We introduce a general framework for analyzing learning algorithms based on the notion of self-regularization, which captures implicit complexity control without requiring explicit regularization. This is motivated by previous observations that many algorithms, such as gradient-descent based learning, exhibit implicit regularization. In a nutshell, for a self-regularized algorithm the complexity of the predictor is inherently controlled by that of the simplest comparator achieving the same empirical risk. This framework is sufficiently rich to cover both classical regularized empirical risk minimization and gradient descent. Building on self-regularization, we provide a thorough statistical analysis of such algorithms including minmax-optimal rates, where it suffices to show that the algorithm is self-regularized -- all further requirements stem from the learning problem itself. Finally, we discuss the problem of data-dependent hyperparameter selection, providing a general result which yields minmax-optimal rates up to a double logarithmic factor and covers data-driven early stopping for RKHS-based gradient descent.

We apply one-dimensional convolutional neural networks to the Frobenius traces of elliptic curves over $\mathbb{Q}$ and evaluate and interpret their predictive capacity. In keeping with similar experiments by Kazalicki--Vlah, Bujanović--Kazalicki--Novak, and Pozdnyakov, we observe high accuracy predictions for the analytic rank across a range of conductors. We interpret the prediction using saliency curves and explore the interesting interplay between murmurations and Mestre--Nagao sums, the details of which vary with the conductor and the (predicted) rank.

Decision-making problems often feature uncertainty stemming from heterogeneous and context-dependent human preferences. To address this, we propose a sequential learning-and-optimization pipeline to learn preference distributions and leverage them to solve downstream problems, for example risk-averse formulations. We focus on human choice settings that can be formulated as (integer) linear programs. In such settings, existing inverse optimization and choice modelling methods infer preferences from observed choices but typically produce point estimates or fail to capture contextual shifts, making them unsuitable for risk-averse decision-making. Using a bounded-variance score function gradient estimator, we train a predictive model mapping contextual features to a rich class of parameterizable distributions. This approach yields a maximum likelihood estimate. The model generates scenarios for unseen contexts in the subsequent optimization phase. In a synthetic ridesharing environment, our approach reduces average post-decision surprise by up to 114$\times$ compared to a risk-neutral approach with perfect predictions and up to 25$\times$ compared to leading risk-averse baselines.

Neural networks are central to modern artificial intelligence, yet their training remains highly sensitive to data contamination. Standard neural classifiers are trained by minimizing the categorical cross-entropy loss, corresponding to maximum likelihood estimation under a multinomial model. While statistically efficient under ideal conditions, this approach is highly vulnerable to contaminated observations including label noises corrupting supervision in the output space, and adversarial perturbations inducing worst-case deviations in the input space. In this paper, we propose a unified and statistically grounded framework for robust neural classification that addresses both forms of contamination within a single learning objective. We formulate neural network training as a minimum-divergence estimation problem and introduce rSDNet, a robust learning algorithm based on the general class of $S$-divergences. The resulting training objective inherits robustness properties from classical statistical estimation, automatically down-weighting aberrant observations through model probabilities. We establish essential population-level properties of rSDNet, including Fisher consistency, classification calibration implying Bayes optimality, and robustness guarantees under uniform label noise and infinitesimal feature contamination. Experiments on three benchmark image classification datasets show that rSDNet improves robustness to label corruption and adversarial attacks while maintaining competitive accuracy on clean data, Our results highlight minimum-divergence learning as a principled and effective framework for robust neural classification under heterogeneous data contamination.

Mirror Descent (MD) is a scalable first-order method widely used in large-scale optimization, with applications in image processing, policy optimization, and neural network training. This paper generalizes MD to optimization on Riemannian manifolds. In particular, we develop a Riemannian Mirror Descent (RMD) framework via reparameterization and further propose a stochastic variant of RMD. We also establish non-asymptotic convergence guarantees for both RMD and stochastic RMD. As an application to the Stiefel manifold, our RMD framework reduces to the Curvilinear Gradient Descent (CGD) method proposed in [26]. Moreover, when specializing the stochastic RMD framework to the Stiefel setting, we obtain a stochastic extension of CGD, which effectively addresses large-scale manifold optimization problems.

The reported use of Claude in recent military operations has shifted the Overton window around A.I. in warfare--and sparked a battle between Anthropic and the Department of War. The staff writer Gideon Lewis-Kraus joins Tyler Foggatt to discuss the escalating standoff between the A.I. company Anthropic and the Department of War. They consider recent reporting on the use of Claude--Anthropic's family of large language models--in military operations in Venezuela and Iran, and how that news has pushed the company's relationship with the Pentagon to a breaking point. They also explore how the tech industry is responding to the conflict between the Trump Administration and Anthropic, and the thorny question of whether A.I. should be subject to greater safeguards and more oversight than previous technological innovations. " The Pentagon Went to War with Anthropic. " The Iran War Is Another Reason to Quit Oil," by Bill McKibben " How Should We Remember the Hippies?

THE LOST WEDDING PHOTOS: See JFK Jr and Carolyn Bessette at their secret nuptials... and read every intimate detail of ultra-private ceremony Tulsi Gabbard lets Iran nuke bombshell slip as Senate hearing spirals for Trump's embattled spy chief Candace Owens's sickening low-blow at Karoline Leavitt as Iran war sparks wild attacks Lunatic Megyn Kelly is FINALLY ruined! Her appalling X-rated smear of my friend proves it... but now I know her truly disturbing plan: JOSH HAMMER Inside the epidemic of midlife women who are repulsed by their husbands, the age and'vital statistics' that make men most at risk - and the telltale signs YOUR marriage is about to die: Special report by SADIE NICHOLAS Meghan gives glimpse of'mama's little helpers' Archie and Lilibet in'behind the scenes' video of her latest As Ever launch Shameful hypocrisy of NASCAR star Daniel Suarez's nepo-baby wife: 'Victim' mask slips as she ignites new Las Vegas drama... and dark family past rears its ugly head Princess Kate dons her favourite tiara and the late Queen's earrings as she arrives at King's banquet for the Nigerian President in country's first state visit in almost 40 years Everything JFK Jr told friends about his love affair with'sexual dynamo' Madonna... her unprintable pillow talk... and his perverse incest request that she couldn't go through with Site of'Jesus' crucifixion' forced to shut for Holy Week in unprecedented move tied to biblical prophecies of the Antichrist Ugly new Nicole Kidman and Keith Urban divorce fight ERUPTS: Her friends share humiliating details of'midlife crisis'... and reveal brutal REAL reason daughter Sunday Rose'snubbed' him Outrage after Seattle museum vandal destroys $250,000 of famous Dale Chihuly glass at city's museum dedicated to him Amanda Bynes, 39, 'is now a size 4 after losing 35lb' thanks to weight-loss medication... after hitting 180lb Chilling unclassified threat report reveals the'most likely' terror attack scenario on US soil Three's Company bombshell Jenilee Harrison who was also on Dallas and The Love Boat still looks great at 67, see her now The discovery of a Roman artifact in the Americas has sparked a debate about who truly discovered the New World. While Christopher Columbus is hailed as the first in 1492, archaeologists uncovered a small terracotta head of a bearded man carved with distinctive European features tucked inside a Mexican tomb. The artifact, known as the Tecaxic-Calixtlahuaca Head, was discovered in 1933 inside a sealed pre-Hispanic burial beneath multiple intact layers, indicating it had not been disturbed after its placement. Experts say its facial features, beard style and craftsmanship bear a striking resemblance to objects from the ancient Mediterranean rather than indigenous Mesoamerican traditions.

Reviving the Claymills Pumping Station in Staffordshire, England has been a labor of love. Restoration work has progressed steadily for over 30 years. Breakthroughs, discoveries, and DIY tips sent six days a week. It's always good to have a passion project, but what's going on in Staffordshire, England, is likely a one-of-a-kind endeavor. In the town of Burton upon Trent, a rotating team of volunteers has spent over 30 years restoring a Victorian pump house.

After OpenAI's Instant Checkout feature fell short, Walmart is instead embedding its Sparky chatbot directly into ChatGPT and Google Gemini. Since November, Walmart has let some ChatGPT users order a limited selection of products without ever leaving OpenAI's chatbot interface. Sales have been disappointing, a Walmart executive vice president exclusively tells WIRED. The results suggest that a future where chatbots and AI agents take over ecommerce is still a way off, if it ever materializes. Last year, OpenAI made a bet that it could boost revenue by charging a commission on purchases made through ChatGPT.

Gooey birch tar helped our distant cousins make weapons and possibly treat wounds. The bark of birch trees has been used to produce tar for more than 150,000 years. The center photo shows birch bark tar condensed onto a rock that borders a hearth. When scraped off the rocks, the viscous tar can be used as both an adhesive and antibiotic. Breakthroughs, discoveries, and DIY tips sent six days a week.