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$\begingroup$ Actually, the objective function is the function (e.g. a linear function) you seek to optimize (usually by minimizing or maximizing) under the constraint of a loss function (e.g. L1, L2). Examples are ridge regression or SVM. You can also optimize the objective function without any loss function, e.g. simple OLS or logit. $\endgroup$
Without trying to give a full primer on PCA, from an optimization standpoint, the primary objective function is the Rayleigh quotient. The matrix that figures in the quotient is (some multiple of) the sample covariance matrix S = 1 n n ∑ i = 1xixTi = XTX / n where each xi is a vector of p features and X is the matrix such that the i th row is ...
The "objective function" is the function that you want to minimise or maximise in your problem. The expression "objective function" is used in several different contexts (e.g. machine learning or linear programming), but it always refers to the function to be maximised or minimised in the specific (optimisation) problem.
Those two formulae are different things: 1 2wTw + C ∑ ξi 1 2 w T w + C ∑ ξ i. is one form of the objective function, the function which is minimized over w w. , b b. , and ξi ξ i. (subject to certain constraints, which are where b b. comes in) to find the best SVM solution. Once you've found the model (defined by w w. and b b.
The help page of XGBoost specifies, for the objective parameter (loss function): reg:gamma: gamma regression with log-link. Output is a mean of gamma distribution. It might be useful, e.g., for modeling insurance claims severity, or for any outcome that might be gamma-distributed.
If it is correct, then minimizing SSE rather than some other objective function can be justified by consistency, which is acceptable, in fact, better than saying the quadratic function is nicer. In pratice, I actually saw many cases where people directly minimize the sum of square errors without first clearly specifying the complete model, e.g ...
Now, in these steps, 1.) where does the objective function of the XGboost (G,H, regularized) fit ? Is it the same as the loss function in step 1 above (using any arbitrary loss with also including regularization ?). 2.) If that's case what do the authors mean by 'best objective function' in 'the structure score section' of docs. 3.)
Ridge regression adds another term to the objective function (usually after standardizing all variables in order to put them on a common footing), asking to minimize $$(y - X\beta)^\prime(y - X\beta) + \lambda \beta^\prime \beta$$ for some non-negative constant $\lambda$. It is the sum of squares of the residuals plus a multiple of the sum of ...
$\begingroup$ @HaitaoDu I am not sure but I guess that the (small) discrepancy is because the algorithm used in optim stops already early before reaching the exact solution (optim is using a gradient method to itteratively improve the estimate, obtaining an estimate with a higher likelihood each step and getting closer to the exact solution each step, and stops when the steps of the ...
$\begingroup$ I'm not sure what you mean by "final" and "original" objective function, Cam. PCA is not (conceptually) an optimization program. PCA is not (conceptually) an optimization program. Its output is a set of principal directions, not just one.