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.

翻译：不变性学习即使在不变结构具有统计可辨识性时也可能失败。我们展示了一个条件性计算障碍：在基于平均情形稀疏恢复归约的黑箱可采样监督稀疏恢复原语假设下，存在具有一维预测不变子空间（k=1）的\emph{可采样}多环境实例，这些实例可通过穷举搜索以多项式样本量学习，而任何多项式时间的恒定精度恢复算法都将与原语相矛盾。我们进一步通过分离参数γ量化环境多样性，该参数控制可辨识性及不变性目标函数的曲率。在充分多样性和局部高斯正则性条件下，极小极大风险为$\mathbb{E}[\dist(\hat{V},V_{\mathrm{inv}})^2]=Θ(k(d-k)/(n|\mathcal{E}|))$，且在标签诱导偏移下，当$n^*\propto k(d-k)/(|\mathcal{E}|γ^2)$时会出现相变，此时估计误差尺度与$1/γ^2$成正比。合成数据集和真实数据集验证了所预测的差距与相变，并启发了简单的多样性诊断方法。