We address the problem of detecting a change in the distribution of a high-dimensional multivariate normal time series. Assuming that the post-change parameters are unknown and estimated using a sliding window of historical data, we extend the framework of quickest change detection (QCD) to the high-dimensional setting in which the observation dimension increases proportionally with the number of samples used to estimate the post-change parameters. Our analysis reveals that the high-dimensional performance of QCD procedures is governed by an information-theoretic quantity: the Kullback-Leibler (KL) divergence between the true and estimated post-change distributions. Using random matrix theory, we express this divergence explicitly in terms of the underlying post-change model and a parameterized family of estimators. We further identify estimators that asymptotically minimize this KL divergence, resulting in a method that provably outperforms QCD procedures relying on classical maximum likelihood estimation of the postchange parameters.

In this work, we present a novel approach for compressing overparameterized models, developed through studying their learning dynamics. We observe that for many deep models, updates to the weight matrices occur within a low-dimensional invariant subspace. For deep linear models, we demonstrate that their principal components are fitted incrementally within a small subspace, and use these insights to propose a compression algorithm for deep linear networks that involve decreasing the width of their intermediate layers. We empirically evaluate the effectiveness of our compression technique on matrix recovery problems. Remarkably, by using an initialization that exploits the structure of the problem, we observe that our compressed network converges faster than the original network, consistently yielding smaller recovery errors. We substantiate this observation by developing a theory focused on deep matrix factorization. Finally, we empirically demonstrate how our compressed model has the potential to improve the utility of deep nonlinear models. Overall, our algorithm improves the training efficiency by more than 2x, without compromising generalization.

Diffusion models, emerging as powerful deep generative tools, excel in various applications. They operate through a two-steps process: introducing noise into training samples and then employing a model to convert random noise into new samples (e.g., images). However, their remarkable generative performance is hindered by slow training and sampling. This is due to the necessity of tracking extensive forward and reverse diffusion trajectories, and employing a large model with numerous parameters across multiple timesteps (i.e., noise levels). To tackle these challenges, we present a multi-stage framework inspired by our empirical findings. These observations indicate the advantages of employing distinct parameters tailored to each timestep while retaining universal parameters shared across all time steps. Our approach involves segmenting the time interval into multiple stages where we employ custom multi-decoder U-net architecture that blends time-dependent models with a universally shared encoder. Our framework enables the efficient distribution of computational resources and mitigates inter-stage interference, which substantially improves training efficiency. Extensive numerical experiments affirm the effectiveness of our framework, showcasing significant training and sampling efficiency enhancements on three state-of-the-art diffusion models, including large-scale latent diffusion models. Furthermore, our ablation studies illustrate the impact of two important components in our framework: (i) a novel timestep clustering algorithm for stage division, and (ii) an innovative multi-decoder U-net architecture, seamlessly integrating universal and customized hyperparameters.

We investigate the landscape of the negative log-likelihood function of Gaussian Mixture Models (GMMs) with a general number of components in the population limit. As the objective function is non-convex, there can exist multiple spurious local minima that are not globally optimal, even for well-separated mixture models. Our study reveals that all local minima share a common structure that partially identifies the cluster centers (i.e., means of the Gaussian components) of the true location mixture. Specifically, each local minimum can be represented as a non-overlapping combination of two types of sub-configurations: (1) fitting a single mean estimate to multiple Gaussian components or (2) fitting multiple estimates to a single true component. These results apply to settings where the true mixture components satisfy a certain separation condition, and are valid even when the number of components is over- or under-specified. We also present a more fine-grained analysis for the setting of one-dimensional GMMs with three components, which provide sharper approximation error bounds with improved dependence on the separation parameter.

Lifelong learning (LL) aims to improve a predictive model as the data source evolves continuously. Most work in this learning paradigm has focused on resolving the problem of ‘catastrophic forgetting,’ which refers to a notorious dilemma between improving model accuracy over new data and retaining accuracy over previous data. Yet, it is also known that machine learning (ML) models can be vulnerable in the sense that even tiny, adversarial input perturbations can deceive the models into producing erroneous predictions. This motivates the research objective of this paper – specification of a new LL framework that can salvage model robustness (against adversarial attacks) from catastrophic forgetting. Specifically, we propose a new memory-replay LL strategy that leverages modern bi-level optimization techniques to determine the ‘coreset’ of the current data (i.e., a small amount of data to be memorized) for ease of preserving adversarial robustness over time. We term the resulting LL framework ‘Data-Efficient Robustness-Preserving LL’ (DERPLL). The effectiveness of DERPLL is evaluated for class-incremental image classification using ResNet-18 over the CIFAR-10 dataset. Experimental results show that DERPLL outperforms the conventional coreset-guided LL baseline and achieves a substantial improvement in both standard accuracy and robust accuracy.

Fr\'echet regression has emerged as a promising approach for regression analysis involving non-Euclidean response variables. However, its practical applicability has been hindered by its reliance on ideal scenarios with abundant and noiseless covariate data. In this paper, we present a novel estimation method that tackles these limitations by leveraging the low-rank structure inherent in the covariate matrix. Our proposed framework combines the concepts of global Fr\'echet regression and principal component regression, aiming to improve the efficiency and accuracy of the regression estimator. By incorporating the low-rank structure, our method enables more effective modeling and estimation, particularly in high-dimensional and errors-in-variables regression settings. We provide a theoretical analysis of the proposed estimator's large-sample properties, including a comprehensive rate analysis of bias, variance, and additional variations due to measurement errors. Furthermore, our numerical experiments provide empirical evidence that supports the theoretical findings, demonstrating the superior performance of our approach. Overall, this work introduces a promising framework for regression analysis of non-Euclidean variables, effectively addressing the challenges associated with limited and noisy covariate data, with potential applications in diverse fields.

Recalibrating probabilistic classifiers is vital for enhancing the reliability and accuracy of predictive models. Despite the development of numerous recalibration algorithms, there is still a lack of a comprehensive theory that integrates calibration and sharpness (which is essential for maintaining predictive power). In this paper, we introduce the concept of minimum-risk recalibration within the framework of mean-squared-error (MSE) decomposition, offering a principled approach for evaluating and recalibrating probabilistic classifiers. Using this framework, we analyze the uniform-mass binning (UMB) recalibration method and establish a finite-sample risk upper bound of order $\tilde{O}(B/n + 1/B^2)$ where $B$ is the number of bins and $n$ is the sample size. By balancing calibration and sharpness, we further determine that the optimal number of bins for UMB scales with $n^{1/3}$, resulting in a risk bound of approximately $O(n^{-2/3})$. Additionally, we tackle the challenge of label shift by proposing a two-stage approach that adjusts the recalibration function using limited labeled data from the target domain. Our results show that transferring a calibrated classifier requires significantly fewer target samples compared to recalibrating from scratch. We validate our theoretical findings through numerical simulations, which confirm the tightness of the proposed bounds, the optimal number of bins, and the effectiveness of label shift adaptation.

We consider the question of learning $Q$-function in a sample efficient manner for reinforcement learning with continuous state and action spaces under a generative model. If $Q$-function is Lipschitz continuous, then the minimal sample complexity for estimating $\epsilon$-optimal $Q$-function is known to scale as $\Omega(\frac{1}{\epsilon^{d_1+d_2+2}})$ per classical non-parametric learning theory, where $d_1$ and $d_2$ denote the dimensions of the state and action spaces respectively. The $Q$-function, when viewed as a kernel, induces a Hilbert-Schmidt operator and hence possesses square-summable spectrum. This motivates us to consider a parametric class of $Q$-functions parameterized by its "rank" $r$, which contains all Lipschitz $Q$-functions as $r\to\infty$. As our key contribution, we develop a simple, iterative learning algorithm that finds $\epsilon$-optimal $Q$-function with sample complexity of $\widetilde{O}(\frac{1}{\epsilon^{\max(d_1, d_2)+2}})$ when the optimal $Q$-function has low rank $r$ and the discounting factor $\gamma$ is below a certain threshold. Thus, this provides an exponential improvement in sample complexity. To enable our result, we develop a novel Matrix Estimation algorithm that faithfully estimates an unknown low-rank matrix in the $\ell_\infty$ sense even in the presence of arbitrary bounded noise, which might be of interest in its own right. Empirical results on several stochastic control tasks confirm the efficacy of our "low-rank" algorithms.

We consider the setup of nonparametric blind regression for estimating the entries of a large m × n matrix, when provided with a small, random fraction of noisy measurements. We assume that all rows u ∈ [m] and columns i ∈ [n] of the matrix are associated to latent features xrow(u) and xcol(i) respectively, and the (u, i)-th entry of the matrix, A(u, i) is equal to f(xrow(u), xcol(i)) for a latent functionf. Given noisy observations of a small, random subset of the matrix entries, our goal is to estimate the unobserved entries of the matrix as well as to “denoise” the observed entries. As the main result of this work, we introduce a nearest-neighbor-based estimation algorithm, and establish its consistency when the underlying latent function f is Lipschitz, the underlying latent space is a bounded diameter Polish space, and the random fraction of observed entries in the matrix is at least max (m-1+δ, n-1/2+δ), for any δ > 0. As an important byproduct, our analysis sheds light into the performance of the classical collaborative filtering algorithm for matrix completion, which has been widely utilized in practice. Experiments with the MovieLens and Netflix datasets suggest that our algorithm provides a principled improvement over basic collaborative filtering and is competitive with matrix factorization methods. Our algorithm has a natural extension to the setting of tensor completion via flattening the tensor to matrix. When applied to the setting of image in-painting, which is a 3-order tensor, we find that our approach is competitive with respect to state-of-art tensor completion algorithms across benchmark images.

Consider the setting of Linear Regression where the observed response variables, in expectation, are linear functions of the p-dimensional covariates. Then to achieve vanishing prediction error, the number of required samples scales faster than pσ2, where σ2 is a bound on the noise variance. In a high-dimensional setting where p is large but the covariates admit a low-dimensional representation (say r ≪ p), then Principal Component Regression (PCR), cf. [36], is an effective approach; here, the response variables are regressed with respect to the principal components of the covariates. The resulting number of required samples to achieve vanishing prediction error now scales faster than rσ2(≪ pσ2). Despite the tremendous utility of PCR, its ability to handle settings with noisy, missing, and mixed (discrete and continuous) valued covariates is not understood and remains an important open challenge, cf. [24]. As the main contribution of this work, we address this challenge by rigorously establishing that PCR is robust to noisy, sparse, and possibly mixed valued covariates. Specifically, under PCR, vanishing prediction error is achieved with the number of samples scaling as r max(σ2, ρ−4 log5(p)), where ρ denotes the fraction of observed (noisy) covariates. We establish generalization error bounds on the performance of PCR, which provides a systematic approach in selecting the correct number of components r in a data-driven manner. The key to our result is a simple, but powerful equivalence between (i) PCR and (ii) Linear Regression with covariate pre-processing via Hard Singular Value Thresholding (HSVT). From a technical standpoint, this work advances the state-of-the-art analysis for HSVT by establishing stronger guarantees with respect to the ∥·∥2,∞-error for the estimated matrix rather than the Frobenius norm/mean-squared error (MSE) as is commonly done in the matrix estimation / completion literature.

We introduce the framework of {\em blind regression} motivated by {\em matrix completion} for recommendation systems: given $m$ users, $n$ movies, and a subset of user-movie ratings, the goal is to predict the unobserved user-movie ratings given the data, i.e., to complete the partially observed matrix. Following the framework of non-parametric statistics, we posit that user $u$ and movie $i$ have features $x_1(u)$ and $x_2(i)$ respectively, and their corresponding rating $y(u,i)$ is a noisy measurement of $f(x_1(u), x_2(i))$ for some unknown function $f$. In contrast with classical regression, the features $x = (x_1(u), x_2(i))$ are not observed, making it challenging to apply standard regression methods to predict the unobserved ratings. Inspired by the classical Taylor's expansion for differentiable functions, we provide a prediction algorithm that is consistent for all Lipschitz functions. In fact, the analysis through our framework naturally leads to a variant of collaborative filtering, shedding insight into the widespread success of collaborative filtering in practice. Assuming each entry is sampled independently with probability at least $\max(m^{-1+\delta},n^{-1/2+\delta})$ with $\delta > 0$, we prove that the expected fraction of our estimates with error greater than $\epsilon$ is less than $\gamma^2 / \epsilon^2$ plus a polynomially decaying term, where $\gamma^2$ is the variance of the additive entry-wise noise term. Experiments with the MovieLens and Netflix datasets suggest that our algorithm provides principled improvements over basic collaborative filtering and is competitive with matrix factorization methods.