Unbalanced optimal transport (UOT) extends classical optimal transport to measures with different total masses, but statistical guarantees for Monge-type estimation remain limited. We study unbalanced transport with quadratic cost and Kullback-Leibler marginal penalties and argue that the natural population target is not a map alone, but a transport-growth pair. Consequently, we develop two estimators for the transport-growth pairs under several setups: an optimal transport plan-based estimator for a general case, and a kernel-based estimator for a case with smooth densities. We also show that an error of the estimator achieves the minimax optimal rate by deriving a matching lower bound of the minimax risk. Our main technical contribution is a value-based stability reduction that converts perturbations of the UOT objective into transport and growth risks through a UOT gap condition. These results provide a statistical foundation for Monge-type estimation in unbalanced optimal transport.

We design Local LMO - a new projection-free gradient-type method for constrained optimization. The key algorithmic idea is to replace the global linear minimization oracle over the constraint set used by Frank-Wolfe (FW) with a local linear minimization oracle over the intersection of the constraint set and a "small" ball centered at the current iterate. In particular, when minimizing $f:\mathbb{R}^d\to \mathbb{R}$ over a constraint $\emptyset\neq\mathcal{X}\subseteq\mathbb{R}^d$, Local LMO performs the iteration \[x_{k+1}\in \arg\min_{z\in\mathcal{X}\cap\mathcal{B}(x_{k},t_k)}\langle\nabla f(x_{k}), z \rangle,\] where $x_0\in\mathcal{X}$, and $t_k>0$ is a suitably chosen radius which can be interpreted as an effective stepsize. While designed as an alternative to FW, Local LMO is perhaps best viewed as a generalization of Gradient Descent (GD) rather than a modification of FW. Indeed, it is easy to see that Local LMO reduces to GD in the unconstrained setting and, more generally, to GD restricted to an affine subspace if the constraint $\mathcal{X}$ is affine. We prove that this simple algorithmic scheme transfers the known (unaccelerated) convergence rates of Projected Gradient Descent (PGD) to the projection-free world in several important regimes, some of which are beyond the reach of FW. In contrast to FW theory, i) our guarantees hold without requiring the feasible set $\mathcal{X}$ to be bounded, ii) our theory does not require the "curvature" assumption, which allows us to establish a standard sublinear rate for convex functions with bounded gradients, iii) we obtain a linear rate in the smooth strongly convex regime. Furthermore, we obtain sharp sublinear rates in the smooth convex and non-convex regimes, in the $(L_0,L_1)$-smooth convex regime, and in stochastic and non-differentiable settings.

This paper relaxes the restrictive symmetry conditions adopted in [4], [5] and extends their universal feature selection framework to accommodate noisy observations as well as attribute structures that may exhibit directional preferences. We introduce the notion of weak spherical symmetry, quantified by second-moment distances, which allows controlled deviations from rotational invariance. Under this relaxed condition, we develop a universal feature selection framework based on the singular value decomposition of the canonical dependence matrix computed from noisy data. Our main result shows that the selected features achieve asymptotically optimal error exponents up to a residual term that depends on the symmetry deviation $δ$ and the noise levels $η_1, η_2$. When $δ, η_1, η_2$ are relatively small, our result recovers that of [5], thereby demonstrating that exact spherical symmetry is unnecessary. Overall, our findings highlight the robustness of the selection framework against second-moment deviations and observation noise, thereby broadening its applicability across diverse inference tasks and providing a theoretically grounded tool for universal feature selection in practical scenarios.

Multicalibration requires predicted scores to agree with label probabilities across rich families of subgroups and score-dependent tests, but existing methods require clean input-label pairs for evaluation and post-processing. This assumption fails in weakly supervised learning (WSL) regimes -- including positive-unlabeled, unlabeled-unlabeled, and positive-confidence learning -- where clean labels are costly or unavailable even though reliable uncertainty estimates may be crucial. We address this gap by developing estimators of multicalibration error and post-hoc correction methods for WSL settings in which clean input-label pairs are unavailable. We propose a unified framework for estimating and correcting multicalibration under weak supervision by combining contamination-matrix risk rewrites with witness-based calibration constraints, yielding corrected multicalibration moments with finite-sample guarantees. We further propose weak-label multicalibration boost (WLMC), a generic post-hoc recalibration algorithm under weak supervision. Finally, we conduct experiments across multiple weak-supervision settings to evaluate multicalibration behavior and offer empirical insight into uncertainty estimation under weak supervision.

Operator learning for partial differential equations (PDEs) aims to learn solution operators on infinite-dimensional function spaces from finite-resolution data. In this setting, it is important for the learned model to be discretization-invariant, or resolution-robust, and to reflect PDE-specific structure. It is therefore natural to ask how such structure should be encoded in the model architecture, hypothesis class, or learning procedure. In this paper, we study operator learning for solution operators of nonlinear parabolic PDEs based on Duhamel--Picard iteration. We formulate Picard iteration as an abstract state-transition model and present a theoretical framework for Picard-type operator learning. We derive implementation-agnostic generalization error bounds that separate the implementation error from the estimation error associated with the abstract state-transition model induced by Picard iteration. A key consequence is that increasing the Picard depth reduces the Picard truncation error without causing an unbounded growth of the entropy-based estimation error. We also extend the analysis to long-time prediction by rolling out the same learned local model over successive time blocks. Finally, we illustrate the theory for nonlinear heat equations on the torus using a Picard-type Fourier neural operator as a concrete implementation.

Constraint-based causal discovery is widely used for learning causal structures, but heavy reliance on conditional independence (CI) testing makes it computationally expensive in high-dimensional settings. To mitigate this limitation, many divide-and-conquer frameworks have been proposed, but most assume causal sufficiency, i.e., no latent variables. In this paper, we show that divide-and-conquer strategies can be theoretically generalized beyond causal sufficiency to settings with latent variables. Specifically, we propose a recursive decomposition framework, termed DiCoLa, that enables divide-and-conquer causal discovery in the presence of latent variables. It recursively decomposes the global learning task into smaller subproblems and integrates their solutions through a principled reconstruction step to recover the global structure. We theoretically establish the soundness and completeness of the proposed framework. Extensive experiments on synthetic data demonstrate that our approach significantly improves computational efficiency across a range of causal discovery algorithms, while experiments on a real-world dataset further illustrate its practical effectiveness.

Large language models are increasingly deployed with test-time strategies: sample $N$ responses, score them with a reward model or verifier, and return the best. This deployment rule exposes a mismatch in post-training: standard objectives optimize the mean reward of a single response, whereas best-of-$N$ performance is governed by the upper tail of the reward distribution. Recent test-time-aware objectives partly address this mismatch, but typically assume that training can use the same per-prompt rollout budget as deployment, which is impractical when post-training must cover many prompts while deployment can allocate much larger per-prompt test-time compute. We study this budget-mismatch regime, where only $m\ll N$ per-prompt rollouts are available during training but the target objective is best-of-$N$ deployment. Under structural assumptions on the reward tails, we show that the policy gradient of the best-of-$N$ objective can be approximated from a much smaller rollout group by extrapolating upper-tail statistics. This yields a family of Tail-Extrapolated estimators for best-of-$N$-oriented post-training: a simple direct estimator, Tail-Extrapolated Advantage (TEA), and a fixed-order debiased Prefix-TEA estimator based on moment cancellation. Experiments on instruction-following tasks show that TEA and Prefix-TEA improve best-of-$N$ performance across different language models, reward models and datasets under various training and test-time budget settings.

Security around Beijing's historic Tiananmen Square has been heightened for days, with rumours on social media swirling of a special parade or some big, choreographed event. Preparations for this major event have started with a whisper, but China appears ready to put on a show for US President Donald Trump. The visit will include talks, a banquet, and a visit to the Temple of Heaven, a complex of imperial temples where emperors would pray for a good harvest. And both Trump and Chinese President Xi Jinping will be hoping the visit will bear fruit. This summit between the world's two most powerful leaders is set to be one of the most consequential encounters for years.

We can now add cybercrimes to the list of growing concerns associated with artificial intelligence. Google's Threat Intelligence Group (GTIG) said it discovered, for the first time ever, a threat actor using a zero-day exploit that it believes was developed by AI. Zero-day vulnerabilities are often the most dangerous since they're unknown to the targets, leaving them with zero days to prepare for the attack. Google said in the report the threat actor was planning to use it in a mass exploitation event, but its proactive discovery may have prevented its use. Google added that it doesn't believe its own Gemini models were used, but still has high confidence an AI model was part of discovering the vulnerability and weaponizing an exploit.

The game, developed by the group of anonymous artists known as Secret Handshake, is available online and in person in Washington, DC. A new video game about President Donald Trump's war in Iran features fights with the pope and New York City mayor Zohran Mamdani . It's impossible to win, and that's the point. The game,, was developed by Secret Handshake, an anonymous group of artists behind a handful of satirical works mocking the Trump administration. The group previously installed a gold statue of Trump and Jeffrey Epstein on the National Mall; it portrayed Trump holding onto Epstein in a pose reminiscent of Jack and Rose from the movie .