arXiv subjects

Pratik Patil

Publications and source records attributed to Pratik Patil.

At least 19 recordsLinked to original sources

arXiv2026

Non-asymptotic bounds for the average singular value of a complex Gaussian matrix

Let $G_{d}$ be a $d \times d$ matrix with independent standard complex Gaussian entries, let $\alpha_{\mathbb{C}}(d)$ be the expected average singular value of $G_{d}/\sqrt{d}$, and set $\Delta_d := \alpha_{\mathbb{C}}(d)-\alpha_{\mathbb{C}}(d+1)$. The statistic $\alpha_{\mathbb{C}}(d)$ admits a variational representation as an expected normalized maximum over the unitary group and governs approximation guarantees for the little Grothendieck problem over the unitary group and related unitary registration problems. We obtain a strictly positive lower bound and an upper bound for $\Delta_d$, both valid in every dimension, together with corresponding bounds for $\alpha_{\mathbb{C}}(d)$ around the Marchenko--Pastur limit. These bounds match the sharp leading behavior of complete asymptotic expansions for both quantities, whose coefficients are explicitly computable. The proof combines a three-term recurrence for $Y_d = d^{3/2}\alpha_{\mathbb{C}}(d)$, obtained from its continuous dual Hahn representation, along with singularity analysis of the underlying Laguerre moment generating function.

math.PR↗

arXiv2026

A Quantum Circuit Framework for Protein Ensemble-Level Energetics

Proteins occupy heterogeneous free-energy landscapes in which high-entropy ensembles converge toward compact, low-energy basins with multiple sub-states. Molecular dynamics can access these landscapes at atomic resolution, but exhaustive sampling remains computationally demanding. Meanwhile, most quantum approaches target only single optimal structures, leaving full ensemble energetic heterogeneity unexplored. We introduce a residue-level, gate-based quantum circuit framework for coarse-graining protein thermodynamics. Each amino acid is represented as a two-state qubit (stabilised vs. excited solvation state) based on residue solvation energetics. A structure-informed entanglement block then encodes covalent and non-covalent contacts using parameterised controlled gates, embedding correlations across the residue-interaction network. Sampling the circuit ($\sim 10^6$ measurements) yields binary thermodynamic microstates used to compute protein energy distributions, residue-level statistical couplings, energetic sensitivities, and information gains relative to total free energy. We showcase the framework on the benchmark Trp-cage miniprotein 1L2Y (TC5b) and 9GDL, a disulfide-stabilised Trp-cage-fortified exenatide chimera. For 1L2Y, the circuit reproduces a structured, folding-funnel-like energy distribution. Comparative analysis with 9GDL reveals shifts in global energy distributions and residue-level stability profiles. Coupling and information-theoretic analyses localise residues associated with ensemble reorganisation, while multi-body couplings show the circuit resolves both direct and indirect statistical correlations. This framework expands quantum protein modelling beyond single-structure optimisation toward ensemble-level characterisation, capturing key features of rugged energy landscapes to guide protein design, mutation mapping, and allosteric pathway identification.

cs.ET↗

arXiv2026

Optimal Self-Distillation for Rectified Flow via Linear Probing

Modern generative models are increasingly trained using model-generated signals, creating both opportunities for self-improvement and risks of collapse. We study optimal self-distillation (SD) for rectified flow (RF): given a suboptimal teacher velocity field, can a student trained on a mixture of true RF velocities and teacher velocities provably improve the teacher? For linear RF with ridge regularization on fixed interpolation pairs, we prove an exact affine path identity, derive the optimal mixing coefficient in closed form, and show strict improvement in integrated velocity risk whenever the teacher risk is nonstationary along the regularization path. The optimal coefficient obeys a sign rule: positive mixing corrects under-regularized teachers, while negative mixing corrects over-regularized teachers. We also give one-shot generalized cross-validation (GCV) and validation tuning procedure that avoids grid search over mixing weights and repeated refitting. Combining this theorem with RF Wasserstein convergence bounds, we show that optimal self-distillation improves the velocity estimation terms controlling continuous-time and finite-step generation error. Experiments with Gaussian models, Gaussian mixtures, and image data show that optimal self-distillation improves velocity risk, mode recovery, and finite-step generation relative to both the teacher and pure distillation.

stat.ML↗

arXiv2026

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.

math.ST↗

arXiv2026

Strategic Feature Selection

When algorithmic predictors inform resource allocation in high-stakes domains such as healthcare, these predictors must account for strategic manipulation of input features. The typical solution is to redesign the predictor itself to explicitly account for strategic interactions. In practice, however, decision makers are often constrained to adjusting coarser levers within existing prediction pipelines. For example, healthcare organizations often select which features to exclude based on perceived manipulability, while using standard regularization procedures to shrink the coefficients of retained features. In this work, we initiate a formal study of strategic classification through feature selection and its interaction with ridge regularization. Our main finding is that excluding individual features based on their manipulability alone is generally suboptimal. We provide a fine-grained characterization of the performance of a feature subset under optimal regularization, yielding new insights for policy design. Motivated by this characterization, we develop a practical algorithm for jointly choosing the feature set and the level of ridge regularization. Through a real-world case study on a healthcare payments benchmark, we illustrate how our algorithm can guide the design of coarse policy levers in practice. Our results provide a principled, practical framework for mitigating the effects of strategic behavior in algorithmic decision-making systems.

cs.LG↗

arXiv2026

Evaluating Stochasticity in Deep Research Agents

Deep Research Agents (DRAs) are promising agentic systems that gather and synthesize information to support research across domains such as financial decision-making, medical analysis, and scientific discovery. Despite recent improvements in research quality (e.g., outcome accuracy when ground truth is available), DRA system design often overlooks a critical barrier to real-world deployment: stochasticity. Under identical queries, repeated executions of DRAs can exhibit substantial variability in terms of research outcome, findings, and citations. In this paper, we formalize the study of stochasticity in DRAs by modeling them as information acquisition Markov Decision Processes. We introduce an evaluation framework that quantifies variance in the system and identify three sources of it: information acquisition, information compression, and inference. Through controlled experiments, we investigate how stochasticity from these modules across different decision steps influences the variance of DRA outputs. Our results show that reducing stochasticity can improve research output quality, with inference and early-stage stochasticity contributing the most to DRA output variance. Based on these findings, we propose strategies for mitigating stochasticity while maintaining output quality via structured output and ensemble-based query generation. Our experiments on DeepSearchQA show that our proposed mitigation methods reduce average stochasticity by 22% while maintaining high research quality.

cs.AI↗

arXiv2026

Optimal Unconstrained Self-Distillation in Ridge Regression: Strict Improvements, Precise Asymptotics, and One-Shot Tuning

Self-distillation (SD) is the process of retraining a student on a mixture of ground-truth labels and the teacher's own predictions using the same architecture and training data. Although SD has been empirically shown to often improve generalization, its formal guarantees remain limited. We study SD for ridge regression in unconstrained setting in which the mixing weight $\xi$ may be outside the unit interval. Conditioned on the training data and without any distributional assumptions, we prove that for any squared prediction risk (including out-of-distribution), the optimally mixed student strictly improves upon the ridge teacher for every regularization level $\lambda > 0$ at which the teacher ridge risk $R(\lambda)$ is nonstationary (i.e., $R'(\lambda) \neq 0$). We obtain a closed-form expression for the optimal mixing weight $\xi^\star(\lambda)$ for any value of $\lambda$ and show that it obeys the sign rule: $\operatorname{sign}(\xi^\star(\lambda))=-\operatorname{sign}(R'(\lambda))$. In particular, $\xi^\star(\lambda)$ can be negative, which is the case in over-regularized regimes. To quantify the risk improvement due to SD, we derive exact deterministic equivalents for the optimal SD risk in the proportional asymptotics regime (where the sample and feature sizes $n$ and $p$ both diverge but their aspect ratio $p/n$ converges) under general anisotropic covariance and deterministic signals. Our asymptotic analysis extends standard second-order ridge deterministic equivalents to their fourth-order analogs using block linearization, which may be of independent interest. From a practical standpoint, we propose a consistent one-shot tuning method to estimate $\xi^\star$ without grid search, sample splitting, or refitting. Experiments on real-world datasets and pretrained neural network features support our theory and the one-shot tuning method.

math.ST↗

arXiv2025

Simultaneous Frequentist Calibration of Confidence Regions for Multiple Functionals in Constrained Inverse Problems

Many scientific analyses require simultaneous comparison of multiple functionals of an unknown signal at once, calling for multidimensional confidence regions with guaranteed simultaneous frequentist under structural constraints (e.g., non-negativity, shape, or physics-based). This paper unifies and extends many previous optimization-based approaches to constrained confidence region construction in linear inverse problems through the lens of statistical test inversion. We begin by reviewing the historical development of optimization-based confidence intervals for the single-functional setting, from "strict bounds" to the Burrus conjecture and its recent refutation via the aforementioned test inversion framework. We then extend this framework to the multiple-functional setting. This framework can be used to: (i) improve the calibration constants of previous methods, yielding smaller confidence regions that still preserve frequentist coverage, (ii) obtain tractable multidimensional confidence regions that need not be hyper-rectangles to better capture functional dependence structure, and (iii) generalize beyond Gaussian error distributions to generic log-concave error distributions. We provide theory establishing nominal simultaneous coverage of our methods and show quantitative volume improvements relative to prior approaches using numerical experiments.

math.ST↗

arXiv2025

Confidence intervals for functionals in constrained inverse problems via data-adaptive sampling-based calibration

We address functional uncertainty quantification for ill-posed inverse problems where it is possible to evaluate a possibly rank-deficient forward model, the observation noise distribution is known, and there are known parameter constraints. We present four constraint-aware confidence intervals extending the work of Batlle et al. (2023) by making the intervals both computationally feasible and less conservative. Our approach first shrinks the potentially unbounded constraint set compact in a data-adaptive way, obtains samples of the relevant test statistic inside this set to estimate a quantile function, and then uses these computed quantities to produce the intervals. Our data-adaptive bounding approach is based on the approach by Berger and Boos (1994), and involves defining a subset of the constraint set where the true parameter exists with high probability. This probabilistic guarantee is then incorporated into the final coverage guarantee in the form of an uncertainty budget. We then propose custom sampling algorithms to efficiently sample from this subset, even when the parameter space is high-dimensional. Optimization-based interval methods formulate confidence interval computation as two endpoint optimizations, where the optimization constraints can be set to achieve different types of interval calibration while seamlessly incorporating parameter constraints. However, choosing valid optimization constraints has been elusive. We show that all four proposed intervals achieve nominal coverage for a particular functional both theoretically and in practice, with numerical examples demonstrating superior performance of our intervals over the OSB interval in terms of both coverage and expected length. In particular, we show the superior performance in a realistic unfolding simulation from high-energy physics that is severely ill-posed and involves a rank-deficient forward model.

stat.ME↗

arXiv2024

Precise Model Benchmarking with Only a Few Observations

How can we precisely estimate a large language model's (LLM) accuracy on questions belonging to a specific topic within a larger question-answering dataset? The standard direct estimator, which averages the model's accuracy on the questions in each subgroup, may exhibit high variance for subgroups (topics) with small sample sizes. Synthetic regression modeling, which leverages the model's accuracy on questions about other topics, may yield biased estimates that are too unreliable for large subgroups. We prescribe a simple yet effective solution: an empirical Bayes (EB) estimator that balances direct and regression estimates for each subgroup separately, improving the precision of subgroup-level estimates of model performance. Our experiments on multiple datasets show that this approach consistently provides more precise estimates of the LLM performance compared to the direct and regression approaches, achieving substantial reductions in the mean squared error. Confidence intervals for EB estimates also have near-nominal coverage and are narrower compared to those for the direct estimator. Additional experiments on tabular and vision data validate the benefits of this EB approach.

cs.LG↗

arXiv2024

Revisiting Optimism and Model Complexity in the Wake of Overparameterized Machine Learning

Common practice in modern machine learning involves fitting a large number of parameters relative to the number of observations. These overparameterized models can exhibit surprising generalization behavior, e.g., ``double descent'' in the prediction error curve when plotted against the raw number of model parameters, or another simplistic notion of complexity. In this paper, we revisit model complexity from first principles, by first reinterpreting and then extending the classical statistical concept of (effective) degrees of freedom. Whereas the classical definition is connected to fixed-X prediction error (in which prediction error is defined by averaging over the same, nonrandom covariate points as those used during training), our extension of degrees of freedom is connected to random-X prediction error (in which prediction error is averaged over a new, random sample from the covariate distribution). The random-X setting more naturally embodies modern machine learning problems, where highly complex models, even those complex enough to interpolate the training data, can still lead to desirable generalization performance under appropriate conditions. We demonstrate the utility of our proposed complexity measures through a mix of conceptual arguments, theory, and experiments, and illustrate how they can be used to interpret and compare arbitrary prediction models.

stat.ML↗

arXiv2024

Precise Asymptotics of Bagging Regularized M-estimators

We characterize the squared prediction risk of ensemble estimators obtained through subagging (subsample bootstrap aggregating) regularized M-estimators and construct a consistent estimator for the risk. Specifically, we consider a heterogeneous collection of $M \ge 1$ regularized M-estimators, each trained with (possibly different) subsample sizes, convex differentiable losses, and convex regularizers. We operate under the proportional asymptotics regime, where the sample size $n$, feature size $p$, and subsample sizes $k_m$ for $m \in [M]$ all diverge with fixed limiting ratios $n/p$ and $k_m/n$. Key to our analysis is a new result on the joint asymptotic behavior of correlations between the estimator and residual errors on overlapping subsamples, governed through a (provably) contractive nonlinear system of equations. Of independent interest, we also establish convergence of trace functionals related to degrees of freedom in the non-ensemble setting (with $M = 1$) along the way, extending previously known cases for squared loss with ridge and lasso regularizers. When specialized to homogeneous ensembles trained with a common loss, regularizer, and subsample size, the risk characterization sheds some light on the implicit regularization effect due to the ensemble and subsample sizes $(M,k)$. For any ensemble size $M$, optimally tuning subsample size yields sample-wise monotonic risk. For the full-ensemble estimator (when $M \to \infty$), the optimal subsample size $k^\star$ tends to be in the overparameterized regime $(k^\star \le \min\{n,p\})$, when explicit regularization is vanishing. Finally, joint optimization of subsample size, ensemble size, and regularization can significantly outperform regularizer optimization alone on the full data (without any subagging).

math.ST↗

arXiv2024

Implicit Regularization Paths of Weighted Neural Representations

We study the implicit regularization effects induced by (observation) weighting of pretrained features. For weight and feature matrices of bounded operator norms that are infinitesimally free with respect to (normalized) trace functionals, we derive equivalence paths connecting different weighting matrices and ridge regularization levels. Specifically, we show that ridge estimators trained on weighted features along the same path are asymptotically equivalent when evaluated against test vectors of bounded norms. These paths can be interpreted as matching the effective degrees of freedom of ridge estimators fitted with weighted features. For the special case of subsampling without replacement, our results apply to independently sampled random features and kernel features and confirm recent conjectures (Conjectures 7 and 8) of the authors on the existence of such paths in Patil et al. We also present an additive risk decomposition for ensembles of weighted estimators and show that the risks are equivalent along the paths when the ensemble size goes to infinity. As a practical consequence of the path equivalences, we develop an efficient cross-validation method for tuning and apply it to subsampled pretrained representations across several models (e.g., ResNet-50) and datasets (e.g., CIFAR-100).

cs.LG↗

arXiv2024

A Framework for Efficient Model Evaluation through Stratification, Sampling, and Estimation

Model performance evaluation is a critical and expensive task in machine learning and computer vision. Without clear guidelines, practitioners often estimate model accuracy using a one-time completely random selection of the data. However, by employing tailored sampling and estimation strategies, one can obtain more precise estimates and reduce annotation costs. In this paper, we propose a statistical framework for model evaluation that includes stratification, sampling, and estimation components. We examine the statistical properties of each component and evaluate their efficiency (precision). One key result of our work is that stratification via k-means clustering based on accurate predictions of model performance yields efficient estimators. Our experiments on computer vision datasets show that this method consistently provides more precise accuracy estimates than the traditional simple random sampling, even with substantial efficiency gains of 10x. We also find that model-assisted estimators, which leverage predictions of model accuracy on the unlabeled portion of the dataset, are generally more efficient than the traditional estimates based solely on the labeled data.

cs.CV↗

arXiv2024

Confidence Intervals for Error Rates in 1:1 Matching Tasks: Critical Statistical Analysis and Recommendations

Matching algorithms are commonly used to predict matches between items in a collection. For example, in 1:1 face verification, a matching algorithm predicts whether two face images depict the same person. Accurately assessing the uncertainty of the error rates of such algorithms can be challenging when data are dependent and error rates are low, two aspects that have been often overlooked in the literature. In this work, we review methods for constructing confidence intervals for error rates in 1:1 matching tasks. We derive and examine the statistical properties of these methods, demonstrating how coverage and interval width vary with sample size, error rates, and degree of data dependence on both analysis and experiments with synthetic and real-world datasets. Based on our findings, we provide recommendations for best practices for constructing confidence intervals for error rates in 1:1 matching tasks.

stat.ME↗

arXiv2024

Corrected generalized cross-validation for finite ensembles of penalized estimators

Generalized cross-validation (GCV) is a widely-used method for estimating the squared out-of-sample prediction risk that employs a scalar degrees of freedom adjustment (in a multiplicative sense) to the squared training error. In this paper, we examine the consistency of GCV for estimating the prediction risk of arbitrary ensembles of penalized least-squares estimators. We show that GCV is inconsistent for any finite ensemble of size greater than one. Towards repairing this shortcoming, we identify a correction that involves an additional scalar correction (in an additive sense) based on degrees of freedom adjusted training errors from each ensemble component. The proposed estimator (termed CGCV) maintains the computational advantages of GCV and requires neither sample splitting, model refitting, or out-of-bag risk estimation. The estimator stems from a finer inspection of the ensemble risk decomposition and two intermediate risk estimators for the components in this decomposition. We provide a non-asymptotic analysis of the CGCV and the two intermediate risk estimators for ensembles of convex penalized estimators under Gaussian features and a linear response model. Furthermore, in the special case of ridge regression, we extend the analysis to general feature and response distributions using random matrix theory, which establishes model-free uniform consistency of CGCV.

math.ST↗

arXiv2024

Optimization-based frequentist confidence intervals for functionals in constrained inverse problems: Resolving the Burrus conjecture

We present an optimization-based framework to construct confidence intervals for functionals in constrained inverse problems, ensuring valid one-at-a-time frequentist coverage guarantees. Our approach builds upon the now-called strict bounds intervals, originally pioneered by Burrus (1965) and Rust and Burrus (1972), which offer ways to directly incorporate any side information about the parameters during inference without introducing external biases. This family of methods allows for uncertainty quantification in ill-posed inverse problems without needing to select a regularizing prior. By tying optimization-based intervals to an inversion of a constrained likelihood ratio test, we translate interval coverage guarantees into type I error control and characterize the resulting interval via solutions to optimization problems. Along the way, we refute the Burrus conjecture, which posited that, for possibly rank-deficient linear Gaussian models with positivity constraints, a correction based on the quantile of the chi-squared distribution with one degree of freedom suffices to shorten intervals while maintaining frequentist coverage guarantees. Our framework provides a novel approach to analyzing the conjecture, and we construct a counterexample employing a stochastic dominance argument, which we also use to disprove a general form of the conjecture. We illustrate our framework with several numerical examples and provide directions for extensions beyond the Rust-Burrus method for nonlinear, non-Gaussian settings with general constraints.

math.ST↗

arXiv2024

Optimal Ridge Regularization for Out-of-Distribution Prediction

We study the behavior of optimal ridge regularization and optimal ridge risk for out-of-distribution prediction, where the test distribution deviates arbitrarily from the train distribution. We establish general conditions that determine the sign of the optimal regularization level under covariate and regression shifts. These conditions capture the alignment between the covariance and signal structures in the train and test data and reveal stark differences compared to the in-distribution setting. For example, a negative regularization level can be optimal under covariate shift or regression shift, even when the training features are isotropic or the design is underparameterized. Furthermore, we prove that the optimally-tuned risk is monotonic in the data aspect ratio, even in the out-of-distribution setting and when optimizing over negative regularization levels. In general, our results do not make any modeling assumptions for the train or the test distributions, except for moment bounds, and allow for arbitrary shifts and the widest possible range of (negative) regularization levels.

math.ST↗