Sharp Minimax Regret for Infinite-Memory Logistic Prediction

We study online prediction for a specific finite-alphabet, exogenously driven source with infinite input memory. Independent Rademacher inputs $(U_t)$ are observed sequentially, and the next binary mark has logit $\sum_{j=1}^{t}θ_jU_{t+1-j}$, where $\abs{θ_j}\leq r_j$ and $\sum_jr_j\leq B$. Regret is expected cumulative excess log loss. Lag $j$ can affect prediction by scale $r_j$ and enters only $n_{T,j}=T-j+1$ prediction rounds, leading to the lag-resolved spectrum $Γ_T(r)=\sum_{j=1}^{T}\log\!\left(1+n_{T,j}r_j^2\right)$. For every summable envelope, a localized Bayesian mixture proves $\cR_T(r)\leq CΓ_T(r)$. For exponential and polynomial envelopes, under the stated finite-sample dimension condition, a Toeplitz-design converse proves $\cR_T(r)\geq cΓ_T(r)$, with constants allowed to depend on the fixed decay parameters and the logit bound. Thus $Γ_T(r)$ is the minimax cumulative-regret scale for this source class in these canonical regimes, giving $Θ(α^{-1}\log^2T)$ for $r_j=Ae^{-αj}$ and $Θ(T^{1/(2s)})$ for $r_j=Aj^{-s}$, $s>1$. The converse is specific to the exogenous lagged model and is not a profile-only theorem for arbitrary stationary infinite-memory sources. Retaining only the most recent $h$ inputs costs order $\sum_{j>h}n_{T,j}θ_j^2$, yet the same worst-case truncation profile can correspond to polynomially different regret. A scaled online Newton predictor attains the spectrum upper bound.

EDGE: a closed-form directed test for the calibration of probabilistic binary classifiers

A probabilistic binary classifier is judged almost everywhere by discrimination - accuracy, the ROC curve, the area under it. Every such criterion is invariant to a monotone distortion of the predicted probabilities, so a classifier can rank perfectly and still return probabilities that are badly wrong. Calibration is the property decisions need, and the field's instrument for it, the binned expected calibration error with its reliability diagram, is descriptive: it has no null distribution, so it cannot say whether the miscalibration it displays is real or noise, and it depends on the binning. We propose EDGE, a calibration test for the canonical probabilistic classifier, logistic regression. EDGE reads the same binned predicted-versus-observed table a reliability diagram plots, and projects its standardized bin residuals onto a small pre-specified basis of smooth calibration-distortion shapes. Its null distribution is a weighted sum of chi-square variables in closed form, costing one pass over the data and one small eigendecomposition: no refit, no resampling, no tuning, so it can run inside cross-validation loops. Binning also makes it robust to the sparsity continuous features create. Across link and feature misspecification the pre-specified default led or tied every rival binned test on the fitted index in 19 of 22 detectable scenarios, and stayed computable where the refit-based Stukel score test separates in 20% to 28% of sparse samples. Its honest limit is rough, high-frequency miscalibration, where omnibus statistics win - a limit an elementary resolution argument shows is shared by every binned instrument, the calibration error included.

Prediction-Only Distillation in Linear and Logistic Regression

Self-distillation (SD) is typically studied when the student is retrained on the teacher's original training inputs. In many practical deployments, however, the labeled training data are no longer available, and one has access only to the trained predictor and fresh unlabeled covariates. We study SD in this prediction-only regime through a fresh-X prediction-mixed scheme: a pure-distilled student is trained on fresh covariates pseudo-labeled by the teacher, and the final predictor is an affine combination of the teacher and student predictions. For ridge regression under proportional asymptotics, we derive deterministic equivalents for the optimally mixed prediction risk under general anisotropic covariance and deterministic signal. We show that this risk is strictly smaller than the teacher risk for almost every pair of teacher and student regularization levels, including when the fresh covariates are out-of-distribution and even when their covariance is isotropic. We further show that the optimal mixing weight cannot be identified from unlabeled data alone, but can be consistently estimated in a single post-training step using a small independent labeled calibration set, without additional model fitting. Finally, for binary logistic regression, we show that prediction mixing can outperform both the teacher and the pure-distilled classifier.

Finite-Sample Performance of Gradient Descent in Logistic Regression with Gaussian Design

We consider the parameter estimation problem in logistic regression with Gaussian design: the estimation of a fixed unknown parameter $θ^*\in \mathbb{R}^d$ ($\|θ^*\|_2\ge 1$) from $n$ i.i.d. samples $\{(x_i,y_i)\}_{i=1}^n$, where $x_i\sim N(0,I_d)$ and $y_i|x_i \sim {\rm Bernoulli}(1/(1+\exp(-x_i^\top θ^*)))$. Our main aim is to characterize the finite-sample estimation performance and convergence behavior of gradient descent (GD) on the maximum likelihood objective (i.e., the logistic loss). Under small $O(1)$ stepsize and $0$ initialization, we show that GD linearly converges to a small neighborhood of $θ^*$ achieving an $\ell_2$ error of order $O(\sqrt{\|θ^*\|_2^5d/n})$. This substantially goes beyond existing theoretical results that lack non-asymptotic estimation error rate and exhibit much slower parameter convergence. We also establish a faster local linear convergence to the same statistical error under a large $Θ(\|θ^*\|_2)$ stepsize. The main technical component is to show that the gradient of the logistic loss satisfies a certain approximate invertibility condition (AIC). To that end, we uniformly control the deviation of the gradient from its population counterpart by covering and peeling arguments, and then show that the population GD is a contraction by a delicate analysis based on the eigenvalues of population Hessian matrices. Finally, we build upon the recent work Matsumoto and Mazumdar (2025) and devise a novel efficient estimator that attains a sharper rate in high dimensions. This indicates that the existing non-asymptotic guarantees exhibit sub-optimal dependence on $\|θ^*\|_2$, and that in many regimes $Θ(\sqrt{\|θ^*\|_2d/n})$ is the tight estimation error rate. Numerical examples are provided to corroborate our theoretical results.