Is LSE^{+}-like function strictly convex?

Question

Consider the following function, $$ f(u) = T\log \left( 1+ \sum_{i=1}^{N} \exp \left( \frac{a_i^{\intercal}u + b_i }{T}\right) \right) , $$ where $T > 0$, $a_i \in \mathbb{R}^m$, $b_i \in \mathbb{R}$ are fixed constants.

Is $f$ strictly convex in $u$? If so, how can one prove this?

Please share your ideas :)

Answer 1

Sorry for the answer by myself. I found a paper about this.

The main point is, it's almost surely strictly convex.

See [1, Proof of Thm 4].

[1] F. Nielsen and G. Hadjeres, “Monte Carlo Information Geometry: The dually flat case,” arXiv:1803.07225 [cs, stat], Mar. 2018, Accessed: Mar. 16, 2022.