WeightedERMDP.CMS: Privacy-preserving Weighted Empirical Risk Minimization
In DPpack: Differentially Private Statistical Analysis and Machine Learning
WeightedERMDP.CMS R Documentation
Privacy-preserving Weighted Empirical Risk Minimization
Description
This class implements differentially private empirical risk
minimization in the case where weighted observation-level losses are
desired (such as weighted SVM \insertCiteYang2005DPpack). Currently,
only the output perturbation method is implemented.
Details
To use this class for weighted empirical risk minimization, 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, data bounds,
weights, and weight 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 weighted empirical
risk minimization, 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, though currently only
output perturbation is implemented. Specifically, the provided loss
function must be convex and differentiable with respect to y.hat,
and the absolute value of the first derivative of the loss function must be
at most 1. If objective perturbation is chosen (not currently implemented),
the loss function must also be doubly differentiable and the absolute value
of the second derivative of the loss function must be bounded above by a
constant c for all possible values of y.hat and y, where
y.hat is the predicted label and y is the true label. The
regularizer must be 1-strongly convex and differentiable. It also must be
doubly differentiable if objective perturbation is chosen. For the data x,
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. Note that because of this, a bias term
cannot be included without appropriate scaling/preprocessing of the
dataset. To ensure privacy, the add.bias argument in the fit and
predict methods should only be utilized in subclasses within this
package where appropriate preprocessing is implemented, not in this class.
Finally, if weights are provided, they should be nonnegative, of the same
length as y, and be upper bounded by a global or public bound which must
also be provided.
Super class
DPpack::EmpiricalRiskMinimizationDP.CMS -> WeightedERMDP.CMS
Methods
Public methods
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-
-
-
Method new()
Create a new WeightedERMDP.CMS object.
Usage
WeightedERMDP.CMS$new(
mapXy,
loss,
regularizer,
eps,
gamma,
perturbation.method = "objective",
c = NULL,
mapXy.gr = NULL,
loss.gr = NULL,
regularizer.gr = NULL
)
Arguments
mapXyMap function of the form mapXy(X, coeff) mapping input
data matrix X and coefficient vector or matrix coeff to
output labels y. Should return a column matrix of predicted labels
for each row of X. See mapXy.sigmoid for an example.
lossLoss function of the form loss(y.hat, y, w), where
y.hat and y are matrices and w is a matrix or vector
of weights of the same length as y. Should be defined such that it
returns a matrix of weighted loss values for each element of y.hat
and y. If w is not given, the function should operate as if
uniform weights were given. See generate.loss.huber for an
example. It must be convex and differentiable, and the absolute value of
the first derivative of the loss function must be at most 1.
Additionally, if the objective perturbation method is chosen, it must be
doubly differentiable and the absolute value of the second derivative of
the loss function must be bounded above by a constant c for all possible
values of y.hat and y.
regularizerString 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
differentiable. If the objective perturbation method is chosen, it must
also be doubly differentiable.
epsPositive real number defining the epsilon privacy budget. If set
to Inf, runs algorithm without differential privacy.
gammaNonnegative real number representing the regularization
constant.
perturbation.methodString indicating whether to use the 'output' or
the 'objective' perturbation methods \insertCitechaudhuri2011DPpack.
Defaults to 'objective'. Currently, only the output perturbation method
is supported.
cPositive real number denoting the upper bound on the absolute
value of the second derivative of the loss function, as required to
ensure differential privacy for the objective perturbation method. This
input is unnecessary if perturbation.method is 'output', but is required
if perturbation.method is 'objective'. Defaults to NULL.
mapXy.grOptional function representing the gradient of the map
function with respect to the values in coeff. If given, must be of
the form mapXy.gr(X, coeff), where X is a matrix and
coeff is a matrix or numeric vector. Should be defined such that
the ith row of the output represents the gradient with respect to the ith
coefficient. See mapXy.gr.sigmoid for an example. If not
given, non-gradient based optimization methods are used to compute the
coefficient values in fitting the model.
loss.grOptional function representing the gradient of the loss
function with respect to y.hat and of the form
loss.gr(y.hat, y, w), where y.hat and y are matrices
and w is a matrix or vector of weights. Should be defined such
that the ith row of the output represents the gradient of the (possibly
weighted) loss function at the ith set of input values. See
generate.loss.gr.huber for an example. If not given,
non-gradient based optimization methods are used to compute the
coefficient values in fitting the model.
regularizer.grOptional 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 WeightedERMDP.CMS object.
Method fit()
Fit the differentially private weighted empirical risk
minimization model. This method runs either the output perturbation or
the objective perturbation algorithm \insertCitechaudhuri2011DPpack
(only output is currently implemented), 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,
weights, and hyperparameters. The built-in optim function
using the "BFGS" optimization method is used. If mapXy.gr,
loss.gr, and regularizer.gr are all 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
WeightedERMDP.CMS$fit(
X,
y,
upper.bounds,
lower.bounds,
add.bias = FALSE,
weights = NULL,
weights.upper.bound = NULL
)
Arguments
XDataframe of data to be fit.
yVector or matrix of true labels for each row of X.
upper.boundsNumeric 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.boundsNumeric 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.biasBoolean indicating whether to add a bias term to X.
Defaults to FALSE.
weightsNumeric vector of observation weights of the same length as
y.
weights.upper.boundNumeric value representing the global or public
upper bound on the weights.
Method predict()
Predict label(s) for given X using the fitted
coefficients.
Usage
WeightedERMDP.CMS$predict(X, add.bias = FALSE)
Arguments
XDataframe of data on which to make predictions. Must be of same
form as X used to fit coefficients.
add.biasBoolean indicating whether to add a bias term to X.
Defaults to FALSE. If add.bias was set to TRUE when fitting the
coefficients, add.bias should be set to TRUE for predictions.
Returns
Matrix of predicted labels corresponding to each row of X.
Method clone()
The objects of this class are cloneable with this method.
Usage
WeightedERMDP.CMS$clone(deep = FALSE)
Arguments
deepWhether to make a deep clone.
References
\insertRefchaudhuri2011DPpack
\insertRefYang2005DPpack
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 weighted linear SVM
mapXy <- function(X, coeff) X%*%coeff
# Huber loss from DPpack
huber.h <- 0.5
loss <- generate.loss.huber(huber.h)
regularizer <- 'l2' # Alternatively, function(coeff) coeff%*%coeff/2
eps <- 1
gamma <- 1
perturbation.method <- 'output'
c <- 1/(2*huber.h) # Required value for SVM
mapXy.gr <- function(X, coeff) t(X)
loss.gr <- generate.loss.gr.huber(huber.h)
regularizer.gr <- function(coeff) coeff
wermdp <- WeightedERMDP.CMS$new(mapXy, loss, regularizer, eps,
gamma, perturbation.method, c,
mapXy.gr, loss.gr,
regularizer.gr)
# Fit with data
# Bounds for X based on construction
upper.bounds <- c( 1, 1)
lower.bounds <- c(-1,-1)
weights <- rep(1, nrow(y)) # Uniform weighting
weights[nrow(y)] <- 0.5 # half weight for last observation
wub <- 1 # Public upper bound for weights
wermdp$fit(X, y, upper.bounds, lower.bounds, weights=weights,
weights.upper.bound=wub)
wermdp$coeff # Gets private coefficients
# Predict new data points
predicted.y <- wermdp$predict(Xtest)
n.errors <- sum(round(predicted.y)!=ytest)
DPpack documentation built on April 8, 2023, 9:09 a.m.
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| WeightedERMDP.CMS | R Documentation |
Privacy-preserving Weighted Empirical Risk Minimization
Description
This class implements differentially private empirical risk minimization in the case where weighted observation-level losses are desired (such as weighted SVM \insertCiteYang2005DPpack). Currently, only the output perturbation method is implemented.
Details
To use this class for weighted empirical risk minimization, 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, data bounds,
weights, and weight 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 weighted empirical
risk minimization, 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, though currently only
output perturbation is implemented. Specifically, the provided loss
function must be convex and differentiable with respect to y.hat,
and the absolute value of the first derivative of the loss function must be
at most 1. If objective perturbation is chosen (not currently implemented),
the loss function must also be doubly differentiable and the absolute value
of the second derivative of the loss function must be bounded above by a
constant c for all possible values of y.hat and y, where
y.hat is the predicted label and y is the true label. The
regularizer must be 1-strongly convex and differentiable. It also must be
doubly differentiable if objective perturbation is chosen. For the data x,
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. Note that because of this, a bias term
cannot be included without appropriate scaling/preprocessing of the
dataset. To ensure privacy, the add.bias argument in the fit and
predict methods should only be utilized in subclasses within this
package where appropriate preprocessing is implemented, not in this class.
Finally, if weights are provided, they should be nonnegative, of the same
length as y, and be upper bounded by a global or public bound which must
also be provided.
Super class
DPpack::EmpiricalRiskMinimizationDP.CMS -> WeightedERMDP.CMS
Methods
Public methods
Method new()
Create a new WeightedERMDP.CMS object.
Usage
WeightedERMDP.CMS$new( mapXy, loss, regularizer, eps, gamma, perturbation.method = "objective", c = NULL, mapXy.gr = NULL, loss.gr = NULL, regularizer.gr = NULL )
Arguments
mapXyMap function of the form
mapXy(X, coeff)mapping input data matrixXand coefficient vector or matrixcoeffto output labelsy. Should return a column matrix of predicted labels for each row ofX. SeemapXy.sigmoidfor an example.lossLoss function of the form
loss(y.hat, y, w), wherey.hatandyare matrices andwis a matrix or vector of weights of the same length asy. Should be defined such that it returns a matrix of weighted loss values for each element ofy.hatandy. Ifwis not given, the function should operate as if uniform weights were given. Seegenerate.loss.huberfor an example. It must be convex and differentiable, and the absolute value of the first derivative of the loss function must be at most 1. Additionally, if the objective perturbation method is chosen, it must be doubly differentiable and the absolute value of the second derivative of the loss function must be bounded above by a constant c for all possible values ofy.hatandy.regularizerString or regularization function. If a string, must be 'l2', indicating to use l2 regularization. If a function, must have form
regularizer(coeff), wherecoeffis a vector or matrix, and return the value of the regularizer atcoeff. Seeregularizer.l2for an example. Additionally, in order to ensure differential privacy, the function must be 1-strongly convex and differentiable. If the objective perturbation method is chosen, it must also be doubly differentiable.epsPositive real number defining the epsilon privacy budget. If set to Inf, runs algorithm without differential privacy.
gammaNonnegative real number representing the regularization constant.
perturbation.methodString indicating whether to use the 'output' or the 'objective' perturbation methods \insertCitechaudhuri2011DPpack. Defaults to 'objective'. Currently, only the output perturbation method is supported.
cPositive real number denoting the upper bound on the absolute value of the second derivative of the loss function, as required to ensure differential privacy for the objective perturbation method. This input is unnecessary if perturbation.method is 'output', but is required if perturbation.method is 'objective'. Defaults to NULL.
mapXy.grOptional function representing the gradient of the map function with respect to the values in
coeff. If given, must be of the formmapXy.gr(X, coeff), whereXis a matrix andcoeffis a matrix or numeric vector. Should be defined such that the ith row of the output represents the gradient with respect to the ith coefficient. SeemapXy.gr.sigmoidfor an example. If not given, non-gradient based optimization methods are used to compute the coefficient values in fitting the model.loss.grOptional function representing the gradient of the loss function with respect to
y.hatand of the formloss.gr(y.hat, y, w), wherey.hatandyare matrices andwis a matrix or vector of weights. Should be defined such that the ith row of the output represents the gradient of the (possibly weighted) loss function at the ith set of input values. Seegenerate.loss.gr.huberfor an example. If not given, non-gradient based optimization methods are used to compute the coefficient values in fitting the model.regularizer.grOptional function representing the gradient of the regularization function with respect to
coeffand of the formregularizer.gr(coeff). Should return a vector. Seeregularizer.gr.l2for an example. Ifregularizeris given as a string, this value is ignored. If not given andregularizeris a function, non-gradient based optimization methods are used to compute the coefficient values in fitting the model.
Returns
A new WeightedERMDP.CMS object.
Method fit()
Fit the differentially private weighted empirical risk
minimization model. This method runs either the output perturbation or
the objective perturbation algorithm \insertCitechaudhuri2011DPpack
(only output is currently implemented), 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,
weights, and hyperparameters. The built-in optim function
using the "BFGS" optimization method is used. If mapXy.gr,
loss.gr, and regularizer.gr are all 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
WeightedERMDP.CMS$fit( X, y, upper.bounds, lower.bounds, add.bias = FALSE, weights = NULL, weights.upper.bound = NULL )
Arguments
XDataframe of data to be fit.
yVector or matrix of true labels for each row of
X.upper.boundsNumeric vector of length
ncol(X)giving upper bounds on the values in each column of X. Thencol(X)values are assumed to be in the same order as the corresponding columns ofX. Any value in the columns ofXlarger than the corresponding upper bound is clipped at the bound.lower.boundsNumeric vector of length
ncol(X)giving lower bounds on the values in each column ofX. Thencol(X)values are assumed to be in the same order as the corresponding columns ofX. Any value in the columns ofXlarger than the corresponding upper bound is clipped at the bound.add.biasBoolean indicating whether to add a bias term to
X. Defaults to FALSE.weightsNumeric vector of observation weights of the same length as
y.weights.upper.boundNumeric value representing the global or public upper bound on the weights.
Method predict()
Predict label(s) for given X using the fitted
coefficients.
Usage
WeightedERMDP.CMS$predict(X, add.bias = FALSE)
Arguments
XDataframe of data on which to make predictions. Must be of same form as
Xused to fit coefficients.add.biasBoolean indicating whether to add a bias term to
X. Defaults to FALSE. If add.bias was set to TRUE when fitting the coefficients, add.bias should be set to TRUE for predictions.
Returns
Matrix of predicted labels corresponding to each row of X.
Method clone()
The objects of this class are cloneable with this method.
Usage
WeightedERMDP.CMS$clone(deep = FALSE)
Arguments
deepWhether to make a deep clone.
References
\insertRefchaudhuri2011DPpack
\insertRefYang2005DPpack
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 weighted linear SVM
mapXy <- function(X, coeff) X%*%coeff
# Huber loss from DPpack
huber.h <- 0.5
loss <- generate.loss.huber(huber.h)
regularizer <- 'l2' # Alternatively, function(coeff) coeff%*%coeff/2
eps <- 1
gamma <- 1
perturbation.method <- 'output'
c <- 1/(2*huber.h) # Required value for SVM
mapXy.gr <- function(X, coeff) t(X)
loss.gr <- generate.loss.gr.huber(huber.h)
regularizer.gr <- function(coeff) coeff
wermdp <- WeightedERMDP.CMS$new(mapXy, loss, regularizer, eps,
gamma, perturbation.method, c,
mapXy.gr, loss.gr,
regularizer.gr)
# Fit with data
# Bounds for X based on construction
upper.bounds <- c( 1, 1)
lower.bounds <- c(-1,-1)
weights <- rep(1, nrow(y)) # Uniform weighting
weights[nrow(y)] <- 0.5 # half weight for last observation
wub <- 1 # Public upper bound for weights
wermdp$fit(X, y, upper.bounds, lower.bounds, weights=weights,
weights.upper.bound=wub)
wermdp$coeff # Gets private coefficients
# Predict new data points
predicted.y <- wermdp$predict(Xtest)
n.errors <- sum(round(predicted.y)!=ytest)
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