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Write $z_i(x)=⟨a_i,x⟩+b_i$. The log-sum-exp function is convex and composition with the affine functions $z_i$ preserves convexity, so $f_{\beta }$ is convex. Directly, its Hessian matrix will also be shown positive semidefinite in part d.

Let $M=f(x)={\max }_i{z}_i(x)$. Factoring $e^{\beta M}$ out of the sum givesAt least one term in the sum is $1$, while every term is at most $1$. Thereforeand henceThus $f_{\beta }$ is a uniform smooth maximum of the affine pieces of $f$.

Chooseso the smooth maximum error is at most $ϵ/2$ and the Lipschitz gradient constant isSuppose a minimizer of $f_{\beta }$ lies within distance $R$ of the starting point. The Nesterov accelerated gradient method can find $x$ such thatiniterations. If $x_{\ast }$ minimizes $f$, then the smoothing inequalities implyThis improves the nonsmooth subgradient method dependence from $O({ϵ}^{-2})$ to $O({ϵ}^{-1})$.