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Suppose we want to solve the following: $$ \min_{w} f(x^Tw, y) + \lambda g(x^Tw) $$

with $f$ a (logistic) loss and $g$ something like a variance.

Is this a convex optimization problem ?

What are the constraints on $g$ to make it convex ?

PS: the original intent is to constrain the problem so that predicted values have low variance; so any hint on variance instead of sparsity regularization are welcome

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If $\lambda \geq 0$, and $g(x^{T}w)=(x^{T}w)^{2}=w^{T}(xx^{T})w$, then $\lambda g(x^{T}w)$ is clearly a convex function of $w$. If you could be more specific about what you want to do with $g()$, then perhaps I could provide a more complete answer.

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  • $\begingroup$ the idea would be that $g(x,w) = var(x^Tw)$ $\endgroup$ – oDDsKooL Dec 23 '14 at 13:34
  • $\begingroup$ $x^{T}w$ is a scalar rather than a random variable, so it isn't clear what you mean by its variance? $\endgroup$ – Brian Borchers Dec 23 '14 at 16:58
  • $\begingroup$ sorry my wording was ambiguous; what I seek is a formulation that minimize variance of the predicted values $\hat{y}$. I know how to estimate the bias and variance of a fitted estimator but I'd like to incorporate the low variance constraint in the optimization. $\endgroup$ – oDDsKooL Dec 31 '14 at 7:32

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