Poisson Loss in Random Forest Mode

With MSE loss, the random forest mode seems to work well. However, when switching to "count:poisson" (and also "reg:gamma") loss, the model is completely off. The distribution of the predictions is heavily biased.

In this example, the R-squared (with respect to MSE as well as Poisson loss) drops from 70%-80% to 0%. `max_delta_step` has a large impact.

```
set.seed(1)
n <- 1000
x1 <- seq_len(n)
x2 <- rnorm(n)
X <- cbind(x1, x2)
y <- rpois(n, x1 / 1000 + x2^2)

library(xgboost)
library(MetricsWeighted)

dtrain_xgb <- xgb.DMatrix(X, label = y)

#=======================================================
# WITH MSE loss
#=======================================================

# xgboost with random forest like parameters
param <- list(max_depth = 10,
learning_rate = 1,
objective = "reg:linear",
subsample = 0.63,
lambda = 0,
alpha = 0,
colsample_bylevel = 1/3)

fit_xgb <- xgb.train(param,
dtrain_xgb,
watchlist = list(train = dtrain_xgb),
nrounds = 1,
num_parallel_tree = 500)

pred_mse <- predict(fit_xgb, X)

summary(pred_mse)
# Min. 1st Qu. Median Mean 3rd Qu. Max.
# 0.01976 0.69959 1.06019 1.51380 1.74806 11.24593

# MSE-based R-squared
r_squared(y, pred_mse) # 0.8228559

# Poisson-based R-squared
r_squared_poisson(y, pred_mse) # 0.6900247

#=======================================================
# WITH Poisson loss
#=======================================================

# xgboost with random forest like parameters
param <- list(max_depth = 20,
learning_rate = 1,
objective = "count:poisson",
subsample = 0.63,
lambda = 0,
alpha = 0,
# max_delta_step = 0.7,
colsample_bylevel = 1/3)

fit_xgb <- xgb.train(param,
dtrain_xgb,
watchlist = list(train = dtrain_xgb),
nrounds = 1,
num_parallel_tree = 500)

pred_poi <- predict(fit_xgb, X)

summary(pred_poi)
# Min. 1st Qu. Median Mean 3rd Qu. Max.
# 0.3325 0.6366 0.8874 0.8054 0.9949 1.0069

# MSE-based R-squared
r_squared(y, pred_poi) # -0.02036563

# Poisson-based R-squared
r_squared_poisson(y, pred_poi) # 0.01646716

sessionInfo()
# R version 3.6.1 (2019-07-05)
# Platform: x86_64-w64-mingw32/x64 (64-bit)
# Running under: Windows 10 x64 (build 18362)
#
# Matrix products: default
#
# locale:
# [1] LC_COLLATE=English_Switzerland.1252 LC_CTYPE=English_Switzerland.1252
# [3] LC_MONETARY=English_Switzerland.1252 LC_NUMERIC=C
# [5] LC_TIME=English_Switzerland.1252
#
# attached base packages:
# [1] stats graphics grDevices utils datasets methods base
#
# other attached packages:
# [1] MetricsWeighted_0.5.0 xgboost_0.90.0.2
#
# loaded via a namespace (and not attached):
# [1] compiler_3.6.1 magrittr_1.5 Matrix_1.2-17 tools_3.6.1 stringi_1.4.6
# [6] grid_3.6.1 data.table_1.12.8 lattice_0.20-38
```