How XGBoost sample_weights alternate the objective function formula?

[geogreko]

From my understanding and reading post #144

when sample_weights are used, then first and second gradient

g_i and h_i are multiplied with sample_weights so that for each instance i:

grad_xgb[i] = grad[i] * case_weight[i]
hess_xgb[i] = hess[i] * case_weight[i]

The 2016 paper ‘XGBoost: A Scalable Tree Boosting System’ states that,
if we remove the constant terms, the objective function at step t is given in eq.3

The question is: How eq.3 is alternated when using sample weights?
Is it as simple as that:

(g_i*case_weight[i])*ft(xi)+1/2(h_i * case_weight[i]) (ft)power of 2(xi) + Ω(ft)

In other words, just substituting g_i, and h_i with grad[i] * case_weight[i] and hess[i] * case_weight[i] respectively?

Unfortunately, the links provided in post 144 are broken.

[trivialfis]

Yes, it's multiplied into gradients. It's also used in some other places like sketching.

[geogreko]

Great, thanks for the response!

[trivialfis]

Feel free to reopen if there are further questions.

[geogreko]

One more question.
In the case of sample_weights, do we also multiply the gradients (gi,hi) to the sample_weights when calculating the Gain function? Like g_i * case_weight_i, h_i * case_weight_i, for all gradients in the gain function below?