In such settings, the inverse scattering transform (IST) of Ablowitz, Kaup, Newell, and Segur [2] has enjoyed a rich and successful history, and is now the standard theoretical framework for deriving reduced-order evolution equations for soliton dynamics. Although these derivations are traditionally of an analytical - rather than data-driven - nature, recent work has employed the IST formalism as a tool for experimental data analysis, using the technique to analyze soliton content from empirical measurements [8, 15, 24]. Moreover, recent approaches using alternative parameterization techniques have demonstrated that the learning of reduced-order, interpretable equations of motion for solitons is tenable in a data-driven setting [6, 26, 27]. Despite the success of this recent work, however, little effort has been devoted to developing a data-driven modeling approach based on the IST itself, most likely due to the fact that the framework is fundamentally problem-specific. In this paper, we address the question of whether effective soliton dynamics can be inferred directly from observed scattering data (as opposed to being derived or approximated analytically).

Machine learning algorithms and their training datasets have grown substantially in both size and complexity over the past decade. This increased model complexity has made it challenging to interpret and predict their behavior in unobserved scenarios. Hence, many applications that involve societal decisions still rely on simple, interpretable models like linear regression, often after feature engineering. Examples of such applications include predicting national housing prices, estimating wages across industries, forecasting loan amounts across banks, predicting life insurance premiums across groups, and projecting energy consumption across communities [CGKMN24]. A shared safety and sometimes legal concern across the above applications is the potential for wildly different model qualities for different distributions, i.e., outputting a notably worse model for some source data distributions [Dat14; BS16; HPS16; VVB18; SBFVV19; BHJKR21; CGNSG23; Cho16; KLMR18; ADW19; CGKMN24; SVWZ24].

We characterize weakly regularized continual classification in homogeneous models as sequential projections onto task margin sets. This result generalizes prior analyses restricted to either stationary (single-task) deep models or continual linear models. We show that global convergence generally fails, even for simple models linear in data but nonlinear in parameters. Nevertheless, by leveraging results from nonconvex projection theory, we identify regularity properties of homogeneous deep networks that guarantee local linear convergence under random and cyclic task sequences. Finally, we extend our analysis to continual regression, unifying the framework for homogeneous models.

Pervasive data contamination -- stemming from measurement errors, outliers, or adversarial corruption -- has motivated the development of robust statistical methods. In this context, we propose a two-stage Adversarial Contamination-resistant Iterative Hard Thresholding (AC-IHT) algorithm for high-dimensional regression with contamination. Our nonconvex algorithm achieves minimax near-optimal (up to logarithmic terms) estimation by iteratively updating the coefficient vector and the contamination vector with different thresholding scales. We further demonstrate that our AC-IHT estimator is signal-adaptive: under proper signal conditions, it adaptively attains a sharper estimation rate and more accurate support recovery. Moreover, it enjoys the strong oracle property, laying a theoretical foundation for asymptotic inference. Numerical experiments confirm its superior finite-sample performance. Finally, we discuss theoretical extensions of the proposed procedure to generalized linear models and to heavy-tailed noise settings.

Finding dense subgraphs is a fundamental problem with applications to community detection, clustering, and data mining. Our work focuses on finding approximate densest subgraphs in directed graphs in computational models for processing massive data. We consider two such models: Massively Parallel Computation (MPC) and semi-streaming. We show how to find a (2+ε)-approximation in O( logn) MPC rounds with sublinear memory per machine.

This adaptive mechanism adjusts the resolution of the predictions over both time and space, yielding refined regret bounds in terms of local regularity. Consequently, in heterogeneous environments, our adaptive guarantees can significantly surpass those obtained by standard global methods.

Modern data marketplaces and data sharing consortia increasingly rely on incentive mechanisms to encourage agents to contribute data. However, schemes that reward agents based on the quantity of submitted data are vulnerable to manipulation, as agents may submit fabricated or low-quality data to inflate their rewards. Prior work has proposed comparing each agent's data against others' to promote honesty: when others contribute genuine data, the best way to minimize discrepancy is to do the same. Yet prior implementations of this idea rely on very strong assumptions about the data distribution (e.g.

We study the problem of offline imitation learning in Markov decision processes (MDPs), where the goal is to learn a well-performing policy given a dataset of state-action pairs generated by an expert policy. Complementing a recent line of work on this topic that assumes the expert belongs to a tractable class of known policies, we approach this problem from a new angle and leverage a different type of structural assumption about the environment. Specifically, for the class of linear Qπ-realizable MDPs, we introduce a new algorithm called saddle-point offline imitation learning (SPOIL), which is guaranteed to match the performance of any expert up to an additive error ε with access to O(ε 2) samples. Moreover, we extend this result to possibly nonlinear Qπ-realizable MDPs at the cost of a worse sample complexity of order O(ε 4). Finally, our analysis suggests a new loss function for training critic networks from expert data in deep imitation learning. Empirical evaluations on standard benchmarks demonstrate that the neural net implementation of SPOIL is superior to behavior cloning and competitive with state-of-the-art algorithms.

We study the problem of differentially private second moment estimation and present a new algorithm that achieve strong privacy-utility trade-offs even for worst-case inputs under subsamplability assumptions on the data. We call an input (m,α,β)-subsamplable if a random subsample of size m(or larger) preserves w.p 1 β the spectral structure of the original second moment matrix up to a multiplicative factor of 1 α. Building upon subsamplability, we give a recursive algorithmic framework similar to Kamath et al. (2019) that abides zero-Concentrated Differential Privacy (zCDP) while preserving w.h.p the accuracy of the second moment estimation upto an arbitrary factor of (1 γ). We then show how to apply our algorithm to approximate the second moment matrix of a distribution D, even when a noticeable fraction of the input are outliers.

We consider the parameter estimation problem in logistic regression with Gaussian design: the estimation of a fixed unknown parameter $θ^*\in \mathbb{R}^d$ ($\|θ^*\|_2\ge 1$) from $n$ i.i.d. samples $\{(x_i,y_i)\}_{i=1}^n$, where $x_i\sim N(0,I_d)$ and $y_i|x_i \sim {\rm Bernoulli}(1/(1+\exp(-x_i^\top θ^*)))$. Our main aim is to characterize the finite-sample estimation performance and convergence behavior of gradient descent (GD) on the maximum likelihood objective (i.e., the logistic loss). Under small $O(1)$ stepsize and $0$ initialization, we show that GD linearly converges to a small neighborhood of $θ^*$ achieving an $\ell_2$ error of order $O(\sqrt{\|θ^*\|_2^5d/n})$. This substantially goes beyond existing theoretical results that lack non-asymptotic estimation error rate and exhibit much slower parameter convergence. We also establish a faster local linear convergence to the same statistical error under a large $Θ(\|θ^*\|_2)$ stepsize. The main technical component is to show that the gradient of the logistic loss satisfies a certain approximate invertibility condition (AIC). To that end, we uniformly control the deviation of the gradient from its population counterpart by covering and peeling arguments, and then show that the population GD is a contraction by a delicate analysis based on the eigenvalues of population Hessian matrices. Finally, we build upon the recent work Matsumoto and Mazumdar (2025) and devise a novel efficient estimator that attains a sharper rate in high dimensions. This indicates that the existing non-asymptotic guarantees exhibit sub-optimal dependence on $\|θ^*\|_2$, and that in many regimes $Θ(\sqrt{\|θ^*\|_2d/n})$ is the tight estimation error rate. Numerical examples are provided to corroborate our theoretical results.