Definition. empirical squared-ratio excess [miao2026when, Theorem 2] [ftip-0066]
Definition. empirical squared-ratio excess [miao2026when, Theorem 2] [ftip-0066]
For token-level current-to-behavior ratios \(r_1,\ldots ,r_T\), DGG defines
\[\widehat \chi ^2=\frac 1T\sum _{i=1}^T(r_i^2-1).\]The source relates this statistic to an upper bound on language-model-head gradient energy through a batch-dependent constant. In a finite batch \(\widehat \chi ^2\) can be negative, so it is not automatically a nonnegative empirical Pearson divergence. The result also gives no converse from a small batch gradient to small policy shift.