Closed-Form Logit Steering

COLLINS WESTNEDGE
OCT 8, 2025

Goal

The goal is to derive the closed-form minimal perturbation to an input x that achieves any target probability p ∈ (0,1) in logistic regression.

Sigmoid: \(\sigma(z)=\dfrac{1}{1+e^{-z}}\)
maps score to probability (e.g., \(z=0 \Rightarrow p=0.5\))
Score: \(z=w^\top x + b\)
linear combination of features plus bias term
Logit (log-odds): \(\operatorname{logit}(p)=\ln\!\left(\dfrac{p}{1-p}\right)\)
maps probability to linear score (inverse of \(\sigma\))

\(H=\{\,x\in\mathbb{R}^n:\; w^\top x + b = 0\,\}\) — decision boundary (\(p=0.5\))
\(w \perp H\) — \(w\) is normal to \(H\)

Move \(x\) along \(w\) by some \(\lambda\): \[ x' = x + \lambda w \] with \[ \operatorname{logit}(p) = w^\top x' + b. \]

Set up the constraint: Since the model’s score is \(z = w^\top x' + b\) and we want probability \(p\), we require \(w^\top x' + b = \operatorname{logit}(p)\).
Plug in \(x'\) (use \(w^\top w=\|w\|^2\)): \[ \begin{aligned} \operatorname{logit}(p) &= w^\top(x+\lambda w)+b \\ &= w^\top x + \lambda\, w^\top w + b \\ &= w^\top x + \lambda\|w\|^{2} + b. \end{aligned} \]
Solve for \(\lambda\): \[ \lambda = \frac{\operatorname{logit}(p) - (w^\top x + b)}{\|w\|^{2}}. \]
Substitute \(\lambda\) into \(x' = x + \lambda w\): \[ x' = x + \frac{\operatorname{logit}(p) - (w^\top x + b)}{\|w\|^{2}}\,w. \]

\[ \boxed{ x' = x + \frac{\operatorname{logit}(p) - (w^\top x + b)}{\|w\|^{2}}\,w } \]

Where \(x' = x + \lambda w\) achieves the target probability \(p\).