LogisticRegressionDP: Privacy-preserving Logistic Regression

In DPpack: Differentially Private Statistical Analysis and Machine Learning

Privacy-preserving Logistic Regression

Description

This class implements differentially private logistic regression \insertCitechaudhuri2011DPpack. Either the output or the objective perturbation method can be used.

Details

To use this class for logistic regression, first use the new method to construct an object of this class with the desired function values and hyperparameters. After constructing the object, the fit method can be applied with a provided dataset and data bounds to fit the model. In fitting, the model stores a vector of coefficients coeff which satisfy differential privacy. These can be released directly, or used in conjunction with the predict method to privately predict the outcomes of new datapoints.

Note that in order to guarantee differential privacy for logistic regression, certain constraints must be satisfied for the values used to construct the object, as well as for the data used to fit. These conditions depend on the chosen perturbation method. The regularizer must be 1-strongly convex and differentiable. It also must be doubly differentiable if objective perturbation is chosen. Additionally, it is assumed that if x represents a single row of the dataset X, then the l2-norm of x is at most 1 for all x. In order to ensure this constraint is satisfied, the dataset is preprocessed and scaled, and the resulting coefficients are postprocessed and un-scaled so that the stored coefficients correspond to the original data. Due to this constraint on x, it is best to avoid using a bias term in the model whenever possible. If a bias term must be used, the issue can be partially circumvented by adding a constant column to X before fitting the model, which will be scaled along with the rest of X. The fit method contains functionality to add a column of constant 1s to X before scaling, if desired.

Super class

DPpack::EmpiricalRiskMinimizationDP.CMS -> LogisticRegressionDP

Methods

Public methods

• LogisticRegressionDP$new()

• LogisticRegressionDP$fit()

• LogisticRegressionDP$predict()

• LogisticRegressionDP$clone()

---

Method new()

Create a new LogisticRegressionDP object.

Usage

LogisticRegressionDP$new(
  regularizer,
  eps,
  gamma,
  perturbation.method = "objective",
  regularizer.gr = NULL
)

Arguments

regularizer

String or regularization function. If a string, must be 'l2', indicating to use l2 regularization. If a function, must have form regularizer(coeff), where coeff is a vector or matrix, and return the value of the regularizer at coeff. See regularizer.l2 for an example. Additionally, in order to ensure differential privacy, the function must be 1-strongly convex and doubly differentiable.

eps

Positive real number defining the epsilon privacy budget. If set to Inf, runs algorithm without differential privacy.

gamma

Nonnegative real number representing the regularization constant.

perturbation.method

String indicating whether to use the 'output' or the 'objective' perturbation methods \insertCitechaudhuri2011DPpack. Defaults to 'objective'.

regularizer.gr

Optional function representing the gradient of the regularization function with respect to coeff and of the form regularizer.gr(coeff). Should return a vector. See regularizer.gr.l2 for an example. If regularizer is given as a string, this value is ignored. If not given and regularizer is a function, non-gradient based optimization methods are used to compute the coefficient values in fitting the model.

Returns

A new LogisticRegressionDP object.

---

Method fit()

Fit the differentially private logistic regression model. This method runs either the output perturbation or the objective perturbation algorithm \insertCitechaudhuri2011DPpack, depending on the value of perturbation.method used to construct the object, to generate an objective function. A numerical optimization method is then run to find optimal coefficients for fitting the model given the training data and hyperparameters. The built-in optim function using the "BFGS" optimization method is used. If regularizer is given as 'l2' or if regularizer.gr is given in the construction of the object, the gradient of the objective function is utilized by optim as well. Otherwise, non-gradient based optimization methods are used. The resulting privacy-preserving coefficients are stored in coeff.

Usage

LogisticRegressionDP$fit(X, y, upper.bounds, lower.bounds, add.bias = FALSE)

Arguments

X

Dataframe of data to be fit.

y

Vector or matrix of true labels for each row of X.

upper.bounds

Numeric vector of length ncol(X) giving upper bounds on the values in each column of X. The ncol(X) values are assumed to be in the same order as the corresponding columns of X. Any value in the columns of X larger than the corresponding upper bound is clipped at the bound.

lower.bounds

Numeric vector of length ncol(X) giving lower bounds on the values in each column of X. The ncol(X) values are assumed to be in the same order as the corresponding columns of X. Any value in the columns of X larger than the corresponding upper bound is clipped at the bound.

add.bias

Boolean indicating whether to add a bias term to X. Defaults to FALSE.

---

Method predict()

Predict label(s) for given X using the fitted coefficients.

Usage

LogisticRegressionDP$predict(X, add.bias = FALSE, raw.value = FALSE)

Arguments

X

Dataframe of data on which to make predictions. Must be of same form as X used to fit coefficients.

add.bias

Boolean indicating whether to add a bias term to X. Defaults to FALSE. If add.bias was set to TRUE when fitting the coefficients, add.bias should be set to TRUE for predictions.

raw.value

Boolean indicating whether to return the raw predicted value or the rounded class label. If FALSE (default), outputs the predicted labels 0 or 1. If TRUE, returns the raw score from the logistic regression.

Returns

Matrix of predicted labels or scores corresponding to each row of X.

---

Method clone()

The objects of this class are cloneable with this method.

Usage

LogisticRegressionDP$clone(deep = FALSE)

Arguments

deep

Whether to make a deep clone.

References

\insertRef

chaudhuri2011DPpack

\insertRef

Chaudhuri2009DPpack

Examples

# Build train dataset X and y, and test dataset Xtest and ytest
N <- 200
K <- 2
X <- data.frame()
y <- data.frame()
for (j in (1:K)){
  t <- seq(-.25, .25, length.out = N)
  if (j==1) m <- stats::rnorm(N,-.2, .1)
  if (j==2) m <- stats::rnorm(N, .2, .1)
  Xtemp <- data.frame(x1 = 3*t , x2 = m - t)
  ytemp <- data.frame(matrix(j-1, N, 1))
  X <- rbind(X, Xtemp)
  y <- rbind(y, ytemp)
}
Xtest <- X[seq(1,(N*K),10),]
ytest <- y[seq(1,(N*K),10),,drop=FALSE]
X <- X[-seq(1,(N*K),10),]
y <- y[-seq(1,(N*K),10),,drop=FALSE]

# Construct object for logistic regression
regularizer <- 'l2' # Alternatively, function(coeff) coeff%*%coeff/2
eps <- 1
gamma <- 1
lrdp <- LogisticRegressionDP$new(regularizer, eps, gamma)

# Fit with data
# Bounds for X based on construction
upper.bounds <- c( 1, 1)
lower.bounds <- c(-1,-1)
lrdp$fit(X, y, upper.bounds, lower.bounds) # No bias term
lrdp$coeff # Gets private coefficients

# Predict new data points
predicted.y <- lrdp$predict(Xtest)
n.errors <- sum(predicted.y!=ytest)

DPpack documentation built on April 8, 2023, 9:09 a.m.