#log-linear-model

A log-linear model $P(y \mid x) \propto \exp(\theta \cdot \phi(x, y))$ is trained by maximizing the log-likelihood minus $C\sum_k \theta_k^2$ with $C > 0$. Its first two features are identical, $\phi_1(x, y) = \phi_2(x, y)$ for all $x, y$. True or false: the optimal weights satisfy $\theta_1^\ast = \theta_2^\ast$.