Optimal Subsampling Method for Softmax (multinomial logistic) Regression Model

Draw subsample from full dataset and fit softmax(multinomial logistic) regression model on the subsample. Refer to vignette for a quick start.

ssp.softmax(
  formula,
  data,
  subset,
  n.plt,
  n.ssp,
  criterion = "MSPE",
  sampling.method = "poisson",
  likelihood = "MSCLE",
  constraint = "summation",
  control = list(...),
  contrasts = NULL,
  ...
)

Arguments

formula

A model formula object of class "formula" that describes the model to be fitted.

data

A data frame containing the variables in the model. Denote \(N\) as the number of observations in data.

subset

An optional vector specifying a subset of observations from data to use for the analysis. This subset will be viewed as the full data.

n.plt

The pilot subsample size (first-step subsample size). This subsample is used to compute the pilot estimator and estimate the optimal subsampling probabilities.

n.ssp

The expected size of the optimal subsample (second-step subsample). For sampling.method = 'withReplacement', The exact subsample size is n.ssp. For sampling.method = 'poisson', n.ssp is the expected subsample size.

criterion

The criterion of optimal subsampling probabilities. Choices include optA, optL, MSPE(default), LUC and uniform.

MSPE Minimizes the mean squared prediction error between subsample estimator and full data estimator.
optA Minimizes the trace of the asymptotic covariance matrix of the subsample estimator.
optL Minimizes the trace of a transformation of the asymptotic covariance matrix, which reduces computational costs than optA.
LUC Local uncertainty sampling method, serving as a baseline subsampling strategy. See Wang and Kim (2022).
uniform Assigns equal subsampling probability \(\frac{1}{N}\) to each observation, serving as a baseline subsampling strategy.

sampling.method

The sampling method to use. Choices include withReplacement and poisson(default). withReplacement draws exactly n.ssp subsamples from size \(N\) full dataset with replacement, using the specified subsampling probabilities. poisson draws observations independently by comparing each subsampling probability with a realization of uniform random variable \(U(0,1)\). Differences between methods:

Sample size: withReplacement draws exactly n.ssp subsamples while poisson draws subsamples with expected size n.ssp, meaning the actual size may vary.
Memory usage: withReplacement requires the entire dataset to be loaded at once, while poisson allows for processing observations sequentially (will be implemented in future version).
Estimator performance: Theoretical results show that the poisson tends to get a subsample estimator with lower asymptotic variance compared to the withReplacement

likelihood

A bias-correction likelihood function is required for subsample since unequal subsampling probabilities introduce bias. Choices include weighted and MSCLE(default).

weighted Applies a weighted likelihood function where each observation is weighted by the inverse of its subsampling probability.
MSCLE It uses a conditional likelihood, where each element of the likelihood represents the density of \(Y_i\) given that this observation was drawn.

constraint

The constraint for identifiability of softmax model. Choices include baseline and summation(default). The baseline constraint assumes the coefficient for the baseline category are \(0\). Without loss of generality, we set the category \(Y=0\) as the baseline category so that \(\boldsymbol{\beta}_0=0\). The summation constraint \(\sum_{k=0}^{K} \boldsymbol{\beta}_k\) is also used in the subsampling method for the purpose of calculating subsampling probability. These two constraints lead to different interpretation of coefficients but are equal for computing \(P(Y_{i,k} = 1 \mid \mathbf{x}_i)\). The estimation of coefficients returned by ssp.softmax() is under baseline constraint.

control

A list of parameters for controlling the sampling process. There are two tuning parameters alpha and b. Default is list(alpha=0, b=2).

alpha \(\in [0,1]\) is the mixture weight of the user-assigned subsampling probability and uniform subsampling probability. The actual subsample probability is \(\pi = (1-\alpha)\pi^{opt} + \alpha \pi^{uni}\). This protects the estimator from extreme small subsampling probability. The default value is 0.
b is a positive number which is used to constaint the poisson subsampling probability. b close to 0 results in subsampling probabilities closer to uniform probability \(\frac{1}{N}\). b=2 is the default value. See relevant references for further details.

contrasts

An optional list. It specifies how categorical variables are represented in the design matrix. For example, contrasts = list(v1 = 'contr.treatment', v2 = 'contr.sum').

...

A list of parameters which will be passed to nnet::multinom().

Value

ssp.softmax returns an object of class "ssp.softmax" containing the following components (some are optional):

model.call: The original function call.
coef.plt: The pilot estimator. See Details for more information.
coef.ssp: The estimator obtained from the optimal subsample.
coef: The weighted linear combination of coef.plt and coef.ssp, under baseline constraint. The combination weights depend on the relative size of n.plt and n.ssp and the estimated covariance matrices of coef.plt and coef.ssp. We blend the pilot subsample information into optimal subsample estimator since the pilot subsample has already been drawn. The coefficients and standard errors reported by summary are coef and the square root of diag(cov).
coef.plt.sum: The pilot estimator under summation constrraint. coef.plt.sum = G %*% as.vector(coef.plt).
coef.ssp.sum: The estimator obtained from the optimal subsample under summation constrraint. coef.ssp.sum = G %*% as.vector(coef.ssp).
coef.sum: The weighted linear combination of coef.plt and coef.ssp, under summation constrraint. coef.sum = G %*% as.vector(coef).
cov.plt: The covariance matrix of coef.plt.
cov.ssp: The covariance matrix of coef.ssp.
cov: The covariance matrix of coef.cmb.
cov.plt.sum: The covariance matrix of coef.plt.sum.
cov.ssp.sum: The covariance matrix of coef.ssp.sum.
cov.sum: The covariance matrix of coef.sum.
index.plt: Row indices of pilot subsample in the full dataset.
index.ssp: Row indices of of optimal subsample in the full dataset.
N: The number of observations in the full dataset.
subsample.size.expect: The expected subsample size.
terms: The terms object for the fitted model.

Details

A pilot estimator for the unknown parameter \(\beta\) is required because MSPE, optA and optL subsampling probabilities depend on \(\beta\). There is no "free lunch" when determining optimal subsampling probabilities. For softmax regression, this is achieved by drawing a size n.plt subsample with replacement from full dataset with uniform sampling probability.

References

Yao, Y., & Wang, H. (2019). Optimal subsampling for softmax regression. Statistical Papers, 60, 585-599.

Han, L., Tan, K. M., Yang, T., & Zhang, T. (2020). Local uncertainty sampling for large-scale multiclass logistic regression. Annals of Statistics, 48(3), 1770-1788.

Wang, H., & Kim, J. K. (2022). Maximum sampled conditional likelihood for informative subsampling. Journal of machine learning research, 23(332), 1-50.

Yao, Y., Zou, J., & Wang, H. (2023). Optimal poisson subsampling for softmax regression. Journal of Systems Science and Complexity, 36(4), 1609-1625.

Yao, Y., Zou, J., & Wang, H. (2023). Model constraints independent optimal subsampling probabilities for softmax regression. Journal of Statistical Planning and Inference, 225, 188-201.

Examples

# softmax regression
d <- 3 # dim of covariates
K <- 2 # K + 1 classes
G <- rbind(rep(-1/(K+1), K), diag(K) - 1/(K+1)) %x% diag(d)
N <- 1e4
beta.true.baseline <- cbind(rep(0, d), matrix(-1.5, d, K))
beta.true.summation <- cbind(rep(1, d), 0.5 * matrix(-1, d, K))
set.seed(1)
mu <- rep(0, d)
sigma <- matrix(0.5, nrow = d, ncol = d)
diag(sigma) <- rep(1, d)
X <- MASS::mvrnorm(N, mu, sigma)
prob <- exp(X %*% beta.true.summation)
prob <- prob / rowSums(prob)
Y <- apply(prob, 1, function(row) sample(0:K, size = 1, prob = row))
n.plt <- 500
n.ssp <- 1000
data <- as.data.frame(cbind(Y, X))
colnames(data) <- c("Y", paste("V", 1:ncol(X), sep=""))
head(data)
#>   Y         V1          V2         V3
#> 1 2 -0.3756189 -0.17727086 -0.9816025
#> 2 2  0.2912939  0.60753208 -0.4489936
#> 3 2 -1.0530547  0.02079337 -1.0146024
#> 4 0  0.1854791  2.45385260  1.2682922
#> 5 0  0.8687332  0.21941612 -0.2810234
#> 6 1 -0.8336174 -0.32556141 -0.8505501
formula <- Y ~ . -1
WithRep.MSPE <- ssp.softmax(formula = formula,
 data = data, 
 n.plt = n.plt,
 n.ssp = n.ssp,
 criterion = 'MSPE', 
 sampling.method = 'withReplacement',
 likelihood = 'weighted',
 constraint = 'baseline')
#> [1] "Message from nnet::multinom: "
#> # weights:  12 (6 variable)
#> initial  value 549.306144 
#> iter  10 value 367.125951
#> final  value 365.836516 
#> converged
#> [1] "Message from nnet::multinom: "
#> # weights:  12 (6 variable)
#> initial  value 11677047.319684 
#> iter  10 value 7149711.444798
#> iter  20 value 6901692.808182
#> final  value 6901684.550943 
#> converged
summary(WithRep.MSPE)
#> Model Summary
#> 
#> 
#> Call:
#> 
#> ssp.softmax(formula = formula, data = data, n.plt = n.plt, n.ssp = n.ssp, 
#>     criterion = "MSPE", sampling.method = "withReplacement", 
#>     likelihood = "weighted", constraint = "baseline")
#> 
#> Subsample Size:
#>                                
#> 1       Total Sample Size 10000
#> 2 Expected Subsample Size  1000
#> 3   Actual Subsample Size  1000
#> 4   Unique Subsample Size   927
#> 5  Expected Subample Rate   10%
#> 6    Actual Subample Rate   10%
#> 7    Unique Subample Rate 9.27%
#> 
#> Coefficients:
#> 
#>         [,1]      [,2]
#> V1 -1.455068 -1.406786
#> V2 -1.647116 -1.492484
#> V3 -1.248385 -1.435806
#> 
#> Std. Errors:
#> 
#>         [,1]      [,2]
#> V1 0.1127325 0.1122913
#> V2 0.1133850 0.1108925
#> V3 0.1202163 0.1178532