characterizes the risk of look-alike estimator in the underparametrized regime.

Theorem 3.2 ( Look-alike estimator, overparametrized regime ) Consider the linear response model (2.1) , where the features are coming from the GMM (2.4) . Also assume that ∥ θ 0 , s ∥ = r s , ∥ θ 0 , ns ∥ = r ns and ∥ U T s θ 0 , s ∥ = √ ρr s , for all n, p, d . Under Assumption 1 with ψ d -ψ p ≥ 1 , and Assumption 2, the out-of-sample prediction risk of look-alike estimator ̂ θ L , defined by (3.2) , converges in probability,

where π = ( π 1 , . . . , π k ) encodes the cluster priors and γ 0 and α ∈ R k are given by the following relations:

As discussed in the introduction, one of the focal interest in this work is to understand cases where look-alike modeling improves generalization. In Section 5 we discuss this by comparing the lookalike estimator ̂ θ L with the min-norm estimator ̂ θ , given by (3.1) which utilizes the full information on the sensitive features. In order to do that, we next derive a precise characterization of the risk of ̂ θ in the asymptotic setting.

Theorem 3.3 ( min-norm estimator with no look-alike clustering ) Consider the linear response model (2.1) , where the features are coming from the GMM (2.4) . Under Assumption 1, the followings hold for the min-norm estimator ̂ θ given by (3.1) :

(a) (underparametrized setting) If ψ d ≤ 1 , we have

(b) (overparametrized setting) If ψ d ≥ 1 , under Assumption 2, the prediction risk of ̂ θ converges in probability

where ˜ γ 0 and ˜ α are given by the following relations:

Example 3.1 (Balanced clusters) In the case of equal cluster prior ( π 1 = . . . = π k = 1 / k ), the risk characterization (3.3) depends on α only through ∥ α ∥ glyph[lscript] 2 (and likewise, the risk (3.4) depends on ˜ α only through its norm). This significantly simplifies these characterizations.

3.2 Extension to imperfect clustering estimation

In our previous results, we assumed that the underlying cluster memberships of users are known to the learner, so we could concentrate our analysis on the impact of training using anonymous cluster centers. However, in practice, clusters should be estimated from the features and thus includes an estimation error. In our next result, we combine our previous result with a perturbation analysis to bound the risk of the look-alike estimator based on estimated clusters.

Recall matrix M ∈ R d × k from (2.3), whose columns are the cluster centers. Also, recall the matrix Λ ∈ R k × n whose columns are the one-hot encoding of the cluster memberships. We let ̃ M and ̃ Λ indicate the estimated matrices, with the cluster estimation error rate δ n ∶= 1 √ n ∥ M s Λ - ̃ M s ̃ Λ ∥ 2 , where ∥ ⋅ ∥ 2 indicates spectral norm. Note that only the cluster estimation error with respect to the sensitive features matters because in the look-alike modeling only those features are anononymized (replaced by the cluster centers).

Proposition 3.4 Let ̃ X T ∶= [( ̃ M s ̃ Λ ) T , X T ns ] be the feature matrix after replacing the sensitive features with the estimated cluster centers of users. We also let ̃ θ L = ( ̃ X L ̃ X T L ) † ̃ X L y be the lookalike estimator based on ̃ X L . Note that ̃ θ L is the counterpart of ̂ θ L given by (3.2) . Define the cluster estimation error rate δ n ∶= 1 √ n ∥ M s Λ - ̃ M s ̃ Λ ∥ 2 , and suppose that either of the following conditions hold:

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L

(

σ

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ns

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(b) Risk of

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(c) Risk of

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L

(

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1

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=

1

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(d) Risk of

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(f) Risk of

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(

r

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1

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( i ) ψ d -ψ p < 0 . 5 and δ < √ 1 -( ψ d -ψ p ) - √ ψ d -ψ p .

( ii ) ψ d -ψ p > 2 and δ < √ ψ d -ψ p -1 -1 .

Then, for some constant C depending on the problem parameters.

4 Numerical experiments

In this section, we validate our theory with numerical experiments. We consider GMM with k clusters, where the centers of clusters are given by µ u glyph[lscript] , for glyph[lscript] ∈ [ k ] , where u glyph[lscript] ∈ R d are of unit glyph[lscript] 2 -norm. Also the vectors u glyph[lscript] are non-zero only on the first p entries, and their restriction to these entries form a random orthogonal constellation. Therefore, defining U = [ u 1 , . . . , u k ] , we have U = [ U s 0 ] , with

U T s U s = I k . In this setting there is no cluster structure on the non-sensitive features and the cluster centers on the sensitive features are orthogonal and of same norm.

Recall the decomposition of the model θ 0 = ( θ 0 , s , θ 0 , ns ) , with θ 0 the true underlying model (2.1) and θ 0 , s , θ 0 , ns the components corresponding to sensitive and non-sensitive features. We generate θ 0 , ns ∈ R d -p to have i.i.d standard normal entries and then normalize it to have ∥ θ 0 , ns ∥ glyph[lscript] 2 = r ns . For θ 0 , s , we generate Z 1 , Z 2 ∼ N ( 0 p , I p ) , independently and let

where P U s ∶= U s U T s is the projection onto column space of U s . Therefore, ∥ θ 0 , s ∥ glyph[lscript] 2 = r s and ∥ U T s θ 0 , s ∥ glyph[lscript] 2 = √ ρr s . Note that ρ quantifies the alignment of the model with the cluster centers, confined to the sensitive features.)

We will vary the values of r s and r ns in our experiments. We also consider the case of balanced clusters, so the cluster prior probabilities are all equal, π glyph[lscript] = 1 / k , for glyph[lscript] ∈ [ k ] . We set the number of

cluster k = 3 , dimension of sensitive features p = 200 and the dimension of entire features vector d = 500 . We also set µ = 5 and ρ = 0 . 3 . In our experiments, we vary the sample size n and plot the risk of ̂ θ L and ̂ θ versus ψ d -ψ p = ( d -p )/ n . We consider different settings, where we vary r s , r ns and σ (noise variance in model (2.1)). In Figure 2, we report the results. Curves correspond to our asymptotic theory and dots to the numerical simulations. (Each dot is obtained by averaging over 20 realizations of that configuration.) As we observe, in all scenarios our theoretical predictions are a perfect match to the empirical performance.

5 When does look-alike clustering improve generalization?

In Section 3, we provided a precise characterization of the risk of look-alike estimator ̂ θ L and its counterpart, the min-norm estimator ̂ θ which utilizes the full information on the sensitive features. By virtue of these characterizations, we would like to understand regimes where the look-alike clustering helps with the model generalization, and the role of different problem parameters in achieving this improvement. Notably, since the look-alike estimator offers m -anonymity on the sensitive features (with m the minimum size of clusters), our discussion here points out instances where data anonymization and model generalization are not in-conflict.

We define the gain of look-alike estimator as ∆ ∶= Risk ( ̂ θ )/ Risk ( ̂ θ L ) to indicate the gain obtained in generalization via look-alike clustering. For ease in presentation, we focus on the case of balanced clusters (equal priors π 1 = . . . = π k = 1 / k ), and consider three cases:

Case 1 ( ψ d ≤ 1 ): In this case, both ̂ θ L and ̂ θ are in the underparametrized regime and Theorems 3.1 and 3.3 (a) provide simple closed-form characterization of the risks of ̂ θ L and ̂ θ , by which we obtain

Define the signal-to-noise ratio SNR = r s / σ . Since ρ ≤ 1 , it is easy to see that ∆ is decreasing in the SNR. In particular, as SNR → 0 , we have ∆ → ( 1 -ψ d + ψ p )/( 1 -ψ d ) > 1 , which means the look-alike estimator ̂ θ L achieves lower risk compared to ̂ θ . In Figure 3a we plot log ( ∆ ) versus SNR, for several values of ψ p . Here we set ψ d = 0 . 9 and ρ = 0 . 3 . As we observe in low SNR, the look-alike estimator has lower risk. Specifically, for each curve there is a threshold for the SNR, below which log ( ∆ ) > 0 . Furthermore, this threshold increases with ψ p , covering a larger range of SNR where ̂ θ L has better generalization.

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5

In Figure 3b we report similar curves, where this time ψ p = 0 . 5 and we consider several values of ρ . As we observe, at fixed SNR the gain ∆ is increasing in ρ . This is expected since ρ measures the alignment of the underlying model θ 0 with the (left eigenvectors of) cluster centers and so higher ρ is to advantage of the look-alike estimator which uses the cluster centers instead of individuals' sensitive features.

Case 2 ( ψ d ≥ 1 , ψ d -ψ p ≤ 1 ): In this case, the look-alike estimator ̂ θ L is in the underparametrized regime, while the min-norm ̂ θ is in the overparametrized regime. The following theorem uses the characterizations in Theorem 3.1 and and 3.3 (b), and shows that in the low SNR = r s / σ , the look-alike estimator ̂ θ L has a positive gain. It further shows the monotonicity of the gain with respect to different problem parameters.

Theorem 5.1 Suppose that ψ d ≥ 1 and ψ d -ψ p ≤ 1 , and consider the case of equal cluster priors. The gain ∆ is increasing in r ns and ρ , and is decreasing in µ 2 / k . Furthermore, under the following condition

we have ∆ ≥ 1 , for all values of other parameters ( µ, k, ρ, r ns ).

An interpretation based on regularization: We next provide an argument to build further insight on the result of Theorem 5.1. Recall the data model (2.1), where substituting from (2.2) and decomposing over sensitive and non-sensitive features we arrive at

Note that ⟨ z s , θ s ⟩ ∼ N ( 0 , ∥ θ s ∥ 2 ) . At low SNR, this term is of order of the noise term ε ∼ N ( 0 , σ 2 ) . Recall that the look-alike clustering approach replaces the sensitive feature x s by the cluster center µ s , and therefore drops the term ⟨ z s , θ s ⟩ from the model during the training process. In other words, look-alike clustering acts as a form of regularization which prevents overfitting to the noisy component ⟨ z s , θ s ⟩ , and this will help with the model generalization, together with anonymizing the sensitive features.

In Figure 4a we plot log ( ∆ ) versus µ for several values of r ns . Here, ψ d = 2 , ψ p = 1 . 7 , σ = 1 , r s = 0 . 5 and so condition (5.1) holds. As we observe log ( ∆ ) is positive, decreasing in µ and also at any fixed µ , it is increasing in r ns , all of which are consistent with the Theorem 5.1. In Figure 4b, we plot similar curves where this time r ns = 0 . 2 and we try several values of ρ . As we see the look-alike estimator has larger gain ∆ at larger values of ρ .

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0.26

0.28

0.3

0.32

= 0.2

= 0.4

= 0.6

= 0.8

(b)

r

ns

=

0

.

2

Case 3 ( ψ d -ψ p ≥ 1 ): In this case, both ̂ θ L and ̂ θ are in the overparametrized regime. Let us first focus on r ns , the energy of the model on the non-sensitive features. Invoking the equations (3.3) and (3.4) and hiding the terms that do not depend on r ns in constants C 1 , C 2 we arrive at

Therefore, lim r ns →∞ ∆ = ( 1 -ψ -1 d )/( 1 - ( ψ d -ψ p ) -1 ) > 1 , indicating a gain for the look-alike estimator over ̂ θ . In Figure 5a, we plot log ( ∆ ) versus r ns for several values of ψ p . As we observe, when r ns is large enough we always have a gain, which is increasing in ψ p .

0

1

2

3

4

5

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

p

= 0.5

p

= 1

p

= 1.5

p

= 2

(a)

r

s

=

0

.

5

0

2

4

6

8

10

-0.2

-0.1

0

0.1

0.2

0.3

0.4

p

= 0.5

p

= 1

p

= 1.5

p

= 2

(b)

r

ns

=

2

0

0.5

1

1.5

2

-0.5

0

0.5

1

We next consider the effect of SNR = r s / σ . In Figure 5b we log ( ∆ ) versus SNR, for several values of ψ p . Similar to the underparametrized regime, we observe that in low SNR, the look-alike estimator has better generalization ( log ( ∆ ) > 0 ) .

6 Beyond linear models

In previous section, we used our theory for linear models to show that at low SNR, look-alike modeling improves model generalization. We also provided an insight for this phenomenon by arguing that look-alike modeling acts as a form of regularization and avoids over-fitting at low SNR regime. In this section we show empirically that this phenomenon also extends to non-linear models.

Consider the following data generative model:

where ε ∼ N ( 0 , σ 2 ) . We construct the model θ 0 = ( θ 0 , s , θ 0 , ns ) similar to the setup in Section 4. We set n = 200 , d = 180 , k = 3 , µ = 5 , σ = 1 , ρ = 0 . 3 , r ns = 2 and N = 1000 (number of trials in Binomial distribution). We vary SNR by changing r s in the set { 0 . 1 , 0 . 3 , . . . , 1 . 9 } . The estimators ̂ θ and ̂ θ L are obtained by fitting a GLM with logit link function and binomial distribution. We compute the risks of ̂ θ and ̂ θ L by averaging over a test set of size 50 K . In Figure 6, we plot the gain log ( ∆ ) versus r s where each data point is by averaging over 50 different realizations of data. As we observe at low SNR, log ( ∆ ) > 0 indicating that the look-alike estimator ̂ θ L obtains a lower risk than the min-norm estimator.

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Supplementary Material

In this supplementary, we first provide an overview of our proof techniques in Appendix A and then in Appendix B provide the proofs of theorems and technical lemmas stated in the main paper.

A Overview of proof techniques

Our analysis of the generalization error is based on an extension of Gordon's Gaussian process inequality [13], called Convex-Gaussian Minimax Theorem (CGMT) [30]. Here, we outline the general steps of this framework and refer to the supplementary for complete details and derivations.

Consider the following two Gaussian processes:

where G ∈ R n × d , g ∈ R n and h ∈ R d , all have i.i.d standard normal entries. Further, ψ ∶ R d × R n → R is a continuous function, which is convex in the first argument and concave in the second argument.

Given the above two processes, consider the following min-max optimization problems, which are respectively referred to as the Primary Optimization (PO) and the Auxiliary Optimization (AO) problems:

The main result of CGMT is to connect the above two random optimization problems. As shown in [30](Theorem 3), if S u and S v are compact and convex then, for any λ ∈ R and t > 0 ,

An immediate corollary of this result (by choosing λ = E [ Φ AO ( g , h ) ]) is that if the optimal cost of AO problem concentrates in probability, then the optimal cost of the corresponding PO problem also concentrates, in probability, around the same value. In addition, as shown in part (iii) of [30](Theorem 3), concentration of the optimal solution of the AO problem implies concentration of the optimal solution of the PO around the same value. Therefore, the two optimization are intimately connected and by analyzing the AO problem, which is substantially simpler, one can derive corresponding properties of the PO problem.

The CGMT framework has been used to infer statistical properties of estimators in certain highdimensional asymptotic regime. The intermediate steps in the CGMT framework can be summarized as follows: First form an PO problem in the form of (A.1) and construct the corresponding AO problem. Second, derive the point-wise limit of the AO objective in terms of a convex-concave optimization problem, over only few scalar variables. This step is called 'scalarization'. Next, it is possible to establish uniform convergence of the scalarized AO to the (deterministic) min-max optimization problem using convexity conditions. Finally, by analyzing the latter deterministic problem, one can derive the desired asymptotic characterizations.

Of course implementing the above steps involved problem-specific intricate calculations. Our proofs of Theorems 3.1, 3.2, 3.3 in the supplementary follow this general strategy.

B Proof of theorems and technical lemmas

B.1 Proof of Lemma 2.1

By substituting for y from (2.1) in the definition of risk we obtain

,

where ( a ) follows from the Gaussian-Mixture model (2.2) and ( b ) holds since ∑ glyph[lscript] ∈[ k ] π glyph[lscript] = 1 .

B.2 Proof of Theorem 3.1 and Theorem 3.2

Recall that the look-alike estimator is defined as the min-norm estimator over the feature matrix X L , where the look-alike representations are used instead of individual sensitive features; see (3.2).

To analyze risk of ̂ θ L , we consider the ridge regression estimator given by

The minimum-norm estimator is given by ̂ θ L = lim λ → 0 + ̂ θ λ .

We follow the CGMT framework explained in Section A. Recall that

and therefore by substituting for y , X , and X L , we get

We define the primary optimization loss as follows:

We continue by deriving the auxiliary optimization (AO) problem. By duality, we have

Note that the above is jointly convex in ( θ s , θ ns ) and concave in v , and the Gaussian matrix Z is independent of everything else. Therefore, the AO problem reads:

where g s , g ns ∈ R n and h s ∈ R p , h ns ∈ R d -p are independent Gaussian random vectors with i.i.d N ( 0 , 1 ) entries.

We next fix norm of ∥ v ∥ glyph[lscript] 2 = β , and maximize over its direction to obtain

where we used that g s , g ns ∈ R n have independent Gaussian entries. Here, g ∈ R n has i.i.d entries from N ( 0 , 1 ) . Next, note that the above optimization over β has a closed form. Using the identity max β ≥ 0 ( βx -β 2 / 2 ) = x 2 + / 2 , with x + = max ( x, 0 ) , we get

Scalarization of the auxiliary optimization (AO) problem. We next proceed to scalarize the AO problem. Consider the singular value decomposition

with U s ∈ R p × r , Σ s ∈ R r × r , V s ∈ R k × r , where r = rank ( M s ) ≤ k . Decompose q s ∶= θ 0 , s -θ s in its projections onto the space spanned by the columns u 1 , s , . . . , u r, s of U s , and the orthogonal component:

where ∥ q ⊥ s ∥ glyph[lscript] 2 = 1 , α 0 ≥ 0 , and U T s q ⊥ s = 0 . Using the shorthand α = ( α 1 , . . . , α r ) , we write

In addition,

Similarly, we define q ns = θ 0 , ns -θ ns and consider the singular value decomposition

with U ns ∈ R ( d -p )× t , Σ ns ∈ R t × t , V ns ∈ R k × t , where t = rank ( M ns ) ≤ k . Decomposing q ns in its projections on the orthogonal columns u 1 , ns , . . . , u r, ns of U ns , and the orthogonal component we write

with ∥ q ⊥ ns ∥ glyph[lscript] 2 = 1 , γ 0 ≥ 0 , and U T ns q ⊥ ns = 0 . Define γ = ( γ 1 , . . . , γ t ) . In this notation, we have

Also, ∥ θ 0 , ns -θ ns ∥ glyph[lscript] 2 = ∥ q ns ∥ glyph[lscript] 2 = √ γ 2 0 +∥ γ ∥ 2 glyph[lscript] 2 . In addition,

Using the above identities in (B.1), we have

By the above characterization, minimization over θ s and θ ns reduces to minimization over α 0 , γ 0 , α , γ , q ⊥ s and q ⊥ ns . Further, these variables are free from each other and can be optimized over separately. For q ⊥ s , there is only one term involving this variable and therefore, minimization over it reduces to

For q ⊥ ns , we note that there are two terms involving this variable, namely ⟨ h ns √ n , q ⊥ ns ⟩ and ⟨( U ⊥ ns ) T θ 0 , ns , q ⊥ ns ⟩ . Since ∥ q ⊥ ns ∥ glyph[lscript] 2 = 1 , it is easy to see that the optimal q ⊥ ns should be in the span of h ⊥ ns and ( U ⊥ ns ) T θ 0 , ns . In addition,

by the law of large numbers. In words, these two vectors are asymptotically orthogonal. Hence, we can consider the following decomposition of the optimal q ⊥ ns :

where ξ ≥ 0 and h ⊥ ns denotes the projection of h ns onto the (left) null space of U ns . This brings us to

Note that at this stage, the AO problem is reduced to an optimization over r + t + 3 scalar variables ( α 0 , γ 0 ≥ 0 , 0 ≤ ξ ≤ 1 and α ∈ R r , γ ∈ R t ).

Convergence of the auxiliary optimization problem. We next continue to derive the point-wise in-probability limit of the AO problem.

First observe that since ε and g are independent with i.i.d N ( 0 , 1 ) entries, we have

where ˜ g ∈ R n has i.i.d N ( 0 , 1 ) entries.

Second, by construction ΛΛ T = diag ( n 1 , . . . , n k ) ∈ R k × k , where n glyph[lscript] denotes the number of examples from cluster glyph[lscript] . Hence,

Next, by using concentration of Lipschitz functions of Gaussian vectors, we obtain

Also, since ∥ θ 0 , s ∥ glyph[lscript] 2 is bounded and ∥ u i, s ∥ glyph[lscript] 2 = 1 , we get

In addition, ∥ h ⊥ ns ∥ glyph[lscript] 2 concentrates around √ d -p -t and ( d -p -t )/ n → ψ d -ψ p , because t ≤ k remains bounded as n diverges, and so

Using the above limits, the objective in (B.4) converges in-probability to

We are now ready to prove the theorems.

B.2.1 Proof of Theorem 3.1

Using Lemma 2.1, we have

Since ψ d -ψ p ≤ 1 , we are in the over- determined (a.k.a underparametrized) regime. As λ → 0 + , the terms involving λ become negligible compared to the first term in (B.5) except those that include α 0 , as α 0 is not present in the first term . Since ( x ) 2 + is increasing, and

the minimum over α and γ is achieved for α = 0 ∈ R r and γ = 0 ∈ R t . The optimization (B.5) then reduces to

The optimal ξ is given by ξ = 1 . Also, setting derivative with respect to α 0 to zero we obtain the optimal α 0 = ∥( U ⊥ s ) T θ 0 , s ∥ glyph[lscript] 2 . Next, by setting derivative with respect to γ 0 we arrive at

Using the optimal variables in (B.6) we obtain the risk of minimum-norm estimator as

Recall that by assumption, r s = ∥ θ 0 , s ∥ glyph[lscript] 2 and ∥ U T s θ 0 , s ∥ glyph[lscript] 2 = √ ρr s , which completes the proof.

B.2.2 Proof of Theorem 3.2

We continue from (B.5). In the case of ψ d -ψ p ≤ 1 , it is easy to see that the derivative of the first term of (B.5), in the active region is decreasing in γ 0 . With the consideration λ → 0 + , minimizing over γ 0 will push us into the non-active region. Therefore the optimization problem (B.5) reduces to

subject to

By Assumption 2, Σ s = µ I k , V s = I k × k , and Σ ns = 0 , U ns = 0 (no cluster structure on non-sensitive features and an orthogonal, equal energy cluster centers on the sensitive features). Therefore, by fixing γ ∶= ∥ γ ∥ glyph[lscript] 2 , the optimization problem (B.8) becomes:

Since α 0 does not appear in the constraint, it is easy to see that its optimal value is given by α 0 = ∥( U ⊥ s ) T θ 0 , s ∥ glyph[lscript] 2 . Also, note that by decreasing γ the objective value decreases and also by the constraint on the other variables become more relaxed. Consequently, the optimal value of γ is γ = 0 . Removing α 0 from the objective function, we are left with

Optimal choice of ξ results in the constraint to become equality. Solving for ξ , the optimization reduces to

Setting derivative with respect to γ 0 to zero, we obtain

Setting derivative with respect to α to zero and using the previous stationary equation, we get

We next square both sides of (B.12) and rearrange the terms to get

which are the same expressions for α and γ 0 given in the theorem statement.

The final step is to write the risk of estimator in terms of α , γ 0 . Invoke equation (B.6), and recall that in the current case, Σ ns = 0 , Σ s = µ I . Also, as we showed in our derivation, γ = ∥ γ ∥ glyph[lscript] 2 = 0 , α 0 = ∥( U ⊥ s ) T θ 0 , s ∥ glyph[lscript] 2 , by which we arrive at

This concludes the proof.

B.3 Proof of Theorem 3.3

We follow the proof strategy used for Theorem 3.1-3.2. Here, we would like to characterize the risk of min-norm estimator ̂ θ . The features matrix has a clustering structure, but the learner is not using that (no look-alike clustering) and is just compute the min-norm estimator for fitting the responses to individual features. Therefore, one can think of this setting as a special case of our previous analysis when there is no sensitive features (so ψ p = 0 ).

( a ) By setting ψ p = 0 and r s = 0 in the result of Theorem 3.1, we get that when ψ d ≤ 1 ,

( b ) In this case, we specialize the proof of Theorem 3.2 to the case that ψ p = 0 . Continuing from (B.8), and removing the terms corresponding to sensitive features, we arrive at

We drop the index 'ns' as it is not relevant in this case. Also by Assumption 2, Σ ns = µ I d , V ns = I d . Therefore, the above optimization can be written as

Optimal ξ makes the constraint equality. Solving for ξ , the above optimization can be written as so we have minimize

Setting the derivative with respect to γ 0 to zero, we get

Setting derivative with respect to γ to zero and using the above equation, we obtain

We next square both sides of equation (B.16), and rearrange the terms to get:

Under the simplifying Assumption 2, there is no cluster structure on the non-sensitive features and so U ns = 0 . Therefore,

We next proceed to compute the risk of estimator in terms of γ , γ 0 . We use equation (B.6), which for the min-norm estimator with no look-alike clustering, reduces to

This concludes the proof. Note that in the theorem statement we made the change of variables γ 0 → ˜ γ 0 and γ → ˜ α , for an easier comparison with the risk of look-alike estimator.)

B.4 Proof of Proposition 3.4

Consider singular value decompositions X L = U Σ V T and ˜ X L = ̃ U ̃ Σ ̃ V T . We then can write the estimators ̃ θ L and ̂ θ L as follows:

We first bound ∥ ̂ θ L - ̃ θ L ∥ . We write

Note that Z T θ 0 + ε ( d ) = √ ∥ θ 0 ∥ 2 + σ 2 g where g ∼ N ( 0 , I n ) . In addition,

Therefore by using concentration of Lipschitz functions of Gaussian vectors, we get

This shows that

We next use the result of [28, Theorem 3.3], by which we obtain

Note that

by the assumption of the theorem statement. We next lower bound σ min ( Σ ) = σ min ( X L ) . Recall that X T L = ( M Λ ) T +[ 0 n × p , Z n ×( d -p ) ] , with Z having i.i.d N ( 0 , 1 ) entries.

Next suppose that Condition ( i ) holds true, namely δ < √ 1 -( ψ d -ψ p )- √ ψ d -ψ p , with ψ d -ψ p < 0 . 5 . Using the result of [31, Theorem 2.1], we have with probability at least 1 -n -1 ,

We have

Furthermore,

using the assumption on the estimation error rate δ . Therefore, using the above bound along with (B.22) in (B.21) we get

Combining the above bound with (B.20), we get

We next note that by triangle inequality, the above bound implies that

Therefore, by invoking Lemma 2.1, we obtain the desired result on Risk ( ̃ θ L ) .

Next suppose that Condition ( ii ) holds, namely δ < √ ψ d -ψ p -1 -1 with ψ d -ψ p > 2 . Using the result of [31, Theorem 2.1] for X T , we have with probability at least 1 -n -1 ,

By following a similar argument we prove the claim under Condition ( ii ) .

B.5 Proof of Theorem 5.1

We use Theorem 3.3 (b) to characterize Risk ( ̂ θ ) in the regime of ψ d ≥ 1 . Specializing to the case of balanced cluster priors, the risk depends on ˜ α only through its norm ˜ α ∶= ∥ ˜ α ∥ glyph[lscript] 2 , and is given by with

In addition, by Theorem 3.1 we have

Note that Risk ( ̂ θ L ) in this regime does not depend on µ 2 / k . Also, it is easy to verify that Risk ( ̂ θ ) is decreasing in µ 2 / k . Therefore the gain ∆ is decreasing in µ 2 / k .

Also observe that Risk ( ̂ θ ) is increasing in r ns , while Risk ( ̂ θ L ) does not depend on r ns . Therefore, the gain ∆ is increasing in r ns .

To understand the dependence of ∆ on ρ , we write

As we see the numerator is increasing in ρ and denominator is decreasing in ρ , which implies that the gain ∆ is increasing in ρ .

We next show that ∆ ≥ 1 if condition (5.1) holds. Since ∆ is decreasing in µ 2 / k and increasing in ρ , it suffices to show the claim assuming µ 2 / k →∞ and ρ = 0 . In this case we have ( µ 2 k + 1 ) ˜ α 2 → 0 and so

where the last step follows from condition (5.1).

Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis work considers linear regression on data $(x_i, y_i)_{i=1}^n$ where each $x_i \\in \\mathbb{R}^d$ is sampled from one of $k$ Gaussian clusters, each with probability $\\pi_k$, and $y_i = x_i ^T \\theta_0 + \\varepsilon_i$ with $\\varepsilon_i \\sim N(0, \\sigma^2)$ representing noise. The goal of this paper is to characterize and compare the out-of-sample expected squared error of the minimum 2-norm regression predicter with that of the so called ``look-alike\" predicter. The look-alike predicter is the minimum 2-norm predicter after one replaces the first $p$ features of each point with the first $p$ features of their respective cluster average: to motivate this consider the first $p$ features as sensitive in some manner, aggregation therefore provides a degree of anonymization for each data point. The min-norm and look-alike predicter are then analyzed in the asymptotic regime $d,p,n \\rightarrow \\infty$ with $d/n \\rightarrow \\varphi_d$ and $p/n \\rightarrow \\varphi_p$. The min-norm and look-alike estimators are said to be in the underparameterized regime if $\\varphi_d\\leq 1$ and $\\varphi_d - \\varphi_p \\leq 1$ respectively, and are otherwise considered overparameterized. In addition, if $\\theta_s$ denotes the first $p$ (sensitive) entries of $\\theta_0$, then the Signal to Noise Ratio SNR is defined as $||\\theta_s || / \\sigma$. This work identifies a number of scenarios in which the look-alike predicter outperforms the min-norm predicter, in particular when the SNR is low then the underparmaterized look-alike predicter outperforms the min-norm predicter.\n\nSTRENGTHS:\nThis paper illustrates, albeit in a specific setting, that anonymization by aggregating sensitive features does not necessarily lead to a drop in the performance of the predictor, and can in certain settings actually lead to improvements. The results appear technically sound and are based on a nice use of the Convex-Gaussian Minimax Theorem. I did not go through all of the appendix or re-derive any of the results line-by-line. Synthetic data is used to support and validate the results numerically. The paper is also well written and structured. I am not an expert in this space so cannot comment really on the novelty of the work.\n\nWEAKNESSES:\nThe biggest weaknesses to my mind is perhaps the specificity of the data model, both in terms of the data model. Little discussion as to the feasibility of extending the takeaways to more general settings is provided. As a small note I also think greater care is perhaps required when discussing the informal notion of user privacy used here versus more formal notions, e.g., differential privacy, which actually provide privacy guarantees.\n\nQUESTIONS:\n1. Can you comment as to the potential avenues and barriers to relaxing your results from asymptotic to big-O?\n\n2. Can you comment as to the potential for your results to be extended to more general data distributions?\n\n3. In Figure 3a) when $\\varphi_p=0.9$ then $p=d=0.9n$ and all features are sensitive. If in addition the SNR is $0$, then unless I am mistaken this implies $\\theta_0 = 0 $ and therefore $y_i = \\varepsilon_i$, i.e., the target is independent of the input features and is random noise. It seems curious then that one classifier has a better generalization than another in this setting, could you comment?\n\n3. Have you observed the conclusions of your results to actually hold on any real-world data? In particular have you identified any real world data problems where you actually see the look-alike estimator out-perform the min-norm estimator?\n\n4. I wonder if the proportional regime is the most appropriate to consider here for actual applications versus say fixed $d$ and $p$, for instance when considering financial or health records in a country. Could you comment on the how your results might change in this setting?\n\nLIMITATIONS:\nA section discussing limitations of the work is lacking. I cannot foresee any negative societal outcomes from this work. Although it seems only fairly basic numerics are presented a link to the code used to generate the plots is lacking and should probably be included.", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-07-04"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nIn this paper the authors propose a look-alike clustering technique that replaces sensitive feature of individuals with the cluster’s average value – the cluster in which that individual belongs to. The authors provide precise analysis of how replacing sensitive features with cluster center value affect the generalization of the linear regression model by comparing under-parameterized and over-parameterized regimes in the asymptotic setting. Numerical experiments are provided that matches the asymptotic theory.\n\nSTRENGTHS:\n1.\tPrecise analysis of how replacing sensitive features with cluster center value affect the generalization of the linear regression model.\n2.\tExpression of risk in the under-parameterized and over-parameterized regimes and analysis of scenarios when the proposed method improves generalization by achieving lower risk while ensuring privacy.\n\nWEAKNESSES:\n1.\tWhile the introduction suggests that the developed theory clearly demonstrates the role of cluster size and number of clusters in controlling generalization error none of the theory statements presented in the main paper explicitly shows that dependence. While looking at the supplementary material, it is clear that after simple algebraic manipulation such effect can be seen (e.g., in part (b) of Theorem 3.3) it may be a better idea to reorganize the theory statement where such dependence is clearly visible.\n2.\tIn theorem 3.1 what is $\\rho$? While later at somewhere else it is mentioned that it captures the alignment of the model with the left singular vectors of the cluster centers can it be interpreted as $U^T_s\\theta_{0,s}=\\sqrt{\\rho}\\theta_{0,s}$ which yields $|| U^T_s\\theta_{0,s}||=\\sqrt{\\rho}r_s$?\n3.\tFor case 1 in section 5, $\\psi_d=0.9$. If we look at figure 3(a), among various values of $\\psi_p$, $\\psi_p=0.9$ tolerates largest SNR while still ensuring $\\Delta>1 (\\log (\\Delta) >0)$. However, since $\\psi_d=0.9, \\psi_p=0.9$ means all the features are sensitive and yet it tolerates the largest SNR and improves generalization. Does that mean, irrespective of sensitiveness, it is always a good strategy to replace all features by their cluster centers to achieve better generalization?\n4.\tIf we compare Figure 3(a) and 3(b) and look at what SNR values the red curve hits zero for $\\rho=0.3$ and $\\psi_p=0.5$, we get very different values. On Fig 3(a), this SNR value is a little less than 2 while on Fig 3(b), this SNR value is close to 2.5. Since both axes in Fig 3(a), 3(b) are identical, why is this discrepancy?\n5.\tIn figure 2, the six graphs are organized in a 2x3 window without clear “left” and “right” panels as mentioned in the figure caption. Reorganizing them in 2x3 window will have two panels and will match the figure caption.\n6.\tFew typos in the statement of Proposition 3.\n       a.\t $\\tilde{X}^T$ should be $\\tilde{X}^T_L$\n       b.\t  It seems $\\delta_n$ and $\\delta$ represent same thing (cluster estimation error rate). This needs to be fixed.\n7.\tIn line 200, “underparameterized” should be “overparameterized”.\n\nQUESTIONS:\n1.\tSee weakness.\n\nLIMITATIONS:\nLimitations are not clearly mentioned. To establish concrete expression of the risk, Assumption 2 is very simplistic or the isotropic Gaussians in the GMM is very simplistic. It may be a good idea to add a specific section/subsection mentioning the limitations.", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 2, "SOUNDNESS_CORRECTNESS": 2, "DATE": "2023-07-06"}

comments: SUMMARY:
This work presents a generalization analysis for look-alike clustering. In this type of clustering, the features in a model are divided into two groups: sensitive and non-sensitive, and the values of the sensitive features are replaced by the mean of the cluster to ensure K-anonymity (if the size of the cluster is K). They provide an analysis of generalization in the under-parameterized and over-parametrized regimes.

STRENGTHS:
The paper brings in generalization analysis for look-alike clustering, i.e., clustering under the assumptions of some features being held fixed at their mean. They provide some simulations as well, showing the under-parametrized and over-parametrized regimes.

WEAKNESSES:
It would be good to provide more intuitive insights on how different factors affect the generalization, e.g., the number of sensitive and nonsensitive features. My intuition is that as the number of sensitive features increases, the generalization should get worse until it fails as they approach the total number of features?

I do not understand this assumption: "We focus on an asymptotic regime where the size of the training set grows in proportion to the features dimension." For a complete picture, it is best to keep them as separate variables, and if required, consider a substitution at the end in a corollary or something. This assumption takes away important insights related to the relationship between number of features and size of training set.

Experiments are mostly on synthetic data but it may be okay since the focus is mostly theoretical.

Clustering assumptions are based on linear models.

QUESTIONS:
Clarify what happens if number of features are not equal to number of samples?

How do number of sensitive features affect generalization?

LIMITATIONS:
NA
TITLE: Official Review
SOUNDNESS_CORRECTNESS: 3
DATE: 2023-07-06

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper extends look-alike clustering to anonymize sensitive features of data points by replacing them with cluster means. It provides a theoretical analysis of the generalization error of models (linear regression estimator) trained using the anonymized sensitive features in the asymptotic regime. The paper also shows that anonymizing sensitive features through look-alike clustering can improve model generalization by acting as a regularization and mitigating overfitting when the signal-to-noise ratio (ratio of the energy of the model on sensitive features to the response noise) is below a certain threshold under different parametrization regimes.\n\nSTRENGTHS:\n1. The paper studies an interesting area of anonymizing private/sensitive user information, drawing on recent research that examines the memorization of training data samples.\n\n2. The theoretical analysis presented in the paper is intriguing as it explores scenarios where learning from look-alike clustering-based anonymized data can potentially improve model generalization.\n\n3. The paper is well-written with the majority of sections being clear and coherent.\n\nWEAKNESSES:\n1. The current draft lacks coverage of related works on other data anonymization methods. It would be helpful for the readers if the authors included a literature survey in either the main paper or the appendix.\n\n2. The paper lacks comprehensive empirical experiments on real-world datasets (of any scale).\n\n3. Limitations of the theoretical analysis presented in the paper are also missing.\n\n4. Perturbation analysis about the upper bound of the look-alike estimator's risk (Proposition 3.4) is not covered in the over-parametrized regime.\n\nQUESTIONS:\n1. Please refer to the comments in the weaknesses section.\n\n2. Besides the generalization bounds, do the authors have any insights about the DP guarantees of their proposed look-alike anonymization approach?\n\nMinor recommendations:\n\n1.  The authors should specify that proposition 3.4 holds true for the look-alike estimator in the under-parametrized regime.\n\nReference for k-anonymity :\n\n1.  Sweeney, Latanya. \"k-anonymity: A model for protecting privacy.\" International journal of uncertainty, fuzziness and knowledge-based systems 10.05 (2002): 557-570.\n\t\nRelated works covering memorization of sensitive information in neural networks and can provide valuable insights into the problem motivation:\n1. Song, Congzheng, and Vitaly Shmatikov. \"Overlearning reveals sensitive attributes.\" arXiv preprint arXiv:1905.11742 (2019).\n\n2. Feldman, Vitaly, and Chiyuan Zhang. \"What neural networks memorize and why: Discovering the long tail via influence estimation.\" Advances in Neural Information Processing Systems 33 (2020): 2881-2891.\n\n3. Stephenson, Cory, et al. \"On the geometry of generalization and memorization in deep neural networks.\" arXiv preprint arXiv:2105.14602 (2021).\n\n4. Malekzadeh, Mohammad, Anastasia Borovykh, and Deniz Gündüz. \"Honest-but-curious nets: Sensitive attributes of private inputs can be secretly coded into the classifiers' outputs.\" Proceedings of the 2021 ACM SIGSAC Conference on Computer and Communications Security. 2021.\n\nLIMITATIONS:\nI could not find a section addressing the limitations and negative societal impact of this work.", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-08-01"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper studies linear regression where some coordinates of the covariate x are not revealed directly to the learner, and only a cluster-wise average is revealed for those coordinates. This change in the covariates x is called look-alike clustering and it has been used for protecting sensitive attributes in prior work. While it seems that with look-alike clustering, less information is provided to the learner, the paper shows that in certain regimes the performance of linear regression improves after look-alike clustering. As the authors explain, the reason behind this surprising phenomenon is that look-alike clustering can reduce overfitting.\n\nMore concretely, this paper assumes that the data points (x_i, y_i) are generated iid from a linear model y_i = < x_i, theta > + eps_i with mean-zero Gaussian noise eps_i. The distribution of the covariate x is a mixture of k Gaussian distributions each with its covariance matrix being the identity matrix. Look-alike clustering then corresponds to replacing a subset of the coordinates of each x_i by the coordinates of the mean of the Gaussian distribution that generates x_i. Those coordinates are called sensitive features.\n\nFor d dimensional x with p sensitive features, the authors study the asymptotic performance of linear regression with n data points as d, p, n all tending to infinity and the ratios d/n, p/n tending to constants. The number of clusters, k, is fixed, and performance is measured using the (population) mean squared error. The learner estimates the parameter theta in the linear model by solving least squares on the data points. When there are multiple solutions (e.g. when n < d) the learner chooses the one with the minimum l_2 norm $\\\\|\\theta\\\\|_2$.\n\nThe authors show closed-form formulas for the asymptotic performance with and without look-alike clustering. This allows the authors to identify regimes when look-alike clustering improves the performance. They also included experimental results verifying the formulas. While the formulas are only shown to hold asymptotically, even when d, p, n are only a few hundreds the experimental results already closely match the formulas. The authors also extend the formulas to the case where we do not know the means of the k Gaussians when performing look-alike clustering and we instead use estimates for these means.\n\nSTRENGTHS:\nThe authors chose an interesting problem to study: the effect of look-alike clustering on generalization. The exact formulas on the performance of linear regression established in this paper are technically challenging to obtain, for which the authors use tools such as the Convex Gaussian Minimax Theorem. The paper is well written. The proofs in the supplementary are easy to follow and they look correct and complete.\n\nWEAKNESSES:\n1. Unless I am missing some parts of the paper, the authors do not provide any (empirical) examples beyond linear regression where look-alike clustering is helpful for performance. It would be great to have such examples as they may suggest that the analysis in this paper could explain a general phenomenon appearing not just in linear regression. Currently, the impact of the paper is somewhat limited to the linear regression setting.\n\n2. In proposition 3.4 where look-alike clustering uses estimated means of the Gaussians, a requirement seems to be that we can accurately estimate the Gaussian mean of every data point. When we do not know which cluster each point comes from, this seems quite challenging especially when the Gaussians are not well separated. It is important for the authors to provide a clear explanation on how to obtain such accurate estimates (perhaps using prior results on learning Gaussian mixtures).\n\n3. Starting at Line 297, the authors provide an explanation for the improved generalization from look-alike clustering. Indeed, with the main results being presented using math formulas, a high-level explanation is much needed. However, I am not able to fully understand the authors' explanations at Line 297. The authors wrote that look-alike clustering drops the <z_s, theta_s> term, but my understanding is that the y in look-alike clustering is generated using the original x and thus still contains the <z_s, theta_s> term, which means that the authors' description is inaccurate. It is also unclear how the authors' explanation is related to regularization.\n\n    Here is my own understanding of why look-alike clustering can improve generalization. The authors can comment on whether this makes sense. After look-alike clustering, each x_i, when restricted to the sensitive features, can only be chosen from k possible vectors, and thus the learned parameter theta, when restricted to the sensitive features, must be in the space spanned by the k vectors (since the learner picks the minimum norm theta). This low-dimension restriction is a form of regularization and can reduce overfitting. This does not explain why lower SNR leads to larger performance gain, which I feel is not explained well in the authors' explanation either.\n\nQUESTIONS:\n1. The explanation at Line 297 seems to be presented only for Case 2, but I don't see why this should be the case.\n2. It seems that the assumed upper bound on delta at Line 237 could be negative, in which case the assumption cannot be satisfied. I would appreciate some comments on this.\n3. It would be great if the authors can provide their feedback on the weaknesses session of the review.\n\nTypos:\n\n- Line 29: leaner -> learner\n\n- Line 30: that that -> that\n\n- Line 133: belong -> belongs\n\nLIMITATIONS:\nThe authors adequately stated all the assumptions needed for obtaining their results. In terms of negative social impact, perhaps one question is whether look-alike clustering may lead to stereotyping each cluster (e.g. when the clusters are demographic groups). It would be great if the authors could discuss this.", "TITLE": "Official Review", "RECOMMENDATION": 7, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 4, "DATE": "2023-08-02"}