Algorithm 2 . Fast binary logistic regression (FBLR).

1: procedure FBLR( $\mathbf{X}$, y, f, λ, γ, ξ, K, Ctolerance)

2:    Initialization

3:     $n,d,r,\hat{\mathbf{U}},\hat{\mathbf{S}},{\hat{\mathbf{V}}}^{\mathrm{\top }}\leftarrow$ LowRankApproximation $(\mathbf{X},\xi )$

4:     $\mathbf{F}=\hat{\mathbf{V}}{\hat{\mathbf{S}}}^{-1}{\hat{\mathbf{U}}}^{\mathrm{\top }}$

5:     $w=\mathbf{F}y$ $\text{⊳}n$ Initializing weights via the pseudo-inverse

6:    if $(\lambda >0$ or $\gamma >0)$ then

7:       $\mathbf{G}\leftarrow \hat{\mathbf{S}}{\hat{\mathbf{V}}}^{\mathrm{\top }}$

8:       $v\leftarrow \frac{1}{n}{\mathbf{G}}^{\mathrm{\top }}\left({\hat{\mathbf{U}}}^{\mathrm{\top }}(y-\frac{\overrightarrow{1}}{2})\right)$

9:   else

10:      $y_q\leftarrow \hat{\mathbf{U}}\left({\hat{\mathbf{U}}}^{\mathrm{\top }}(y-\frac{\overrightarrow{1}}{2})\right)$

11:  end if

12:  Iteration

13:  for $k\leftarrow 0$ to K do

14:      $\hat{w}\leftarrow w$

15:      $\mathbf{Z}\leftarrow$ diag(z) $\text{⊳}z$ = vector $(z_i)$, Eq. (6)

16:     if $(\lambda >0$ or $\gamma >0)$ then

17:        $\mathbf{H}\leftarrow$ diag(h)  $\text{⊳}h$ = vector $(h_i)$, Eq. (11)

18:        $\mathbf{A}\leftarrow \frac{2}{n}{\mathbf{G}}^{\mathrm{\top }}\left({\hat{\mathbf{U}}}^{\mathrm{\top }}\mathbf{Z}\hat{\mathbf{U}}\right)\mathbf{G}+\frac{\lambda }{d}\mathbf{I}+\frac{\gamma }{d}\mathbf{H}$

19:        $b\leftarrow \frac{\lambda }{d}{w}^{(k)}+v$

20:        $w\leftarrow$ solve $(\mathbf{A},b)$ $\text{⊳}h$ Applying Cholesky decomposition for solving $Ax=b$

21:     else

22:         $w\leftarrow \frac{1}{2}\mathbf{F}{\mathbf{Z}}^{-1}{y}_q$

23:     end if

24:     if $(k\ge 3$ and $||w-\hat{w}|{|}_{\mathrm{\infty }}\le C_{tolerance})$ then

25:        break

26:     end if

27:  end for

28:  Finalization

29:  return w

30: end procedure