POST-PROCESSING DIFFERENTIALLY PRIVATE SYNTHETIC DATA

An embodiment generates, using a probability distribution of synthetic data, a first value of a utility measure function, and a second value of the utility measure function, a value of an optimization variable, the synthetic data generated from a source dataset using a differential privacy technique, the utility measure function measuring a characteristic of a dataset. An embodiment computes, using the value of the optimization variable, a sampling weight, the sampling weight comprising a probability of selecting a portion of data from the synthetic data. An embodiment samples, according to the sampling weight, the synthetic data, the sampling resulting in a sampled synthetic dataset. An embodiment trains, using the sampled synthetic dataset, a machine learning model, the training resulting in a trained machine learning model.

WANG et al., Post-processing Private Synthetic Data for Improving Utility on Selected Measures, 24 May 2023.

BACKGROUND

The present invention relates generally to data privacy. More particularly, the present invention relates to a method, system, and computer program for post-processing differentially private synthetic data.

A dataset or database is a logical container used to organize and control access to resources such as stored data. A dataset typically includes one or more tables. A table stores data values using a model of labelled columns (also referred to as variables or fields) and rows (also referred to as records). A cell of the table is an intersection of a row and a column. Typically, column labels designate a particular type of data (for example, a table might have columns labelled “Customer ID”, “Name”, “Address”, and “Telephone Number”), and rows hold data for particular individuals (e.g., data for Customer A might be stored in row 1 and data for Customer B might be stored in row 2).

Data simulation, or artificial data generation, or synthetic data generation, is a process of generating artificial data that mimics the characteristics and patterns of real-world data. Synthetic data generation is often used to generate training and testing data for use in developing machine learning models and in other situations where insufficient real-world data is available for use. Data simulation is typically performed by fitting a parametric statistical distribution to the observed data, and generating new data points from the fitted distribution. However, statistical analyses of data in a dataset can reveal information about a single individual in the dataset, particularly if an adversary knows information about other individuals in the dataset. Thus, privacy preserving data analysis and data simulation techniques, which attempt to make a dataset usable for analysis or generate artificial data using statistical information about a dataset, without compromising the privacy of any individuals with records in the dataset, have been developed.

One method of implementing privacy preserving data analysis is differential privacy, which hides the presence of an individual in a dataset from a user of the dataset by making two output distributions, one with and the other without the individual, be computationally indistinguishable (for all individuals). Data generated under a differential privacy mechanism or differential privacy technique is referred to as private synthetic data or a private synthetic dataset. A differential privacy mechanism or technique guarantees that even if an adversary observes a private synthetic dataset, the adversary cannot distinguish if a specific individual's data were ever used to generate the synthetic data, and prevents disclosure of a specific individual's data. As a result, a machine learning model can be trained on private synthetic data but then deployed, once trained, for use on real data.

A utility measure function measures a characteristic of a dataset. Some non-limiting examples of utility measure functions are a mean or average of the values in a particular column of a dataset, and a correlation coefficient between data in two columns of a dataset.

SUMMARY

The illustrative embodiments provide for post-processing differentially private synthetic data. An embodiment includes generating, using a probability distribution of synthetic data, a first value of a utility measure function, and a second value of the utility measure function, a value of an optimization variable, the synthetic data generated from a source dataset using a first differential privacy technique, the utility measure function measuring a characteristic of a dataset. An embodiment includes computing, using the value of the optimization variable, a sampling weight, the sampling weight comprising a probability of selecting a portion of data from the synthetic data. An embodiment includes sampling, according to the sampling weight, the synthetic data, the sampling resulting in a sampled synthetic dataset. An embodiment includes training, using the sampled synthetic dataset, a machine learning model, the training resulting in a trained machine learning model. Other embodiments of this aspect include corresponding computer systems, apparatus, and computer programs recorded on one or more computer storage devices, each configured to perform the actions of the embodiment.

An embodiment includes a computer usable program product. The computer usable program product includes a computer-readable storage medium, and program instructions stored on the storage medium.

An embodiment includes a computer system. The computer system includes a processor, a computer-readable memory, and a computer-readable storage medium, and program instructions stored on the storage medium for execution by the processor via the memory.

DETAILED DESCRIPTION

The illustrative embodiments recognize that those responsible for training machine learning models use a standardized model training pipeline that is difficult to adapt to the use of private synthetic data. While there are synthetic data generation techniques available for use under differential privacy, these techniques often optimize a generic class of loss functions during data generation. As a result, the generated data might not preserve a data characteristic (measured by a specific utility measure) needed to train a particular model, thus resulting in a model that does not meet a performance criterion and necessitating additional model training and testing. For example, some standardized model training pipelines rely on correlation coefficients between a model's output and one or more inputs, while some synthetic data generation techniques alter correlation coefficient values of the generated data. Workload-aware differential privacy methods aim to improve the privacy-utility trade-off by considering the specific workload or queries that will be applied to the data being generated, but these methods are inefficient as they need to fit a graphical model or a neural network to generate synthetic data. In addition, workload-aware differential privacy methods only evaluate the utility of the synthetic data by how well the synthetic data preserves known-in-advance statistics of the real data (e.g., 3-way marginals) of the real data. As a result, if the specific workload or queries that will be applied to the data being generated changes, the synthetic data will have to be regenerated. Public data-assisted methods generate private synthetic data by using public data. However, public data with similar characteristics to the private data must be available. In addition, some synthetic data generation techniques are computationally expensive for large datasets as they either require solving an integer program multiple times or need to solve a large-scale optimization problem. As well, some synthetic data generation techniques can introduce more noise to outlier data than to other data, potentially resulting in a disparate impact on an underrepresented population in the trained model.

Thus, the illustrative embodiments recognize that there is an unmet need to generate synthetic private data that preserves a data characteristic needed to train a particular model without requiring data regeneration when an end use of the data changes, that is more efficient than existing techniques when used on comparatively large datasets, while allowing for mitigation of the noise introduced to outlier data.

The present disclosure addresses the deficiencies described above by providing a process (as well as a system, method, machine-readable medium, etc.) that generates, using a probability distribution of synthetic data, a first value of a utility measure function, and a second value of the utility measure function, a value of an optimization variable; computes, using the value of the optimization variable, a sampling weight; samples, according to the sampling weight, the synthetic data; and trains, using the sampled synthetic dataset, a machine learning model. Thus, the illustrative embodiments provide for post-processing differentially private synthetic data.

An illustrative embodiment receives private synthetic data, generated from a source dataset (i.e., real data as opposed to synthetic data) using a presently available differential privacy technique. Which presently available differential privacy technique is used does not matter to an embodiment. An embodiment also receives one or more utility measure functions, each specifying an intended characteristic of the private synthetic data. An embodiment also receives one or more utility measure function values. Each utility measure value is a value of a utility measure function, computed using a presently available differential privacy technique on the source dataset. A utility measure value need not be computed using the same differential privacy technique as the private synthetic data. Thus, utility measure values are goals to which private synthetic data output from an embodiment should conform. In the mathematical expressions described herein, there are K utility measure functions (where K is an integer greater than or equal to one. Each utility measure function k is represented by qk(X), where the source dataset includes a single record x=(x1 . . . ,xd)∈X from each individual. Values of utility measure functions computed on the source dataset are denoted by a=(a1, . . . ,ak).

An embodiment uses a probability distribution of private synthetic data, a first value of a utility measure function, and a second value of the utility measure function to generate a value of an optimization variable. The first value of the utility measure function is computed on the source dataset using the differential privacy technique, and the second value of the utility measure function is computed on the synthetic data. In particular, if Psyn represents a probability distribution of input private synthetic data and Ppost represents a probability distribution of post-processed private data (an output of an embodiment, conforming to one or more utility measure functions), an embodiment formulates an optimization problem, in which the objective function is to minimize a discrepancy, or distance, between Ppost and Psyn, with constraints guaranteeing that the post-processed private data meet the utility measure functions up to a tolerance level greater than or equal to 0 and denoted by γ. This optimization problem is depicted as optimization problem 410 in FIG. 4, in which distance between Ppost and Psyn is computed using KL-divergence, a presently available technique. However, the optimization problem as formulated becomes more difficult to solve as the number of features (i.e., columns in the input dataset) grows, as the number of variables grows exponentially with the number of features. Thus, an embodiment uses a reformulated optimization instead, depicted as reformulated optimization problem 420 in FIG. 4. Within reformulated optimization problem 420, the ∥ ∥1 notation denotes computation of the L1 norm of a vector. The L1 norm is the sum of the absolute value of all of a vector's components. Specifically, for a vector with components (λ1, . . . , λK), the vector's L1 norm is defined as |λ1|+ . . . +|λK|. The dual variable(s) λ (a vector with K components) in the expressions are also referred to as optimization variables, and the number of optimization variables is equal to the number of utility measure functions.

An embodiment uses a presently available technique to denoise the first value(s) of the utility measure function (computed on the source dataset using a differential privacy technique, and denoted by ak in the depicted expressions) before using the denoised first value(s) to compute values of one or more optimization variables. One embodiment denoises the first value(s) by solving a linear program, while another embodiment denoises the first value(s) using a quadratic program, both presently available techniques

An embodiment generates one or more values of the optimization variables using the expression depicted as reformulated optimization problem 420 in FIG. 4. One embodiment generates one or more values of the optimization variables using the technique depicted in pseudocode in optimization variable computation 500 in FIG. 5. In the depicted technique, there are four inputs: (1) utility difference vector set bar (qi), with each component representing the difference between the k-th utility measure of the i-th synthetic data point and its average counterpart from the source data; (2) the maximum number of iterations T; (3) mini-batch index Bt, which defines the data points to be used in the t-th iteration; and (4) step size αt, a hyperparameter governing the magnitude of updates to the optimization variable(s) in every iteration (often referred to as the learning rate). During each iteration, the computation selects a mini-batch of data from the utility difference vector set bar (qi) based on the mini-batch index Bt, and uses this selected mini-batch to update both the dual variable(s) λ and auxiliary variable τ using the depicted expressions. In the depicted expressions, (t) and (t+1) represent a value of a variable at the t-th and (t+1)-th iterations. Additionally, αt denotes the step size at the t-th iteration and τ(t) denotes the value of τ at the t-th iteration. Another embodiment, to avoid underflow and overflow problems, performs the computations in log-domain instead, using the technique depicted in log-domain optimization variable computation 600 in FIG. 6. In log-domain optimization variable computation 600, the inputs, outputs, and notation are the same as in optimization variable computation 500 in FIG. 5; however an additional auxiliary variable llj (a vector with component j) is also used. Another embodiment uses another presently known technique to generates one or more values of the optimization variables.

An embodiment uses the value(s) of the optimization variable to compute corresponding sampling weight(s). A sampling weight is a probability of selecting a particular record (i.e., a portion of data) from the synthetic data. In particular, there are n sampling weights, where n denotes the number of records in the private synthetic data being sampled. One embodiment constructs a function defined by computing a sum, for j=1 to K, of λj*(qj(x)−aj), and computing e raised to a negative of the sum, with λj, qj(x), and aj as defined elsewhere herein. An embodiment applies the function to a synthetic data point, or record (denoted by x) in the private synthetic dataset, and the result of each application becomes a sampling weight for that record.

An embodiment samples the synthetic data according to the computed sampling weight(s). Thus, a record with a higher sampling weight will more likely to be sampled (or possibly be sampled multiple times), and a record with a lower sampling weight will be less likely to be sampled, and thus removed from the sampled synthetic dataset. As a result, the sampled synthetic dataset has a characteristic that matches a first characteristic of the source dataset within a tolerance, where the characteristic is measured according to an input utility measure function.

An embodiment trains, using the sampled synthetic dataset, a machine learning model. Another embodiment sends the sampled synthetic dataset to an existing model training implementation, such as a standardized pipeline, and causes training of a machine learning model using the sampled synthetic dataset.

A correlation matrix is a table that shows the correlation coefficients between pairs of features (i.e., columns) of input data. A correlation matrix is often used as a reference when selecting features for use in machine learning model training. In one use case, an embodiment improves alignment between the correlation matrix of the sampled synthetic dataset and the real data, by using the first-order moment of one or more features, or columns, and second-order moments of one or more pairs of features, or columns, as utility measure functions, thus improving model training using the sampled synthetic dataset. First and second order moments are presently known. In particular, if the dataset includes d features x1, . . . , xd for each record, then the first-order moment utility measure functions are x1, . . . , xd, and the second-order moment utility measure functions are xi xj for 1<=i<=j<=d.

Some differential privacy mechanisms introduce more noise to sparsely sampled data regions than to data regions with higher sample density, potentially resulting in inaccurate training of models using the noisy data. Thus, in one use case, an embodiment uses one or more presently known group fairness metrics as utility measure functions, thus mitigating data biases by filtering out synthetic data that does not accurately represent real data.

Furthermore, simplified diagrams of the data processing environments are used in the figures and the illustrative embodiments. In an actual computing environment, additional structures or components that are not shown or described herein, or structures or components different from those shown but for a similar function as described herein may be present without departing the scope of the illustrative embodiments.

Furthermore, the illustrative embodiments are described with respect to specific actual or hypothetical components only as examples. Any specific manifestations of these and other similar artifacts are not intended to be limiting to the invention. Any suitable manifestation of these and other similar artifacts can be selected within the scope of the illustrative embodiments.

Measured service: cloud systems automatically control and optimize resource use by leveraging a metering capability at some level of abstraction appropriate to the type of service (e.g., storage, processing, bandwidth, and active user accounts). Resource usage can be monitored, controlled, reported, and invoiced, providing transparency for both the provider and consumer of the utilized service.

With reference to FIG. 2, this figure depicts a flowchart of an example process for loading of process software in accordance with an illustrative embodiment. The flowchart can be executed by a device such as computer 101, end user device 103, remote server 104, or a device in private cloud 106 or public cloud 105 in FIG. 1.

Step 202 begins the deployment of the process software. An initial step is to determine if there are any programs that will reside on a server or servers when the process software is executed (203). If this is the case, then the servers that will contain the executables are identified (229). The process software for the server or servers is transferred directly to the servers' storage via FTP or some other protocol or by copying though the use of a shared file system (230). The process software is then installed on the servers (231).

Next, a determination is made on whether the process software is to be deployed by having users access the process software on a server or servers (204). If the users are to access the process software on servers, then the server addresses that will store the process software are identified (205).

A determination is made if a proxy server is to be built (220) to store the process software. A proxy server is a server that sits between a client application, such as a Web browser, and a real server. It intercepts all requests to the real server to see if it can fulfill the requests itself. If not, it forwards the request to the real server. The two primary benefits of a proxy server are to improve performance and to filter requests. If a proxy server is required, then the proxy server is installed (221). The process software is sent to the (one or more) servers either via a protocol such as FTP, or it is copied directly from the source files to the server files via file sharing (222). Another embodiment involves sending a transaction to the (one or more) servers that contained the process software, and have the server process the transaction and then receive and copy the process software to the server's file system. Once the process software is stored at the servers, the users via their client computers then access the process software on the servers and copy to their client computers file systems (223). Another embodiment is to have the servers automatically copy the process software to each client and then run the installation program for the process software at each client computer. The user executes the program that installs the process software on his client computer (232) and then exits the process (210).

In step 206 a determination is made whether the process software is to be deployed by sending the process software to users via e-mail. The set of users where the process software will be deployed are identified together with the addresses of the user client computers (207). The process software is sent via e-mail to each of the users' client computers (224). The users then receive the e-mail (225) and then detach the process software from the e-mail to a directory on their client computers (226). The user executes the program that installs the process software on his client computer (232) and then exits the process (210).

Lastly, a determination is made on whether the process software will be sent directly to user directories on their client computers (208). If so, the user directories are identified (209). The process software is transferred directly to the user's client computer directory (227). This can be done in several ways such as, but not limited to, sharing the file system directories and then copying from the sender's file system to the recipient user's file system or, alternatively, using a transfer protocol such as File Transfer Protocol (FTP). The users access the directories on their client file systems in preparation for installing the process software (228). The user executes the program that installs the process software on his client computer (232) and then exits the process (210).

With reference to FIG. 3, this figure depicts a block diagram of an example configuration for post-processing differentially private synthetic data in accordance with an illustrative embodiment. Application 300 is the same as application 200 in FIG. 1.

In the illustrated embodiment, application 300 receives private synthetic data, generated from a source dataset (i.e., real data as opposed to synthetic data) using a presently available differential privacy technique. Application 300 also receives one or more utility measure functions, each specifying an intended characteristic of the private synthetic data. Application 300 also receives one or more utility measure function values. Each utility measure value is a value of a utility measure function, computed using a presently available differential privacy technique on the source dataset. A utility measure value need not be computed using the same differential privacy technique as the private synthetic data. Thus, utility measure values are goals to which private synthetic data output from an embodiment should conform. In the mathematical expressions described herein, there are K utility measure functions (where K is an integer greater than or equal to one. Each utility measure function k is represented by qk(X), where the source dataset includes a single record x=(x1 . . . xd)∈X from each individual. Values of utility measure functions computed on the source dataset are denoted by a=(a1, . . . ,aK).

Optimization variable generation module 310 uses a probability distribution of private synthetic data, a first value of a utility measure function, and a second value of the utility measure function to generate a value of an optimization variable. The first value of the utility measure function is computed on the source dataset using the differential privacy technique, and the second value of the utility measure function is computed on the synthetic data. In particular, if Psyn represents a probability distribution of input private synthetic data and Ppost represents a probability distribution of post-processed private data (an output of application 300, conforming to one or more utility measure functions), module 310 formulates an optimization problem, in which the objective function is to minimize a discrepancy, or distance, between Ppost and Psyn, with constraints guaranteeing that the post-processed private data meet the utility measure functions up to a tolerance level greater than or equal to 0 and denoted by γ. This optimization problem is depicted as optimization problem 410 in FIG. 4, in which distance between Ppost and Psyn is computed using KL-divergence, a presently available technique. However, the optimization problem as formulated becomes more difficult to solve as the number of features (i.e., columns in the input dataset) grows, as the number of variables grows exponentially with the number of features. Thus, module 310 uses a reformulated optimization instead, depicted as reformulated optimization problem 420 in FIG. 4. Within reformulated optimization problem 420, the ∥ ∥1 notation denotes computation of the L1 norm of a vector. The L1 norm is the sum of the absolute value of all of a vector's components. Specifically, for a vector with components (λ1, . . . , λK), the vector's L1 norm is defined as |λ1|+ . . . +|λK|. The dual variable(s) λ (a vector with K components) in the expressions are also referred to as optimization variables, and the number of optimization variables is equal to the number of utility measure functions.

Module 310 uses a presently available technique to denoise the first value(s) of the utility measure function (computed on the source dataset using a differential privacy technique, and denoted by ak in the depicted expressions) before using the denoised first value(s) to compute values of one or more optimization variables. One implementation of module 310 denoises the first value(s) by solving a linear program, while another embodiment denoises the first value(s) using a quadratic program.

Module 310 generates one or more values of the optimization variables using the expression depicted as reformulated optimization problem 420 in FIG. 4. One implementation of module 310 generates one or more values of the optimization variables using the technique depicted in pseudocode in optimization variable computation 500 in FIG. 5. In the depicted technique, there are four inputs: (1) utility difference vector set bar (qi), with each component representing the difference between the k-th utility measure of the i-th synthetic data point and its average counterpart from the source data; (2) the maximum number of iterations T; (3) mini-batch index Bt, which defines the data points to be used in the t-th iteration; and (4) step size αt, a hyperparameter governing the magnitude of updates to the optimization variable(s) in every iteration (often referred to as the learning rate). During each iteration, the computation selects a mini-batch of data from the utility difference vector set bar (qi) based on the mini-batch index Bt, and uses this selected mini-batch to update both the dual variable(s) λ and auxiliary variable τ using the depicted expressions. In the depicted expressions, (t) and (t+1) represent a value of a variable at the t-th and (t+1)-th iterations. Additionally, αt denotes the step size at the t-th iteration and τ(t) denotes the value of t at the t-th iteration. Another implementation of module 310, to avoid underflow and overflow problems, performs the computations in log-domain instead, using the technique depicted in log-domain optimization variable computation 600 in FIG. 6. In log-domain optimization variable computation 600, the inputs, outputs, and notation are the same as in optimization variable computation 500 in FIG. 5; however an additional auxiliary variable llj (a vector) is also used. Another implementation of module 310 uses another presently known technique to generates one or more values of the optimization variables.

Sampling weight module 320 uses the value(s) of the optimization variable to compute corresponding sampling weight(s). A sampling weight is a probability of selecting a particular record (i.e., a portion of data) from the synthetic data. In particular, there are n sampling weights, where n denotes the number of records in the private synthetic data being sampled. One implementation of module 320 constructs a function defined by computing a sum, for j=1 to K, of λj*(qj(x)−aj), and computing e raised to a negative of the sum, with λj, qj(x), and aj as defined elsewhere herein. Module 320 applies the function to a synthetic data point, or record (denoted by x) in the private synthetic dataset, and the result of each application becomes a sampling weight for that record.

Sampling module 330 samples the synthetic data according to the computed sampling weight(s). Thus, a record with a higher sampling weight will more likely to be sampled (or possibly be sampled multiple times), and a record with a lower sampling weight will be less likely to be sampled, and thus removed from the sampled synthetic dataset. As a result, the sampled synthetic dataset has a characteristic that matches a first characteristic of the source dataset within a tolerance, where the characteristic is measured according to an input utility measure function.

Model training module 340 trains, using the sampled synthetic dataset, a machine learning model. Another implementation of module 340 sends the sampled synthetic dataset to an existing model training implementation, such as a standardized pipeline, and causes training of a machine learning model using the sampled synthetic dataset.

With reference to FIG. 4, this figure depicts optimization problems for use in post-processing differentially private synthetic data in accordance with an illustrative embodiment. In particular, the optimization problems can be used by optimization variable generation module 310 in FIG. 3.

In the depicted optimization problem 410, the objective function is to minimize a discrepancy, or distance, between Ppost and Psyn, with constraints guaranteeing that the post-processed private data meet the utility measure functions up to a tolerance level greater than or equal to 0 and denoted by γ. A distance between Ppost and Psyn is computed using KL-divergence, a presently available technique. Reformulated optimization problem 420 depicts a reformulated version of problem 410. Within reformulated optimization problem 420, the ∥ ∥1 notation denotes computation of the L1 norm of a vector. The L1 norm is the sum of the absolute value of all of a vector's components. Specifically, for a vector with components (λ1, . . . , λK), the vector's L1 norm is defined as |λ1|+ . . . +|λK|. The dual variable(s) λ (a vector with K components) in the expressions are also referred to as optimization variables, and the number of optimization variables is equal to the number of utility measure functions.

With reference to FIG. 5, this figure depicts a computation technique for use in post-processing differentially private synthetic data in accordance with an illustrative embodiment. In particular, the computation technique can be used by optimization variable generation module 310 in FIG. 3.

In particular, optimization variable computation 500 depicts a technique usable by module 310 to generate one or more values of the optimization variables. In the depicted technique, there are four inputs: (1) utility difference vector set 502 (bar (qi)), with each component representing the difference between the k-th utility measure of the i-th synthetic data point and its average counterpart from the source data; (2) the maximum number of iterations T; (3) mini-batch index Bt, which defines the data points to be used in the t-th iteration; and (4) step size at (learning rate 504), a hyperparameter governing the magnitude of updates to the optimization variable(s) in every iteration. During each iteration, the computation selects a mini-batch of data from the utility difference vector set 502, based on the mini-batch index Bt, and uses this selected mini-batch to update both the dual variable(s) λ (506) and auxiliary variable t (508) using the depicted expressions. In the depicted expressions, (t) and (t+1) represent a value of a variable at the t-th and (t+1)-th iterations. Additionally, αt denotes the step size at the t-th iteration and τ(t) denotes the value of t at the t-th iteration.

With reference to FIG. 6, this figure depicts another computation technique for use in post-processing differentially private synthetic data in accordance with an illustrative embodiment. In particular, the computation technique can be used by optimization variable generation module 310 in FIG. 3. Utility difference vector set 502, learning rate 504, and 508 are the same as utility difference vector set 502, learning rate 504, and 508 in FIG. 5.

In particular, optimization variable computation 600 depicts a technique usable by module 310 to generate one or more values of the optimization variables. In the depicted technique, there are four inputs: (1) utility difference vector set 502 (bar(qi)), with each component representing the difference between the k-th utility measure of the i-th synthetic data point and its average counterpart from the source data; (2) the maximum number of iterations T; (3) mini-batch index Bt, which defines the data points to be used in the t-th iteration; and (4) step size at (learning rate 504), a hyperparameter governing the magnitude of updates to the optimization variable(s) in every iteration. During each iteration, the computation selects a mini-batch of data from the utility difference vector set 502, based on the mini-batch index Bt, and uses this selected mini-batch to update both the auxiliary variable(s) λ (606) and auxiliary variable τ (508) using the depicted expressions. After the iterations, dual variable(s) 610 are computed. In the depicted expressions, (t) and (t+1) represent a value of a variable at the t-th and (t+1)-th iterations. Additionally, αt denotes the step size at the t-th iteration and τ(t) denotes the value of t at the t-th iteration.

With reference to FIG. 7, this figure depicts an example of post-processing differentially private synthetic data in accordance with an illustrative embodiment. The example can be executed using application 300 in FIG. 3.

As depicted, optimization variable generation module 310 uses a probability distribution of private synthetic data 730, utility measure function value(s) 720 (computed on the source dataset using a differential privacy technique), and a value of utility measure function 710 (computed on private synthetic data 730) to generate value(s) of an optimization variable 740. Sampling weight module 320 uses value(s) 740 to compute corresponding sampling weight(s) 750. Sampling module 330 samples private synthetic data 730 according to the computed sampling weight(s) 750, resulting in sampled synthetic data 760. Model training module 340 uses sampled synthetic data 760 to train machine learning model 770, resulting in trained machine learning model 780.

With reference to FIG. 8, this figure depicts a flowchart of an example process for post-processing differentially private synthetic data in accordance with an illustrative embodiment. Process 800 can be implemented in application 200 in FIG. 3.

In the illustrated embodiment, at block 802, the process, using a probability distribution of synthetic data, a first value of a utility measure function, and a second value of the utility measure function, generates a value of an optimization variable, the synthetic data generated from a source dataset using a differential privacy technique, the utility measure function measuring a characteristic of a dataset. At block 804, the process, using the value of the optimization variable, computes a sampling weight, the sampling weight comprising a probability of selecting a portion of data from the synthetic data. At block 806, the process samples, according to the sampling weight, the synthetic data. At block 808, the process, using the sampled synthetic dataset, trains a machine learning model. Then the process ends.

Embodiments of the present invention may also be delivered as part of a service engagement with a client corporation, nonprofit organization, government entity, internal organizational structure, or the like. Aspects of these embodiments may include configuring a computer system to perform, and deploying software, hardware, and web services that implement, some or all of the methods described herein. Aspects of these embodiments may also include analyzing the client's operations, creating recommendations responsive to the analysis, building systems that implement portions of the recommendations, integrating the systems into existing processes and infrastructure, metering use of the systems, allocating expenses to users of the systems, and billing for use of the systems. Although the above embodiments of present invention each have been described by stating their individual advantages, respectively, present invention is not limited to a particular combination thereof. To the contrary, such embodiments may also be combined in any way and number according to the intended deployment of present invention without losing their beneficial effects.