A finite-horizon composition law for Gaussian certification from M independent coordinates, a familywise certifier protecting every coordinate and time up to T pays optimal normalized squared half-width of order log(eM) + log log(e^eT), within the sample-mean-centered rectangular class.
Abstract
Adaptive procedures must work without nuisance information an oracle may use, such as a gradient scale or smoothness index, and robust procedures may have to answer queries whose coordinate and inspection time are chosen only after the data are seen. Such comparisons are meaningful only when the oracle advantage and validity contract are stated explicitly. We formalize nuisance adaptation via a slice-normalized minimax ratio retaining the worst-case instance within each nuisance slice, and separately define the robustness cost of expanding from one preannounced Gaussian query to arbitrary post-hoc inspection. Our main result is a finite-horizon composition law for Gaussian certification: from M independent coordinates, a familywise certifier protecting every coordinate and time up to T pays optimal normalized squared half-width of order log(eM) + log log(e^eT), within the sample-mean-centered rectangular class. Epoch stitching gives the upper bound; independent Gaussian block increments across coordinates and geometric time scales give a matching lower bound, already holding on a geometric checkpoint grid, forcing quantiles of the realized maximum width so selection and stopping taxes add. Two benchmark regimes complete the picture: unknown gradient scale in online convex optimization has constant cost, while pointwise adaptation over nested Holder classes costs order (log n / log log n)^(s1/(2s1+1)). Cast as model monitoring, the law lets an analyst inspect any of M slice metrics at any data-dependent time: the naive fixed-query band's selected coverage degrades sharply, to 0.30 at M=1 and to zero for M>=10, while the epoch-stitched certifier holds familywise coverage at an additive iterated-logarithm width cost. Experiments put both sharp predictions at risk of refutation; both survive.
Low-Rank Adaptation (LoRA) has become the standard mechanism for fine-tuning large pretrained models, yet its statistical properties remain only partially understood. Existing generalization results provide upper bounds of the form O~(sqrt(rd/n)) or O~(rd/n), but a matching lower bound is missing, and the question of how to choose the LoRA rank r has no formal answer. Both gaps are closed here. A local Rademacher argument establishes an upper bound of O~(rd/n) on the excess risk of the empirical risk minimizer over rank-r LoRA, whenever the target adaptation has rank at most r. A matching minimax lower bound of Omega(rd/n) is then proved via a Fano-type packing of the rank-r subspace of R^{d x d}; the bound applies to any estimator whose output lies in the rank-r LoRA class. Combining the two yields a rank-selection dichotomy. For the constrained empirical risk minimizer, the optimal rank equals the intrinsic rank r*, and over-ranking strictly hurts. For adaptive estimators of the nuclear-norm-then-truncate type, over-ranking is harmless and the rate saturates at Theta~(r* d / n) regardless of r. Taken together, the three results characterize the statistical complexity of LoRA fine-tuning within the well-specified locally quadratic regime, and identify the empirically observed over-parameterization penalty as a property of unregularized empirical risk minimization rather than of the LoRA class itself. Predictions of the theory are verified on a synthetic trace-regression benchmark and on real LoRA fine-tuning across three (model, task) configurations covering DistilBERT and RoBERTa on SST-2 and MRPC. All configurations exhibit the predicted U-shape in validation loss, with two showing statistically significant loss inflation at large ranks (paired permutation p = 0.016).
We develop a unified information-theoretic framework for lower bounding minimax quantiles. The starting point is a loss-adapted Neyman--Pearson metaconverse that bounds the minimax success probability at every loss threshold and confidence level. The bound separates the small-ball behaviour of the prior under the loss from the statistical distinguishability of the observation model, and is optimised over an auxiliary output distribution. Different relaxations of this Neyman--Pearson bound yield converses based on \(f\)-informativity, Sibson mutual information \(I_\alpha\), Maximal Leakage, and Amemiya norms. Classical Fano and Le Cam lower bounds are recovered as special cases. The framework also clarifies why different information measures are suited to different recovery criteria. Maximal Leakage is exact for a class of symmetric exact-recovery problems. We use this identity to derive finite-sample bounds on the full minimax exact-recovery risk in the balanced Gaussian weighted stochastic block model, as well as two-sided finite-sample minimax-quantile bounds for low-rank matrix estimation under isotropic bounded-energy noise. For approximate Hamming recovery, we exhibit a heterogeneous binary model in which an optimised finite Sibson order yields a strong converse while the Maximal Leakage specialisation is trivial. Finally, for one-coordinate Poisson localisation, a Bennett-type Young function used through its Amemiya norm recovers the exact success-probability scale, whereas classical Fano and fixed-power relaxations are strictly weaker. These results show that sharp converses for minimax quantiles require adapting the information measure to the recovery resolution, whether exact or approximate, and to the tail behaviour of the likelihood ratio.
Wasserstein distributionally robust optimization (DRO) is commonly built around the empirical distribution, with the ambiguity radius selected from a concentration bound. Although this construction provides useful statistical guarantees, it can be conservative and does not fully exploit predictive information about the underlying distribution or the difficulty of a particular decision problem. We develop a more flexible framework in which a predictive model determines the nominal distribution and a separate model estimates a data-dependent radius. The key requirement is not that the ambiguity set be centered at the empirical distribution, but that it contain the unknown data-generating distribution with the desired probability. We establish finite-sample guarantees and asymptotic consistency for arbitrary learned centers, derive tractable reformulations for non-uniform discrete predictive distributions, separate predictive-model and scenario-discretization errors, and prove stability under simultaneous perturbations of the center and radius. We further characterize the oracle conditional-quantile radius as the smallest conditionally valid rule and introduce a split-conformal procedure for finite-sample marginal calibration. Experiments on newsvendor problems, synthetic portfolios, distribution shifts, and real financial data show that learned and calibrated ambiguity sets can improve reliability, but do not automatically yield smaller radii or better decisions. Overall, the proposed framework treats calibration as a practical mechanism for reliable decision making rather than a universal guarantee of improved optimization performance.
Scaling LLM-based applications to millions of users is bottlenecked by the inference cost and latency of modern foundation models. A natural fix is to cluster the inputs and call the LLM only on cluster representatives, letting other members inherit the output -- but this is only safe if each member is measurably close to its representative. Existing clustering methods do not offer such per-sample quality control at scale: none jointly guarantee a minimal within-cluster similarity, exact matching of categorical attributes, and scalability to tens of millions of samples. We propose a two-stage algorithm that generates initial clusters with Mini-batch K-Means, then greedily selects representatives within each initial cluster -- a step equivalent to the Johnson-Chvatal heuristic for Set Cover over alpha-balls in embedding space. The algorithm enforces the similarity and attribute guardrails exactly by construction, and runs in $O(nd + n^2 d/K)$ time and $O(nd + n^2/K^2)$ memory for $n$ samples, feature dimension $d$, and $K$ initial clusters -- linear in $n$ when $K$ grows proportionally with $n$. We provide benchmarks against common clustering methods on internal and public datasets: our method not only delivers per-sample guardrails but also runs 10-1000x faster and scales to data sizes where most standard methods become intractable. Deployed on 38 million customers for a persona-based recommender, the clustering method cut downstream cost and latency by 50-fold while preserving personalization and unblocked the production launch.
Prior analyses by Derezinski and Warmuth established all-size sampling identities, selected-OLS unbiasedness, and inverse moments for ordinary volume sampling, while their exact arbitrary-fixed-response loss and prediction-covariance formulas are at the rank-size endpoint s=d. We establish a Loewner envelope for centered coefficient covariance for every full-rank fixed pool, response, and legal budget d<= s<= m under ordinary indexed fixed-size volume sampling followed by selected unweighted least squares; its coefficient is globally sharp over the full-rank class. Global sharpness does not determine attainability on the pool in hand. Under positive loss, strict-interior budgets, and no coloops, a feature-only margin nu_A gives the exact fixed-design spectral phase: nu_A>0 if and only if the normalized spectral envelope is strict for every compatible residual, whereas nu_A = 0 if and only if some compatible residual is spectrally tight; the same zero-margin residual is tight at every strict-interior budget. A residual-augmented change of measure supplies the response-aware mechanism and a one-sided quantitative slack bound, while support saturation proves the attainment direction. Critical equal-leverage geometry interprets the boundary, and sound lower certificates yield conservative same-primitive cardinality decisions. Frozen-feature examples show that the certificate is nonvacuous and measure the fixed-pool cost of its authorized reduction. The claims concern conditional centered, full-Gram-whitened coefficient covariance, not population generalization.