In machine learning, the I.I.D. assumption (also written iid or i.i.d.) states that all samples in a dataset are Independent and Identically Distributed.

This means that each data point is unrelated to any other (independent), and all data points come from the same underlying probability distribution (identically distributed).

Although many machine learning algorithms, including XGBoost, theoretically assume I.I.D. data for optimal performance and for many of their theoretical guarantees to hold, in practice, XGBoost can often still perform well even when this assumption is violated to some degree.

Here’s a simple example using a synthetic dataset from scikit-learn that violates the I.I.D. assumption:

```
from sklearn.datasets import make_friedman1
from sklearn.model_selection import train_test_split
from sklearn.metrics import mean_squared_error
import xgboost as xgb
# Generate a synthetic dataset that violates I.I.D.
X, y = make_friedman1(n_samples=1000, noise=0.5, random_state=42)
X[:500] += 1 # Add a constant to the first half of the data
# Split data into train and test sets
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2, random_state=42)
# Train an XGBoost model
model = xgb.XGBRegressor(n_estimators=100, learning_rate=0.1, random_state=42)
model.fit(X_train, y_train)
# Evaluate the model's performance
y_pred = model.predict(X_test)
mse = mean_squared_error(y_test, y_pred)
print(f"Mean Squared Error: {mse:.4f}")
```

In this example, we use `make_friedman1`

from scikit-learn to generate a synthetic dataset. We then violate the I.I.D. assumption by adding a constant to the first half of the data, creating a different distribution for those samples.

Despite this violation, when we train an XGBoost model and evaluate its performance on the test set, we still achieve a reasonably low Mean Squared Error, demonstrating that XGBoost can often handle non-I.I.D. data.

However, it’s important to note that while XGBoost may be robust to some violations of the I.I.D. assumption, severe violations can still negatively impact its performance. In practice, it’s always a good idea to strive for data that is as close to I.I.D. as possible.