Is Spurious Correlation Removal Always Learnable?

• URL: http://arxiv.org/abs/2606.12930v1
• Date: Thu, 11 Jun 2026 05:49:43 GMT
• Title: Is Spurious Correlation Removal Always Learnable?
• Abstract summary: Invariant learning can fail even when the structure is statistically identifiable.<n>We show a conditional computational barrier: under a black-box samplable supervised sparse recovery primitive, there exist emphamplable multi-dimensional environments.<n>Synthetic and real datasets illustrate the predicted gaps and transitions and motivate simple diversity diagnostics.
• Score: 56.28155520961125
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Invariant learning can fail even when the invariant structure is statistically identifiable. We show a conditional computational barrier: under a black-box samplable supervised sparse recovery primitive motivated by average-case sparse-recovery reductions, there exist \emph{samplable} multi-environment instances with a one-dimensional predictive invariant subspace ($k=1$) that are learnable with polynomial samples by exhaustive search, while any polynomial-time constant-accuracy recovery algorithm would contradict the primitive. We further quantify environment diversity by a separation parameter $γ$, which controls identifiability and the curvature of invariance objectives. Under sufficient diversity and local Gaussian regularity, the minimax risk is $\mathbb{E}[\dist(\hat{V},V_{\mathrm{inv}})^2]=Θ(k(d-k)/(n|\mathcal{E}|))$, and under label-induced shifts a phase transition occurs at $n^*\propto k(d-k)/(|\mathcal{E}|γ^2)$ with refined estimation error scaling proportional to $1/γ^2$. Synthetic and real datasets illustrate the predicted gaps and transitions and motivate simple diversity diagnostics.

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