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Suppose that I'm building a binary classifier parameterized by $\theta \in \mathbb{R}^k$ that maps some observed features $x_i \in \mathbb{R}^l$ to a decision of whether or not to play a game with an uncertain payoff: $$ f(\centerdot;\theta): \mathbb{R}^l \to \{0,1\}.$$ The payoff of the $i$th game is $y_i \sim N(0,1)$ and my objective is to maximize my expected profit after $n$ games: $$\max_{\theta} \mathrm{E}\left[\sum_{i=1}^n y_i f(x_i;\theta)\right].$$ The classifier might be a linear model, neural network, etc.

From reading https://en.wikipedia.org/wiki/Loss_functions_for_classification, my understanding is that I should construct a loss function that uses the output of a model before discretization. I imagine the reason is so that there is some sense of "how close was the prediction to reality" rather than a binary right/wrong. But if I use one of their suggested loss functions, I lose all of the information from the size of the payoff, i.e., misclassifying a tiny payoff is just as bad misclassifying a large payoff.

So my question is why shouldn't I just use my objective function directly as my loss function? Why are squared loss, cross entropy loss, etc. suggested instead?

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  • $\begingroup$ Did you ever figure out an answer to this? I have the same question! $\endgroup$ – Matt Aug 7 '17 at 9:26
  • $\begingroup$ This answer on CrossValidated gives some clues. $\endgroup$ – Xpector Feb 20 at 14:11

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