In recent years, *task arithmetic* has garnered increasing attention. This approach edits pre-trained models directly in weight space by combining the fine-tuned weights of various tasks into a *unified model*. Its efficiency and cost-effectiveness stem from its training-free combination, contrasting with traditional methods that require model training on large datasets for multiple tasks. However, applying such a unified model to individual tasks can lead to interference from other tasks (lack of *weight disentanglement*). To address this issue, Neural Tangent Kernel (NTK) linearization has been employed to leverage a ''kernel behavior'', facilitating weight disentanglement and mitigating adverse effects from unrelated tasks. Despite its benefits, NTK linearization presents drawbacks, including doubled training costs, as well as reduced performance of individual models. To tackle this problem, we propose a simple yet effective and efficient method that is to finetune the attention modules only in the Transformer. Our study reveals that the attention modules exhibit kernel behavior, and fine-tuning the attention modules only significantly improves weight disentanglement. To further understand how our method improves the weight disentanglement of task arithmetic, we present a comprehensive study of task arithmetic by differentiating the role of the representation module and task-specific module. In particular, we find that the representation module plays an important role in improving weight disentanglement whereas the task-specific modules such as the classification heads can degenerate the weight disentanglement performance.

Self-play methods have demonstrated remarkable success in enhancing model capabilities across various domains. In the context of Reinforcement Learning from Human Feedback (RLHF), self-play not only boosts Large Language Model (LLM) performance but also overcomes the limitations of traditional Bradley-Terry (BT) model assumptions by finding the Nash equilibrium (NE) of a preference-based, two-player constant-sum game. However, existing methods either guarantee only average-iterate convergence, incurring high storage and inference costs, or converge to the NE of a regularized game, failing to accurately reflect true human preferences. In this paper, we introduce Magnetic Preference Optimization (MPO), a novel approach capable of achieving last-iterate convergence to the NE of the original game, effectively overcoming the limitations of existing methods. Building upon Magnetic Mirror Descent (MMD), MPO attains a linear convergence rate, making it particularly suitable for fine-tuning LLMs. To ensure our algorithm is both theoretically sound and practically viable, we present a simple yet effective implementation that adapts the theoretical insights to the RLHF setting. Empirical results demonstrate that MPO can significantly enhance the performance of LLMs, highlighting the potential of self-play methods in alignment.

One of the key technologies for the success of Large Language Models (LLMs) is preference alignment. However, a notable side effect of preference alignment is poor calibration: while the pre-trained models are typically well-calibrated, LLMs tend to become poorly calibrated after alignment with human preferences. In this paper, we investigate why preference alignment affects calibration and how to address this issue. For the first question, we observe that the preference collapse issue in alignment undesirably generalizes to the calibration scenario, causing LLMs to exhibit overconfidence and poor calibration. To address this, we demonstrate the importance of fine-tuning with domain-specific knowledge to alleviate the overconfidence issue. To further analyze whether this affects the model’s performance, we categorize models into two regimes: calibratable and non-calibratable, defined by bounds of Expected Calibration Error (ECE). In the calibratable regime, we propose a calibration-aware fine-tuning approach to achieve proper calibration without compromising LLMs’ performance. However, as models are further fine-tuned for better performance, they enter the non-calibratable regime. For this case, we develop an EM-algorithm-based ECE regularization for the fine-tuning loss to maintain low calibration error. Extensive experiments validate the effectiveness of the proposed methods.

Bowen Jiang, Yangxinyu Xie, Xiaomeng Wang, Yuan Yuan, Zhuoqun Hao, Xinyi Bai, Weijie J Su, Camillo Jose Taylor, Tanwi Mallick. Proceedings of the 2025 Conference of the Nations of the Americas Chapter of the Association for Computational Linguistics: Human Language Technologies (Volume 1: Long Papers). 2025.

Large language models (LLMs) raise concerns about content authenticity and integrity because they can generate human-like text at scale. Text watermarks, which embed detectable statistical signals into generated text, offer a provable way to verify content origin. Many detection methods rely on pivotal statistics that are i.i.d. under human-written text, making goodness-of-fit (GoF) tests a natural tool for watermark detection. However, GoF tests remain largely underexplored in this setting. In this paper, we systematically evaluate eight GoF tests across three popular watermarking schemes, using three open-source LLMs, two datasets, various generation temperatures, and multiple post-editing methods. We find that general GoF tests can improve both the detection power and robustness of watermark detectors. Notably, we observe that text repetition, common in low-temperature settings, gives GoF tests a unique advantage not exploited by existing methods. Our results highlight that classic GoF tests are a simple yet powerful and underused tool for watermark detection in LLMs.

Differentially private (DP) decentralized Federated Learning (FL) allows local users to collaborate without sharing their data with a central server. However, accurately quantifying the privacy budget of private FL algorithms is challenging due to the co-existence of complex algorithmic components such as decentralized communication and local updates. This paper addresses privacy accounting for two decentralized FL algorithms within the $f$-differential privacy ($f$-DP) framework. We develop two new $f$-DP–based accounting methods tailored to decentralized settings: Pairwise Network $f$-DP (PN-$f$-DP), which quantifies privacy leakage between user pairs under random-walk communication, and Secret-based $f$-Local DP (Sec-$f$-LDP), which supports structured noise injection via shared secrets. By combining tools from $f$-DP theory and Markov chain concentration, our accounting framework captures privacy amplification arising from sparse communication, local iterations, and correlated noise. Experiments on synthetic and real datasets demonstrate that our methods yield consistently tighter $(\epsilon, \delta)$ bounds and improved utility compared to Rényi DP–based approaches, illustrating the benefits of $f$-DP in decentralized privacy accounting.

The problem of learning one task using samples from another task is central to transfer learning. In this paper, we focus on answering the following question: when does combining the samples from two related tasks perform better than learning with one target task alone? This question is motivated by an empirical phenomenon known as negative transfer often observed in transfer learning practice. While the transfer effect from one task to another depends on factors such as their sample sizes and the spectrum of their covariance matrices, precisely quantifying this dependence has remained a challenging problem. In order to compare a transfer learning estimator to single-task learning, one needs to compare the risks between the two estimators precisely. Further, the comparison depends on the distribution shifts between the two tasks. This paper applies recent developments of random matrix theory to tackle this challenge in a high-dimensional linear regression setting with two tasks. We provide precise high-dimensional asymptotics for the bias and variance of a classical hard parameter sharing (HPS) estimator in the proportional limit, when the sample sizes of both tasks increase proportionally with dimension at fixed ratios. The precise asymptotics apply to various types of distribution shifts, including covariate shifts, model shifts, and combinations of both. We illustrate these results in a random-effects model to mathematically prove a phase transition from positive to negative transfer as the number of source task samples increases. One insight from the analysis is that a rebalanced HPS estimator, which downsizes the source task when the model shift is high, achieves the minimax optimal rate. The finding regarding phase transition also applies to multiple tasks when feature covariates are shared across all tasks. Simulations validate the accuracy of the high-dimensional asymptotics for finite dimensions.

In the setting of conference peer review, the conference aims to accept high-quality papers and reject low-quality papers based on noisy review scores. A recent work proposes the isotonic mechanism, which can elicit the ranking of paper qualities from an author with multiple submissions to help improve the conference's decisions. However, the isotonic mechanism relies on the assumption that the author's utility is both an increasing and a convex function with respect to the review score, which is often violated in realistic settings (e.g.~when authors aim to maximize the number of accepted papers). In this paper, we propose a sequential review mechanism that can truthfully elicit the ranking information from authors while only assuming the agent's utility is increasing with respect to the true quality of her accepted papers. The key idea is to review the papers of an author in a sequence based on the provided ranking and conditioning the review of the next paper on the review scores of the previous papers. Advantages of the sequential review mechanism include: 1) eliciting truthful ranking information in a more realistic setting than prior work; 2) reducing the reviewing workload and increasing the average quality of papers being reviewed; 3) incentivizing authors to write fewer papers of higher quality.

Training Deep Neural Networks (DNNs) with adversarial examples often results in poor generalization to test-time adversarial data. This paper investigates this issue, known as adversarially robust generalization, through the lens of Rademacher complexity. Building upon the studies by Khim and Loh (2018); Yin et al. (2019), numerous works have been dedicated to this problem, yet achieving a satisfactory bound remains an elusive goal. Existing works on DNNs either apply to a surrogate loss instead of the robust loss or yield bounds that are notably looser compared to their standard counterparts. In the latter case, the bounds have a higher dependency on the width $m$ of the DNNs or the dimension $d$ of the data, with an extra factor of at least $\mathcal{O}(\sqrt{m})$ or $\mathcal{O}(\sqrt{d})$. This paper presents upper bounds for adversarial Rademacher complexity of DNNs that match the best-known upper bounds in standard settings, as established in the work of Bartlett et al. (2017), with the dependency on width and dimension being $\mathcal{O}(\ln(dm))$. The central challenge addressed is calculating the covering number of adversarial function classes. We aim to construct a new cover that possesses two properties: 1) compatibility with adversarial examples, and 2) precision comparable to covers used in standard settings. To this end, we introduce a new variant of covering number called the \emph{uniform covering number}, specifically designed and proven to reconcile these two properties. Consequently, our method effectively bridges the gap between Rademacher complexity in robust and standard generalization.

Bowen Jiang, Yangxinyu Xie, Zhuoqun Hao, Xiaomeng Wang, Tanwi Mallick, Weijie J Su, Camillo Jose Taylor, Dan Roth. Proceedings of the 2024 Conference on Empirical Methods in Natural Language Processing. 2024.

Noisy gradient descent and its variants are the predominant algorithms for differentially private machine learning. It is a fundamental question to quantify their privacy leakage, yet tight characterizations remain open even in the foundational setting of convex losses. This paper improves over previous analyses by establishing (and refining) the “privacy amplification by iteration” phenomenon in the unifying framework of $f$-differential privacy---which tightly captures all aspects of the privacy loss and immediately implies tighter privacy accounting in other notions of differential privacy, e.g., $(\varepsilon,\delta)$-DP and Rényi DP. Our key technical insight is the construction of *shifted interpolated processes* that unravel the popular shifted-divergences argument, enabling generalizations beyond divergence-based relaxations of DP. Notably, this leads to the first *exact* privacy analysis in the foundational setting of strongly convex optimization. Our techniques extend to many settings: convex/strongly convex, constrained/unconstrained, full/cyclic/stochastic batches, and all combinations thereof. As an immediate corollary, we recover the $f$-DP characterization of the exponential mechanism for strongly convex optimization in Gopi et al. (2022), and moreover extend this result to more general settings.

A recent study by De et al. (2022) shows that large-scale representation learning through pre-training on a public dataset significantly enhances differentially private (DP) learning in downstream tasks. To explain this, we consider a layer-peeled model in representation learning, resulting in Neural Collapse (NC) phenomena. Within NC, we establish that the misclassification error is independent of dimension when the distance between actual and ideal features is below a threshold. We empirically evaluate feature quality in the last layer under different pre-trained models, showing that a more powerful pre-trained model improves feature representation. Moreover, we show that DP fine-tuning is less robust compared to non-DP fine-tuning, especially with perturbations. Supported by theoretical analyses and experiments, we suggest strategies like feature normalization and dimension reduction methods such as PCA to enhance DP fine-tuning robustness. Conducting PCA on last-layer features significantly improves testing accuracy.

In this paper, we study the implicit regularization of stochastic gradient descent (SGD) through the lens of dynamical stability (Wu et al., 2018). We start by revising existing stability analyses of SGD, showing how the Frobenius norm and trace of Hessian relate to different notions of stability. Notably, if a global minimum is linearly stable for SGD, then the trace of Hessian must be less than or equal to $2/\eta$, where $\eta$ denotes the learning rate. By contrast, for gradient descent (GD), the stability imposes a similar constraint but only on the largest eigenvalue of Hessian. We then turn to analyze the generalization properties of these stable minima, focusing specifically on two-layer ReLU networks and diagonal linear networks. Notably, we establish the equivalence between these metrics of sharpness and certain parameter norms for the two models, which allows us to show that the stable minima of SGD provably generalize well. By contrast, the stability-induced regularization of GD is provably too weak to ensure satisfactory generalization. This discrepancy provides an explanation of why SGD often generalizes better than GD. Note that the learning rate (LR) plays a pivotal role in the strength of stability-induced regularization. As the LR increases, the regularization effect becomes more pronounced, elucidating why SGD with a larger LR consistently demonstrates superior generalization capabilities. Additionally, numerical experiments are provided to support our theoretical findings.

Hyperparameter optimization, also known as hyperparameter tuning, is a widely recognized technique for improving model performance. Regrettably, when training private ML models, many practitioners often overlook the privacy risks associated with hyperparameter optimization, which could potentially expose sensitive information about the underlying dataset. Currently, the sole existing approach to allow privacy-preserving hyperparameter optimization is to uniformly and randomly select hyperparameters for a number of runs, subsequently reporting the best-performing hyperparameter. In contrast, in non-private settings, practitioners commonly utilize "adaptive" hyperparameter optimization methods such as Gaussian Process-based optimization, which select the next candidate based on information gathered from previous outputs. This substantial contrast between private and non-private hyperparameter optimization underscores a critical concern. In our paper, we introduce DP-HyPO, a pioneering framework for "adaptive" private hyperparameter optimization, aiming to bridge the gap between private and non-private hyperparameter optimization. To accomplish this, we provide a comprehensive differential privacy analysis of our framework. Furthermore, we empirically demonstrate the effectiveness of DP-HyPO on a diverse set of real-world datasets.

Differentially private (DP) machine learning algorithms incur many sources of randomness, such as random initialization, random batch subsampling, and shuffling. However, such randomness is difficult to take into account when proving differential privacy bounds because it induces mixture distributions for the algorithm's output that are difficult to analyze. This paper focuses on improving privacy bounds for shuffling models and one-iteration differentially private gradient descent (DP-GD) with random initializations using $f$-DP. We derive a closed-form expression of the trade-off function for shuffling models that outperforms the most up-to-date results based on $(\epsilon,\delta)$-DP. Moreover, we investigate the effects of random initialization on the privacy of one-iteration DP-GD. Our numerical computations of the trade-off function indicate that random initialization can enhance the privacy of DP-GD. Our analysis of $f$-DP guarantees for these mixture mechanisms relies on an inequality for trade-off functions introduced in this paper. This inequality implies the joint convexity of $F$-divergences. Finally, we study an $f$-DP analog of the advanced joint convexity of the hockey-stick divergence related to $(\epsilon,\delta)$-DP and apply it to analyze the privacy of mixture mechanisms.

A widely recognized difficulty in federated learning arises from the statistical heterogeneity among clients: local datasets often originate from distinct yet not entirely unrelated probability distributions, and personalization is, therefore, necessary to achieve optimal results from each individual’s perspective. In this paper, we show how the excess risks of personalized federated learning using a smooth, strongly convex loss depend on data heterogeneity from a minimax point of view, with a focus on the FedAvg algorithm (McMahan et al., 2017) and pure local training (i.e., clients solve empirical risk minimization problems on their local datasets without any communication). Our main result reveals an approximate alternative between these two baseline algorithms for federated learning: the former algorithm is minimax rate optimal over a collection of instances when data heterogeneity is small, whereas the latter is minimax rate optimal when data heterogeneity is large, and the threshold is sharp up to a constant. As an implication, our results show that from a worst-case point of view, a dichotomous strategy that makes a choice between the two baseline algorithms is rate-optimal. Another implication is that the popular FedAvg following by local fine tuning strategy is also minimax optimal under additional regularity conditions. Our analysis relies on a new notion of algorithmic stability that takes into account the nature of federated learning.

Understanding the iterative behavior of stochastic optimization algorithms for minimizing nonconvex functions remains a crucial challenge in demystifying deep learning. In particular, it is not yet understood why certain simple techniques are remarkably effective for tuning the learning rate in stochastic gradient descent (SGD), arguably the most basic optimizer for training deep neural networks. This class of techniques includes learning rate decay, which begins with a large initial learning rate and is gradually reduced. In this paper, we present a general theoretical analysis of the effect of the learning rate in SGD. Our analysis is based on the use of a learning-rate-dependent stochastic differential equation (LR-dependent SDE) as a tool that allows us to set SGD distinctively apart from both gradient descent and stochastic gradient Langevin dynamics (SGLD). In contrast to prior research, our analysis builds on the analysis of a partial differential equation that models the evolution of probability densities, drawing insights from Wainwright and Jordan (2006); Jordan (2018). From this perspective, we derive the linear convergence rate of the probability densities, highlighting its dependence on the learning rate. Moreover, we obtain an explicit expression for the optimal linear rate by analyzing the spectrum of the Witten-Laplacian, a special case of the Schrödinger operator associated with the LR-dependent SDE. This expression clearly reveals the dependence of the linear convergence rate on the learning rate—the linear rate decreases rapidly to zero as the learning rate tends to zero for a broad class of nonconvex functions, whereas it stays constant for strongly convex functions. Based on this sharp distinction between nonconvex and convex problems, we provide a mathematical interpretation of the benefits of using learning rate decay for nonconvex optimization.

Stochastic gradient descent (SGD) is an immensely popular approach for online learning in settings where data arrives in a stream or data sizes are very large. However, despite an ever-increasing volume of work on SGD, much less is known about the statistical inferential properties of SGD-based predictions. Taking a fully inferential viewpoint, this paper introduces a novel procedure termed HiGrad to conduct statistical inference for online learning, without incurring additional computational cost compared with SGD. The HiGrad procedure begins by performing SGD updates for a while and then splits the single thread into several threads, and this procedure hierarchically operates in this fashion along each thread. With predictions provided by multiple threads in place, a $t$-based confidence interval is constructed by decorrelating predictions using covariance structures given by a Donsker-style extension of the Ruppert--Polyak averaging scheme, which is a technical contribution of independent interest. Under certain regularity conditions, the HiGrad confidence interval is shown to attain asymptotically exact coverage probability. Finally, the performance of HiGrad is evaluated through extensive simulation studies and a real data example. An R package \texttt{higrad} has been developed to implement the method.

Commonsense causality reasoning (CCR) aims at identifying plausible causes and effects in natural language descriptions that are deemed reasonable by an average person. Although being of great academic and practical interest, this problem is still shadowed by the lack of a well-posed theoretical framework; existing work usually relies on deep language models wholeheartedly, and is potentially susceptible to confounding co-occurrences. Motivated by classical causal principles, we articulate the central question of CCR and draw parallels between human subjects in observational studies and natural languages to adopt CCR to the potential-outcomes framework, which is the first such attempt for commonsense tasks. We propose a novel framework, ROCK, to Reason O(A)bout Commonsense K(C)ausality, which utilizes temporal signals as incidental supervision, and balances confounding effects using temporal propensities that are analogous to propensity scores. The ROCK implementation is modular and zero-shot, and demonstrates good CCR capabilities.

In high-dimensional sparse regression, would increasing the signal-to-noise ratio while fixing the sparsity level always lead to better model selection? For high-dimensional sparse regression problems, surprisingly, in this paper we answer this question in thenegativein the regime of linear sparsity for the Lasso method, relying on a new concept we termeffect size heterogeneity. Roughly speaking, a regression coefficient vector has high effect size heterogeneity if its nonzero entries have significantly different magnitudes. From the viewpoint of this new measure, we prove that the false and true positive rates achieve the optimal trade-offuniformlyalong the Lasso path when this measure is maximal in a certain sense, and the worst trade-off is achieved when it is minimal in the sense that all nonzero effect sizes are roughly equal. Moreover, we demonstrate that the first false selection occurs much earlier when effect size heterogeneity is minimal than when it is maximal. The underlying cause of these two phenomena is, metaphorically speaking, the “competition” among variables with effect sizes of the same magnitude in entering the model. Taken together, our findings suggest that effect size heterogeneity shall serve as an important complementary measure to the sparsity of regression coefficients in the analysis of high-dimensional regression problems. Our proofs use techniques from approximate message passing theory as well as a novel technique for estimating the rank of the first false variable.

Federated learning (FL) is a training paradigm where the clients collaboratively learn models by repeatedly sharing information without compromising much on the privacy of their local sensitive data. In this paper, we introduce \emph{federated $f$-differential privacy}, a new notion specifically tailored to the federated setting, based on the framework of Gaussian differential privacy. Federated $f$-differential privacy operates on \emph{record level}: it provides the privacy guarantee on each individual record of one client’s data against adversaries. We then propose a generic private federated learning framework \fedsync that accommodates a large family of state-of-the-art FL algorithms, which provably achieves {federated $f$-differential privacy}. Finally, we empirically demonstrate the trade-off between privacy guarantee and prediction performance for models trained by \fedsync in computer vision tasks.

Algorithmic stability is a key characteristic to ensure the generalization ability of a learning algorithm. Among different notions of stability, \emph{uniform stability} is arguably the most popular one, which yields exponential generalization bounds. However, uniform stability only considers the worst-case loss change (or so-called sensitivity) by removing a single data point, which is distribution-independent and therefore undesirable. There are many cases that the worst-case sensitivity of the loss is much larger than the average sensitivity taken over the single data point that is removed, especially in some advanced models such as random feature models or neural networks. Many previous works try to mitigate the distribution independent issue by proposing weaker notions of stability, however, they either only yield polynomial bounds or the bounds derived do not vanish as sample size goes to infinity. Given that, we propose \emph{locally elastic stability} as a weaker and distribution-dependent stability notion, which still yields exponential generalization bounds. We further demonstrate that locally elastic stability implies tighter generalization bounds than those derived based on uniform stability in many situations by revisiting the examples of bounded support vector machines, regularized least square regressions, and stochastic gradient descent.

Being able to efficiently and accurately select the top-$k$ elements with differential privacy is an integral component of various private data analysis tasks. In this paper, we present the oneshot Laplace mechanism, which generalizes the well-known Report Noisy Max \cite{dwork2014algorithmic} mechanism to reporting noisy top-$k$ elements. We show that the oneshot Laplace mechanism with a noise level of $\widetilde{O}(\sqrt{k}/\eps)$ is approximately differentially private. Compared to the previous peeling approach of running Report Noisy Max $k$ times, the oneshot Laplace mechanism only adds noises and computes the top $k$ elements once, hence much more efficient for large $k$. In addition, our proof of privacy relies on a novel coupling technique that bypasses the composition theorems so without the linear dependence on $k$ which is inherent to various composition theorems. Finally, we present a novel application of efficient top-$k$ selection in the classical problem of ranking from pairwise comparisons.

Perhaps the single most important use case for differential privacy is to privately answer numerical queries, which is usually achieved by adding noise to the answer vector. The central question is, therefore, to understand which noise distribution optimizes the privacy-accuracy trade-off, especially when the dimension of the answer vector is high. Accordingly, an extensive literature has been dedicated to the question and the upper and lower bounds have been successfully matched up to constant factors (Bun et al.,2018; Steinke & Ullman, 2017). In this paper, we take a novel approach to address this important optimality question. We first demonstrate an intriguing central limit theorem phenomenon in the high-dimensional regime. More precisely, we prove that a mechanism is approximately Gaussian Differentially Private (Dong et al., 2021) if the added noise satisfies certain conditions. In particular, densities proportional to $\mathrm{e}^{-\|x\|_p^\alpha}$, where $\|x\|_p$ is the standard $\ell_p$-norm, satisfies the conditions. Taking this perspective, we make use of the Cramer--Rao inequality and show an "uncertainty principle"-style result: the product of privacy parameter and the $\ell_2$-loss of the mechanism is lower bounded by the dimension. Furthermore, the Gaussian mechanism achieves the constant-sharp optimal privacy-accuracy trade-off among all such noises. Our findings are corroborated by numerical experiments.

I consider the setting where reviewers offer very noisy scores for a number of items for the selection of high-quality ones (e.g., peer review of large conference proceedings) whereas the owner of these items knows the true underlying scores but prefers not to provide this information. To address this withholding of information, in this paper, I introduce the Isotonic Mechanism, a simple and efficient approach to improving on the imprecise raw scores by leveraging certain information that the owner is incentivized to provide. This mechanism takes as input the ranking of the items from best to worst provided by the owner, in addition to the raw scores provided by the reviewers. It reports adjusted scores for the items by solving a convex optimization problem. Under certain conditions, I show that the owner's optimal strategy is to honestly report the true ranking of the items to her best knowledge in order to maximize the expected utility. Moreover, I prove that the adjusted scores provided by this owner-assisted mechanism are indeed significantly moreaccurate than the raw scores provided by the reviewers. This paper concludes with several extensions of the Isotonic Mechanism and some refinements of the mechanism for practical considerations.

Understanding the training dynamics of deep learning models is perhaps a necessary step toward demystifying the effectiveness of these models. In particular, how do training data from different classes gradually become separable in their feature spaces when training neural networks using stochastic gradient descent? In this paper, we model the evolution of features during deep learning training using a set of stochastic differential equations (SDEs) that each corresponding to a training sample. As a crucial ingredient in our modeling strategy, each SDE contains a drift term that reflects the impact of backpropagation at an input on the features of all samples. Our main finding uncovers a sharp phase transition phenomenon regarding the intra-class impact: if the SDEs are locally elastic in the sense that the impact is more significant on samples from the same class as the input, the features of training data become linearly separable---meaning vanishing training loss; otherwise, the features are not separable, no matter how long the training time is. In the presence of local elasticity, moreover, an analysis of our SDEs shows the emergence of a simple geometric structure called neural collapse of the features. Taken together, our results shed light on the decisive role of local elasticity underlying the training dynamics of neural networks. We corroborate our theoretical analysis with experiments on a synthesized dataset of geometric shapes as well as on CIFAR-10.

SLOPE is a relatively new convex optimization procedure for high-dimensional linear regression via the sorted ℓ1penalty: the larger the rank of the fitted coefficient, the larger the penalty. This non-separable penalty renders many existing techniques invalid or inconclusive in analyzing the SLOPE solution. In this paper, we develop an asymptotically exact characterization of the SLOPE solution under Gaussian random designs through solving the SLOPE problem using approximate message passing (AMP). This algorithmic approach allows us to approximate the SLOPE solution via the much more amenable AMP iterates. Explicitly, we characterize the asymptotic dynamics of the AMP iterates relying on a recently developed state evolution analysis for non-separable penalties, thereby overcoming the difficulty caused by the sorted ℓ1penalty. Moreover, we prove that the AMP iterates converge to the SLOPE solution in an asymptotic sense, and numerical simulations show that the convergence is surprisingly fast. Our proof rests on a novel technique that specifically leverages the SLOPE problem. In contrast to prior literature, our work not only yields an asymptotically sharp analysis but also offers an algorithmic, flexible, and constructive approach to understanding the SLOPE problem.

An acknowledged weakness of neural networks is their vulnerability to adversarial perturbations to the inputs. To improve the robustness of these models, one of the most popular defense mechanisms is to alternatively maximize the loss over the constrained perturbations (or called adversaries) on the inputs using projected gradient ascent and minimize over weights. In this paper, we analyze the dynamics of the maximization step towards understanding the experimentally observed effectiveness of this defense mechanism. Specifically, we investigate the non-concave landscape of the adversaries for a two-layer neural network with a quadratic loss. Our main result proves that projected gradient ascent finds a local maximum of this non-concave problem in a polynomial number of iterations with high probability. To our knowledge, this is the first work that provides a convergence analysis of the first-order adversaries. Moreover, our analysis demonstrates that, in the initial phase of adversarial training, the scale of the inputs matters in the sense that a smaller input scale leads to faster convergence of adversarial training and a “more regular” landscape. Finally, we show that these theoretical findings are in excellent agreement with a series of experiments.

Datasets containing sensitive information are often sequentially analyzed by many algorithms and, accordingly, a fundamental question in differential privacy is concerned with how the overall privacy bound degrades under composition. To address this question, we introduce a family of analytical and sharp privacy bounds under composition using the Edgeworth expansion in the framework of the recently proposed $f$-differential privacy. In short, whereas the existing composition theorem, for example, relies on the central limit theorem, our new privacy bounds under composition gain improved tightness by leveraging the refined approximation accuracy of the Edgeworth expansion. Our approach is easy to implement and computationally efficient for any number of compositions. The superiority of these new bounds is confirmed by an asymptotic error analysis and an application to quantifying the overall privacy guarantees of noisy stochastic gradient descent used in training private deep neural networks.

As a popular approach to modeling the dynamics of training overparametrized neural networks (NNs), the neural tangent kernels (NTK) are known to fall behind real-world NNs in generalization ability. This performance gap is in part due to the \textit{label agnostic} nature of the NTK, which renders the resulting kernel not as \textit{locally elastic} as NNs~\citep{he2019local}. In this paper, we introduce a novel approach from the perspective of \emph{label-awareness} to reduce this gap for the NTK. Specifically, we propose two label-aware kernels that are each a superimposition of a label-agnostic part and a hierarchy of label-aware parts with increasing complexity of label dependence, using the Hoeffding decomposition. Through both theoretical and empirical evidence, we show that the models trained with the proposed kernels better simulate NNs in terms of generalization ability and local elasticity.

A fundamental problem in high-dimensional regression is to understand the tradeoff between type I and type II errors or, equivalently, false discovery rate (FDR) and power in variable selection. To address this important problem, we offer the first complete diagram that distinguishes all pairs of FDR and power that can be asymptotically realized by the Lasso from the remaining pairs, in a regime of linear sparsity under random designs. The tradeoff between the FDR and power characterized by our diagram holds no matter how strong the signals are. In particular, our results complete the earlier Lasso tradeoff diagram in previous literature by recognizing two simple constraints on the pairs of FDR and power. The improvement is more substantial when the regression problem is above the Donoho-Tanner phase transition. Finally, we present extensive simulation studies to confirm the sharpness of the complete Lasso tradeoff diagram.

SLOPE is a relatively new convex optimization procedure for high-dimensional linear regression via the sorted $\ell_1$ penalty: the larger the rank of the fitted coefficient, the larger the penalty. This non-separable penalty renders many existing techniques invalid or inconclusive in analyzing the SLOPE solution. In this paper, we develop an asymptotically exact characterization of the SLOPE solution under Gaussian random designs through solving the SLOPE problem using approximate message passing (AMP). This algorithmic approach allows us to approximate the SLOPE solution via the much more amenable AMP iterates. Explicitly, we characterize the asymptotic dynamics of the AMP iterates relying on a recently developed state evolution analysis for non-separable penalties, thereby overcoming the difficulty caused by the sorted $\ell_1$ penalty. Moreover, we prove that the AMP iterates converge to the SLOPE solution in an asymptotic sense, and numerical simulations show that the convergence is surprisingly fast. Our proof rests on a novel technique that specifically leverages the SLOPE problem. In contrast to prior literature, our work not only yields an asymptotically sharp analysis but also offers an algorithmic, flexible, and constructive approach to understanding the SLOPE problem.

We study first-order optimization algorithms obtained by discretizing ordinary differential equations (ODEs) corresponding to Nesterov’s accelerated gradient methods (NAGs) and Polyak’s heavy-ball method. We consider three discretization schemes: symplectic Euler (S), explicit Euler (E) and implicit Euler (I) schemes. We show that the optimization algorithm generated by applying the symplectic scheme to a high-resolution ODE proposed by Shi et al. [2018] achieves the accelerated rate for minimizing both strongly convex function and convex function. On the other hand, the resulting algorithm either fails to achieve acceleration or is impractical when the scheme is implicit, the ODE is low-resolution, or the scheme is explicit.

We derive a second-order ordinary differential equation (ODE) which is the limit of Nesterov's accelerated gradient method. This ODE exhibits approximate equivalence to Nesterov's scheme and thus can serve as a tool for analysis. We show that the continuous time ODE allows for a better understanding of Nesterov's scheme. As a byproduct, we obtain a family of schemes with similar convergence rates. The ODE interpretation also suggests restarting Nesterov's scheme leading to an algorithm, which can be rigorously proven to converge at a linear rate whenever the objective is strongly convex.

As deploying Vehicular Ad Hoc NETworks (VANETs) costs large amounts of resources, it is crucial that governments and companies make a thorough estimation and comparison of the benefits and the costs. The network connectivity is an important factor we should take care of, because it can greatly affect the performance of VANETs and further affect how much we can benefit from VANETs. We use percolation theory to analyze the connectivity of VANETs. Through theoretical deduction, we discover the quantitative relationship among network connectivity, vehicle density and transmission range. We show that there is a jump of the network connectivity when vehicle density or transmission range is big enough. Simulations conducted in a large scenario validate our theoretical results. Our results have great meanings in the deployment of VANETs in real world. Given vehicle density, our theorem can be used to calculate the minimum transmission range to achieve good network connectivity. As a large transmission range can cause serious collisions in wireless links, it is a tradeoff to choose a proper transmission range. Our analysis can give a hand to this tradeoff and guide the deployment of VANETs in real world.

Nowadays, Vehicular Ad hoc NETworks (VANETs) attract more and more attentions both from academia and industry. Although it has achieved much success in the research field, large-scale deployments of VANETs are still lacking. One barrier facing us is that both governments and companies are doubtful about the performance of VANETs in large-scale deployment in real world. The connectivity of VANETs is a factor closely related to the performance of VANETs. We give a thorough theoretical analysis of the VANET connectivity using bond percolation model and Bollobas model in different scenarios. We discover the quantitative relationship among network connectivity, vehicle density and transmission range. Given the vehicle density, we can calculate the minimum transmission range to achieve good network connectivity. Simulations conducted in a large scenario validate our analysis. Our results not only give us insights about the properties of the network topology, but also have great meanings in real world, which can guide the deployment of VANETs.