Github user MLnick commented on a diff in the pull request:
https://github.com/apache/spark/pull/13139#discussion_r63522591
--- Diff: docs/ml-classification-regression.md ---
@@ -374,6 +374,197 @@ regression model and extracting model summary
statistics.
</div>
+## Generalized linear regression
+
+When working with data that has a relatively small number of features (<
4096), Spark's GeneralizedLinearRegression interface
+allows for flexible specification of [generalized linear
models](https://en.wikipedia.org/wiki/Generalized_linear_model) (GLMs) which
can be used for various types of
+prediction problems including linear regression, Poisson regression,
logistic regression, and others.
+
+Contrasted with linear regression where the output is assumed to follow a
Gaussian
+distribution, GLMs are specifications of linear models where the response
variable $Y_i$ may take on _any_
+distribution from the [exponential family of
distributions](https://en.wikipedia.org/wiki/Exponential_family).
+
+$$
+Y_i \sim f\left(\cdot|\theta_i, \phi, w_i\right)
+$$
+
+An exponential family distribution is any probability distribution of the
form
+
+$$
+f\left(y|\theta, \phi, w\right) = \exp{\left(\frac{y\theta -
b(\theta)}{\phi/w} - c(y, \phi)\right)}
+$$
+
+where the parameter of interest $\theta_i$ is related to the expected
value of the response variable
+$\mu_i$ by
+
+$$
+\theta_i = h(\mu_i)
+$$
+
+Here, $h(\mu_i)$ is defined by the form of the exponential family
distribution used. GLMs also allow specification
+of a link function, which defines the relationship between the expected
value of the response variable $\mu_i$
+and the so called _linear predictor_ $\eta_i$:
+
+$$
+g(\mu_i) = \eta_i = \vec{x_i}^T \cdot \vec{\beta}
+$$
+
+Often, the link function is chosen such that $h(\mu) = g(\mu)$, which
yields a simplified relationship
+between the parameter of interest $\theta$ and the linear predictor
$\eta$. In this case, the link
+function $g(\mu)$ is said to be the "canonical" link function.
+
+$$
+\theta_i = h(g^{-1}(\eta_i)) = \eta_i
+$$
+
+A GLM finds the regression coefficients $\vec{\beta}$ which maximize the
likelihood function.
+
+$$
+\min_{\vec{\beta}} \mathcal{L}(\vec{\theta}|\vec{y},X) =
+\prod_{i=1}^{N} \exp{\left(\frac{y_i\theta_i - b(\theta_i)}{\phi/w_i} -
c(y_i, \phi)\right)}
+$$
+
+where the parameter of interest $\theta_i$ is related to the regression
coefficients $\vec{\beta}$
+by
+
+$$
+\theta_i = h(g^{-1}(\vec{x_i} \cdot \vec{\beta}))
+$$
+
+Spark's generalized linear regression interface also provides summary
statistics for diagnosing the
+fit of GLM models, including residuals, p-values, deviances, the Akaike
information criterion, and
+others.
+
+### Available families
+
+<table class="table">
+ <thead>
+ <tr>
+ <th></th>
+ <th>PDF</th>
+ <th>Response Type</th>
+ <th>Supported Links</th></tr>
+ </thead>
+ <tbody>
+ <tr>
+ <td>Gaussian</td>
+ <td>$\frac{1}{\sigma \sqrt{2\pi}} \exp \left( -\frac{(x -
\mu)^2}{2\sigma^2}\right)$</td>
+ <td>Continuous</td>
+ <td>Identity*, Log, Inverse</td>
+ </tr>
+ <tr>
+ <td>Binomial</td>
+ <td>$\binom{n}{k}p^k (1-p)^{n-k}$</td>
+ <td>Binary</td>
+ <td>Logit*, Probit, CLogLog</td>
+ </tr>
+ <tr>
+ <td>Poisson</td>
+ <td>$\frac{\lambda^k e^{-\lambda}}{k!}$</td>
+ <td>Count</td>
+ <td>Log*, Identity, Sqrt</td>
+ </tr>
+ <tr>
+ <td>Gamma</td>
+ <td>$\frac{\beta^{\alpha}}{\Gamma(\alpha)} x^{\alpha - 1} e^{-\beta
x}$</td>
+ <td>Continuous</td>
+ <td>Inverse*, Idenity, Log</td>
+ </tr>
+ <tfoot><tr><td colspan="4">* Canonical Link</td></tr></tfoot>
+ </tbody>
+</table>
+
+### Optimization
+
+The `spark.ml` GLM implements the method of
+[iteratively reweighted least
squares](https://en.wikipedia.org/wiki/Iteratively_reweighted_least_squares)
(IRLS) for finding
+the optimal regression coefficients. GLMs seek to find a maximum
likelihood estimate of the
+regression coefficients by finding zeros of the [score
equation](https://en.wikipedia.org/wiki/Score_(statistics)).
+The IRLS solver casts a first-order Taylor approximation of the score
equation to a weighted least squares regression and solves it
+iteratively until convergence.
+
+### Input Columns
+
+<table class="table">
--- End diff --
Yeah this is a good point. I think we should come up with a clear standard
on what goes into the user guide. Do we include input/output columns?
parameters?
The MLlib guide often included more detail on parameters in many cases.
You've suggested elsewhere that including these makes it more difficult to
maintain to ensure the params (and defaults etc) in the code match the use
guide - and I tend to agree. These rather live in the API docs (to which we
typically link in the ml guide).
I'd probably say that if we go with the "check the API docs for all param
details" then we should not bother with documenting the various input/output
columns in the user guide, for the same reasons of maintainability.
cc @jkbradley @mengxr @yanboliang @srowen for comment
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