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What is the difference in R in xgboost between binary:logistic and reg:logistic? Is it only in evaluation metric?

If yes, how does RMSE on binary classification compare to error rate? Is the relationship between the metrics more or less monotonic, output from tuning on one metric should not differ significantly between those two approaches?

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There is no difference in the algorithm, they are the same. When you use reg:logistic the default evaluation metric is rmse, but you can change this parameter.

The error rate and the rmse may differ depending on the distribution of your output, as the error rate uses a limit of $0.5$ if you have the output values concentrated in $0$ or $1$ it will be much smaller than rmse, even though its correlated metric the model can be very different, the application will depend on your problem.

To evaluate performance of logistic regression models is usual to calculate auc, logloss, accuracy, pseudo-$R^2$ and some others.

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