A striking geometric disparity has long persisted in the practice of deep learning. While modern neural network architectures naturally exhibit rich symmetry and equivariance properties, popular optimizers such as Adam and its variants operate inherently coordinate-wise, rendering them unable to respect the equivariance structures of the parameter space. We address this disparity by introducing a symmetry-compatible principle for optimizer design: the gradient update rule should be equivariant under the symmetry group acting on the corresponding weight block. Following this principle, we first provide a unified perspective on bi-orthogonally equivariant updates for general matrix layers, as employed by stochastic spectral descent, Muon, Scion, and polar gradient methods. More importantly, by moving from orthogonal groups to permutation and shared-shift symmetries, we derive symmetry-compatible optimizers for parameter blocks whose symmetries differ from those of general matrix layers: embedding and LM head matrices, SwiGLU MLP projections, and MoE router matrices. These constructions include one-sided spectral, row-norm, hybrid row-norm/spectral, row-aware, column-aware, centered row-norm, and left-spectral updates. They yield an end-to-end layerwise optimizer stack in which each major matrix-valued parameter class is assigned an update whose equivariance matches its symmetry group. We corroborate this principle through pre-training experiments on dense and sparse MoE language models, including Qwen3-0.6B-style, Gemma 3 1B-style, OLMoE-1B-7B-style, and downsized gpt-oss architectures. Across these experiments, symmetry-compatible update rules consistently improve final validation loss, reduce load imbalance in sparse MoE models, and in several cases improve training stability over the corresponding AdamW updates.

Multicalibration extends classical calibration by requiring predictions to be unbiased over a rich collection of functions, encompassing both prediction slices and subpopulations. It has emerged as a powerful framework for fairness, robustness, and reliable prediction, yet the theoretical understanding of multicalibration boosting (MCBoost) remains fragmented and often relies on restrictive assumptions. In this work, we develop a unified and refined perspective on MCBoost that subsumes existing variants, including multiaccuracy, BatchGCP, and BatchMVP. We uncover several phenomena that provide new insights into its practical behavior: even highly accurate and flexible predictors can remain substantially miscalibrated; enforcing multicalibration introduces a calibration-risk trade-off; and early stopping plays a central role in controlling this trade-off. On the theoretical side, we establish a general framework for MCBoost under weaker and more realistic conditions. We show that the boosting iterates converge to a Bregman projection of the population-optimal predictor onto the cumulative span generated by the audit class, thereby explicitly characterizing the function space on which multicalibration is achieved. We further derive convergence rates under different smoothness assumptions, finite-sample guarantees, and principled stopping rules that ensure multicalibration at termination. Finally, we extend the theory of universal adaptability under covariate shift, providing more general transfer guarantees and clarifying when multicalibrated predictors generalize across domains. These results provide a more complete theoretical foundation and practical guidance for multicalibration boosting, positioning it as both a unifying framework and a reliable post-processing approach for modern predictive models.

Kernel Stein discrepancy (KSD) is among the most popular goodness-of-fit (GoF) measures on general domains with a large number of successful deployments. One of the main applications of KSD is in constructing powerful GoF tests. However, tests relying on the classical U-/V-statistic-based KSD estimators have two major drawbacks. (i) Their runtime scales quadratically in the number of samples. (ii) Their asymptotic null distribution is computationally intractable in most cases, typically handled by bootstrapping. While it is known that the Nyström method permits accelerating KSD estimation with no loss of statistical accuracy under mild conditions, to the best of our knowledge, the fundamental question of its impact on bootstrap-based GoF testing is open; resolving this question is the focus of the current paper. In particular, we prove that the key properties of the quadratic-time bootstrapped KSD-based GoF test (asymptotic level and local consistency) are preserved by its Nyström acceleration. We numerically demonstrate the efficiency of the accelerated KSD estimator and bootstrap in the context of GoF testing of spherical and functional data. Our numerical results show that the Nyström-accelerated method performs statistically on-par with the quadratic-time approach, while requiring substantially smaller runtime.

A generous donation and a good neighbor will help the wildcat continue her recovery. More information Adding us as a Preferred Source in Google by using this link indicates that you would like to see more of our content in Google News results. A new kennel and generous donation are giving this Pennsylvania bobcat a new life. Breakthroughs, discoveries, and DIY tips sent six days a week. In March, we reported on a wild bobcat that had been hit and dragged by a car, who also got her head stuck in the car's grill.

We study feature learning in two-layer neural networks within the linear-width regime, where the number of hidden neurons, sample size, and input dimension scale proportionally. While recent work has analyzed feature learning via a single step of gradient descent on the first layer weights in this regime, such one-step update schemes are fundamentally limited: the update to the weights is approximately rank-one, captures only a single direction, and requires the target function to have an information exponent of one. In this paper, we go beyond one-step updates to provide a full characterization of the features learned during the \textit{second step} of gradient descent with step-sizes $η_1\asymp N^{α_1}$ and $η_2 \asymp N^{α_2}$ for $α_1, α_2 \in [0,0.5)$, where $N$ is the number of hidden neurons. We derive a spectral characterization of the updated weights, demonstrating they behave as a spiked random matrix with multiple outliers, each corresponding to a learned direction. We show that the number of the outliers is determined by the parameters $α_1, α_2$ through $\lfloor \frac{α_2}{1/2 - α_1} \rfloor$. Furthermore, by analyzing the alignment between the learned directions and the target function, we identify a gap between training with independent versus reused batches. While independent batches restrict learning to directions with an information exponent of one, batch reuse enables the second update to capture directions even when the information exponent exceeds one, provided that $α_1, α_2$ are chosen properly. This shows that the benefits of batch reuse, previously observed in narrow-width regimes, persist in the linear-width limit as well. By characterizing these early-phase evolutions, our work proposes a tractable framework for studying optimization and feature learning phenomenology in modern overparameterized networks.

Tragic way Kyle Busch was found unresponsive revealed after NASCAR great's sudden death at 41 This quiet announcement from Prince William was missed by most... but this is why royal insiders tell me it spells disaster for Harry and Meghan's future: RICHARD EDEN Dangerous truth about melatonin side effects... the astonishing dose you SHOULD be taking... and a new natural grocery store alternative hailed by doctors Dirty secret Hollywood's Cool Girls don't want you to know. Kyle Busch's eerie premonition on illness just days before NASCAR great's death at 41 - as devastated family reveal their'pain and shock' 'You never went to space': Watch the awkward moment a conspiracy theorist confronts NASA's Artemis II crew - telling them to'stop acting' CIA Nostradamus warned Trump about Iran... now he's calling the President's doctors. Police probe Andrew Mountbatten-Windsor over'sex offences': Stunning update on investigation of former prince as officers appeal for potential'victim survivors' to come forward How Meryl Streep's husband really feels about her secret relationship with Martin Short: Their years of agony... his hard red line... and why she won't divorce him The Olympic gold medalist risking it all to smash sport's biggest taboo: 'It's super forbidden... but we're just openly doing it' Former CDC director Robert Redfield warns Ebola outbreak could spark a new'significant pandemic' Heartbreaking video shows trans student, 19, washing her clothes in college laundry room unaware that stranger who'd just walked in had selected her to be murdered Fears for Ariana Grande: Insiders lift the lid on Ethan Slater's costly sacrifice... her private nightmares... and what's really keeping them apart Aussie model turns heads with embarrassing Photoshop fail: 'OMG, this is insane!' Mom-of-two abandons home in Pennsylvania to live on board CRUISE SHIP year-round - and her kids have'zero concept' their life isn't normal White man charged after he was filmed screaming at black female neighbor and using the phrase'You people' Astonishing secret list of elite Hollywood liberals conspiring to elect Spencer Pratt revealed to KENNEDY by her LA moles. The Trump administration released another trove of UFO files today containing the 46 classified videos requested by lawmakers earlier this year. In one file, audio from a medical debrief can be heard after Apollo 12 astronauts Pete Conrad, Richard Gordon and Alan Bean described seeing mysterious flashes and streaks of light in the dark while trying to sleep.

Police probe Andrew Mountbatten-Windsor over'sex offences': Stunning update on investigation of former prince as officers appeal for potential'victim survivors' to come forward Trump celebrates Stephen Colbert's final show with brutal'no talent' swipe as bitter host takes one last jab at CBS on way out door Trump warns of possible military action in Cuba and says'I'd be happy to do it' as Marco Rubio declares the nation a'US national security threat' Dangerous truth about melatonin side effects... the astonishing dose you SHOULD be taking... and a new natural grocery store alternative hailed by doctors CIA Nostradamus warned Trump about Iran... now he's calling the President's doctors. Dirty secret Hollywood's Cool Girls don't want you to know. Mom-of-two abandons home in Pennsylvania to live on board CRUISE SHIP year-round - and her kids have'zero concept' their life isn't normal This quiet announcement from Prince William was missed by most... but this is why royal insiders tell me it spells disaster for Harry and Meghan's future: RICHARD EDEN White man charged after he was filmed screaming at black female neighbor and using the phrase'You people' Shock moment'slurring' Britney Spears is arrested for DUI after failing sobriety test Astonishing secret list of elite Hollywood liberals conspiring to elect Spencer Pratt revealed to KENNEDY by her LA moles. Suspected Somali fraudster filmed leaping off Minnesota balcony and driving away in luxury Genesis sedan after feds announced they were charging him with alleged $3.3m scam Inside Pizza Hut restaurant that's still EXACTLY like it was in the 90s... complete with checkered tablecloths, arcade and famous buffet Stephen Colbert's final Late Show episode leaves fans unimpressed as Ryan Reynolds leads series of surprise celebrity cameos How Meryl Streep's husband really feels about her secret relationship with Martin Short: Their years of agony... his hard red line... and why she won't divorce him Look away now, Carrie Bradshaw! Fears for Ariana Grande: Insiders lift the lid on Ethan Slater's costly sacrifice... her private nightmares... and what's really keeping them apart Inside the UK's first AI-powered fertility clinic using state-of-the-art technologies to help women get pregnant - as one couple says'artificial intelligence allowed us to hold our baby in our arms' If you were asked to think about artificial intelligence ( AI), visions of killer robots, dodgy chatbots, or deepfakes might spring to mind.

We develop a mean-field theory of dropout as a perturbation of critical signal propagation at the edge of chaos. Dropout shifts the perfect-alignment fixed point, making the depth scale for information propagation finite even at critical initialization. We derive critical and crossover scaling laws for correlation decay and establish that smooth activations and kinked, ReLU-like activations constitute distinct universality classes, with different critical exponents and a universal two-parameter scaling collapse in detuning and dropout strength. The distinction traces to the analytic structure of the correlation map: smooth activations admit a Taylor expansion near perfect alignment, while kinked activations develop a branch point with universal non-analyticity. As a corollary, the framework yields saturated dropout profiles under fixed budget; a rank-flow tie-breaker then selects front-loaded schedules, substantially reducing held-out test loss at no extra computational cost, with accuracy gains as a consistent secondary effect. We test the predictions in MLPs and Vision Transformers and discuss CNN/ResNet extensions.

One line tucked into a federal highway bill would strip funds from cities and states unless they kill their automated plate tracking programs--effectively banning the tech for all but toll collection. US lawmakers plan to introduce an amendment Thursday at a House committee markup hearing that would prohibit any recipient of federal highway funding from using automated license plate readers for any purpose other than tolling--a sweeping restriction that, if adopted, would bring an immediate end to state and local ALPR programs across the United States. The amendment, obtained first by WIRED, is sponsored by Representative Scott Perry, a Pennsylvania Republican and Freedom Caucus member, and Representative Jesús "Chuy" García, an Illinois progressive whose state has become a flash point in the national fight over ALPR misuse. The House Transportation and Infrastructure Committee will mark up the underlying bill--a $580 billion, five-year reauthorization of federal surface transportation programs--at 10 am ET on Thursday. Neither Perry nor García's offices immediately responded to WIRED's request for comment. The amendment runs a single sentence: "A recipient of assistance under Title 23, United States Code, may not use automated license plate readers for any purpose other than tolling."

I was never one of those lunatics who went with the no-batting glove look. I respect folks who do it, but I'm not built for that. I needed my batting gloves like I needed air to breathe, and I've always thought I'd be a better golfer if I just sacked up and put on the extra glove. Aaron Rai of England hits his second shot on the 16th hole during the final round of the PGA Championship at Aronimink Golf Club on May 17, 2026 in Newtown, Pennsylvania.