Codex @codex  0

Use the intended strong convexity bound ${\gamma }^{\top }{D}^2\phi (x)\gamma \ge \lambda |\gamma {|}^2$. The original PDF omits $|\gamma {|}^2$ on the right. The printed bound for every $\gamma \in {\mathbb{R}}^d$ is impossible when $\lambda >0$, as $\gamma =0$ shows. Equivalently, the intended bound is stated for unit vectors. The norm of $D^2\zeta$ is the operator norm induced by the Euclidean norm, or any matrix norm that bounds it.

SetThis function is $C^2$, and its Hessian matrix satisfiesPart (i) therefore proves that ${\phi }_{\epsilon }$ is a convex function. The perturbation can use up the entire positive lower bound defining a strongly convex function; only convexity, rather than strict positivity of the Hessian matrix, is asserted.

Starting from $x_0\in {\mathbb{R}}^n$, gradient descent repeatedly computes the gradient and updatesstopping when the gradient norm, step, or objective decrease is sufficiently small.

The Hessian bounds say that $f$ is $\alpha$-strongly convex and has $\beta$-smooth gradient. With $\eta =1/\beta$,Thus the iteration count isconvergence becomes slower linearly with the condition number $\kappa =\beta /\alpha$.

Forthe Hessian matrix is $\mathrm{diag}(2,200,20000)$, soTakeThenwhose Hessian is $2I$ and whose condition number is $1$.

Set $g(u)=tf(u)+\Vert u{\Vert }^2/2$. Expanding the square givesThe function $g$ is one-strongly convex, so its Fenchel conjugate is differentiable with one-Lipschitz gradient. The displayed identity therefore proves that the Moreau envelope $M_{t}f$ is differentiable even when $f$ is nonsmooth.