Large Language Models (LLMs) have the potential to transform humanities and social science research, yet their history knowledge and comprehension at a graduate level remains untested. Benchmarking LLMs in history is particularly challenging, given that human knowledge of history is inherently unbalanced, with more information available on Western history and recent periods. We introduce the History Seshat Test for LLMs (HiST-LLM), based on a subset of the Seshat Global History Databank, which provides a structured representation of human historical knowledge, containing 36,000 data points across 600 historical societies and over 2,700 scholarly references. This dataset covers every major world region from the Neolithic period to the Industrial Revolution and includes information reviewed and assembled by history experts and graduate research assistants.

We consider the problem of answering queries about formulas of first-order logic based on background knowledge partially represented explicitly as other formulas, and partially represented as examples independently drawn from a fixed probability distribution. PAC semantics, introduced by Valiant, is one rigorous, general proposal for learning to reason in formal languages: although weaker than classical entailment, it allows for a powerful model theoretic framework for answering queries while requiring minimal assumptions about the form of the distribution in question. To date, however, the most significant limitation of that approach, and more generally most machine learning approaches with robustness guarantees, is that the logical language is ultimately essentially propositional, with finitely many atoms. Indeed, the theoretical findings on the learning of relational theories in such generality have been resoundingly negative. This is despite the fact that first-order logic is widely argued to be most appropriate for representing human knowledge. In this work, we present a new theoretical approach to robustly learning to reason in first-order logic, and consider universally quantified clauses over a countably infinite domain. Our results exploit symmetries exhibited by constants in the language, and generalize the notion of implicit learnability to show how queries can be computed against (implicitly) learned first-order background knowledge.

We consider alternating gradient descent (AGD) with fixed step size applied to the asymmetric matrix factorization objective.

We study the scaling limits of stochastic gradient descent (SGD) with constant step-size in the high-dimensional regime. We prove limit theorems for the trajectories of summary statistics (i.e., finite-dimensional functions) of SGD as the dimension goes to infinity. Our approach allows one to choose the summary statistics that are tracked, the initialization, and the step-size. It yields both ballistic (ODE) and diffusive (SDE) limits, with the limit depending dramatically on the former choices. We find a critical scaling regime for the step-size below which this ``effective dynamics matches gradient flow for the population loss, but at which, a new correction term appears which changes the phase diagram. About the fixed points of this effective dynamics, the corresponding diffusive limits can be quite complex and even degenerate. We demonstrate our approach on popular examples including estimation for spiked matrix and tensor models and classification via two-layer networks for binary and XOR-type Gaussian mixture models.

Arsenal are sitting top of the Premier League at Christmas, but it is BBC Sport football expert Chris Sutton who leads the way when it comes to predictions. I'm number one at Christmas - again, said Sutton, who as a player led the Premier League table at this point with Norwich in 1992 and Blackburn in 1994, and went on to win the title with Rovers. It's a big deal for me to be top, as well as for Arsenal . AI is the go-to for virtually everyone in the world whenever they have to ask anything, so the fact that I am beating it - and let's face it, I have stuffed it for half a season now - is pretty incredible. It says a lot about me, and I'm delighted. Rather than asking AI about everything, maybe people should come to me to tap into my intelligence instead?

Three people - including two police officers - have been killed in an explosion in Moscow, Russian authorities have said. Two traffic police officers saw a suspicious individual near a police car on the city's Yeletskaya Street, and when they approached the suspect to detain him, an explosive device was detonated, Russia's Investigative Committee has said. The two police officers died from their injuries, along with another individual who was standing nearby. The attack comes two days after a senior Russian general was killed in a car bombing in the capital on Monday. Lt Gen Fanil Sarvarov died after an explosive device - which had been planted under a car - was detonated.

Stranger Things: What could happen next as the show's finale looms? Spoiler warning: This contains some details about what has happened in the show so far, but does not reveal anything about the final four episodes. A Christmas feast may be around the corner, or perhaps another chocolate (no strawberry creams, thanks), but for fans of Stranger Things, another gift is waiting to be consumed. The grand finale of Netflix's hugely popular sci-fi fantasy horror series, which also showcases some questionable 80s fashion choices, is looming. Fans last saw the inhabitants of Hawkins in a perilous place as season five opened, with Demogorgons running rampant, along with the monstrous Vecna.

Could Ukraine hold a presidential election right now? Will Europe use frozen Russian assets to fund war? How can Ukraine rebuild China ties? 'Ukraine is running out of men, money and time' Russian forces began a "massive attack" on Ukraine on Monday night, killing three people and targeting 13 regions with 650 drones and 30 missiles, Ukrainian President Volodymyr Zelenskyy said in a post on X. Those killed in the overnight attack included a four-year-old girl in the central Zhytomyr region, Governor Vitalii Bunechko said on Telegram.

We study the problem of learning a low-degree spherical polynomial of degree $\ell_0 = Θ(1) \ge 1$ defined on the unit sphere in $\RR^d$ by training an over-parameterized two-layer neural network (NN) with channel attention in this paper. Our main result is the significantly improved sample complexity for learning such low-degree polynomials. We show that, for any regression risk $\eps \in (0,1)$, a carefully designed two-layer NN with channel attention and finite width of $m \ge Θ({n^4 \log (2n/δ)}/{d^{2\ell_0}})$ trained by the vanilla gradient descent (GD) requires the lowest sample complexity of $n \asymp Θ(d^{\ell_0}/\eps)$ with probability $1-δ$ for every $δ\in (0,1)$, in contrast with the representative sample complexity $Θ\pth{d^{\ell_0} \max\set{\eps^{-2},\log d}}$, where $n$ is the training daata size. Moreover, such sample complexity is not improvable since the trained network renders a sharp rate of the nonparametric regression risk of the order $Θ(d^{\ell_0}/{n})$ with probability at least $1-δ$. On the other hand, the minimax optimal rate for the regression risk with a kernel of rank $Θ(d^{\ell_0})$ is $Θ(d^{\ell_0}/{n})$, so that the rate of the nonparametric regression risk of the network trained by GD is minimax optimal. The training of the two-layer NN with channel attention consists of two stages. In Stage 1, a provable learnable channel selection algorithm identifies the ground-truth channel number $\ell_0$ from the initial $L \ge \ell_0$ channels in the first-layer activation, with high probability. This learnable selection is achieved by an efficient one-step GD update on both layers, enabling feature learning for low-degree polynomial targets. In Stage 2, the second layer is trained by standard GD using the activation function with the selected channels.

We introduce two nonlinear sufficient dimension reduction methods for regressions with tensor-valued predictors. Our goal is two-fold: the first is to preserve the tensor structure when performing dimension reduction, particularly the meaning of the tensor modes, for improved interpretation; the second is to substantially reduce the number of parameters in dimension reduction, thereby achieving model parsimony and enhancing estimation accuracy. Our two tensor dimension reduction methods echo the two commonly used tensor decomposition mechanisms: one is the Tucker decomposition, which reduces a larger tensor to a smaller one; the other is the CP-decomposition, which represents an arbitrary tensor as a sequence of rank-one tensors. We developed the Fisher consistency of our methods at the population level and established their consistency and convergence rates. Both methods are easy to implement numerically: the Tucker-form can be implemented through a sequence of least-squares steps, and the CP-form can be implemented through a sequence of singular value decompositions. We investigated the finite-sample performance of our methods and showed substantial improvement in accuracy over existing methods in simulations and two data applications.