Research map / Learning theory and algorithms / Sample complexity
Logistic Regression for Massive Data with Rare Events
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Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Measure estimation error relative to the number of cases rather than total sample size | Full-data logistic-regression MLE · Rare-events data | Asymptotic convergence rate: n_1^{-1/2} instead of n^{-1/2} | Helped |
| Under-sample controls while retaining all cases, with the number of sampled controls much larger than the number of cases | Full-data logistic-regression MLE · Rare-events data | Asymptotic estimation efficiency | No clear effect |
| Under-sample controls until their number is of the same order as cases | Full-data logistic-regression MLE · Rare-events data | Asymptotic estimation efficiency | Hurt |
| Use an unweighted objective with intercept bias correction instead of a weighted objective on the same under-sampled data | Under-sampled logistic regression · Rare-events data | Asymptotic estimation efficiency | Helped |
| Replicate cases while retaining all controls, using the weighted estimator | Full-data logistic-regression MLE · Rare-events data | Asymptotic estimation efficiency | Hurt |
| Use an unweighted objective with bias correction instead of a weighted objective for case over-sampling | Over-sampled logistic regression · Rare-events data | Asymptotic estimation efficiency | Hurt |
| Scale empirical MSE by expected case count rather than total sample size | Full-data logistic-regression MLE · Full-data simulation | Scaled empirical MSE: 2.51, 2.06, 2.22, 2.16 (alpha); 1.21, 1.09, 1.00, 1.08 (beta) | No clear effect |
| Scale empirical MSE by total sample size rather than expected case count | Full-data logistic-regression MLE · Full-data simulation | Scaled empirical MSE: 125.7, 515.5, 2774.4, 13474.9 (alpha); 60.6, 271.9, 1248.8, 6731.6 (beta) | Hurt |
| Under-sample controls so their number is roughly equal to the number of cases | Under-sampled estimators · Simulation with n=10^5 and Pr(y=1)≈0.004 | Empirical MSE | Hurt |
| Increase the control sampling probability above 0.1 | Under-sampled estimators · Simulation with n=10^5 and Pr(y=1)≈0.004 | Empirical MSE | No clear effect |
| Over-sample cases at an intermediate rate | Over-sampled estimators · Simulation with n=10^5 and Pr(y=1)≈0.004 | Empirical MSE | Hurt |
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