Between 2018 and 2020, the field witnessed asurgein the development of these models (Devlin etal.,2019;Liu

Sudan's Rapid Support Forces (RSF) have launched a series of drone attacks targeting humanitarian aid convoys and fuel trucks across North Kordofan, killing at least one person and wounding several others, officials and medical organisations said. The North Kordofan state government condemned Friday's strikes on a convoy linked to the World Food Programme (WFP), urging the international community and United Nations bodies to impose sanctions on the RSF paramilitary group's leadership. The attacks occurred along the key road connecting the state capital, el-Obeid, with Kosti in neighbouring White Nile state. Fighting between the government-aligned Sudanese Armed Forces (SAF) and the RSF has intensified across the Kordofan region since October 2025 after el-Fasher fell to the RSF, where the group committed atrocities - a "crime scene" according to the UN. According to the UN Office for the Coordination of Humanitarian Affairs (OCHA), the first strike at dawn hit three trucks in Er-Rahad.

Understanding the limitations of gradient methods, and stochastic gradient descent (SGD) in particular, is a central challenge in learning theory. To that end, a commonly used tool is the Statistical Queries (SQ) framework, which studies performance limits of algorithms based on noisy interaction with the data. However, it is known that the formal connection between the SQ framework and SGD is tenuous: Existing results typically rely on adversarial or specially-structured gradient noise that does not reflect the noise in standard SGD, and (as we point out here) can sometimes lead to incorrect predictions. Moreover, many analyses of SGD for challenging problems rely on non-trivial algorithmic modifications, such as restricting the SGD trajectory to the sphere or using very small learning rates. To address these shortcomings, we develop a new, non-SQ framework to study the limitations of standard vanilla SGD, for single-index and multi-index models (namely, when the target function depends on a low-dimensional projection of the inputs). Our results apply to a broad class of settings and architectures, including (potentially deep) neural networks.

Matrix completion studies the problem of reconstructing a matrix from a (typically random) subset of its entries by exploiting prior structural assumptions such as low rank and incoherence. Roughly speaking, when the underlying n n matrix has low rank and its eigenvectors are sufficiently incoherent, observing Ω(n log n) entries sampled uniformly at random is sufficient for exact recovery via efficient optimization methods [Keshavan et al., 2009, 2010, Candes and Tao, 2010, Candes and Plan, 2010, Recht, 2011, Candes and Recht, 2012, Jain et al., 2013]. This sample complexity is nearly optimal, since specifying a rank-r matrix requires only O(n) degrees of freedom. Tensor completion generalizes this problem to higher-order arrays, aiming to recover a low-rank tensor from a limited set of observed entries, for example, under uniform random sampling. As a natural higher-order analogue of matrix completion, tensor completion has found broad applications in areas such as recommendation systems [Frolov and Oseledets, 2017], signal and image processing [Govindu, 2005, Liu et al., 2012], and data science [Song et al., 2019]. Despite this close analogy, tensor completion behaves fundamentally differently from its matrix counterpart. In contrast to the classical matrix setting, tensor completion exhibits a pronounced trade-off between computational and statistical complexity: while information-theoretic considerations suggest that relatively few samples suffice for recovery, all currently known polynomial-time algorithms require substantially more observations than this optimal limit. Polynomial-time methods A widely used polynomial-time approach to tensor completion is to reduce the problem to matrix completion via matricization.