import numpy as np
import xgboost as xgb
from typing import Tuple

def gradient(predt: np.ndarray, dtrain: xgb.DMatrix) -> np.ndarray:
    y = dtrain.get_label()
    return (np.log1p(predt) - np.log1p(y)) / (predt + 1)

def hessian(predt: np.ndarray, dtrain: xgb.DMatrix) -> np.ndarray:
    y = dtrain.get_label()
    return ((-np.log1p(predt) + np.log1p(y) + 1) / np.power(predt + 1, 2))

def squared_log(predt: np.ndarray, dtrain: xgb.DMatrix) -> Tuple[np.ndarray, np.ndarray]:
    predt[predt < -1] = -1 + 1e-6
    grad = gradient(predt, dtrain)
    hess = hessian(predt, dtrain)
    return grad, hess

xgb.train({'tree_method': 'hist', 'seed': 1994},
           dtrain=dtrain,
           num_boost_round=10,
           obj=squared_log)  # Using the custom objective function