ssp.softmax: Subsampling for Softmax (Multinomial) Regression Model

# `ssp.softmax`: Subsampling for Softmax (Multinomial) Regression Model

This vignette introduces the usage of `ssp.softmax`, which draws optimal subsample from full data and fit softmax (multinomial) regression on the subsample. The statistical theory and algorithms in this implementation can be found in the relevant reference papers.

Denote \(y\) as multi-category response variable and \(K+1\) is the number of categories. \(N\) is the number of observations in the full dataset. \(X\) is the \(N \times d\) covariates matrix. Softmax regression model assumes that \[ P(y_{i,k} = 1 \mid \mathbf{x}_i) = \frac{\exp(\mathbf{x}_i^\top \boldsymbol{\beta}_k)}{\sum_{l=0}^{K} \exp(\mathbf{x}_i^\top \boldsymbol{\beta}_l)} \] for \(i = 1, \ldots, N\) and \(k = 0, 1, \ldots, K\), where \(\boldsymbol{\beta}_k\)’s are \(d\)-dimensional unknown coefficients.

The log-likelihood function of softmax regression is

\[ \max_{\beta} L(\beta) = \frac{1}{N} \sum_{i=1}^{N} \left[ \sum_{k=0}^{K} y_{i,k} \mathbf{x}_i^\top \boldsymbol{\beta}_k - \ln \left\{ \sum_{l=0}^{K} \exp(\mathbf{x}_i^\top \boldsymbol{\beta}_l) \right\} \right]. \]

The idea of subsampling methods is as follows: instead of fitting the model on the size \(N\) full dataset, a subsampling probability is assigned to each observation and a smaller, informative subsample is drawn. The model is then fitted on the subsample to obtain an estimator with reduced computational cost.

## Terminology

- Full dataset: The whole dataset used as input.

- Full data estimator: The estimator obtained by fitting the model on the full dataset.

- Subsample: A subset of observations drawn from the full dataset.

- Subsample estimator: The estimator obtained by fitting the model on the subsample.

- Subsampling probability (\(\pi\)): The probability assigned to each observation for inclusion in the subsample.

## Example

We introduce the usage of `ssp.softmax` with simulated data. \(X\) contains \(d=3\) covariates drawn from multinormal distribution and \(Y\) is the multicategory response variable with \(K+1=3\) categories. The full data size is \(N = 1 \times 10^4\).

    library(subsampling)

    set.seed(1)
    d <- 3
    K <- 2
    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))
    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))
    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

## Key Arguments

The function usage is

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

The core functionality of `ssp.softmax` revolves around three key questions:

- How are subsampling probabilities computed? (Controlled by the `criterion` argument)

- How is the subsample drawn? (Controlled by the `sampling.method` argument)

- How is the likelihood adjusted to correct for bias? (Controlled by the `likelihood` argument)

### `criterion`

The choices of `criterion` include `optA`, `optL`, ,`MSPE`(default), `LUC` and `uniform`. The default criterion `MSPE` minimizes the mean squared prediction error between subsample estimator and full data estimator. Criterion `optA` and `optL` are derived by minimizing the asymptotic covariance of subsample estimator. `LUC` and `uniform` are baseline methods. See Yao, Zou, and Wang (2023) and Wang and Kim (2022) for details.

### `sampling.method`

The options for `sampling.method` include `withReplacement` and `poisson` (default). `withReplacement.` stands for drawing \(n.ssp\) subsamples from full dataset with replacement, using the specified subsampling probability. `poisson` stands for drawing subsamples one by one by comparing the subsampling probability with a realization of uniform random variable \(U(0,1)\). The expected number of drawn samples are \(n.ssp\).

### `likelihood`

The available choices for `likelihood` include `weighted` and `MSCLE`(default). `MSCLE` stands for maximum sampled conditional likelihood. Both of these likelihood functions can derive an unbiased optimal subsample estimator. See Wang and Kim (2022) for details about `MSCLE`.

### `constraint`

Softmax model needs constraint on unknown coefficients for identifiability. The options for `constraint` include `summation` and `baseline` (default). The baseline constraint assumes the coefficient for the baseline category are \(0\). Without loss of generality, `ssp.softmax` sets the category \(Y=0\) as the baseline category so that \(\boldsymbol{\beta}_0=0\). The summation constraint \(\sum_{k=0}^{K} \boldsymbol{\beta}_k\) can also used in the subsampling method for the purpose of calculating optimal 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.

## Outputs

After drawing subsample, `ssp.softmax` utilizes `nnet::multinom` to fit the model on the subsample. Arguments accepted by `nnet::multinom` can be passed through `...` in `ssp.softmax`.

Below are two examples demonstrating the use of `ssp.softmax` with different configurations.

    n.plt <- 200
    n.ssp <- 600
    formula <- Y ~ . -1
    ssp.results1 <- 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 219.722458
    #> iter  10 value 143.304799
    #> final  value 143.304778
    #> converged
    #> [1] "Message from nnet::multinom: "
    #> # weights:  12 (6 variable)
    #> initial  value 5789394.068697
    #> iter  10 value 4219770.981211
    #> final  value 4219003.340996
    #> converged
    summary(ssp.results1)
    #> 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   600
    #> 3   Actual Subsample Size   600
    #> 4   Unique Subsample Size   563
    #> 5  Expected Subample Rate    6%
    #> 6    Actual Subample Rate    6%
    #> 7    Unique Subample Rate 5.63%
    #>
    #> Coefficients:
    #>
    #>         [,1]      [,2]
    #> V1 -1.175907 -1.281935
    #> V2 -1.619482 -1.502412
    #> V3 -1.231662 -1.414723
    #>
    #> Std. Errors:
    #>
    #>         [,1]      [,2]
    #> V1 0.1457377 0.1459830
    #> V2 0.1442445 0.1461619
    #> V3 0.1498036 0.1492071

`summary(ssp.results1)` shows that it draws 600 observations out of 10000, where the number of unique indices is less than 600 since we use `sampling.method = 'withReplacement'`. After fitting softmax model on subsample using the chosen `weighted` likelihood function, we get coefficients estimation and standard errors as above.

    ssp.results2 <- ssp.softmax(formula = formula,
                                data = data,
                                n.plt = n.plt,
                                n.ssp = n.ssp,
                                criterion = 'MSPE',
                                sampling.method = 'poisson',
                                likelihood = 'MSCLE',
                                constraint = 'baseline'
                                )
    #> [1] "Message from nnet::multinom: "
    #> # weights:  12 (6 variable)
    #> initial  value 219.722458
    #> iter  10 value 126.651711
    #> final  value 126.637171
    #> converged
    #> [1] "Message from nnet::multinom: "
    #> # weights:  21 (6 variable)
    #> initial  value 908.207717
    #> iter  10 value 651.150361
    #> final  value 650.943777
    #> converged
    summary(ssp.results2)
    #> Model Summary
    #>
    #>
    #> Call:
    #>
    #> ssp.softmax(formula = formula, data = data, n.plt = n.plt, n.ssp = n.ssp,
    #>     criterion = "MSPE", sampling.method = "poisson", likelihood = "MSCLE",
    #>     constraint = "baseline")
    #>
    #> Subsample Size:
    #>
    #> 1       Total Sample Size 10000
    #> 2 Expected Subsample Size   600
    #> 3   Actual Subsample Size   702
    #> 4   Unique Subsample Size   702
    #> 5  Expected Subample Rate    6%
    #> 6    Actual Subample Rate 7.02%
    #> 7    Unique Subample Rate 7.02%
    #>
    #> Coefficients:
    #>
    #>         [,1]      [,2]
    #> V1 -1.450790 -1.434135
    #> V2 -1.764387 -1.726621
    #> V3 -1.448671 -1.607003
    #>
    #> Std. Errors:
    #>
    #>         [,1]      [,2]
    #> V1 0.1314360 0.1353568
    #> V2 0.1441155 0.1413029
    #> V3 0.1389344 0.1411973

The returned object contains estimation results and index of drawn subsamples in the full dataset.

    names(ssp.results1)
    #>  [1] "model.call"            "coef.plt"              "coef.ssp"
    #>  [4] "coef"                  "coef.plt.sum"          "coef.ssp.sum"
    #>  [7] "coef.sum"              "cov.plt"               "cov.ssp"
    #> [10] "cov"                   "cov.plt.sum"           "cov.sum"
    #> [13] "cov.ssp.sum"           "index.plt"             "index.ssp"
    #> [16] "N"                     "subsample.size.expect" "terms"

Some key returned variables:

- `index.plt` and `index` are the row indices of drawn pilot subsamples and optimal subsamples in the full data.

- `coef.ssp` is the subsample estimator for \(\beta\) and `coef` is the linear combination of `coef.plt` (pilot estimator) and `coef.ssp`.

- `cov.ssp` and `cov` are estimated covariance matrices of `coef.ssp` and `coef`.

## References

Wang, HaiYing, and Jae Kwang Kim. 2022. “Maximum Sampled Conditional Likelihood for Informative Subsampling.” Journal of Machine Learning Research 23 (332): 1–50.
Yao, Yaqiong, Jiahui Zou, and HaiYing Wang. 2023. “Model Constraints Independent Optimal Subsampling Probabilities for Softmax Regression.” Journal of Statistical Planning and Inference 225: 188–201.