In this paper, we study a system in which a sensor forwards status updates to a receiver through an error-prone channel, while the receiver sends the transmission results back to the sensor via a reliable channel. Both channels are subject to random delays. To evaluate the timeliness of the status information at the receiver, we use the Age of Information (AoI) metric. The objective is to design a sampling policy that minimizes the expected time-average AoI, even when the channel statistics (e.g., delay distributions) are unknown. We first review the threshold structure of the optimal offline policy under known channel statistics and then reformulate the design of the online algorithm as a stochastic approximation problem. We propose a Robbins-Monro algorithm to solve this problem and demonstrate that the optimal threshold can be approximated almost surely. Moreover, we prove that the cumulative AoI regret of the online algorithm increases with rate $\mathcal{O}(\ln K)$, where $K$ is the number of successful transmissions. In addition, our algorithm is shown to be minimax order optimal, in the sense that for any online learning algorithm, the cumulative AoI regret up to the $K$-th successful transmissions grows with the rate at least $\Omega(\ln K)$ in the worst case delay distribution. Finally, we improve the stability of the proposed online learning algorithm through a momentum-based stochastic gradient descent algorithm. Simulation results validate the performance of our proposed algorithm.

Motivated by the settings where sensing the entire tensor is infeasible, this paper proposes a novel tensor compressed sensing model, where measurements are only obtained from sensing each lateral slice via mutually independent matrices. Leveraging the low tubal rank structure, we reparameterize the unknown tensor ${\boldsymbol {\mathcal X}}^\star$ using two compact tensor factors and formulate the recovery problem as a nonconvex minimization problem. To solve the problem, we first propose an alternating minimization algorithm, termed \textsf{Alt-PGD-Min}, that iteratively optimizes the two factors using a projected gradient descent and an exact minimization step, respectively. Despite nonconvexity, we prove that \textsf{Alt-PGD-Min} achieves $\epsilon$-accuracy recovery with $\mathcal O\left( \kappa^2 \log \frac{1}{\epsilon}\right)$ iteration complexity and $\mathcal O\left( \kappa^6rn_3\log n_3 \left( \kappa^2r\left(n_1 + n_2 \right) + n_1 \log \frac{1}{\epsilon}\right) \right)$ sample complexity, where $\kappa$ denotes tensor condition number of $\boldsymbol{\mathcal X}^\star$. To further accelerate the convergence, especially when the tensor is ill-conditioned with large $\kappa$, we prove \textsf{Alt-ScalePGD-Min} that preconditions the gradient update using an approximate Hessian that can be computed efficiently. We show that \textsf{Alt-ScalePGD-Min} achieves $\kappa$ independent iteration complexity $\mathcal O(\log \frac{1}{\epsilon})$ and improves the sample complexity to $\mathcal O\left( \kappa^4 rn_3 \log n_3 \left( \kappa^4r(n_1+n_2) + n_1 \log \frac{1}{\epsilon}\right) \right)$. Experiments validate the effectiveness of the proposed methods.

Coordinate Descent (CD) methods have gained significant attention in machine learning due to their effectiveness in solving high-dimensional problems and their ability to decompose complex optimization tasks. However, classical CD methods were neither designed nor analyzed with data privacy in mind, a critical concern when handling sensitive information. This has led to the development of differentially private CD methods, such as DP-CD (Differentially Private Coordinate Descent) proposed by Mangold et al. (ICML 2022), yet a disparity remains between non-private CD and DP-CD methods. In our work, we propose a differentially private random block coordinate descent method that selects multiple coordinates with varying probabilities in each iteration using sketch matrices. Our algorithm generalizes both DP-CD and the classical DP-SGD (Differentially Private Stochastic Gradient Descent), while preserving the same utility guarantees. Furthermore, we demonstrate that better utility can be achieved through importance sampling, as our method takes advantage of the heterogeneity in coordinate-wise smoothness constants, leading to improved convergence rates.

For gradient-based machine learning (ML) methods commonly adopted in practice such as stochastic gradient descent, the de facto differential privacy (DP) technique is perturbing the gradients with random Gaussian noise. Data valuation attributes the ML performance to the training data and is widely used in privacy-aware applications that require enforcing DP such as data pricing, collaborative ML, and federated learning (FL). Can existing data valuation methods still be used when DP is enforced via gradient perturbations? We show that the answer is no with the default approach of injecting i.i.d.~random noise to the gradients because the estimation uncertainty of the data value estimation paradoxically linearly scales with more estimation budget, producing estimates almost like random guesses. To address this issue, we propose to instead inject carefully correlated noise to provably remove the linear scaling of estimation uncertainty w.r.t.~the budget. We also empirically demonstrate that our method gives better data value estimates on various ML tasks and is applicable to use cases including dataset valuation and~FL.

Test-time adaptation (TTA) aims to fine-tune a trained model online using unlabeled testing data to adapt to new environments or out-of-distribution data, demonstrating broad application potential in real-world scenarios. However, in this optimization process, unsupervised learning objectives like entropy minimization frequently encounter noisy learning signals. These signals produce unreliable gradients, which hinder the model ability to converge to an optimal solution quickly and introduce significant instability into the optimization process. In this paper, we seek to resolve these issues from the perspective of optimizer design. Unlike prior TTA using manually designed optimizers like SGD, we employ a learning-to-optimize approach to automatically learn an optimizer, called Meta Gradient Generator (MGG). Specifically, we aim for MGG to effectively utilize historical gradient information during the online optimization process to optimize the current model. To this end, in MGG, we design a lightweight and efficient sequence modeling layer -- gradient memory layer. It exploits a self-supervised reconstruction loss to compress historical gradient information into network parameters, thereby enabling better memorization ability over a long-term adaptation process. We only need a small number of unlabeled samples to pre-train MGG, and then the trained MGG can be deployed to process unseen samples. Promising results on ImageNet-C, R, Sketch, and A indicate that our method surpasses current state-of-the-art methods with fewer updates, less data, and significantly shorter adaptation iterations. Compared with a previous SOTA method SAR, we achieve 7.4% accuracy improvement and 4.2 times faster adaptation speed on ImageNet-C.

We study statistical inverse learning in the context of nonlinear inverse problems under random design. Specifically, we address a class of nonlinear problems by employing gradient descent (GD) and stochastic gradient descent (SGD) with mini-batching, both using constant step sizes. Our analysis derives convergence rates for both algorithms under classical a priori assumptions on the smoothness of the target function. These assumptions are expressed in terms of the integral operator associated with the tangent kernel, as well as through a bound on the effective dimension. Additionally, we establish stopping times that yield minimax-optimal convergence rates within the classical reproducing kernel Hilbert space (RKHS) framework. These results demonstrate the efficacy of GD and SGD in achieving optimal rates for nonlinear inverse problems in random design.

In recent years, the application of deep networks has revolutionized the field of machine learning, particularly in tasks involving complex data such as images and natural language. These models, typically trained using stochastic gradient descent, have demonstrated remarkable performance on various benchmarks, raising questions about the underlying mechanisms that contribute to their success. Despite their practical efficacy, the theoretical understanding of these models remains relatively limited, creating a pressing need for deeper insights into their generalization abilities. The classical theory shows that the latter is a consequence of regularization, which is the way to impose a priori knowledge into the model and to favour "simple" solutions. While usually regularization is achieved either by choosing simple models or explicitly adding a penalty term to the empirical risk during training, this is not the case for deep neural networks, which are trained simply by minimizing the empirical risk. A new perspective has then emerged in the recent literature, which relates regularization directly to the optimization procedure (gradient based methods). The main idea is to show that the training dynamics themselves exhibit self regularizing properties, by inducing an implicit regularization (bias) which prefers generalizing solutions (see [Vardi, 2023] for an extended review of the importance of implicit bias in machine learning). In this context, a common approach is to study simplified models that approximate the networks used in practice. Analyzing the implicit bias of optimization algorithms for such networks is facilitated but still might give some insights on the good performance of neural networks in various scenarios.

As 6G and beyond networks grow increasingly complex and interconnected, federated learning (FL) emerges as an indispensable paradigm for securely and efficiently leveraging decentralized edge data for AI. By virtue of the superposition property of communication signals, over-the-air FL (OtA-FL) achieves constant communication overhead irrespective of the number of edge devices (EDs). However, training neural networks over the air still incurs substantial communication costs, as the number of transmitted symbols equals the number of trainable parameters. To alleviate this issue, the most straightforward approach is to reduce the number of transmitted symbols by 1) gradient compression and 2) gradient sparsification. Unfortunately, these methods are incompatible with OtA-FL due to the loss of its superposition property. In this work, we introduce federated zeroth-order estimation (Fed-ZOE), an efficient framework inspired by the randomized gradient estimator (RGE) commonly used in zeroth-order optimization (ZOO). In FedZOE, EDs perform local weight updates as in standard FL, but instead of transmitting full gradient vectors, they send compressed local model update vectors in the form of several scalar-valued inner products between the local model update vectors and random vectors. These scalar values enable the parameter server (PS) to reconstruct the gradient using the RGE trick with highly reduced overhead, as well as preserving the superposition property. Unlike conventional ZOO leveraging RGE for step-wise gradient descent, Fed-ZOE compresses local model update vectors before transmission, thereby achieving higher accuracy and computational efficiency. Numerical evaluations using ResNet-18 on datasets such as CIFAR-10, TinyImageNet, SVHN, CIFAR-100, and Brain-CT demonstrate that Fed-ZOE achieves performance comparable to Fed-OtA while drastically reducing communication costs.

Training data attribution (TDA) methods aim to attribute model outputs back to specific training examples, and the application of these methods to large language model (LLM) outputs could significantly advance model transparency and data curation. However, it has been challenging to date to apply these methods to the full scale of LLM pretraining. In this paper, we refine existing gradient-based methods to work effectively at scale, allowing us to retrieve influential examples for an 8B-parameter language model from a pretraining corpus of over 160B tokens with no need for subsampling or pre-filtering. Our method combines several techniques, including optimizer state correction, a task-specific Hessian approximation, and normalized encodings, which we find to be critical for performance at scale. In quantitative evaluations on a fact tracing task, our method performs best at identifying examples that influence model predictions, but classical, model-agnostic retrieval methods such as BM25 still perform better at finding passages which explicitly contain relevant facts. These results demonstrate a misalignment between factual *attribution* and causal *influence*. With increasing model size and training tokens, we find that influence more closely aligns with factual attribution. Finally, we examine different types of examples identified as influential by our method, finding that while many directly entail a particular fact, others support the same output by reinforcing priors on relation types, common entities, and names. We release our prompt set and model outputs, along with a web-based visualization tool to explore influential examples for factual predictions, commonsense reasoning, arithmetic, and open-ended generation for an 8B-parameter LLM.

In the context of uncertainty quantification (UQ) the curse of dimensionality, whereby quantification efficiency degrades drastistically with parameter dimension, is particular extreme with highly parameterized models such as neural networks (NNs). Fortunately, in many cases, these models are overparameterized in the sense that the number of parameters can be reduced with negligible effects on accuracy and sometimes improvements in generalization [1]. Furthermore, NNs often have parameterizations that have fungible parameters such that permutations of the values and connections lead to equivalent output responses. This suggests methods that simultaneously sparsify and characterize the uncertainty of a model, while handling and taking advantage of the symmetries inherent in the model, are potentially advantageous approaches. Although Markov chain Monte Carlo (MCMC) methods [2] have been the reference standard to generate samples for UQ methods, they can be temperamental and do not scale well for high dimensional models. More recently, there has been widespread use of variational inference methods (VI), which cast the parameter posterior sampling problem as an optimization of a surrogate posterior guided by a suitable objective, such as the Kullback-Liebler (KL) divergence between the predictive posterior and true posterior induced by the data. In particular, there is now a family of model ensemble methods based on Stein's identity [3], such as Stein variational gradient descent (SVGD) [4], projected SVGD [5], and Stein variational Newton's method [6]. These methods have advantages over MCMC methods by virtue of propagating in parallel a coordinated ensemble of particles that represent the empirical posterior.