Yanhao Jin

Yanhao Jin, Krishnakumar Balasubramanian, Debashis Paul

Meta-learning involves training models on a variety of training tasks in a way that enables them to generalize well on new, unseen test tasks. In this work, we consider meta-learning within the framework of high-dimensional multivariate random-effects linear models and study generalized ridge-regression based predictions. The statistical intuition of using generalized ridge regression in this setting is that the covariance structure of the random regression coefficients could be leveraged to make better predictions on new tasks. Accordingly, we first characterize the precise asymptotic behavior of the predictive risk for a new test task when the data dimension grows proportionally to the number of samples per task. We next show that this predictive risk is optimal when the weight matrix in generalized ridge regression is chosen to be the inverse of the covariance matrix of random coefficients. Finally, we propose and analyze an estimator of the inverse covariance matrix of random regression coefficients based on data from the training tasks. As opposed to intractable MLE-type estimators, the proposed estimators could be computed efficiently as they could be obtained by solving (global) geodesically-convex optimization problems. Our analysis and methodology use tools from random matrix theory and Riemannian optimization. Simulation results demonstrate the improved generalization performance of the proposed method on new unseen test tasks within the considered framework.

Yanhao Jin, Krishnakumar Balasubramanian, Lifeng Lai

We investigate the in-context learning capabilities of transformers for the $d$-dimensional mixture of linear regression model, providing theoretical insights into their existence, generalization bounds, and training dynamics. Specifically, we prove that there exists a transformer capable of achieving a prediction error of order $\mathcal{O}(\sqrt{d/n})$ with high probability, where $n$ represents the training prompt size in the high signal-to-noise ratio (SNR) regime. Moreover, we derive in-context excess risk bounds of order $\mathcal{O}(L/\sqrt{B})$ for the case of two mixtures, where $B$ denotes the number of training prompts, and $L$ represents the number of attention layers. The dependence of $L$ on the SNR is explicitly characterized, differing between low and high SNR settings. We further analyze the training dynamics of transformers with single linear self-attention layers, demonstrating that, with appropriately initialized parameters, gradient flow optimization over the population mean square loss converges to a global optimum. Extensive simulations suggest that transformers perform well on this task, potentially outperforming other baselines, such as the Expectation-Maximization algorithm.

Haodong Liang, Yanhao Jin, Krishnakumar Balasubramanian et al.

We study instrumental variable regression (IVaR) under differential privacy constraints. Classical IVaR methods (like two-stage least squares regression) rely on solving moment equations that directly use sensitive covariates and instruments, creating significant risks of privacy leakage and posing challenges in designing algorithms that are both statistically efficient and differentially private. We propose a noisy two-state gradient descent algorithm that ensures $ρ$-zero-concentrated differential privacy by injecting carefully calibrated noise into the gradient updates. Our analysis establishes finite-sample convergence rates for the proposed method, showing that the algorithm achieves consistency while preserving privacy. In particular, we derive precise bounds quantifying the trade-off among privacy parameters, sample size, and iteration-complexity. To the best of our knowledge, this is the first work to provide both privacy guarantees and provable convergence rates for instrumental variable regression in linear models. We further validate our theoretical findings with experiments on both synthetic and real datasets, demonstrating that our method offers practical accuracy-privacy trade-offs.

Yanhao Jin, Tesi Xiao, Krishnakumar Balasubramanian

Statistical machine learning models trained with stochastic gradient algorithms are increasingly being deployed in critical scientific applications. However, computing the stochastic gradient in several such applications is highly expensive or even impossible at times. In such cases, derivative-free or zeroth-order algorithms are used. An important question which has thus far not been addressed sufficiently in the statistical machine learning literature is that of equipping stochastic zeroth-order algorithms with practical yet rigorous inferential capabilities so that we not only have point estimates or predictions but also quantify the associated uncertainty via confidence intervals or sets. Towards this, in this work, we first establish a central limit theorem for Polyak-Ruppert averaged stochastic zeroth-order gradient algorithm. We then provide online estimators of the asymptotic covariance matrix appearing in the central limit theorem, thereby providing a practical procedure for constructing asymptotically valid confidence sets (or intervals) for parameter estimation (or prediction) in the zeroth-order setting.