Suppose you're a surgeon and you are making a decision based on the output of a model. The variables in this model $x_1, x_2,...$ are clinically interpretable, like age, blood sugar, or hemoglobin. Every patient has her own set of these variables, and the decision is binary (e.g., either surgery or no surgery). The model's parameters are $\theta_i$, and the decision is made by the following rule:
If $\theta_1x_1 + \theta_2x_2 + ... <a$, carry out Decision 1
Otherwise ($\theta_1x_1 + \theta_2x_2 + ... \geq a$), carry out Decision 2
I am interesting in discovering the most important variable(s) that favored Decision1 over Decision2. How would I go about doing this? One way is to discover the smallest $\Delta x_j$, such that
$...+\theta_{j-1}x_{j-1} + \theta_j(x_j + \Delta x_j) + ...\geq a$ , but
$...+\theta_{j-1}x_{j-1} + \theta_jx_j + ...< a$
This would be interpreted counterfactually: had $x_j$ been $x_j + \Delta x_j$, Decision2 would be chosen rather than Decision1. Is there a better way to do this, given the fact that there are multiple variables? For example, maybe it was a combination of two different variables being slightly too large that pushed Decision1 into Decision2, rather than just one variable.
EDIT:
The problem is essentially this.
Let $S$ be the sum $\theta_1x_1 + ... + \theta_nx_n$. Find the smallest magnitude vector $\Delta \boldsymbol{x}_{normalized}$ such that
$S < a$, but
$S \geq a - \boldsymbol{\theta}^\intercal\boldsymbol{\Delta x}$ ,
where $\boldsymbol{\Delta x} = (\Delta x_1, \Delta x_2, ..., \Delta x_n)$
and $\Delta \boldsymbol{x}_{normalized} = (\frac{\Delta x_1 - \bar{x_1}}{\sigma_1}, \frac{\Delta x_2 - \bar{x_2}}{\sigma_2} , ..., \frac{\Delta x_n - \bar{x_n}}{\sigma_n})$
How do I solve this problem?