Local Differential Privacy for Mixtures of Experts

Venue: NeurIPS_cc_2024_Conference
Published: 2024-05-14 17:08:37

Abstract: We introduce a new approach to the mixture of experts model that consists in imposing local differential privacy on the gating mechanism. This is theoretically justified by statistical learning theory. Notably, we provide generalization bounds specifically tailored for mixtures of experts, leveraging the one-out-of-$n$ gating mechanism rather than the more common $n$-out-of-$n$ mechanism.  Moreover, through experiments, we show that our approach improves the generalization ability of mixtures of experts.

Main paper: 
 introduction: 
 Mixtures of experts, initially introduced by Jacobs et al. [1991] , have found widespread use in modeling sequential data, including applications in classification, regression, pattern recognition and feature selection tasks (Städler et al. [2010] and Khalili and Lin [2013] ). One of the fundamental motivations behind mixtures of experts is their ability to break down complex problems into more manageable sub-problems, potentially simplifying the overall task. The structure of these models is well suited to capturing unobservable heterogeneity in the data generation process, dealing with this problem by splitting the data into homogeneous subsets (with the gating network) and associating each subset with an expert. This intuitive architecture has led to significant interest in mixture of experts models, resulting in a wealth of research (Yuksel et al. [2012] ), ranging from simple mixtures of experts ( Jacobs et al. [1991] , Jordan and Jacobs [1993] ) to sparsely gated models (Shazeer et al. [2017] ). Moreover, this architecture has inspired the development of various other models, such as switch transformers (Fedus et al. [2022] ). However, despite the considerable attention mixtures of experts have received, advancements in their theoretical analysis have been relatively limited. Azran and Meir [2004] proved data-dependent risk bounds for mixtures of experts (with the n-out-of-n gating mechanism) using Rademacher complexity, but they exhibit a dependence on the complexity of the class of gating networks and the sum of the complexities of the expert classes, which reflects the complex structure of mixtures of experts but unfortunately leads to potentially large bounds. We are not aware of other work proving generalization bounds specifically tailored to mixtures of experts.To make theoretical progress, we utilize a well-known privacy-preserving technique called Local Differential Privacy (LDP). It was initially introduced by Dwork [2006] and has since been widely used to preserve privacy for individual data points as in Kasiviswanathan et al. [2010] . This is achieved by introducing stochasticity in algorithm outputs to control their dependence on specific inputs. This stochasticity is generally quantified by a positive real number ϵ. In this case, we write ϵ-LDP instead of just LDP. The parameter ϵ quantifies the level of privacy protection in the local differential privacy mechanism. A smaller value indicates stronger privacy protection, which requires the addition of more noise.In this work, we exploit this noise for regularization in our models by imposing the ϵ-LDP condition on their gating networks. This method allows us to leverage the numerous benefits of the most complex architectures, such as neural networks, without compromising theoretical guarantees on risk.By relying on LDP, we offer tight theoretical guarantees on the risk of mixtures of experts models, Submitted to 38th Conference on Neural Information Processing Systems (NeurIPS 2024). Do not distribute.provided with the one-out-of-n gating mechanism. Unlike the very few existing guarantees, these bounds depend only logarithmically on the number of experts we have, and the complexity of the gating network only appears in our bounds through the parameter ϵ of the LDP condition.Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?Answer: [Yes] Justification: Our claims are supported by theorems, which we prove, and by experiments.Guidelines:• The answer NA means that the abstract and introduction do not include the claims made in the paper.• The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers.• The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings.• It is fine to include aspirational goals as motivation as long as it is clear that these goals are not attained by the paper. 
preliminaries: 
 Let X be the instance space, Y the label space, and Y ′ the output space (which can be different from Y). As is usual in supervised learning, we assume that data (x, y) ∈ X × Y are generated independently from an unknown probability distribution D. We consider a training set of m examples S = ((x 1 , y 1 ), . . . , (x m , y m )) ∼ D m and a bounded loss function ℓ :Y ′ × Y → [0, 1].We consider classes H i of experts h i : X → Y ′ for i = 1, . . . , n. Let G be a set of gating functions g : X → [0, 1] n such that, given any x ∈ X , we have thatn i=1 g i (x) = 1, where g i (x) is the i-th component of g(x). This means that each gating function defines a probability distribution on[n] = {1, . . . , n} for each x ∈ X , where g i (x) is the probability of i.In this work, a mixture of experts consists of n experts,h = (h 1 , . . . , h n ) ∈ H 1 × • • • × H n , agating function g ∈ G and a gating mechanism that combines the outputs of the experts and the output of the gating function to produce the final output. Our models use the stochastic one-out-of-n gating mechanism, as described in Jacobs et al. [1991] . It is defined as follows: to make a prediction with (g, h) ∈ G × n i=1 H i given an instance x, draw i ∼ g(x) and output h i (x). This stochastic predictor has risk and empirical risk defined by, respectively,R(g, h) = E (x,y)∼D E i∼g(x) ℓ(h i (x), y), and R S (g, h) = 1 m m j=1 E i∼g(xj ) ℓ(h i (x j ), y j ).The preference for the one-out-of-n gating mechanism over the n-out-of-n mechanism in mixtures of experts is justified by its ability to induce sparsity and noise, enhancing computational efficiency and robustness to overfitting. This sparsity also offers scalability benefits, particularly in large-scale applications, where activating all experts for each input can lead to increased computational and memory requirements as explained in Shazeer et al. [2017] and Jacobs et al. [1991] . Moreover, the one-out-of-n mechanism is more amenable to certain kinds of theoretical analysis, including ours.Definition 2.1. Let I be a finite set, consider a mechanism that produces an output i ∈ I, given an input x ∈ X , with probability P(i | x), and let ϵ be a nonnegative real number. Then, the mechanism satisfies the ϵ-Local Differential Privacy (ϵ-LDP) property if and only ifP(i | x) ≤ e ϵ P(i | x ′) for all x, x ′ ∈ X and all i ∈ I.Unless stated otherwise, we assume that each g ∈ G satisfies ϵ-LDP, for some fixed nonnegative real number ϵ. Since we can interpret g as a random mechanism that, given x ∈ X , selects i ∈ [n] with probability g i (x), the condition of ϵ-LDP amounts to the following:g i (x) ≤ e ϵ g i (x ′ ) for all x, x ′ ∈ X and all i ∈ [n].Since ϵ-LDP is an important condition for all of our theoretical results, we provide a practical way of obtaining gating functions satisfying ϵ-LDP from an arbitrary set F of bounded functions, in the form of the following theorem.Theorem 2.2. Let b > 0 and β ≥ 0 be real numbers, and suppose that F is a set of functionsf : X → [-b, b] n . Let G be the set of functions g : X → [0, 1] n defined by g i (x) = exp(βf i (x) + c i ) n k=1 exp(βf k (x) + c k ) , where f = (f 1 , . . . , f n ) ∈ F and (c 1 , . . . , c n ) ∈ R n .Then, each g ∈ G satisfies 4βb-LDP.Proof. The proof is obtained by performing simple calculations, bounding the ratio g i (x)/g i (x ′ ), for all x, x ′ ∈ X and all i ∈ [n]. The detailed proof is given in Appendix A.Question: Does the paper discuss the limitations of the work performed by the authors?Answer: [Yes] Justification: We discuss the limitations of the work in the conclusion section.Guidelines:• The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those are not discussed in the paper.• The authors are encouraged to create a separate "Limitations" section in their paper.• The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification, asymptotic approximations only holding locally). The authors should reflect on how these assumptions might be violated in practice and what the implications would be.• The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on a few datasets or with a few runs. In general, empirical results often depend on implicit assumptions, which should be articulated.• The authors should reflect on the factors that influence the performance of the approach.For example, a facial recognition algorithm may perform poorly when image resolution is low or images are taken in low lighting. Or a speech-to-text system might not be used reliably to provide closed captions for online lectures because it fails to handle technical jargon.• The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size.• If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and fairness.• While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, a worse outcome might be that reviewers discover limitations that aren't acknowledged in the paper. The authors should use their best judgment and recognize that individual actions in favor of transparency play an important role in developing norms that preserve the integrity of the community. Reviewers will be specifically instructed to not penalize honesty concerning limitations. 
pac-bayesian bounds for mixtures of experts: 
 To apply the PAC-Bayes theory, we need to add a level of stochasticity to our predictors: instead of training experts h i , we train probability measures Q i on each expert set H i . For convenience, we writeQ = Q 1 ⊗ • • • ⊗ Q n .Now, putting everything together, a mixture of experts (g, Q) makes predictions as follows: given x ∈ X , draw i ∼ g(x), then draw h ∼ Q i , and finally output h(x). Such a predictor has risk and empirical risk defined by, respectively,R(g, Q) = E h∼Q R(g, h) and R S (g, Q) = E h∼Q R S (g, h).Notice that, though probability distributions have replaced the individual experts, there is no need to define a probability distribution on the gating functions to get a PAC-Bayesian bound. Training a single gating function will do, and, remarkably, Lemma 3.1 below shows that it can be obtained from a very complicated function, such as a neural network, provided we impose ϵ-LDP (for example, with Theorem 2.2).Finally, let us recall the notion of Kullback-Leibler (KL) divergence. Given probability distributions Q i and P i on H i , it is defined byKL(Q i ∥ P i ) =    E h∼Qi ln dQ i dP i (h) if Q i ≪ P i ∞ otherwise,where dQ i /dP i is a Radon-Nikodym derivative.Lemma 3.1. We consider mixtures of experts as defined in section 2.1 and provided with the oneout-of-n routing mechanism. Let ∆ : R 2 → R be a convex function that is decreasing in its first argument and increasing in its second argument, and let ϵ be a nonnegative real number. Then, for any g ∈ G that satisfies the ϵ-LDP property, for anyQ = Q 1 ⊗ • • • ⊗ Q n on H 1 × • • • × H n , and for any x ′ ∈ X : ∆ e ϵ R S (g, Q), e -ϵ R(g, Q) ≤ E i∼g(x ′ ) ∆ R S (Q i ), R(Q i ) where R(Q i ) = E x∼D E h∼Qi ℓ(h(x), y) and R S (Q i ) = 1 m m j=1 E h∼Qi ℓ(h(x j ), y j ).Proof. Since the gating function satisfies ϵ-LDP, we have that e -ϵ g i (x ′ ) ≤ g i (x) ≤ e ϵ g i (x ′ ) for all x, x ′ ∈ X and all i ∈[n]. It follows that e ϵ R S (g, Q) ≥ E i∼g(x ′ ) R S (Q i ) and e -ϵ R(g, Q) ≤ E i∼g(x ′ ) R(Q i ).Given that ∆ is decreasing in its first argument and increasing in its second argument, we find that∆ e ϵ R S (g, Q), e -ϵ R(g, Q) ≤ ∆ E i∼g(x ′ ) R S (Q i ), E i∼g(x ′ ) R(Q i )Since ∆ is a convex function, we can apply Jensen's inequality to the expression on the right-hand side, yielding the desired result.Different choices of function ∆ will allow us to obtain different PAC-Bayes bounds:• Let ∆(u, v) = v -u. This is compatible with typical PAC-Bayes bounds on the difference between the true and empirical risks.• Given λ > 1/2, let ∆ be defined by ∆(u, v) = v -2λ 2λ-1 u. This choice is compatible with a Catoni-type bound, as we will see below.• Let ∆ be defined by∆(u, v) = kl(u ∥ v) = u ln u v + (1 -u) ln 1-u 1-v .This choice is compatible with a Langford-Seeger-type bound. However, note that the function ∆ defined here does not quite obey the hypotheses of lemma 3.1. Indeed, it is only defined for (u, v) ∈ [0, 1] 2 , and only has the right monotonicity properties on the set{ (u, v) ∈ [0, 1] 2 | u ≤ v }.We can remedy those defects through small adjustments to the proof.We prove a generalization bound of Catoni-type as an illustration of the machinery just described.Theorem 3.2 (Theorem 2 in McAllester [2013] ). Let δ ∈ (0, 1) and λ > 1/2. Fix i ∈ [n], and let P i be a probability measure on H i (chosen without seeing the training data). Then, with probability at least 1 -δ over the draws of S, for all probability measures Q i on H i , we have thatR(Q i ) ≤ 2λ 2λ -1 R S (Q i ) + λ m KL(Q i ∥ P i ) + ln 1 δ .Theorem 3.3. Let δ ∈ (0, 1), ϵ ≥ 0, and λ > 1/2. For each i ∈ [n], let P i be a probability measure on H i (chosen without seeing the training data). Then, with probability at least 1 -δ over the draws of S, for all probability measuresQ = Q 1 ⊗ • • • ⊗ Q n onH, all g ∈ G that satisfy ϵ-LDP, and allx ′ ∈ X , we have thatR(g, Q) ≤ 2λe ϵ 2λ -1 e ϵ R S (g, Q) + λ m E i∼g(x ′ ) KL(Q i ∥ P i ) + ln n δ .Proof. By n applications of Theorem 3.2, we have that, for each i ∈ [n], with probability at least1 -δ/n, for all Q i , R(Q i ) ≤ 2λ 2λ -1 R S (Q i ) + λ m KL(Q i ∥ P i ) + ln n δ .We can make all these inequalities (for each i ∈ [n]) hold simultaneously with a union bound. Now,applying Lemma 3.1 with ∆(u, v) = v -2λ2λ-1 u, we find that, with probability at least 1 -δ, for all Q, all g ∈ G and all x ′ ∈ X , we have thate -ϵ R(g, Q) - 2λe ϵ 2λ -1 R S (g, Q) ≤ E i∼g(x ′ ) R(Q i ) - 2λ 2λ -1 R S (Q i ) ≤ 2λ 2 (2λ -1)m E i∼g(x ′ ) KL(Q i ∥ P i ) + ln n δ .We also give a bound of Langford-Seeger type, since they are generally recognized as among the tightest PAC-Bayes bounds available, and to prove the flexibility of our approach.Theorem 3.4. Let δ ∈ (0, 1), ϵ ≥ 0, and m ≥ 8. For each i ∈ [n], let P i be a probability measure on H i (chosen without seeing the training data). Then, with probability at least 1 -δ over the draws of S, for all probability measuresQ = Q 1 ⊗ • • • ⊗ Q n onH, all g ∈ G that satisfy ϵ-LDP, and allx ′ ∈ X , we have that, either R(g, Q) < e 2ϵ R S (g, Q), orkl(e ϵ R S (g, Q) ∥ e -ϵ R(g, Q)) ≤ 1 m E i∼g(x ′ ) KL(Q i ∥ P i ) + ln 2n √ m δ .Proof. The proof, which is similar to that of Theorem 3.3, is available in Appendix A.Very few generalizations bound tailored specifically to mixtures of experts appear in the literature, and those we could find do not apply to mixtures of experts with the one-out-of-n gating mechanism.We can, however, compare our bounds to those obtained by naively applying generic PAC-Bayes generalization bounds to mixtures of experts. In this case, we need to consider classifiers of the form(Q G , Q), where Q G is a probability measure on G, and Q = Q 1 ⊗ • • • ⊗ Q n is a probability measure on H 1 × • • • × H n as before. Then, note that KL(Q G ⊗ Q 1 ⊗ • • • ⊗ Q n ∥ P G ⊗ P 1 ⊗ • • • ⊗ P n ) = KL(Q G ∥ P G ) + n i=1 KL(Q i ∥ P i ).This means that a generic PAC-Bayes bound applied to mixtures of experts will depend on the sum of the KL divergences corresponding to the gating functions and each of the experts. Obviously, this sum could be very large. By imposing ϵ-LDP to the gating functions as in our approach, we can eliminate the stochasticity associated to the gating functions, and rid our bounds of the (potentially very large) KL(Q G ∥ P G ) term. Instead, it is ϵ-LDP which controls our gating functions to ensure generalization. Furthermore, our bounds replace the sum of the KL divergences of the experts by a g(x ′ )-weighted average, which means we can have many more experts with almost no penalty from the theoretical point of view. Indeed, our bounds only depend on the number n of experts logarithmically, through the use of the union bound.Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof?Answer: [Yes]• The factors of variability that the error bars are capturing should be clearly stated (for example, train/test split, initialization, random drawing of some parameter, or overall run with given experimental conditions).• The method for calculating the error bars should be explained (closed form formula, call to a library function, bootstrap, etc.)• The assumptions made should be given (e.g., Normally distributed errors).• It should be clear whether the error bar is the standard deviation or the standard error of the mean.• It is OK to report 1-sigma error bars, but one should state it. The authors should preferably report a 2-sigma error bar than state that they have a 96% CI, if the hypothesis of Normality of errors is not verified.• For asymmetric distributions, the authors should be careful not to show in tables or figures symmetric error bars that would yield results that are out of range (e.g. negative error rates).• If error bars are reported in tables or plots, The authors should explain in the text how they were calculated and reference the corresponding figures or tables in the text. 
rademacher bounds for mixtures of experts: 
 Let us start with a slight modification of Lemma 3.1.Lemma 4.1. We consider mixtures of experts as defined in section 2.1 and provided with the oneout-of-n routing mechanism. Let ∆ : R 2 → R be a convex function that is decreasing in its first argument and increasing in its second argument, and let ϵ be a nonnegative real number. Then, for any g ∈ G that satisfies the ϵ-LDP property, for any h ∈ H , and for any x ′ ∈ X :∆ e ϵ R S (g, h), e -ϵ R(g, h) ≤ E i∼g(x ′ ) ∆ R S (h i ), R(h i ) where R(h i ) = E x∼D ℓ(h i (x), y) and R S (h i ) = 1 m m j=1 ℓ(h i (x j ), y j ).Proof. The proof is similar to that of Lemma 3.1 and is provided in Appendix A.Let us now recall the following definition. R(ℓ • H) = E S∼D m E σ σ σ sup h∈H 1 m m j=1 σ j ℓ(h(x j ), y j ),whereσ σ σ = (σ 1 , . . . , σ m ) is distributed uniformly on {-1, 1} m .Our main theorem will make use of the following well-known risk bound.Theorem 4.3 (Basic Rademacher risk bound). Given a [0, 1]-valued loss function ℓ, with probability at least 1 -δ, for all h ∈ H, we have thatR(h) ≤ R S (h) + 2R(ℓ • H) + 2 ln(2/δ) m .Theorem 4.4. Let δ ∈ (0, 1) and ϵ ≥ 0. Given a [0, 1]-valued loss function ℓ, then, with probability at least 1 -δ over the draws of S, for all h ∈ H 1 × • • • × H n , for all g ∈ G that satisfy ϵ-LDP, and all x ′ ∈ X , we have thatR(g, h) ≤ e ϵ e ϵ R S (g, h) + 2 E i∼g(x ′ ) R(ℓ • H i ) + 2 ln(2n/δ) m .Proof. By n applications of Theorem 4.3, we have that, for each i ∈ [n], with probability at least1 -δ/n, for all h i ∈ H i , R(h i ) ≤ R S (h i ) + 2R(ℓ • H i ) + 2 ln(2n/δ) m .We can make all these inequalities (for each i ∈ [n]) hold simultaneously with a union bound. Now, applying Lemma 4.1 with ∆(u, v) = v -u, we find that, with probability at least 1 -δ, for allh ∈ H 1 × • • • × H n , all g ∈ G and all x ′ ∈ X , we have that e -ϵ R(g, h) -e ϵ R S (g, h) ≤ E i∼g(x ′ ) R(h i ) -R S (h i ) ≤ E i∼g(x ′ ) 2R(ℓ • H i ) + 2 ln(2n/δ) m .Note, that the risk bound of Theorem 4.4 depends only on the average Rademacher complexity of the classes of experts instead of the sum of their Rademacher complexities. Note also that, as in the previous section, the complexity of G does not affect the risk bound. Finally, the risk bound does not hold uniformly for all values of ϵ. However, by the union bound, the theorem holds for any fixed set {ϵ 1 , . . . , ϵ k } if we replace δ by δ/k. Consequently, this suggests a learning algorithm that minimizesR S (g, h) for ϵ ∈ {ϵ 1 , . . . , ϵ k }.Also note that Lemma 4.1 allows us to obtain risk bounds for mixtures of experts as long as we have bounds on ∆ R S (h i ), R(h i ) which hold with high probability, whether they are based on Rademacher complexity, margins, VC dimension, or algorithmic stability.Following these theoretical results, we may be tempted to use a gating network satisfying ϵ-LDP to accomplish a learning task all by itself using non-adaptive experts, that is, experts h i each taking a constant value, no matter the input: h i (x) = i for all x ∈ X . In that case, each Rademacher complexity R(ℓ•H i ) is zero and we can show that Theorem 4.4 can become vacuous under reasonable circumstances.Consider, for example, the binary classification case with the 0-1 loss. In that case, we have two experts h +1 and h -1 such that h +1 (x) = +1 and h -1 (x) = -1 for all x ∈ X , and a gating network g = (g +1 , g -1 ). Then, the following holds:R S (g, h) = 1 m m j=1 E i∼g(xj ) ℓ 0-1 (h i (x j ), y j ) = 1 m m j=1 E i∼g(xj ) 1(h i (x j ) ̸ = y j ) ≥ 1 m m j=1 i∈I e -ϵ max x ′ ∈X g i (x ′ )1(h i (x j ) ̸ = y j ), with I = {+1, -1} = e -ϵ 1 m m j=1 max x ′ ∈X g -yj (x ′ ).Under the assumption that the classes are balanced, meaning that the (marginal) probability of a positive label is equal to the (marginal) probability of a negative label, we have the following:lim m→∞ 1 m m j=1 max x ′ ∈X g -yj (x ′ ) = 1 2 max x ′ ∈X g -1 (x ′ ) + max x ′ ∈X g +1 (x ′ ) ≥ 1 2 max x ′ ∈X g -1 (x ′ ) + g +1 (x ′ ) = 1 2 .It follows that, in the limit m → ∞, the risk bound of Theorem 4.4 for any g has a value of at least e ϵ /2 ≥ 1/2. Consequently, the risk bound becomes large or even vacuous in this regime, highlighting the importance of having adaptive experts of finite complexity that can drive the empirical risk to zero when they are selected by the gating network. 
experiments and results: 
 In what follows, we consider mixtures of n linear experts in binary classification tasks. Let X = R d for some positive integer d. Let S be a training set of m examples. Each expert, denoted by h i , where i ranges from 1 to n, is characterized by a weight vector w i . Given an input x ∈ X , the output of the expert h i is given by h i (x) = w i • x. We use the probit loss function ℓ = Φ, which can be seen as a smooth surrogate to the 0-1 loss function, when it is used with an argument of the form ywi•x ∥x∥ . In this case, R(g, Q) and R S (g, Q) are given by:EQUATIONwhereΦ(x) = 1 √ 2π +∞ xe -t 2 /2 dt provides the probability that a standard normal random variable is greater than a given value x.To illustrate the regularizing effect of the LDP condition, we carried out several experiments, on different datasets, by minimizing the empirical risk as defined in Equation 1. For all experiments, our models consist of mixtures of n = 100 linear experts and a gating network. The gating network is a neural network having 2 hidden layers. It is parameterized by weights W 1 ∈ R 64×d , where d is the dimension of input vectors, W 2 ∈ R 64×64 , and W 3 ∈ R n×64 , and biases b 1 ∈ R 64 , b 2 ∈ R 64 and b 3 ∈ R n . Given an input x ∈ R d , the output of the gating network g(x) = (g 1 (x), . . . , g n (x)) is computed as follows: first, we compute f 0 (x) = tanh(W 2 ReLU(W 1 x + b 1 ) + b 2 ). Then, when we want the ϵ-LDP condition to be satisfied, we ensure that the outputs are between -ϵ/4 and ϵ/4:f (x) = ϵW3f0(x) 4∥f0(x)∥∥W3∥ F if the gating network must satisfy ϵ-LDP W 3 f 0 (x)otherwise.Note that tanh is the hyperbolic tangent activation function, ReLU the Rectified Linear Unit function, ∥W 3 ∥ F the Frobenius norm of the matrix W 3 , and ∥f 0 (x)∥ the euclidean norm of the vector f 0 (x).Indeed, if we let W i 3 denote the i-th row of W 3 , then the i-th component ofW 3 f 0 (x) is W i 3 • f 0 (x) ≤ ∥W i 3 ∥∥f 0 (x)∥ ≤ ∥W 3 ∥ F ∥f 0 (x)∥, by the Cauchy-Schwarz inequality and the definition of the Frobenius norm. The reason we use the Frobenius norm instead of directly using ∥W i 3 ∥ is to preserve the proportions between the components of W 3 f 0 (x) when setting up ϵ-LDP.The final output of the gating network is given byg i (x) = exp(f i (x) + (b 3 ) i ) n k=1 exp(f k (x) + (b 3 ) k ) for all i ∈ [n].In our experiments, we ran the Stochastic Gradient Descent algorithm 10 times with a learning rate fixed to 0.1. In each experiment, we trained the model for 1000 epochs, except for the MNIST dataset, where the training duration was shortened to 300 epochs due to dataset size. We allocated approximately 75% of the data to the training set and the remaining 25% to the test set. At the outset of each experiment, the weights of our neural networks were reinitialized to ensure a fresh starting point. After each training run, we computed both the training and test loss values to evaluate the model's performance. We first ran the training without imposing any constraints on the gating network, except for the architecture. Then, we ran several experiments with a gating mechanism satisfying ϵ-LDP, with ϵ ∈ {0.5, 2, 4, 5, 10}. A summary of the results is shown in Table 1 . One can observe that regularization with ϵ-LDP improves results in practice, and this regularization is even more evident when the models employing a gating network not satisfying LDP overfit heavily, as in the Breast Cancer and Heart experiments. The regularization effect is slightly less pronounced on MNIST, where the overfitting is not as severe as with the previous datasets. We can also observe the importance of choosing the right hyperparameter ϵ. Indeed, if the value is too small, the output of the gating network becomes insufficiently dependent on the input x. In this case, the experts have to do all the work, and the gating network does not allow them to specialize in well-defined subsets of the instance space. This makes our model closer to a weighted sum of linear classifiers and significantly reduces its performance. Conversely, if ϵ is overly large, our model tends towards a situation where the LDP condition does not hold, making it prone to overfitting.Note that our experiments are executed on GPUs in order to parallelize computations and take advantage of the sparsity of our model, but they can also be performed without GPUs. The duration of experiments can range from a few minutes for small datasets such as Breast Cancer to around 3 hours for large datasets like MNIST. 
conclusion: 
 In this work, we introduce a new way to regularize mixtures of experts. We provide both theoretical and algorithmic contributions in this regard. Our approach offers a significant advantage in that it allows us to harness the remarkable performances of neural networks by using them as gating networks, without being constrained by their architecture or their complexity from the theoretical point of view. By imposing LDP, we obtain nonvacuous bounds on the mixture of experts' risk. Our bounds can become significantly tighter than those presented in section 3.1 and those presented in 2 If N denotes the number of runs, R k denotes the training or test empirical risk during the k-th run, and R denotes the average, then standard deviation is given by1 N N k=1 (R k -R) 2 .Table 1 : Experiment results for mixtures of 100 linear models applied to binary classification tasks: Ads, Breast Cancer [Zwitter and Soklic, 1988] , Heart [Janosi et al., 1988] and MNIST [Deng, 2012] .The objective is to minimize the empirical risk as defined in Equation 1. We report the mean training loss (R S ) and mean test loss (R T ), averaged over ten runs, along with their associated standard deviations. and Meir [2004] , especially in cases where the empirical risk is close to zero and ϵ < ln n.However, as the empirical risk is multiplied by e ϵ , the bounds can become loose when ϵ is large and the empirical risk is significant.Even though the ϵ-LDP condition is easy to set up, a challenge arises in striking a balance between the parameter ϵ and the KL divergence or the Rademacher complexity of our experts. Our method introduces an extra hyperparameter ϵ to optimize but does not provide theoretical guidance on configuring it. This forces us to navigate a trade-off between the value of ϵ, which measures the extent to which the output of the gating network can depend on a given x ∈ X , and the complexity of our experts, which reflects how well our model captures the data distribution.  
experiments compute resources: 
 Question: For each experiment, does the paper provide sufficient information on the computer resources (type of compute workers, memory, time of execution) needed to reproduce the experiments?Answer: [Yes] Justification: We provide information on the type of compute workers (GPU) and time of execution in section 5.Guidelines:• The answer NA means that the paper does not include experiments.• The paper should indicate the type of compute workers CPU or GPU, internal cluster, or cloud provider, including relevant memory and storage.• The paper should provide the amount of compute required for each of the individual experimental runs as well as estimate the total compute.• The paper should disclose whether the full research project required more compute than the experiments reported in the paper (e.g., preliminary or failed experiments that didn't make it into the paper). 
code of ethics: 
 Question: Does the research conducted in the paper conform, in every respect, with the NeurIPS Code of Ethics https://neurips.cc/public/EthicsGuidelines?Answer: [Yes] Justification: In this work, we provide theoretical results that do not have any direct negative impact on society and do not pose any obvious ethical problem. We have ensured to cite our sources and adhere to the NeurIPS Code of Ethics.Guidelines:• The answer NA means that the authors have not reviewed the NeurIPS Code of Ethics.• If the authors answer No, they should explain the special circumstances that require a deviation from the Code of Ethics.• The authors should make sure to preserve anonymity (e.g., if there is a special consideration due to laws or regulations in their jurisdiction). 
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 Appendix: Proof of theorem 2.2. Given x ∈ X , let Z(x) = n i=1 exp(βf i (x) + c i ), for convenience.For all x, x ′ ∈ X and all i ∈ [n], we have thatg i (x) g i (x ′ ) = exp β(f i (x) -f i (x ′ )) 1 Z(x) n k=1 exp(βf k (x ′ ) + c k ) = exp β(f i (x) -f i (x ′ )) 1 Z(x) n k=1 exp(βf k (x) + c k ) exp β(f k (x ′ ) -f k (x)) ≤ max i∈[n]; x1,x2∈X exp 2β(f i (x 1 ) -f i (x 2 )) 1 Z(x) n k=1 exp(βf k (x) + c k )≤ exp(4βb). Theorem A.1 (Jensen's inequality, proposition 1.1 in Perlman [1974] ). Let Ω be a probability space, let A be a convex subset of R k , let X : Ω → A be an integrable vector-valued random variable, and let ϕ : A → R be a convex function. Then, E X ∈ A, and ϕ(E X) ≤ E ϕ(X) (in particular, the right-hand side of this inequality exists, though it may be infinite).Theorem A.2 (Theorem 5 in Maurer [2004] ). Let δ ∈ (0, 1) and m ≥ 8. Fix i ∈ [n], and let P i be a probability measure on H i (chosen without seeing the training data). Then, with probability at least 1 -δ over the draws of S, for all probability measures Q i on H i , we have thatkl(R S (Q i ) ∥ R(Q i )) ≤ 1 m KL(Q i ∥ P i ) + ln 2 √ m δ .Proof of theorem 3.4. As remarked earlier, the function(u, v) - → kl(u ∥ v) : [0, 1] 2 → R does not exactly satisfy the hypotheses of lemma 3.1, but it is convex. Moreover, on { (u, v) ∈ [0, 1] 2 | u ≤ v },it is decreasing in its first argument and increasing in its second argument. Also note that, assuming that R(g, Q) ≥ e 2ϵ R S (g, Q), then we also have the following inequalities:0 ≤ E i∼g(x ′ ) R S (Q i ) ≤ e ϵ R S (g, Q) ≤ e -ϵ R(g, Q) ≤ E i∼g(x ′ ) R(Q i ) ≤ 1. It follows that kl(e ϵ R S (g, Q) ∥ e -ϵ R(g, Q)) ≤ kl E i∼g(x ′ ) R S (Q i ) e -ϵ R(g, Q) ≤ kl E i∼g(x ′ ) R S (Q i ) E i∼g(x ′ ) R(Q i ) ,and thereforekl(e ϵ R S (g, Q) ∥ e -ϵ R(g, Q)) ≤ E i∼g(x ′ ) kl R S (Q i ) ∥ R(Q i )by Jensen's inequality. Now, by theorem A.2, for a fixed i, with probability at least 1 -δ/n, we have thatkl(R S (Q i ) ∥ R(Q i )) ≤ 1 m KL(Q i ∥ P i ) + ln 2n √ m δ .We can make the above inequality hold for all i ∈ [n] simultaneously with the union bound. Then, with probability at least 1 -δ, for all (g, Q), given that R(g, Q) ≥ e 2ϵ R S (g, Q), we have thatkl(e ϵ R S (g, Q) ∥ e -ϵ R(g, Q)) ≤ 1 m E i∼g(x ′ ) KL(Q i ∥ P i ) + ln 2n √ m δ .Proof of Lemma 4.1. Since the gating function satisfies ϵ-LDP, we have that e -ϵ g i (x ′ ) ≤ g i (x) ≤ e ϵ g i (x ′ ) for all x, x ′ ∈ X and all i ∈ [n]. It follows that e ϵ R S (g, h) ≥ E i∼g(x ′ ) R S (h i ) and e -ϵ R(g, h) ≤ E i∼g(x ′ ) R(h i ).Given that ∆ is decreasing in its first argument and increasing in its second argument, we find that∆ e ϵ R S (g, h), e -ϵ R(g, h) ≤ ∆ E i∼g(x ′ ) R S (h i ), E i∼g(x ′ ) R(h i )Since ∆ is a convex function, we can apply Jensen's inequality to the expression on the right-hand side, yielding the desired result.NeurIPS Paper Checklist 


Decision: This paper proposes a new regularization for mixture of experts (MoE) model based on the ideas from differential privacy. In particular, the gating function (which assigns probabilities to the experts) is forced to satisfy $\epsilon$-local differential privacy ($\epsilon$-LDP), meaning that changing the input does not change each probability by more than a factor of $e^{\epsilon}$. The paper shows that this regularization allows one to prove generalization bounds for MoEs, and that, empirically, it also performs well (compared to no such regularization) on shallow NNs.

Although this paper tackles an interesting problem, the current paper's presentation leaves a majority of the reviewers confused. Namely, the proposed method does *not* yield any meaningful privacy guarantee, but rather is purely to give generalization bounds. This should be highlighted more clearly both in the title and the main paper text. Furthermore, no discussion is given as to why this regularization is used as opposed to others; similarly, no other baselines are used in the experiments. With a major revision, this paper could be a very good contribution to the field. However, as is, the paper is not yet ready to be published at NeurIPS.

Reviews: review: paper_summary: paper_summary: The authors  take the first step (I though so at first) to MoE under LDP theoretically.  I read again and found that the author seems to have raised the utility lower bound of existing studies. Few experiments could be found. Perhaps, I am not an expert in MoE, but it really leave a  hard time.
strength_weakness: strength_weakness: 1. Perhaps I am not an expert in MoE, and I cannot tell from the author's introduction that there are any challenges. 
2. The experimental results and application scenarios are not clear.
questions: questions: 1. According to my rough understanding, the proposed approach seems to have some similarity with ensemble learning So what is the difference between the proposed approach and existing  ensemble learning under LDP?
2. MoE seems to be widely used in deep learning, so what are the differences between the proposed method and DP-SGD or LDP-SGD? Is there LDP-SGD-MoE?
limitations: limitations: see weakness.
paper_summary: paper_summary: The authors consider the mixtures of experts models, in particular the one-out-of-n gating mechanism for ease of theoretical analysis, and show that applying a soft-max, which is also the exponential mechanism, on the gating mechanism gives LDP and can improve generalization. The privacy techniques are largely the same as previous work, PATE, but specifically applied to mixtures of experts. The authors then provide theoretical analysis showing generalization bounds for this approach.


Unfortunately, I’m not familiar enough with the mixture of experts literature to evaluate the novelty of applying the soft-max and the corresponding theoretical guarantees. I am rather surprised though that the soft-max has never been applied and am still somewhat confused upon what the previous techniques were in the one-out-of-n mixtures of experts.
strength_weakness: strength_weakness: Unless I am mistaken (please correct me if I’m wrong) the authors are not providing privacy guarantees for the model itself, but only one inference call to the model. In particular, if feature vector x is input to the model, then the expert is chosen randomly according to the soft-max / exponential mechanism, which is \epsilon-LDP. If inference was then run again, suppose even on the same feature vector, then the random draw from experts would occur again. This is known as composition in the privacy literature, and the privacy guarantees	would now be 2*\epsilon. 

Providing privacy guarantees on only one inference call to a model is both not very useful nor interesting due to the composition properties.
questions: questions: See weaknesses section.
limitations: limitations: See weaknesses section.
paper_summary: paper_summary: This paper provides generalization bounds for a particular type of mixture of experts (MoE) networks. They focus on MoE architectures where an input $x$ first goes through a gating function $g$, and then gets routed to a single (one out of n) expert $i \in [n]$ according to the $g(x) \in [0,1]^n$ distribution. The final output of the MoE network is the output of the expert $h_i(x)$.

The authors observe that when the gating function $g$ has certain regularization properties, which correspond to local differential privacy (LDP), then the resulting network has better generalization bounds than what appears in the existing MoE litterature. The authors provide such bounds. 

Finally, the authors evaluate LDP-regularized routing on binary classification tasks with mixtures of linear models. The results show that LDP regularization outperforms an un-regularized baseline.
strength_weakness: strength_weakness: First, it is worth emphasizing that the MoE networks in this paper do not satisfy local differential privacy themselves. The gating network is not even *trained* with differential privacy. This paper only uses local differential privacy as a regularization condition on the *gating network only* at *inference* time, which does not provide meaningful privacy guarantees. That is not immediately clear from the title of the paper. To be fair, this work does not try to achieve any privacy goals, and is entirely focused on generalization bounds. But if privacy is not needed, then it is not clear why DP is the right tool for the job. The paper directly uses LDP (it could have been called something like "exponentially regularized routing") but does not motivate this choice. Are there other forms of regularization that could achieve similar or better bounds? While there are some known connections between robustness, differential privacy, and generalization, they are not mentioned here. In the context of MoEs, some large models already regularize their gating functions (e.g., the Switch Transformer adds some "jitter" noise to the routing logits). 

Next, the paper motivates the study of MoE models by mentioning recent progress with LLMs such as the Switch Transformer, which uses multiple layers of experts (deep MoE) and combines the output of different experts. All the modern LLM MoE models I am aware of are such deep MoEs. Meanwhile, the paper focuses on simple shallow MoE models with a single gating network followed a single layer of experts, which limits the potential impact of the paper in my opinion. While theoretical bounds may be of interest even on shallow MoE models, I would appreciate at least some discussion about whether the authors' approach can generalize to deeper models. 

Finally, the experiments have some limitations, which mostly stem from the two previous concerns. 
* The authors only evaluate a single, rather simplistic (3-layer MLP gating network followed by linear experts), MoE architecture. More concerningly, they use a fixed number of experts (n = 100), thereby missing an opportunity to evaluate their claim that "we can have many more experts with almost no penalty from the theoretical point of view".
* The only baseline is "No LDP", which I think is a quite weak baseline. It is not entirely surprising that adding some regularization, in the form of LDP routing, improves generalization compared to a completely un-regularized baseline. How about other forms of regularization, such as dropout, clipping, or jitter noise (which already exists in the context of MoEs)?
* Another baseline would be a non-MoE model, with a comparable number of parameters, e.g., even a simple, dense, multi-layer perceptron. Showing that shallow MoEs outperform dense models would alleviate concerns about the practical relevance of this work.

Minor comments:
* Table 1 might be more readable as a graph.
* It is quite surprising to see MNIST being qualified as a "large" dataset, for which a 4-layer network takes 3 hours to train on a GPU, in 2024.
* Also, it is unclear why MNIST has to be broken down into 3 binary classification tasks.
questions: questions: Have you considered other forms of regularization (including the noisy routing techniques that are already used in practice by MoE)? Is there any justification why local differential privacy would be the type of regularization that gives the most desirable generalization bounds?
limitations: limitations: The authors adequately addressed the limitation they identified (difficulty of tuning epsilon), even though this is not the main limitation of this work in my opinion.
paper_summary: paper_summary: This paper introduces a novel approach to regularize mixtures of experts by imposing local differential privacy (LDP) on the gating mechanism. The authors provide theoretical justifications and derive PAC-Bayesian and Rademacher bounds tailored to this approach. Experiments conducted on various datasets demonstrate that using LDP as a regularizer improves the generalization ability of the models, especially in cases prone to overfitting. The method offers a balance between leveraging neural networks for gating and maintaining robust theoretical guarantees, making it a valuable contribution to the field of machine learning.
strength_weakness: strength_weakness: The primary concern regarding this paper lies in its significance. There have been previous works that incorporated differential privacy (DP) into the construction of mixture of experts models with privacy considerations. This paper, however, utilizes local differential privacy (LDP) to analyze the theoretical aspects of mixture of experts models. The introduction of LDP significantly alters the generalization behavior of these models because both LDP and DP are methods that inherently enhance algorithm robustness, thereby affecting generalization. If the main goal of the paper is to enhance privacy, it is imperative to compare this approach with existing DP-based methods and highlight what specific aspects LDP protects that traditional DP cannot. Without this comparison, the added value of using LDP over existing DP methods remains unclear. On the other hand, if the focus is on analyzing the generalization of mixture of experts models, the paper must justify the rationale behind incorporating LDP for this analysis, as LDP is not inherently required for mixture of experts models. The paper needs to elaborate on why LDP is a suitable and necessary tool for this analysis and how it fundamentally impacts the generalization properties of the models in a meaningful way. Additionally, while the theoretical contributions are substantial, the practical implications need to be demonstrated more robustly through experiments. Comparing the results directly with models using traditional DP methods would strengthen the paper by showing the practical improvements and specific scenarios where LDP outperforms DP. Furthermore, using a broader range of datasets could better illustrate the claimed benefits in robustness and scalability. By addressing these concerns, the paper can more convincingly argue the necessity and advantages of using LDP in mixture of experts models, thereby enhancing its significance in the field.
questions: questions: Please refer to weakness.
limitations: limitations: No.
paper_summary: paper_summary: This paper introduces a novel mixture of experts model that applies local differential privacy to the gating mechanism. Their methods leverages the one-out-of-n gating mechanism and provides specific generalization bounds.
strength_weakness: strength_weakness: 1. The section 3 talks about PAC-Bayesian bounds for mixtures of experts, but it lacks insight into why PAC-Bayesian bounds are applied instead of other bounds. More explanation is needed here, similar to the explanation needed in section 4 regarding Rademacher bounds.

2. In the experiment section, it is unclear why the chosen dataset is used for the experiments and why only 5 epsilon values were selected.

3. Even though there are very few existing guarantees, the experiment should include other methods as baselines and compare the results.
4. For the experiment section, only consider mixtures of n linear experts in binary classification tasks seems easy. Need to add other classification tasks.
5. There is no description about the datasets used in experiments.
questions: questions: 1. For section 3, why there is no reference for PAC-Bayes theory when you first mention it?
2. Why there is no reference about Kullback-Leibler divergence?
limitations: limitations: 1. Lack many reference as I motioned in questions part. 
2. The paper lacks a smooth flow, making it difficult to follow. Specifically, there is no clear insight or reasoning provided to explain why the existing mechanisms or bounds were chosen, as highlighted in weakness 1.

Comments: comment: I thank the authors for their efforts in clarifying and revising their manuscript.

That said, I will keep my original score. I do not feel that the rebuttal is able to resolve my concerns in a meaningful way.

I'm happy to discuss my opinion with other reviewers and area chairs.
Thks for your response. I would keep my score.
Thank you for your insightful feedback. We acknowledge your remark regarding the baselines, and we will compare our regularization method with other techniques such as dropout and adding noise in future revisions. 

Regarding the use of the LDP condition, we recognize the importance of justifying our choice and would like to provide some clarification: Our motivation was to find a middle ground between input-independent mechanisms, such as weighted majority vote classifiers, which offer tight theoretical guarantees on risk, and input-dependent aggregation mechanisms, such as the best-known versions of mixtures of experts, which excel in practical performance. Our analysis of existing bounds on mixtures of experts revealed that they can be loose in certain cases—particularly when the weights provided by the gating network do not depend, or depend only *minimally*, on the input. We aimed to bridge these two approaches by introducing a unified framework that quantifies this dependence and incorporates it into the theoretical bounds.
Thanks to the authors for their comments and clarifications! Regarding point 1), I can appreciate that this can be part of ensuring LDP, but ensuring the each expert is LDP is the much harder difficulty here, not the exponential mechanism for choosing the expert. Further, the idea of using exponential mechanism or noisy max on \epsilon-DP (or LDP) experts/models to choose one for inference is not a new concept at all. Also the \epsilon parameter here is essentially just acting as a regularizer for the softmax function which is also not a novel concept.

The authors have some nice novel theoretical contributions! But I think the necessary re-write is too substantial at this time and in my opinion the connection to LDP is more of a remark given the well understood connection between exponential mechanism and softmax with a regularizer. I will keep my score.
Thank you for the answer. I appreciate the clarification regarding the title and the use of differential privacy. 

While composing LDP experts with a LDP gating function does indeed give LDP guarantees for the whole network, these guarantees only hold for a single inference. Since inference-time DP is rarely used and less relevant than DP training of neural networks, I think that this is an interesting side remark but not a particularly strong contribution from the paper, so I agree with the new framing/title/abstract you propose. 

I still believe that other baselines or more context about why LDP regularization is a natural choice would strengthen the paper, but I am no longer strongly opposed to accepting this paper.