Q:

Underset spacing

I've got the following equation 
\begin{align*}
  \hat{\boldsymbol{\eta}} = \underset{(\boldsymbol{\beta}_{1}, \ldots,
  \boldsymbol{\beta}_{d}, \boldsymbol{\theta}) \in B_{1} \times
  \cdots \times B_{d} \times \Theta}{\text{arg max }}
  \ell(\boldsymbol{\beta}_{1}, \ldots, \boldsymbol{\beta}_{d}, \boldsymbol{\theta})
\end{align*}

resulting in 

I would like for the expression under "arg max" to be positioned a bit lower so that the large horizontal spaces disappear. 

A:

In the example below, the first equation reproduces what you have right now, while the second equation achieves your stated objective (by encasing the subscript material in a \mathclap directive). Note the use of \strut to force a bit more whitespace between the two lines. Even so, the reader may have to study the equation for more time than should be required to figure out what the material in the second line pertains to. 
The third equation below uses the \substack macro to insert a judiciously-chosen line break in the subscript material. IMNSHO, breaking up the subscript material into two parts strikes a reasonable balance between compact appearance and readability of the equation.
Note that the code also defines a dedicated "math operator" named \argmax.

\documentclass{article}
\usepackage{mathtools,bm}
\DeclareMathOperator*{\argmax}{arg\,max} 

\begin{document}
\[
  \hat{\bm{\eta}} = \argmax_{(\bm{\beta}_{1}, \dots,
  \bm{\beta}_{d}, \bm{\theta}) \in B_{1} \times
  \cdots \times B_{d} \times \Theta}
  \ell(\bm{\beta}_{1}, \dots, \bm{\beta}_{d}, \bm{\theta})
\]

\bigskip
\[
  \hat{\bm{\eta}} = \argmax_{\strut\mathclap{(\bm{\beta}_{1}, \dots,
  \bm{\beta}_{d}, \bm{\theta}) \in B_{1} \times
  \cdots \times B_{d} \times \Theta}}
  \ell(\bm{\beta}_{1}, \dots, \bm{\beta}_{d}, \bm{\theta})
\]

\bigskip
\[
  \hat{\bm{\eta}} = \argmax_{\substack{(\bm{\beta}_{1}, \dots,
  \bm{\beta}_{d}, \bm{\theta}) \in \\ (B_{1} \times
  \cdots \times B_{d} \times \Theta)}}
  \ell(\bm{\beta}_{1}, \dots, \bm{\beta}_{d}, \bm{\theta})
\]

\end{document} 

Addendum: The \substack and \mathclap approaches may be combined, to create the following look:

\documentclass{article}
\usepackage{mathtools,bm}
\DeclareMathOperator*{\argmax}{arg\,max}

\begin{document}

\[
  \hat{\bm{\eta}} = \argmax_{\mathclap{\substack{
     (\bm{\beta}_{1}, \dots,\bm{\beta}_{d}, \bm{\theta}) \in \\ 
     (B_{1} \times \cdots \times B_{d} \times \Theta)}}} \,
  \ell(\bm{\beta}_{1}, \dots, \bm{\beta}_{d}, \bm{\theta})
\]

\end{document} 

A:

I would suggest an alternative notation:

\documentclass{article}

\usepackage{amsmath}

\begin{document}

\begin{equation*}
  \hat{\boldsymbol{\eta}} = \operatorname{arg\,max}\bigl\{
  \ell(\boldsymbol{\beta}_{1}, \ldots, \boldsymbol{\beta}_{d},
  \boldsymbol{\theta}) : (\boldsymbol{\beta}_{1}, \dots,
  \boldsymbol{\beta}_{d}, \boldsymbol{\theta}) \in B_{1} \times
  \cdots \times B_{d} \times \Theta \bigl\}
\end{equation*}

\end{document}

