arXiv 论文：预测正确不等于记忆充分
Prediction Is Not a Sufficient Memory State
一篇理论论文，证明只存预测结果会丢掉继续学习所需的信息，岭回归那组维度对比很直观，做持续学习的值得看看。
arXiv 2610.13881 论文刻画了精确累积学习中记忆状态与当前预测的区别。结论指出，对平滑凸残差损失，固定维度连续循环状态存在当且仅当损失是多项式，次数 D 需要 r=binom(d+D,D)-1 个坐标。以岭回归为例，完整隐藏证据（设计矩阵）需要 dp+d(d+1)/2 维状态，而仅存预测器只需 dp 维。论文还给出核注意力机制的第二个精确闭包分类。
Prediction Is Not a Sufficient Memory State
A memory that reproduces every current prediction may still have forgotten how to learn from the next observation. We characterize this distinction for exact cumulative learning and attention. For smooth convex residual losses with unrestricted real features and labels, a continuous recurrent state of fixed dimension exists exactly when the loss is polynomial. With ordinary, unweighted observations, degree $D$ requires precisely $r=\binom{d+D}{D}-1$ coordinates; even among histories with one fixed current predictor, exactly $r-d$ coordinates can remain necessary for continuation. Nonpolynomial losses have unbounded requirements on such equal-prediction histories. Bounded scalar logistic examples retain this obstruction after fixing any finite derivative jet. For ridge regression, the hidden evidence is the design matrix: the exact state costs $dp+d(d+1)/2$, versus $dp$ for the predictor. On a well-conditioned zero-prediction family, optimal $B$-bit compression has average one-update error of order $d^{-1}2^{-2B/s_d}$, $s_d=d(d+1)/2$, even with arbitrary randomized codes. Kernel attention gives a second exact closure classification. The results identify what future updates reveal, rather than proposing a new regression algorithm or asserting trained-model gains.
