arXiv:2608.26515v3 Announce Type: replace Abstract: We study sequential prediction of binary marks whose log-odds are a linear function of all past inputs. The inputs are independent uniform signs. The coefficient of the input observed $j$ steps earlier is unknown, but its absolute value is at most a known number $r_j$, and $\sum_j r_j\le B$. A learner is judged by its expected cumulative log loss in excess of the loss of the true conditional probabilities. We show that for every such sequence $r$ and every horizon $T$, the minimax value of this regret lies between $c\,\Gamma_T(r)/\max\{1,B^2\}$ and $\frac12\Gamma_T(r)$, where $\Gamma_T(r)=\sum_{j\le T}\log(1+(T-j+1)r_j^2)$ and $c$ is an absolute constant. The term for lag $j$ combines the size bound $r_j$ with the number $T-j+1$ of rounds in which that lag influences a prediction, and no monotonicity or other regularity of $r$ is needed. The upper bound is attained by a Bayesian mixture with independent Gaussian priors. For the lower bound, we apply the van Trees inequality at each prediction step and then bound the expected log-determinant of the random matrix whose rows are the lagged input vectors. We also characterize when $\frac12$ is the exact leading constant: for envelopes with a common bound $B$, possibly changing with $T$, the regret is asymptotic to $\frac12\Gamma_T(r)$ if and only if $\sum_{j\le T}\min\{1,(T-j+1)r_j^2\}=o(\Gamma_T(r))$. This gives, for example, $\log^2(A^2T)/(8\alpha)$ for $r_j=Ae^{-\alpha j}$, whereas for $r_j=Aj^{-s}$ with $s>1$ the regret is of order $T^{1/(2s)}$ and its ratio to $\Gamma_T(r)$ stays below $\frac12$. Finally, for a known horizon, an online Newton predictor with features scaled by $r_j$ attains regret of order $\Gamma_T(r)$ with polynomial computation per round.