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is there a way to express the logistic function $$\frac{1}{1+\exp(-x)}$$ as the difference of two convex functions?


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Can you provide some background for this question? Although your question is a good question, there is an aversion to questions in which no effort is shown. If you provide some background or, if this is a homework question, your work so far, your question may not be closed, or if it is closed, it may be able to be reopened. – robjohn Feb 24 '14 at 10:49
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up vote 0 down vote accepted

A convex function has an increasing derivative and we can write any function of bounded variation as a difference of two increasing functions. So let us look at the derivative $$ \begin{align} \frac{\mathrm{d}}{\mathrm{d}x}\frac1{1+e^{-x}} &=\frac{e^{-x}}{(1+e^{-x})^2}\\ &=\frac1{(e^{x/2}+e^{-x/2})^2}\\ &=\tfrac14\,\mathrm{sech}^2(x/2) \end{align} $$ Thus, we can break this up into the difference of two increasing functions: $$ f'(x)=\left\{\begin{array}{l} \tfrac14&\text{when }x\ge0\\ \tfrac14\,\mathrm{sech}^2(x/2)\hphantom{-4}&\text{when }x\lt0 \end{array}\right. $$ and $$ g'(x)=\left\{\begin{array}{l} \tfrac14-\tfrac14\,\mathrm{sech}^2(x/2)&\text{when }x\ge0\\ 0&\text{when }x\lt0 \end{array}\right. $$ Then if $$ f(x)=\left\{\begin{array}{l} \tfrac14x+\tfrac12&\text{when }x\ge0\\ \tfrac12\tanh(x/2)+\tfrac12&\text{when }x\lt0 \end{array}\right. $$ and $$ g(x)=\left\{\begin{array}{l} \tfrac14x-\tfrac12\tanh(x/2)&\text{when }x\ge0\\ 0&\text{when }x\lt0 \end{array}\right. $$ $f$ and $g$ are convex and $f(x)-g(x)=\frac12+\frac12\tanh(x/2)=\dfrac1{1+e^{-x}}$

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$$ \frac{d^2}{dx^2} (1+e^x)^{-1} = \frac{d}{dx} \left(-(1+e^x)^{-2}e^x\right) = e^x\left( -(1+e^x)^{-2}+2(1+e^x)^{-3}e^x \right) $$ $$ = \frac{e^x (-(1+e^x)+2e^x)}{(1+e^x)^3} = \frac{ e^x(e^x-1) }{(1+e^x)^3}. $$ This is a bounded function because it is everywhere continuous and goes to $0$ as $x\to\pm\infty$.

So let $f(x) = Ax^2 + \dfrac{1}{1+e^x}$ with $A$ big enough so that the second derivative $f''$ is always positive. Then the logistic function is the difference between the convex function $f$ and the convex function $x\mapsto Ax^2$.

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Thanks, but I meant if there was a "fundamental" way. Let me explain. For instance, consider $$g(x) \triangleq \frac{1}{t} (x)^+$$, where $t>0$ and $(\cdot)^+$ denotes the positive part. We have something similar to the logistic function by $g(x+\frac{t}{2})-g(x-\frac{t}{2})$. Is there a similar way to write the logistic (perhaps as difference of two exponential functions, just guessing)? Thanks – Ita Atz Feb 23 '14 at 19:29
This certainly answers the question. Did you compute the minimum of $A$? – robjohn Feb 24 '14 at 16:46

A different approach.

If $f$ is any function with continuous second derivative let $$ f''_+=\max(f'',0)\quad f''_-=-\min(f'',0). $$ Then $f''_+$ and $f''_-$ are continuous, non-negative and $f''=f''_+-f''_-$. Now let $F_+$ and $F_-$ be such that $F_+''=f''_+$ and $F_-''=f''_-$. $F_+$ and $F_-$ are convex, and the constants of integration can be chosen so that $f=F_+-F_-$.

This is analogous to writing any $C^1$ function as the difference of two increasing functions.

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