Knowledge Removal in Sampling-based Bayesian Inference

1 INTRODUCTION . “ The right to be forgotten ” refers to the right of individuals to request data controllers such as tech giants to delete the data collected from them . It has been recognized in many countries through legislation , including the European Union ’ s General Data Protection Regulation ( 2016 ) and the California Consumer Privacy Act ( 2018 ) . In the field of machine learning , legal experts suggest that companies may need to delete and re-train the machine learning models to meet the legal requirements of the right to be forgotten ( Li et al. , 2018 ; Garg et al. , 2020 ) . However , training a machine learning model would usually cost massive amounts of resources , including data , energy , and time . Thus , the enforcement of the right to be forgotten in the AI industry may result in unbearable costs . Toward this end , recent works ( Cao & Yang , 2015 ; Guo et al. , 2020 ) propose machine unlearning algorithms to efficiently characterize and remove the knowledge learned from specific data . To enable knowledge removal analysis , existing unlearning approaches usually require the machine learning models to be explicitly parameterized , which however do not cover an important family of machine learning algorithm , the Markov chain Monte Carlo ( MCMC ) . MCMC is a sampling-based Bayesian inference method that aims to draw samples along a Markov chain to approximate a target posterior distribution ( Hastings , 1970 ; Geman & Geman , 1984 ; Welling & Teh , 2011 ) . Distributions inferred by MCMC are usually implicitly parameterized , which makes it difficult to directly apply existing unlearning algorithms to the MCMC unlearning problem . In this paper , we propose an MCMC unlearning algorithm to realize the right to be forgotten in the sampling-based Bayesian inference for the first time . The key of tackling the MCMC unlearning problem is to design methods to directly analyze the implicit parameter of the learned distribution . To this end , we propose to employ the learned implicit distribution to approximate the target posterior under the measure of KL-divergence , which then converts the MCMC unlearning problem to an optimization problem with an explicit distribution parameter for knowledge removal analysis . Based on this conversion , an MCMC influence function is designed ( Huber , 2004 ; Koh & Liang , 2017 ) to characterize the knowledge learned from single data samples . The new influence function then delivers the MCMC unlearning algorithm , which removes data knowledge by directly subtracting the corresponding influence from the learned distribution . Theoretical analyses are conducted for the proposed MCMC unlearning algorithm . For the knowledge removal analysis , we prove that the proposed MCMC unlearning algorithm can provably realize a ε-knowledge removal , which is a new notation defined for evaluating the machine unlearning performance in Bayesian inference . For the generalization analysis , a PAC-Bayesian generalization upper bound ( McAllester , 1998 ; 1999 ; 2003 ) is established for the distribution processed by the MCMC unlearning algorithm . The upper bound shows that the difference introduced by MCMC unlearning in the generalization upper bound is no larger than the order of O ( 1/N ) . This result demonstrates that the proposed algorithm would not compromise the generalization ability of the learned distribution . In summary , our work has four main contributions : ( 1 ) We enable the knowledge removal analysis in MCMC by converting the MCMC unlearning problem to an optimization problem . ( 2 ) Based on this problem conversion , we design an MCMC influence function to characterize the knowledge learned from data , which then delivers the MCMC unlearning algorithm . ( 3 ) We conduct theoretical analyses for the proposed algorithm . We prove that the MCMC unlearning can realize an εknowledge removal and has little impact on the generalization ability of the learned distribution . ( 4 ) We conduct comprehensive experiments to verify the effectiveness of the proposed algorithm , which include Gaussian mixture models for clustering on synthetic data , and Bayesian neural networks for classification on real-world dataset . The results suggest that the MCMC unlearning algorithm can effectively remove the knowledge learned from the requested data while retaining the information learned from other data intact . The code has been submitted and will be released publicly . 2 RELATED WORKS . Markov chain Monte Carlo . MCMC is a Bayesian inference method that would draw samples along a Markov chain , in which the drawn samples are proved to converge to that from a targeted distribution ( Hastings , 1970 ) . Although many improvements have been made such as Hamiltonian Monte Carlo ( HMC ) ( Duane et al. , 1987 ; Neal et al. , 2011 ) , Gibbs sampling ( Geman & Geman , 1984 ; George & McCulloch , 1993 ) and slice sampling ( Neal , 2003 ; Walker , 2007 ) , MCMC still suffers from huge computational cost when inferring on large-scale dataset since each sampling step in MCMC requires computation over the whole dataset . To tackle this issue , Welling & Teh ( 2011 ) introduce a scalable MCMC sampler named stochastic gradient Langevin dynamics ( SGLD ) that by adding a proper noise to a standard stochastic gradient optimization algorithm ( Robbins & Monro , 1951 ) , which can help the update iterations converge to samples from the true posterior distribution . Since that , a variety of stochastic gradient MCMC samplers ( SG-MCMC ) have been proposed , including stochastic gradient HMC ( SGHMC ) ( Chen et al. , 2014 ) , stochastic gradient Riemannian Langevin dynamics ( SGRLD ) ( Patterson & Teh , 2013 ) , Ahn et al . ( 2012 ) , Ding et al . ( 2014 ) , and Zhang et al . ( 2020 ) . Ma et al . ( 2015 ) further design a complete framework for constructing SG-MCMC . Machine unlearning . Machine unlearning aims to remove the knowledge learned from specific data in efficient manners ( Cao & Yang , 2015 ) . Existing unlearning methods can be roughly divided into two categories , exact methods and approximate methods . Exact methods aim to recover the model that trained without the removed data exactly , which therefore are usually realized by reducing the retraining cost as much as possible . For example , Ginart et al . ( 2019 ) propose an efficient unlearning method for k-means clustering via model parameter quantization . Bourtoule et al . ( 2021 ) propose a SISA framework that would only retraining the affected part of the model during data deletion . Other advances of exact machine unlearning include Ullah et al . ( 2021 ) and Brophy & Lowd ( 2021 ) . On the other hand , the approximate method aims to modify the trained model to approximate that trained only on the remaining data . A series of works are designed via characterizing the influence of data on the trained model with influence functions ( Cook & Weisberg , 1982 ; Huber , 2004 ; Koh & Liang , 2017 ) , including Guo et al . ( 2020 ) , Golatkar et al . ( 2020 ) , and Golatkar et al . ( 2021 ) . Other approaches include Baumhauer et al . ( 2020 ) and Izzo et al . ( 2021 ) . Some works have also studied machine unlearning in Bayesian inference ( Nguyen et al. , 2020 ; Gong et al. , 2021 ) . However , these works only focus on variational inference , an optimization-based Bayesian inference method , and could not be applied to sampling-based Bayesian inference , MCMC . In contrast , our approach is the first machine unlearning method for MCMC , with a theoretical knowledge removal guarantee . 3 PRELIMINARIES . Suppose p ( z|θ ) is a parameterized distribution over the sample space Z , where θ ∈ Θ ⊂ Rd is the parameter with a prior distribution p ( θ ) . Suppose S = { z1 , z2 , · · · , zN } is a dataset consists of N samples , where each sample zi ∈ Z is drawn from the parameterized distribution p ( z|θ ) , i.e. , zi ∼ p ( z|θ ) . Bayesian inference aims to infer the posterior distribution p ( θ|S ) of the parameter θ , p ( θ|S ) = p ( θ ) ∏N i=1 p ( zi|θ ) ∫ p ( θ ) ∏N i=1 p ( zi|θ ) dθ . For simplicity , for the given dataset S , we let pS denotes the true posterior distribution p ( θ|S ) , and p̂S denotes the posterior distribution that inferred by Bayesian inference . Usually , the denominator of the posterior pS has no closed-form solution , which barriers directly calculating the posterior distribution . Thereby , researchers have designed methods to approximate the target posterior . A canonical approach is the sampling-based Bayesian inference method , Markov chain Monte Carlo ( MCMC ) ( Hastings , 1970 ) . MCMC infers a posterior distribution via first constructs a Markov chain and then draws samples according to the state of the chain . When the target posterior is the stationary distribution of the Markov chain , it is proved that the drawn samples will converge to that from the target posterior . However , MCMC methods usually suffer from scalable issues on large-scale datasets . To this end , stochastic gradient MCMC ( SG-MCMC ) ( Ma et al. , 2015 ) leverage mini-batch estimation ( Robbins & Monro , 1951 ) to enable scalable sampling . In SG-MCMC , the posterior distribution p ( θ|S ) can be rewritten as p ( θ|S ) ∝ exp ( −U ( θ ) ) , where U ( θ ) is the potential function defined as U ( θ ) = − ∑ zi∈S log p ( zi|θ ) − log p ( θ ) . In each sampling step , drawing samples with SG-MCMC requires one to estimate the potential function U ( θ ) with mini-batch data S̃ ⊂ S as follows , Ũ ( θ ) = −|S| |S̃| ∑ zi∈S̃ log p ( zi|θ ) − log p ( θ ) . Typical SG-MCMC methods include SGLD and SGHMC . Stochastic gradient Langevin dynamics ( SGLD ) ( Welling & Teh , 2011 ) inserts Gaussian noise to stochastic gradients . In each sampling step , SGLD updates the posterior sample θt as follows , θt+1 = θt − ηt∇θŨ ( θt ) + 2 √ ηtW , where W is drawn from the standard Gaussian distribution N ( 0 , I ) and ηt > 0 is the step size . To improve the sampling efficiency of SGLD , stochastic gradient Hamiltonian Monte Carlo ( SGHMC ) ( Chen et al. , 2014 ) inserts a momentum vt into the posterior sample update as below , θt+1 = θt + vt , vt+1 = ( 1− α ) vt − ηt∇θŨ ( θt ) + √ 2αηtW , where W is also drawn from the Gaussian distribution N ( 0 , I ) , ηt > 0 is the step size , and α ∈ ( 0 , 1 ) is the momentum factor . The initial momentum term v0 is drawn from the Gaussian distribution N ( 0 , η0I ) . To ensure the drawn samples converge to the targeted posterior , the step size ηt for both SGLD and SGHMC needs to satisfy ( Welling & Teh , 2011 ) : ( 1 ) ∑∞ t=1 ηt = ∞ ; and ( 2 ) ∑∞ t=1 η 2 t < ∞ . A typical step size schedule is ηt = a ( b+ t ) −r , where a , b > 0 and r ∈ ( 0.5 , 1 ] .

Summary: 1. Summary  The paper proposes a novel method for knowledge removal in a MCMC context. The contribution is four-fold: First, a theoretical analysis relates the MCMC unlearning problem to an optimiziation problem introduced. Second, the conversed problem is related to a MCMC influence function. Third an abduction of theoretical results is done. Finally real world data experiments are performed showing that the proposed approach can indeed tackle the problem introduced.