# Foundations of Bayesian Learning from Synthetic Data - Proceedings of Machine Learning Research

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Foundations of Bayesian Learning from Synthetic Data Harrison Wilde Jack Jewson Sebastian Vollmer Chris Holmes Department of Statistics Barcelona GSE Department of Statistics, Department of Statistics University of Warwick Universitat Pompeu Fabra Mathematics Institute University of Oxford; University of Warwick The Alan Turing Institute Abstract (Dwork et al., 2006), to define working bounds on the probability that an adversary may identify whether a particular observation is present in a dataset, given There is significant growth and interest in the that they have access to all other observations in the use of synthetic data as an enabler for machine dataset. DP’s formulation is context-dependent across learning in environments where the release of the literature; here we amalgamate definitions regard- real data is restricted due to privacy or avail- ing adjacent datasets from Dwork et al. (2014); Dwork ability constraints. Despite a large number of and Lei (2009): methods for synthetic data generation, there are comparatively few results on the statisti- Definition 1 ((ε, δ)-differential privacy) A ran- cal properties of models learnt on synthetic domised function or algorithm K is said to be data, and fewer still for situations where a (ε, δ)-differentially private if for all pairs of adjacent, researcher wishes to augment real data with equally-sized datasets D and D0 that differ in one another party’s synthesised data. We use observation and all S ⊆ Range(K), a Bayesian paradigm to characterise the up- dating of model parameters when learning Pr[K(D) ∈ S] ≤ eε × Pr [K (D0 ) ∈ S] + δ (1) in these settings, demonstrating that caution should be taken when applying conventional learning algorithms without appropriate con- Current state-of-the-art approaches involve the privati- sideration of the synthetic data generating sation of generative modelling architectures such as process and learning task at hand. Recent Generative Adversarial Networks (GANs), Variational results from general Bayesian updating sup- Autoencoders (VAEs) or Bayesian Networks. This is port a novel and robust approach to Bayesian achieved through adjustments to their learning pro- synthetic-learning founded on decision theory cesses such that their outputs fulfil a DP guarantee that outperforms standard approaches across specified at the point of training (e.g. Zhang et al., repeated experiments on supervised learning 2017; Xie et al., 2018; Jordon et al., 2018; Rosenblatt and inference problems. et al., 2020). Despite these contributions, a fundamen- tal question remains regarding how, from a statistical perspective, one should learn from privatised synthetic data. Progress has been made for simple exponential 1 Introduction family and regression models (Bernstein and Sheldon, 2018, 2019), but these model classes are of limited use Privacy enhancing technologies comprise an area of in modern machine learning applications. rapid growth (The Royal Society, 2019). An important aspect of this field concerns publishing privatised ver- We characterise this problem for the first time via an sions of datasets for learning; it is known that simply adoption of the M -open world viewpoint (Bernardo and anonymising the data is not sufficient to guarantee Smith, 2001) associated with model misspecification; individual privacy (e.g. Rocher et al., 2019). We in- unifying the privacy and synthetic data generation lit- stead adopt the Differential Privacy (DP) framework erature alongside recent results in generalised Bayesian updating (Bissiri et al., 2016) and minimum divergence Proceedings of the 24th International Conference on Artifi- inference (Jewson et al., 2018) to ask what it means cial Intelligence and Statistics (AISTATS) 2021, San Diego, to learn from synthetic data, and how can we improve California, USA. PMLR: Volume 130. Copyright 2021 by upon our inferences and predictions given that we ac- the author(s). knowledge its privatised synthetic nature?

Foundations of Bayesian Learning from Synthetic Data This characterisation results in generative models that density f0 (x) with respect to the Lebesque measure, are ‘misspecificed by design’, owing to the constraints such that x1:n ∼ F0 (x); we suppose xi ∈ Rd . These imposed upon their design by requiring the fulfilment observations are held privately by a data keeper K. of a DP guarantee. This inevitably leads to discrepancy • K uses data x1:n to produce an (ε, δ)-differentially between the learner’s final model and the one that they private synthetic data generating mechanism (S- would otherwise have formulated if not for this DP DGP). With a slight abuse of notation we use restriction. In real-world, finite data contexts where Gε,δ (x1:n ) to denote the S-DGP, noting that Gε,δ could synthesis methods are often ‘black-box’ in nature, it is be a fully generative model, or a private release mech- difficult for a learner to fully capture and understand anism that acts directly on the finite data x1:n (see the inherent differences in the underlying distributions discussion on the details of the S-DGP below). We of the real and synthetic data that they have access to. denote the density of this S-DGP as gε,δ . • Let fθ (x) denote a learner L’s model likeli- There are two key insights that we explore in this pa- hood for F0 (x), parameterised by θ with prior per following the characterisation above: Firstly, when π̃(θ), and marginal (predictive) likelihood p(x) = left unchecked, the Bayesian inference machine learns R f (x)π̃(θ)dθ. model parameters minimising the Kullback-Leibler di- θ θ • L’s prior may already encompass some other set of vergence (KLD) to the synthetic data generating process real-data drawn from F0 leading to π̃(θ) = π(θ | (S-DGP) (Berk et al., 1966; Walker, 2013) rather than xL1:nL ), for nL ≥ 0 prior observations. the true data generating process (DGP); Secondly, ro- bust inference methods offer improved performance We adopt a decision theoretic framework (Berger, 2013), by acknowledging this misspecification, where in some in assuming that L wishes to take some optimal action cases synthetic data can otherwise significantly hinder â in a prediction or inference task; satisfying: learning rather than helping it. Z In order to investigate these behaviours, we experiment â = arg max U (x, a)F0 (x)dx. (2) with models based on a mix of simulated-private and a∈A real-world data to offer empirical insights on the learn- This is with respect to a user-specified utility-function ing procedure when a varying amount of real data is U (x, a) that evaluates actions in the action space A, available, and explore the optimalities in the amounts and makes precise L’s desire to learn about F0 in order of synthetic data with which to augment this real data. to accurately identify â. The contributions of our work are summarised below: Details of the synthetic data generation mech- 1. Learning from synthetic data can lead to unpre- anism. In defining Gε,δ , we believe it is important to dictable and negative outcomes, due to varying levels differentiate between its two possible forms: of model misspecification introduced by its genera- tion and associated privacy constraints. 1. Gε,δ (x1:n ) = Gε,δ (z | x1:n ): G is a privacy- 2. Robust Bayesian inference offers improvements over preserving generative model fit on the real data, classical Bayes when learning from synthetic data. such as the PATE-GAN (Jordon et al., 2018), DP- GAN (Xie et al., 2018) or PrivBayes (Zhang et al., 3. Real and synthetic data can be used in tandem to achieve practical effectiveness through the discovery 2017). Privatised synthetic data is produced by in- of desirable stopping points for learning, and optimal jecting potentially heavy-tailed noise into gradient- model configurations. based learning and/or through partitioned training 4. Consideration of the preferred properties of the in- leveraging marginal distributions, aggregations and ference procedure are critical; the specific task at subsets of the data. The S-DGP provides conditional hand can determine how best to use synthetic data. independence between z1:m and x1:m and therefore no longerR queries the real data after training. 2. Gε,δ = Kε,δ (x, dz)F0 (dx): A special case of this We adopt a Bayesian standpoint throughout this paper, integral comprises the convolution of F0 with some but note that many of the results also hold in the noise distribution H, such that Gε,δ = F0 ?Hε,δ . The frequentist setting. sampling distribution is therefore not a function of the private data x1:n . In this case, the number of 2 Problem Formulation samples that we can draw is limited to m ≤ n as drawing one data item requires using one sample We outline the inference problem as follows, of K’s data. Examples of this formulation include the Laplace mechanism (Dwork et al., 2014) and • Let x1:n denote a training set of n exchangeable transformation-based privatisation (Aggarwal and observations from Nature’s true DGP, F0 (x) with Yu, 2004).

Harrison Wilde, Jack Jewson, Sebastian Vollmer, Chris Holmes The fundamental problem of synthetic learning. (KLD) of the model from the DGP of the data (Berk L wants to learn about F0 but only has access to their et al., 1966; Walker, 2013; Bissiri et al., 2016), where prior π̃(θ) and to z1:m ∼ Gε,δ , where Gε,δ 6≡ F0 . That log (gε,δ/f ) dGε,δ . R KLD(gε,δ k f ) = is, the S-DGP Gε,δ (·) is ‘misspecified by design’. This As a result, if L updates their model fθ (x) using syn- claim is supported by a number of observations: thetic data z1:m ∼ Gε,δ (x1:n ), then as m → ∞ they will be learning about the limiting parameter that min- • L specifies a model p(x) using beliefs about the tar- imises the KLD to the S-DGP: get F0 to be built using real data x1:n , they are then constrained by a subsequently imposed requirement θGKLD = arg min KLD (gε,δ (·) k fθ (·)) , (3) of guaranteeing DP which instead requires consider- ε,δ θ∈Θ ation of the resulting S-DGP Gε,δ ; this leads to an inevitable change in their beliefs such that the re- and under regularity conditions the posterior distri- sulting model would be misspecified relative to the bution concentrates around that point, π(θ | z1:m ) → original ‘best’ model for the true DGP F0 . 1θKLD as m → ∞. Gε,δ • Correctly modelling a privacy preserving mechanism as part of an S-DGP such as a ‘black-box’ GAN or Furthermore, this posterior will concentrate away from complex noise convolution is often intractable. the model that is closest to F0 in KLD, corresponding • There is inherent finiteness to real-world sensitive to the limiting model that would be learnt given an data contexts that makes it difficult for a gener- infinite real sample x1:∞ from F0 : ative model to capture the true DGP of even the θGKLD 6≡ θFKLD = arg min KLD (f0 (·) k fθ (·)) . (4) non-private data it seeks to emulate. Considering ε,δ 0 θ∈Θ sufficiently large quantities of data, and sufficiently flexible models, would in theory allow for the true This problem is exacerbated by the fact that S-DGP’s DGP to be learnt, but relying on asymptotic be- can be prone to generating ‘outliers’ relative to the haviour in this setting is at odds with the definition private data as a result of the injection of additional of DP given that the identifiability of individuals noise into their training or generation to ensure (ε, δ)- would also naturally diminish as data availability DP, combined with the fact that θG KLD is known to be increases. Moreover, for non-trivial high-dimensional ε,δ non-robust to outliers (e.g. Basu et al., 1998). There- models, the amount of data required to properly cap- fore, given that as we collect more synthetic data our ture the DGP often becomes infeasible, regardless of inference is no longer minimising the KLD towards F0 , the magnitude of DP constraints. we must carefully consider and investigate whether our inference is still ‘useful’ for learning about F0 at all. Therefore, the posterior predictive converges to a differ- ent distribution under real and synthetic data generat- ing processes such that p(x | z1:m→∞ ) 6≡ p(x | x1:n→∞ ). 2.2 The approximation to F0 Learning from synthetic data is an intricate example of learning under model misspecification, where the Before we proceed any further we must consider what misspecification is by K’s design. It is important, as it means for data from Gε,δ to be ‘useful’ for learning shown below, that this is recognised in the learning about F0 . We can do so using the concepts of scoring of models. Fortunately we can adapt recent advances rules and statistical divergence. in Bayesian inference under model misspecification to help optimise learning with respect to L’s task. Definition 2 (Proper Scoring Rule) The function s : X × P is a strictly proper scoring rule provided its difference function D satisfies 2.1 Bayesian Inference under model misspecification D(f0 k f ) = Ex∼f0 [s(x, f (·))] − Ex∼f0 [s(x, f0 (·))] Bayesian inference under model misspecification has recently been formalised (Walker, 2013; Bissiri et al., D(f1 k f2 ) ≥ 0, D(f k f ) = 0 for all f, f1 , f2 ∈ P(x) 2016) and represents a growing area of research, see Z Watson and Holmes (2016); Jewson et al. (2018); Miller P(x) := f (x) : f (x) ≥ 0 ∀x ∈ X , f (x)dx = 1 , and Dunson (2018); Lyddon et al. (2018); Grünwald X et al. (2017); Knoblauch et al. (2019) to name but a few. Traditional Bayes rule updating in this context Where the function D measures a distance between can be seen as an approach that learns about the pa- two probability distributions. s(x, f ) arises as the di- rameters of the model that minimises the logarithmic vergence and is minimised when f0 = f (Gneiting and score, or equivalently, the Kullback-Leibler divergence Raftery, 2007; Dawid, 2007). A further advantage of

Foundations of Bayesian Learning from Synthetic Data this representation is that it allows for the minimisation rule updating. The predictive distribution associated of D(f0 k ·) using only samples from F0 , with such a posterior and the model fθ is: Z arg min D(f0 k f ) = arg min Ex∼f0 [s(x, f (·))] p` (x|z1:m ) = fθ (x)π ` (θ|z1:m )dθ (7) f ∈F f ∈F n 1 X ←−−−− s(xi , f (·)), xi ∼ F0 (5) n→∞ n 3.2 Robust Bayes and dealing with outliers i=1 Henceforth, we define any concepts of closeness (or In the absence of the ability to correctly model the S-DGP, robust statistics (see e.g. Berger et al., 1994) ‘usefulness’) in terms of a chosen divergence D and associated scoring rule s, where the approximating provide an alternative option to guard against artefacts density f is given by the predictive inferences resulting of the generated synthetic data. We can gain increased from synthetic z1:m ∼ Gε,δ . Given that inference using robustness in our learning procedure to data z1:m by Gε,δ is no longer able to exactly capture f0 , L can use changing the loss function `(z, fθ ) used for inference in this notion of closeness to define what aspects of F0 they Eq. (6). We consider two alternative loss functions to are most concerned with capturing. The importance the standard logarithmic score underpinning standard of this specification is illustrated in Section 4. Bayesian statistics, `w (z, fθ ) := −w log fθ (z) (8) 3 Improved learning from the S-DGP 1 Z 1 `(β) (z, fθ ) := fθ (y)β+1 dy − fθ (z)β . (9) β+1 β The classical assumptions underlying statistics are that minimising the KLD is the optimal way to learn about Here, `w (z, fθ ) introduces a learning parameter w > 0 the DGP, and that more observations provide more into the Bayesian update (e.g. Lyddon et al., 2018; information about this underlying DGP; such logic does Grünwald et al., 2017; Miller and Dunson, 2018; Holmes not necessarily apply here. L wishes to learn about the and Walker, 2017). Down-weighting, w < 1 will gener- private DGP, F0 , but must rely on observations from the ally produce a less confident posterior than in the case S-DGP Gε,δ to do so. In this section we acknowledge this of traditional Bayes’ rule, with a greater dependence setting to propose a framework for improved learning on the prior. Conversely, w > 1 will have the opposite from synthetic data. In so doing we pose the following effect. The value of w can have ramifications for infer- question and detail our solutions in turn: Given the ence and prediction (Rossell and Rubio, 2018; Grün- scoring criteria D, is θGKLD ε,δ the best the learner can do? wald et al., 2017). However, we note that as the sample size grows using the weighted likelihood posterior will 1. Can the robustness of the learning procedure be im- still learn about θG ∗ if w is fixed. Choosing w = e/m proved to better approximate F0 by acknowledging instead can be seen to average the log-likelihood, with the misspecification and outlier prone nature of z1:m ? e providing a notion of effective sample size. 2. Starting from the prior predictive, p(x), for a given Alternatively, minimising `(β) (x, f (·)) in expectation learning method, when does learning using z ∼ Gε,δ over the DGP is equivalent to minimising the β- stop improving inference for F0 (x)? That is, when divergence (βD) (Basu et al., 1998). Therefore, analo- Ez [D (f0 (·) k p(· | z1:j+1 ))] > Ez [D (f0 (·) k p(· | z1:j ))] gously to the KLD and the log-score, using `(β) (x, f (·)) (Bissiri et al., 2016; Jewson et al., 2018; Ghosh and Basu, 2016) produces a Bayesian update targeting: 3.1 General Bayesian Inference θGβD : = arg min βD (gε,δ (·) k fθ ) . (10) In order to address these issues we adopt a general ε,δ θ∈Θ Bayesian, minimum divergence paradigm for inference (Bissiri et al., 2016; Jewson et al., 2018) inspired by As β → 0, then βD → KLD, but as β increases it model misspecification, where L can coherently update provides increased robustness through skepticism of beliefs about their model parameter θ from prior π(θ) new observations relative to the prior. We demonstrate to posterior π(θ | z1:m ) using: the robustness properties of the βD in some simple scenarios in the Supplementary material (see A.1) and refer the reader to e.g. Knoblauch et al. (2019, 2018) Pm π̃(θ) exp (− i=1 `(zj , fθ )) ` π (θ | z1:m ) ∝ R Pm , (6) for further examples. We note there are many possible π̃(θ) exp (− i=1 `(zj , fθ )) dθ divergences providing greater robustness properties where `(z, fθ ) is the loss function used by L for infer- than the KLD, e.g. Wasserstein or Stein discrepancy ence. Note that in this formulation, the logarithmic (Barp et al., 2019), but for our exposition we focus on score `0 (z, fθ ) = − log fθ (z) recovers traditional Bayes the βD for its convenience and simplicity.

Harrison Wilde, Jack Jewson, Sebastian Vollmer, Chris Holmes A key difference between the two robust loss func- We provide a proposition that says using more data tions considered above is that while `w (z, fθ ) down- and approaching the limit, θGKLD ε,δ is not necessarily the weights the log-likelihood of each observation equally, optimal target to learn about according to criteria D. `(β) (x, f (·)) does so adaptively, based on how likely the new observation is under the current inference (Ci- Proposition 1 (Suboptimality of the S-DGP) chocki et al., 2011). It is this adaptive down-weighting For S-DGP Gε,δ , model fθ (·), and divergence D, there that allows the βD to target a different limiting param- exists prior π̃(θ), private DGP F0 and at least one eter to θGKLDε,δ . This, in particular, allows the βD to be value of 0 ≤ m < ∞ such that, robust to outliers and/or heavy tailed contaminations. As a result, we believe that T`0 (m; D, f0 , gε,δ ) ≤ D F0 k fθKLD , (11) Gε,δ D F0 k fθβD < D F0 k fθKLD Gε,δ Gε,δ where θGKLD := arg minθ∈Θ KLD(Gε,δ k fθ ) and T`0 is the learning trajectory of the Bayesian posterior predictive across a wide range of S-DGPs, i.e. the βD minimising distribution (using `0 ) based on (synthetic) data z1:m , approximation to Gε,δ is a better approximation of F0 see Eq. (7). than the KLD minimising approximation. Proving that this is the case in general proves complicated The proof of Proposition 1 involves a simple counter and is hampered by the intractability both of the βD example in which the prior is a better approximation minimisation and of many popular S-DGP’s. However, to F0 according to divergence D than Gε,δ . While we further justify this claim in A.1.2 where we show trivial, this could reasonably occur if L has strong, well- that for the prevalent Laplace DP mechanism, this calibrated expert belief judgements, or if they have a holds uniformly over the privacy level parameterised considerable amount of their own data before beginning by the scaling parameter of the Laplace distribution λ to incorporate synthetic data. Furthermore, we argue for D = KLD. next that by considering this learning trajectory path A strength of the βD is that, unlike standard robust between the prior and the S-DGP Gε,δ , in terms of the methods using heavier tailed models or losses (Berger number of synthetic observations m, it is possible to et al., 1994; Huber and Ronchetti, 1981; Beaton and get even ‘closer’ to F0 in terms of a chosen D. Tukey, 1974), `(β) (x, f (·)) does not change the model Changing the divergence used for inference as suggested used for inference. In the absence of any specific knowl- in Section 3.2 changes these trajectories by chang- edge about the S-DGP, updating using the βD maintains ing their limiting parameter. However, Proposition the model L would have used to estimate F0 , but up- 1, which considered learning minimising the KLD, can dates its parameters robustly. This also has advantages equally be shown for learning minimising the βD (see in the data combination scenario where L is combining A.2.1). In the following sections we talk generally inferences from their own private data xL 1:nL with syn- about optimising such a trajectory for a given learn- thetic data z1:m . They can maintain the same model ing method, before focusing on comparisons in these for both datasets, with the same model parameters, methods and the resulting trajectories in Section 4. yet update robustly about z1:m whilst still using the log-score for xL 1:nL (i.e. to condition their prior π̃(θ)). 3.4 Optimising the Learning Trajectory 3.3 The Learning Trajectory The insights from the previous section raise two ques- tions for a learner L using synthetic data: The concept of closeness provided by D allows us to consider how L’s approximation to F0 changes as more data is collected from the S-DGP. To do so we consider 1. Is synthetic data able to improve L’s inferences the ‘learning trajectory’, a function in m of the expected about F0 according to divergence D? If so, divergence to F0 of inference using m observations from 2. What is the optimal quantity to use for getting the Gε,δ : learner closest to F0 according to D? T` (m; D, f0 , gε,δ ) = Ez D f0 (·) k p` (· | z1:m ) , Both questions can be solved by the estimation of where p` (· | z1:m ) is the general Bayesian posterior m∗ := arg min T` (m; D, f0 , gε,δ ) , (12) predictive distribution (using `) based on (synthetic) 0≤m≤M data z1:m . This ‘learning trajectory’ traces the path taken by Bayesian inference from its prior predictive However, clearly the learner never has access to the (m = 0) towards the synthetic data’s DGP under in- data generating density. Instead we take advantage of creasing amounts of data (e.g. Figure 1; Section A.4). the representation of proper scoring rules and propose

Foundations of Bayesian Learning from Synthetic Data using a ‘test set’ x01:N ∼ F0 to estimate study. Here we recommend that alongside releasing synthetic data, K optimises the learning trajectory B N 1 1 XX (b) themselves, under some default model, loss and prior m̂ : = arg min s(x0j , p` (·|z1:m )) (13) 0≤m≤M N B setting by repeatedly partitioning x1:n into test and j=1 b=1 training sets. For example, when releasing classification (b) with {z1:m }b1:B ∼ Gε,δ . data, K could release an m̂ associated with logistic regression and BART, for the log-score, under some As such we use a small amount of data from F0 to guide vaguely informative priors, providing learners an idea the synthetic data inference towards F0 . We consider of what to expect in terms of utility from the synthetic two procedures for doing so: Firstly, by tailoring to data. Whilst this is less tailored to any specific inference L’s specific inference problem, and when this is not problem, it still allows K to communicate a broad possible, putting the onus on K to evaluate the general measure of the quality of its released data for learning ability of their S-DGP to capture F0 . Before so doing about F0 , and is advisable given our results. the following remark briefly considers optimising the learning trajectory for a specific stream of data rather 3.5 Posthoc improvement through averaging than average over the S-DGP. Once m̂ has been estimated, the question then remains Remark 1 We note that although previously the learn- of how to conduct inference given m̂. In particular, ing trajectory was defined to average across samples if more synthetic data is available (e.g. m̂

Harrison Wilde, Jack Jewson, Sebastian Vollmer, Chris Holmes justments discussed in Section 3.2. In order to draw A.6.1.1 for explicit formulations): comparisons between these methods, we study the tra- jectories’ dependence on values spanning a grid of data 1. The standard likelihood adjusted with an additional quantities nL and m, robustness parameters w and β, reweighting parameter w as in Eq. (8). prior values, and the parameters of chosen DP mecha- 2. The posterior under the βD loss as in Eq. (9). nisms (see A.6.3 for full experimental specifications). 3. The ‘Noise-Aware’ likelihood where the S-DGP can The varying amounts of non-private data available to be tractably modelled using the Normal-Laplace L were used to construct increasingly informative pri- convolution (Reed, 2006; Amini and Rabbani, 2017). ors π̃(θ) = π(θ | xL1:nL ) through the use of standard Bayesian updating, as robustness is not required when 4.1.1 Results and Discussion learning using data drawn from F0 . Learning trajec- tories are then estimated utilising an unseen dataset We observe that three different categories of learning x01:N (mimicking either K’s data or some subset of L’s trajectory occur across the models; these are illustrated data not used in training). in the ‘branching’ plots in Figure 1 (explained and To this end we use optimised MCMC sampling schemes analysed further in A.6.4 and A.6.5): (e.g. Hoffman and Gelman, 2014) to sample from all of the considered posteriors in each experiment’s case; 1. The prior π̃ is sufficiently inaccurate or uninforma- drawing comparisons across the grid laid out above and tive (in this case due to low nL ) such that the syn- repeating experiments to mitigate any sources of noise. thetic data continues to be useful across the range This results in an extensive computational task, made of m we consider. As a result the learning trajectory feasible through a mix of Julia’s Turing PPL (Ge et al., is a monotonically decreasing curve in the criteria 2018), MLJ (Blaom et al., 2020) and Stan (Carpenter of interest. et al., 2017). 2. A turning point is observed; synthetic data initially brings us closer to F0 before the introduction of The majority of the experiments are carried out with further synthetic observations moves the inference ε = 6, which is seen to be a realistic value respective of away. We see that in the majority of cases these practical applications (Lee and Clifton, 2011; Erlings- trajectories lie under the limiting KLD and βD ap- son et al., 2014; Tang et al., 2017; Differential Privacy proximations to Gε,δ demonstrating the efficacy of Team at Apple, 2017) and upon observation of the re- ‘optimising the learning trajectory’ through the ex- lationship between privacy and misspecification shown istence of these optimal turning points. in Figure A.4. We evaluate our experiments according 3. The final scenario occurs under a sufficiently infor- to the divergences and scoring rules discussed in detail mative prior π̃ (here due to a large nL ) such that in A.6.2, presented across a range of the figures in the synthetic data is not observably of any use at all; it main text and supplementary material. can be seen to immediately cause model performance to deteriorate. 4.1 Simulated Gaussian Models We can further quantify the turning points which are We first introduce a simple but illustrative simulated perhaps the most interesting characteristic of these example in which we infer the parameters of a Gaus- experiments. To do this we formulate bootstrapped sian model fθ = N (µ, σ 2 ) where θ = (µ, σ 2 ). We place averages of the number of ‘effective real samples’ that conjugate priors on θ with σ 2 ∼ InverseGamma(αp , βp ) correspond to the estimated optimal quantity of syn- and µ ∼ N (µp , σp × σ) respectively. We consider x1:n thetic data. This is done by comparing these minima drawn from DGP F0 = N (0, 12 ) and adopt the Laplace with the black curves in the ‘branching’ plots represent- mechanism (Dwork et al., 2014) to define our S-DGP. ing the learning trajectory under an increasing nL and This mechanism works by perturbing samples drawn m = 0 (see A.6.4 for more details). These calculations from the DGP with noise drawn from the Laplace dis- are shown in Figure 2 (and discussed further in A.6.5). tribution of scale λ, calibrated via the sensitivity S of In general, we observe a significant increase in perfor- the DGP in order to provide (ε, 0)-DP per the Laplace mance from the βD (see Figures 1, 3), indicated by mechanism’s definition with ε = S/λ. To achieve finite- its proximity to even the noise-aware model at lower ness of S in this case, we adjust our model to be that values of nL , alongside more modest improvements of a truncated Gaussian; restricting its range to ±3σ from reweighting methods. The βD achieves more de- to allow for meaningful ε’s to be calculated under the sirable minimum-trajectory log score, KLD and Wasser- Laplace mechanism (See A.3 for a proof of this result). stein values compared to other model types; exhibiting We then compare and evaluate the empirical perfor- greater robustness to larger amounts of synthetic data mance of the competing methods defined below (see where other approaches lose out significantly.

Foundations of Bayesian Learning from Synthetic Data w = 0.5 β = 0.5 Number of 0.3 Real Samples 2 4 6 Kullback-Leibler Distance 8 0.2 10 13 16 19 22 25 0.1 30 35 40 50 75 0.0 100 0 25 50 75 100 0 25 50 75 100 Total Number of Samples (Real + Synth) Figure 1: Shows how the KLD to F0 changes as we add more synthetic data, starting with increasing amount of real data. This demonstrates the effectiveness of the optimal discovered βD configuration compared to the the closest alternative traditional model in terms of performance with down-weighting w = 0.5; the black dashed line illustrates KLD(F0 k fθ∗ ) for θ∗ = θGKLD ε,δ (left) and θ∗ = θGβD ε,δ (right), representing the approximation to F0 given an infinite sample from Gε,δ under the two learning methods, and exhibiting the superiority of the βD. AUROC Log Score Maximum Effective Real Sample Gain Throough the Use of Synthetic Data 50 400 Model Type w = 0.5 w = 1 (Naive) β = 0.25 0 200 β = 0.5 β = 0.75 -50 0 0 1000 2000 3000 4000 0 1000 2000 3000 4000 Number of Real Samples Figure 2: Shows the effective number of real samples gained through optimal m̂ synthetic observations alongside varying amounts of real data usage with respect to the AUROC and log-score performance criteria. These are calculated and presented here via bootstrapped averaging under a logistic regression model learnt on the Framingham dataset. The amount of effective real samples is significantly affected by the learning task’s criteria. 0.85 Model 0.3 Model Configuration Configuration Wasserstein Distance 0.84 Resampled Noise-Aware w = 0 (No Syn) AUROC w = 0 (No Syn) 0.83 w = 0.5 w = 0.5 w = 1 (Naive) 0.2 w = 1 (Naive) β = 0.25 β = 0.5 0.82 β = 0.5 β = 0.8 β = 0.75 0.81 0.1 0 50 100 150 200 0 100 200 300 Number of Synthetic Samples Number of Synthetic Samples (a) Simulated Gaussian, real n = 10 (b) UCI Heart Dataset, real n = 30 (10%) Figure 3: Given a fixed real amount of data, we can compare model performances directly by focusing on one of the ‘branches’ in the class of diagrams shown in Figures 1 & 4, to see that the βD’s performance falls between that of the noise-aware model and the other models, exhibiting robust and desirable behaviour across a range of β. Naïve and reweighting-based approaches fail to gain significantly over not using synthetic data (shown by w = 0’s flat trajectory); the resampled model in the logistic case can also be seen to perform very poorly in comparison to models that leverage the synthetic data.

Harrison Wilde, Jack Jewson, Sebastian Vollmer, Chris Holmes Referring to Figures 2, 3 and 4 we see that the learn- ing trajectories observed in this more realistic example mirror those observed in our simulated Gaussian ex- periments. There are however some cases in which the reweighted posterior outperforms the βD, and we see large discrepancies in m̂ when comparing log score to AUROC values, reinforcing the importance of carefully defining the learning task to prioritise. Additionally, experiments using synthetic data from a GAN offer the unique observation that performance can actually improve as ε decreases. We believe this Figure 4: This plot illustrates an interesting and impor- is due to potential mode collapse in the GAN learning tant observation made when varying ε for a GAN based process on imbalanced datasets, and concentrations of model, we observe that there is a privacy ‘sweet-spot’ realisations of Gε,δ as the injected noise increases such around ε = 1 whereby more private data performs bet- that a small number of synthetic samples can actually ter than less private data (see the curves for ε = 100 be more representative of F0 than even the real data. which essentially represent non-private data).1 This effect is short-lived as more synthetic observations are used in learning, as presumably these samples then become over-represented through the posterior distribu- 4.2 Logistic Regression tion and performance begins to deteriorate, see Figure 4 for performance comparisons across ε. We now move on to a more prevalent and practical class of models that also exhibit the potentially dangerous behaviours of synthetic data in real-world contexts, 5 Conclusions via datasets concerning subjects that have legitimate potential privacy concerns. Namely, we build logistic We consider foundations of Bayesian learning from regression models for the UCI Heart Disease dataset synthetic data that acknowledge the intrinsic model (Dua and Graff, 2017) and the Framingham Cohort misspecification and learning task at hand. Contrary dataset (Splansky et al., 2007). Clearly, we are now only to traditional statistical inferences, conditioning on in- able to access the empirical distribution F n∗ , where n∗ creasing amounts of synthetic data is not guaranteed to is the total amount of data present in each dataset. We help you learn about the true data generating process use x1:nT to train an instance of the aforementioned or make better decisions. Down-weighting the infor- PATE-GAN Gε,δ and keep back x1:(n∗ −nT ) for evaluation; mation in synthetic data (either using a weight w or we then draw synthetic data samples z1:m ∼ Gε,δ . As divergence βD) provides a principled approach to robust before, we investigate how the learning trajectories are optimal information processing and warrants further affected across the experimental parameter grid. investigation. Further work could consider augmenting these general robust techniques with tailored adjust- Again, we consider learning using `w and `β applied ments to inference based on the specific synthetic data to the logistic regression likelihood, fθ (see A.6.1.2 for acknowledging the discrepancies between its generating exact formulations). In this case we cannot formulate density and the Learner’s true target. a ‘Noise-Aware’ model due to the black-box nature of the GAN, highlighting the reality of the model mis- specification setting we find ourselves in aside from Acknowledgements simple or simulated examples. We can instead define a ‘resampled’ model that recycles the real data used in HW is supported by the Feuer International Scholar- formulating the prior. ship in Artificial Intelligence. JJ was funded by the Ayudas Fundación BBVA a Equipos de Investigación 4.2.1 Results and Discussion Cientifica 2017 and Government of Spain’s Plan Na- cional PGC2018-101643-B-I00 grants whilst working Here the learning trajectories are defined with respect on this project. SJV is supported by The Alan Turing to the AUROC as well as the log score; whilst not tech- Institute (EPSRC grant EP/N510129/) and the Uni- nically a divergence, this gives us a decision theoretic versity of Warwick IAA funding. CH is supported by criteria to quantify the closeness of our inference to F0 . The Alan Turing Institute, Health Data Research UK, 1 This plot exhibits the effect under the βD model on the the Medical Research Council UK, the Engineering and Framingham dataset with β = 0.5, but is observable across Physical Sciences Research Council (EPSRC) through all model types in both AUROC and log score. the Bayes4Health programme Grant EP/R018561/1,

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