Differentially Private Generative Models Through Optimal Transport

1 INTRODUCTION . As the full value of data comes to fruition through a growing number of data-centric applications ( e.g . recommender systems ( Gomez-Uribe & Hunt , 2016 ) , personalized medicine ( Ho et al. , 2020 ) , face recognition ( Wang & Deng , 2020 ) , speech synthesis ( Oord et al. , 2016 ) , etc . ) , the importance of privacy protection has become apparent to both the public and academia . At the same time , recent Machine Learning ( ML ) algorithms and applications are increasingly data hungry and the use of personal data will eventually be a necessity . Differential Privacy ( DP ) is a rigorous definition of privacy that quantifies the amount of information leaked by a user participating in any data release ( Dwork et al. , 2006 ; Dwork & Roth , 2014 ) . DP was originally designed for answering queries to statistical databases . In a typical setting , a data analyst ( party wanting to use data , such as a healthcare or marketing company ) sends a query to a data curator ( party in charge of safekeeping the database , such as a hospital ) , who makes the query on the database and replies with a semi-random answer that preserves privacy . Differentially Private Stochastic Gradient Descent ( DPSGD ) 1 ( Abadi et al. , 2016 ) is the most popular method for training general machine learning models with DP guarantees . DPSGD involves large numbers of queries , in the form of gradient computations , to be answered quickly by the curator . This requires technology transfer of model design from analyst to curator , and strong computational capacity be present at the curator . Furthermore , if the analyst wants to train on multiple tasks , the curator must subdivide the privacy budget to spend on each task . As few institutions have simultaneous access to private data , computational resources , and expertise in machine learning , these requirements significantly limit adoption of DPSGD for learning with privacy guarantees . To address this challenge , generative models—models with the capacity to synthesize new data— can be applied as a general medium for data-sharing ( Xie et al. , 2018 ; Augenstein et al. , 2020 ) . The 1Including any variants that use gradient perturbation for ensuring privacy . curator first encodes private data into a generative model ; then , the model is used by the analysts to synthesize similar yet different data that can train other ML applications . So long as the generative model is learned ” privately ” , the user can protect their privacy by controlling how specific the generative model is to their own data . Differentially private learning of generative models has been studied mostly under the Generative Adversarial Networks ( GAN ) framework ( Xie et al. , 2018 ; Torkzadehmahani et al. , 2019 ; Frigerio et al. , 2019 ; Yoon et al. , 2019 ; Chen et al. , 2020 ) . While GANs in the non-private setting have demonstrated the ability to synthesize complex data like high definition images ( Brock et al. , 2019 ; Karras et al. , 2020 ) , their application in the private setting is more challenging . This is in part because GANs suffer from training instability problems ( Arjovsky & Bottou , 2017 ; Mescheder et al. , 2018 ) , which can be exacerbated when adding noise to the network ’ s gradients during training , a common technique to implement DP . Because of that , GANs typically require careful hyperparameter tuning and supervision during training to avoid model collapse . This goes against the principle of privacy , where repeated interactions with data need to be avoided ( Chaudhuri & Vinterbo , 2013 ) . Optimal Transport ( OT ) is another method to train generative models . In the optimal transport setting , the problem of learning a generative model is framed as minimizing the optimal transport distance , a type of Wasserstein distance , between the generator-induced distribution and the real data distribution ( Bousquet et al. , 2017 ; Peyré & Cuturi , 2019 ) . Unfortunately , exactly computing the OT distance is generally expensive . Nevertheless , Wasserstein distance-based objectives are actually widely used to train GANs ( Arjovsky et al. , 2017 ; Gulrajani et al. , 2017b ) . However , these approaches typically estimate a Wasserstein distance using an adversarially trained discriminator . Hence , training instabilities remain ( Mescheder et al. , 2018 ) . An alternative to adversarial-based OT estimation is provided by the Sinkhorn divergence ( Genevay et al. , 2016 ; Feydy et al. , 2019 ; Genevay et al. , 2018 ) . The Sinkhorn divergence is an entropyregularized version of the exact OT distance , for which the optimal transport plan can be computed efficiently via the Sinkhorn algorithm ( Cuturi , 2013 ) . In this paper , we propose DP-Sinkhorn , a novel method to train differentially private generative models using the Sinkhorn divergence as objective . Since the Sinkhorn approach does not intrinsically rely on adversarial components , it avoids any potential training instabilities and removes the need for early stopping . This makes our method easy to train and deploy in practice . As a side , we also develop a simple yet effective way to perform conditional generation in the Sinkhorn framework , by forcing the optimal transport plan to couple same-label data closer together . To the best of our knowledge , DP-Sinkhorn is the first fully OT-based approach for differentially private generative modeling . Experimentally , despite its simplicity DP-Sinkhorn achieves state-of-the-art results on image-based classification benchmarks that use data generated under differential privacy for training . We can also generate informative RGB images , which , to the best of our knowledge , has not been demonstrated by any generative models trained with differential privacy and without auxiliary public data . We make the following contributions : ( i ) We propose DP-Sinkhorn , a flexible and robust optimal transport-based framework for training differentially private generative models . ( ii ) We introduce a simple technique to perform label-conditional synthesis in the Sinkhorn framework . ( iii ) We achieve state-of-the-art performance on widely used image modeling benchmarks . ( iv ) We present informative RGB images generated under strict differential privacy without the use of public data . 2 BACKGROUND . 2.1 NOTATIONS AND SETTING . Let X denote a sample space , P ( X ) all possible measures on X , and Z ⊆ Rd the latent space . We are interested in training a generative model g : Z 7→ X such that its induced distribution µ = g ◦ ξ with noise source ξ ∈ P ( Z ) is similar to observed ν through an independently sampled finite sized set of observations D = { y } N . In our case , g is a trainable parametric function with parameters θ . 2.2 GENERATIVE LEARNING WITH OPTIMAL TRANSPORT . Optimal Transport-based generative learning considers minimizing variants of the Wasserstein distance between real and generated distributions ( Bousquet et al. , 2017 ; Peyré & Cuturi , 2019 ) . Two key advantages of the Wasserstein distance over standard GANs , which optimize the JensenShannon divergence ( Goodfellow et al. , 2014 ) , are its definiteness on distributions with nonoverlapping supports , and its weak metrization of probability spaces ( Arjovsky et al. , 2017 ) . This prevents collapse during training caused by discriminators that are overfit to the training examples . Methods implementing the OT framework use either the primal or the dual formulation . For generative learning , the dual formulation has been more popular . Under the dual formulation , the distance between the generator-induced and the data distribution is computed as the expectation of potential functions over the sample space . In WGAN and variants ( Arjovsky et al. , 2017 ; Gulrajani et al. , 2017a ; Miyato et al. , 2018 ) , the dual potential is approximated by an adversarially trained discriminator network . While theoretically sound , these methods still encounter instabilities during training since the non-optimality of the discriminator can produce arbitrarily large biases in the generator gradient ( Bousquet et al. , 2017 ) . The primal formulation involves solving for the optimal transport plan—a joint distribution over the real and generated sample spaces . The distance between the two distributions is then measured as the expectation of a point-wise cost function between pairs of samples as distributed according to the transport plan . Thus , the properties of the point-wise cost function are of great importance . In practice , sufficiently convex cost functions allow for an efficient optimization of the generator . It is also possible to learn cost functions adversarially ( Szabó & Sriperumbudur , 2017 ) . However , as discussed earlier , adversarial training often comes with additional challenges , which can be especially problematic in the differentially private setting . In general , finding the optimal transport plan is a difficult optimization problem due to the constraints of equality between its marginals and the real and generated distributions . Entropy Regularized Wasserstein Distance ( ERWD ) imposes a strongly convex regularization term on the Wasserstein distance , making the OT problem between finite samples solvable in linear time ( Peyré et al. , 2019 ) . Given a positive cost function c : X × X 7→ R+ and ≥ 0 , the EERWD is defined as : Wc , ( µ , ν ) = min π∈Π ∫ c ( x , y ) π ( x , y ) + ∫ log π ( x , y ) dµ ( x ) dν ( y ) dπ ( x , y ) ( 1 ) where Π = { π ( x , y ) ∈ P ( X × X ) | ∫ π ( x , · ) dx = ν , ∫ π ( · , y ) dy = µ } . Sinkhorn divergence uses cross correlation terms to cancel out the entropic bias introduced by ERWD . This results in faithful matching between the generator and real distributions . In this paper , we use the Sinkhorn divergence as defined in Feydy et al . ( 2019 ) . Definition 2.1 ( Sinkhorn Divergence ) The Sinkhorn divergence between measures µ and ν is defined as : Sc , ( µ , ν ) = 2Wc , ( µ , ν ) −Wc , ( µ , µ ) −Wc , ( ν , ν ) ( 2 ) Works on the efficient computation of the Sinkhorn divergence have yielded algorithms that converge to the optimal transport plan within tens of iterations ( Cuturi , 2013 ; Feydy et al. , 2019 ) . Gradient computation follows by taking the Jacobian vector product between the cost matrix Jacobian and the transport weights , which is implemented in many auto-differentiation frameworks . When compared to WGAN , learning with the Sinkhorn divergence has distinct differences . First , the Sinkhorn divergence is computed under the primal formulation of OT , whereas WGAN ’ s loss is computed under the dual formulation . While both are approximations to the exact Wasserstein distance , the source of the approximation error differs . The Sinkhorn divergence uses entropic regularization to ensure linear convergence when finding the optimal transport plan π ( Cuturi , 2013 ) . Its two sources of error are the suboptimality of the transport plan and bias introduced by entropic regularization . With the guarantee of linear convergence , π can converge to optimality by using enough iterations , thereby allowing control of the first error . The second source of error can be controlled by using small values of , which we found to work well in practice . In contrast , WGAN ’ s source of error lies in the sub-optimality of the dual potential function . Since this potential function is parameterized by an adversarially trained deep neural network , it enjoys neither convergence guarantees nor feasibility guarantees . Furthermore , the adversarial training scheme can produce oscillatory behavior , where the discriminator and generator change abruptly every iteration to counter the strategy of the other player from the previous iteration ( Mescheder et al. , 2017 ) . These shortcomings contribute to WGAN ’ s problems of non-convergence , which in turn can lead to mode dropping . In contrast , training with the Sinkhorn divergence does not involve any adversarial training at all , converges more stably , and reaps the benefits of OT metrics at covering modes .

Summary: The paper proposes a method for training OT GANs using differentially private sinkhorn algorithm. The idea is very simple - train GANs with sinkhorn divergences and add Gaussian noise to the gradient of output wrt generated samples. So, the novelty by itself is minimal as sinkhorn GANs have previously been proposed. The method merely adds Gaussian noise to gradients. The DP analysis is also not very different.