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L1-penalized logistic regression minimizes empirical logistic loss plus $\lambda \Vert \beta {\Vert }_1$. Its subgradient optimality conditions set each logistic score coordinate equal to $-\lambda z_j$, where $z_j=\mathrm{sgn}({\beta }_j)$ off zero and $z_j\in [-1,1]$ at zero.

The scalar logistic loss $u\mapsto -u+\log (1+e^u)$ is strictly convex because its second derivative is $e^u/(1+e^u{)}^2>0$. If $\hat{\beta }$ and $\tilde{\beta }$ are minimizers but $X\hat{\beta }\ne X\tilde{\beta }$, strict convexity of the loss as a function of the fitted vector and convexity of the L1 norm make the objective at their midpoint strictly smaller than the common minimum. This contradiction proves thatfor every pair of solutions.

A binary logistic regression setsand classifies by the sign of ${\gamma }_0+x^T\beta$. Its unpenalized maximum-likelihood estimator minimizes the empirical logistic lossThe plotted data are complete separation data: there is a vector $v$ with every signed margin $Y_i{X}_i^{\ast T}v>0$. For every finite $c>0$, increasing $c$ strictly decreases each term of $L(cv)$, and $L(cv)\to 0$ as $c\to \infty$. No finite parameter attains zero, so the unpenalized optimization has no solution.

Adding an $L^2$ penalty $\lambda \Vert \gamma {\Vert }_2^2$ with $\lambda >0$, constraining $\Vert \gamma {\Vert }_2$, or using a finite stopping rule makes the problem attain a finite approximate solution. The penalized option is preferable because cross-validation can select the strength of regularization.

The hypothesis class and margin loss areandThus this is a positive semidefinite quadratic-form classifier with logistic loss. Its empirical risk, viewed as a function of the matrix parameter, isDifferentiating under the sum shows thatHence the displayed $g_i$ is exactly $\nabla F(M^{(i)})$, and the algorithm is projected gradient descent on the positive semidefinite trace ball.

The scalar factor multiplying each $Y_j{X}_j{X}_j^T$ lies in $[0,1]$. Sincewe have the uniform gradient boundLet $\hat{M}$ parametrize the empirical minimizer $\hat{h}$. Positive semidefiniteness givesThus the initial distance from $M^{(1)}=0$ to $\hat{M}$ is at most $D=s$, while the gradient bound is $G=C^2$.

The averaged projected-gradient bound, in its slightly looser formapplies because $F$ is convex. ChooseSubstitution gives