Shrinkage Estimation

Large retail chains must tailor assortment and pricing decisions to local customer preferences, yet even retailers with extensive sales records often have limited data for estimating demand at the store or local market level, particularly in product categories with short selling windows. Local estimation (fitting a demand model for each store on its own) can therefore be unreliable, while pooling data across stores improves estimation precision at the cost of ignoring local demand heterogeneity. We study how retailers can balance these competing effects through transfer learning under a multinomial logit (MNL) demand model. We propose a shrinkage estimator that combines store-specific demand estimates with information pooled from other stores. Unlike the linear or quadratic settings underlying much of the existing theoretical literature on information transfer in operations, the nonlinear MNL likelihood does not admit a straightforward bias-variance decomposition. By exploiting its self-concordant-like properties, we establish finite-sample performance guarantees that characterize when information transfer improves demand estimation and how the appropriate degree of shrinkage depends on local sample size, feature dimensionality, and cross-store heterogeneity. We further derive regret bounds for downstream assortment and pricing decisions. Numerical experiments demonstrate the operational benefits of shrinkage across different data environments. An empirical study using data from several hundred stores of a European apparel retailer shows that shrinkage improves out-of-sample product prioritization and revenue capture relative to both local estimation and full pooling. Our findings demonstrate how retailers can leverage information across their store networks to improve revenue management decisions while preserving local demand differences.

Evaluating the causal effect of new features is a central goal for online platforms. While recent literature addresses limited testing traffic via centralized portfolio optimization, this perspective abstracts away a critical institutional reality: experimentation is operationally decentralized. The experimenters who develop new features also dictate which hypotheses to test, and they are typically rewarded based on empirical average treatment effects that are prone to upward bias. Left unchecked, this principal-agent conflict can severely erode platform value, a structural failure that conventional centralized levers, such as significance thresholds and traffic budgets, cannot resolve. By reframing experimentation as an incentive design problem, we demonstrate that two practical mechanisms, sample splitting and shrinkage, can effectively bridge this gap. Sample splitting aligns incentives perfectly at a bounded traffic cost, while shrinkage consumes no additional traffic and guarantees that interventions with negative expected effects are strictly unprofitable to field.

One-shot federated graph learning generally aims to train Graph Neural Networks (GNNs) across clients with disconnected subgraphs in a single communication round. Existing methods predominantly design advanced optimization strategies under the premise that local GNN training is indispensable. However, empirical observations reveal that under extreme non-IID conditions, local GNN training suffers from severe cross-client representation misalignment, becoming a major source of error rather than a remedy. Motivated by this, we reformulate one-shot FGL as a statistical estimation problem. We propose SPEAR (Statistical Prototype Estimation with Adaptive Reliability), a completely training-free framework that directly computes topology-smoothed class prototypes from local graphs in the original feature space. The server then aggregates these prototypes using a sample-size-adaptive shrinkage estimator that down-weights unreliable local estimates, producing robust global class prototypes. Extensive experiments across seven benchmarks demonstrate that SPEAR consistently achieves state-of-the-art accuracy under extreme heterogeneity. Moreover, SPEAR delivers at least an order-of-magnitude speedup over all baselines, reaching several orders of magnitude against generative and distillation-based methods. Our findings suggest that training-free statistical estimation, rather than local GNN optimization, provides the key to robust and efficient one-shot federated graph learning. The code is available at https://github.com/Yodeesy/SPEAR .

We present a simple Gaussian approximation to the finite-sample distribution of the classical ridge regression estimator. Our approximation captures the fact that, in finite samples, the ridge regression estimator trades off bias and variance to reduce estimation and prediction error. Our approximation is based on nonstandard asymptotics where $i)$ we let the estimator's regularization parameter grow proportionally to the sample size; and $ii)$ we treat the population regression coefficients as local to the reference vector that defines the estimator's direction of shrinkage. In contrast to other asymptotic approximations in the literature, we allow for general forms of heteroskedasticity and autocorrelation in the data generating process (at the cost of considering a low-dimensional model where the number of covariates is not allowed to grow with the sample size). We use our simple Gaussian approximation to propose two new strategies to select the regularization parameter for the ridge regression estimator. The suggested strategies select the regularization parameter to minimize either average or worst-case excess prediction risk, where risk is computed using our suggested Gaussian approximation.

Empirical Bayes (EB) estimators can match the first-order asymptotic risk of maximum likelihood (ML) while behaving very differently at second order: recent excess mean squared error (XMSE) analysis shows that kernel-based EB estimation may be worse than ML when the kernel is poorly aligned with the true parameter. This paper turns that diagnostic into a design principle. We propose an XMSE-aware mixed estimator that interpolates between ML and EB shrinkage. Its fixed-weight XMSE is a scalar quadratic, yielding a closed-form oracle mixing weight that is no worse than both ML and the base EB estimator at the XMSE scale. A plug-in implementation based on finite-sample XMSE approximations is proved consistent, with a second-order oracle regret rate for an interior oracle weight. We further establish a transfer of the regret bound to the fixed-weight risk curve evaluated at the selected weight, a thresholded boundary rule, and extensions to compact kernel families and to finite and growing kernel dictionaries with high-probability oracle bounds. Finite impulse response simulations with SURE-tuned, hard-selection, and trace-corrected baselines, together with the public Silverbox and Cascaded Tanks benchmarks, show that the proposed estimator retains most of the benefit of regularization when it is helpful and retreats toward ML under kernel misspecification, with an identified finite-de analyzed on the benchmarks.

Physical-structure priors such as conservation laws, Hamiltonian forms, and symmetries can improve scientific machine learning when correct, but can degrade predictions when misspecified. Existing methods usually enforce a chosen structure or tune a soft penalty, without a calibrated rule for deciding whether to impose a prior, how strongly to impose it, which prior to use, or which subset of candidate laws holds. We introduce SPADE, Structure-Prior Adaptive Decision Estimation, a closed-form framework that treats this problem as shrinkage of the structure-violating block of an unconstrained estimator. SPADE uses one exact specification test and one estimand: the test decides whether the prior is supported by data, Stein-unbiased James-Stein shrinkage sets the enforcement strength with an $O(σ^2/n)$ oracle guarantee, and a gate commits to the hard prior only when the test certifies it. The same test yields consistent nested structure selection and Benjamini-Hochberg control for subset discovery in non-nested constraint families. Across a linear-subspace prior, a reservoir conservation law, and a nonlinear Hamiltonian prior on Duffing dynamics, SPADE tracks the oracle, beats a neural-network baseline, reduces correct-prior regret from $10.3\%$ to $2.6\%$, matches cross-validation with $1/71$ of the solves, selects the correct structure with $100\%$ accuracy, and recovers partial laws with controlled false relaxation.

We recast classical shrinkage of high-dimensional covariance estimators as empirical risk minimization over a parametric stochastic interpolant between a source and a target distribution. This formalism recovers known shrinkage estimators as special cases and reveals three distinct mechanisms for reducing statistical risk: (i) Scheduling: the interpolant schedule determines the class of admissible covariances, and hence the achievable risk. (ii) Flow maps and couplings: whereas naive constructions amount to assuming independence between the distributions, specific coupling structures (e.g., solutions of optimal transport problems) can lower the empirical risk. Moreover, non-linear flow maps realizing such couplings free the interpolant covariance from the eigenbasis of the empirical estimate, enabling eigenvector regularization. (iii) Early stopping: estimators defined by integrating a regressed vector field afford an additional bias-variance trade-off through approximation of the true interpolant distribution. We then propose a neural estimator of the interpolant, together with an upper bound on its quadratic risk in terms of the interpolant approximation error, and validate both on synthetic experiments. Finally, we apply the estimator to real neuroimaging data, demonstrating the additional regularization power this approach offers in practice.

Self-distillation has emerged as a promising technique for improving model performance in modern machine learning systems. We develop the statistical foundations of self-distillation in spiked covariance models, by introducing and analyzing a broad class of estimators, namely spectral shrinkage estimators. We establish that for spiked covariance matrices with $s$ spikes, $s$-step self-distillation achieves optimal performance among spectral shrinkage estimators, outperforming well-known estimators in statistics and machine learning. Moreover, we show that $s$ steps are necessary for optimality: any $(s-k)$-step distilled estimator is strictly suboptimal for $1 \leq k \leq s$. For the special subclass of isotropic covariances, we show that optimally tuned Ridge regression performs best among spectral shrinkage estimators. We also study a federated approach where multiple data centers share spectral shrinkage estimators and a common server seeks to aggregate them to achieve optimal performance. In this case, we find that the best local rule again takes the form of self-distillation, though it differs from the optimal rule when data are hosted centrally on a single server. Together, our results elucidate why self-distillation improves predictive performance and provide a broader statistical framework connecting it with classical shrinkage-based methods.

We derive the asymptotic risk function of regularized empirical risk minimization (ERM) estimators tuned by $n$-fold cross-validation (CV). The out-of-sample prediction loss of such estimators converges in distribution to the squared-error loss (risk function) of shrinkage estimators in the normal means model, tuned by Stein's unbiased risk estimate (SURE). This risk function provides a more fine-grained picture of predictive performance than uniform bounds on worst-case regret, which are common in learning theory: it quantifies how risk varies with the true parameter. As key intermediate steps, we show that (i) $n$-fold CV converges uniformly to SURE, and (ii) while SURE typically has multiple local minima, its global minimum is generically well separated. Well-separation ensures that uniform convergence of CV to SURE translates into convergence of the tuning parameter chosen by CV to that chosen by SURE.

Heterogeneous treatment effects (HTEs) are increasingly estimated using machine learning models that produce highly personalized predictions of treatment effects. In practice, however, predicted treatment effects are rarely interpreted, reported, or audited at the individual level but, instead, are often aggregated to broader subgroups, such as demographic segments, risk strata, or markets. We show that such aggregation can induce systematic bias of the group-level causal effect: even when models for predicting the individual-level conditional average treatment effect (CATE) are correctly specified and trained on data from randomized experiments, aggregating the predicted CATEs up to the group level does not, in general, recover the corresponding group average treatment effect (GATE). We develop a unified statistical framework to detect and mitigate this form of group bias in randomized experiments. We first define group bias as the discrepancy between the model-implied and experimentally identified GATEs, derive an asymptotically normal estimator, and then provide a simple-to-implement statistical test. For mitigation, we propose a shrinkage-based bias-correction, and show that the theoretically optimal and empirically feasible solutions have closed-form expressions. The framework is fully general, imposes minimal assumptions, and only requires computing sample moments. We analyze the economic implications of mitigating detected group bias for profit-maximizing personalized targeting, thereby characterizing when bias correction alters targeting decisions and profits, and the trade-offs involved. Applications to large-scale experimental data at major digital platforms validate our theoretical results and demonstrate empirical performance.

We endeavour to estimate numerous multi-dimensional means of various probability distributions on a common space based on independent samples. Our approach involves forming estimators through convex combinations of empirical means derived from these samples. We introduce two strategies to find appropriate data-dependent convex combination weights: a first one employing a testing procedure to identify neighbouring means with low variance, which results in a closed-form plug-in formula for the weights, and a second one determining weights via minimization of an upper confidence bound on the quadratic risk. Through theoretical analysis, we evaluate the improvement in quadratic risk offered by our methods compared to the empirical means. Our analysis focuses on a dimensional asymptotics perspective, showing that our methods asymptotically approach an oracle (minimax) improvement as the effective dimension of the data increases. We demonstrate the efficacy of our methods in estimating multiple kernel mean embeddings through experiments on both simulated and real-world datasets.