# Is logistic regression a generalized linear model for more than two classes?

Suppose that $K>2$ denote the number of classes and $Y$ be the class label that follows the logistic regression model for a given $X \in \mathbb{R}^d$ as follows: $$P(Y=k|X)= \frac{e^{\beta_k^\top X}}{1+\sum_{i=1}^{K-1} e^{\beta_i^\top X}}, \quad k=1,\ldots,K-1 \\ P(Y=K|X)=\frac{1}{1+\sum_{i=1}^{K-1} e^{\beta_i^\top X}}.$$ It is well known that for $K=2$, the logistic regression model is a GLM (generalized linear model) and the logit is the cannonical link function for the exponential family. My question is does the same conlcusion still hold for multi-class?

In other words, can we find an exponential family distribution, a link function and suitable set of parameters such that the logistic regression model above falls into a GLM category?

My attempt: With a slight abuse of notation, denote $Y \in\mathbb{R}^{K-1}$ as the one-hot encoding of the class labels in $\{1,\ldots,K-1\}$. Then we have $$P(Y|X)= \exp \{ Y^\top(\log(\frac{p_1}{p_K}),\ldots,\log(\frac{p_{K-1}}{p_K})) +\log p_K \},$$ where $p_i$ denotes the class probabilities for each $X$. I am not sure if this satisfies the criteria for being a GLM.