Fixed incorrect "translation invariance" in problem in softmax-regression.md

[Daniel-Sinkin]

Hello, when working through the exercises inside of softmax-regression.md I've noticed that one of the problem seems to have a small error, namely the function $g$ is not translation invariant, but rather translation equivariant. This means that there should be a $+b$ term outside of the function as well.

Also I've changed the sign of the b to align it closer to what should be done in the last subproblem: "Show that if we choose $b = \mathrm{max}_i x_i$ we end up with a numerically stable implementation."

As such the corrected version of the problem is $g(\underline{x} - b) = g(\underline{x}) - b$.

Proof

$$\begin{aligned} g(\underline{x} - b) &= \log \sum_i \exp(x_i - b) \\\ &= \log \sum_i \exp(x_i) \exp(-b) \\\ &= \log \left( \left( \sum_i \exp(x_i) \right) \exp(-b) \right) \\\ &\overset{1}{=} \log \left(\sum_i \exp(x_i)\right) + \log \exp(-b) \\\ &\overset{2}{=} \log \left(\sum_i \exp(x_i)\right) - b = g(\underline{x}) - b. \end{aligned}$$

Where we have used that (1) $\log(xy) = \log(x) + \log(y)$ and (2) $\log\exp x = x$.

By submitting this pull request, I confirm that you can use, modify,
copy, and redistribute this contribution, under the terms of your
choice.