Sales of the iPhone hit an all-time high in the final three months of last year, tech firm Apple reported on Thursday. Revenue rose by 16% compared to the same period last year to $144bn (£82.5bn) - the strongest growth since 2021 - thanks to a jump in sales in China, as well as Europe, the Americas, and Japan. However, sales in other parts of the company were less positive. Wearables and accessories, which include things like the Apple Watch and AirPods, fell by roughly 3%. Apple chief executive Tim Cook said the iPhone's boost in sales meant the firm was in supply chase mode.

Such failure can be caused by distribution shift (also known as dataset shift) between the training and test datasets. For this reason, distribution shift and domain adaptation (a notion comprising techniques for tackling distribution shift) has been a major research topic in machine learning for some time. This paper takes the perspective of Kouw and Loog (2021) and studies the case where feature observations from the test dataset are available for analysis but observations of labels are missing. Under these circumstances, without any assumptions on the nature of the distribution shift between the training and test datasets meaningful prediction of the labels in the test dataset or of their distribution is not feasible. See Kouw and Loog (2021) for a survey of approaches to domain adaptation and their related assumptions. Arguably, covariate shift (also known as population drift) and label shift (also known as prior probability shift or target shift) are the most popular specific distribution shift assumptions, both for their intuiveness as well as their computational manageability. However, exclusive covariate and label shift assumptions have been criticised for being insufficient for common domain adaptation tasks (e.g.

We study the learning dynamics of a multi-pass, mini-batch Stochastic Gradient Descent (SGD) procedure for empirical risk minimization in high-dimensional multi-index models with isotropic random data. In an asymptotic regime where the sample size $n$ and data dimension $d$ increase proportionally, for any sub-linear batch size $κ\asymp n^α$ where $α\in [0,1)$, and for a commensurate ``critical'' scaling of the learning rate, we provide an asymptotically exact characterization of the coordinate-wise dynamics of SGD. This characterization takes the form of a system of dynamical mean-field equations, driven by a scalar Poisson jump process that represents the asymptotic limit of SGD sampling noise. We develop an analogous characterization of the Stochastic Modified Equation (SME) which provides a Gaussian diffusion approximation to SGD. Our analyses imply that the limiting dynamics for SGD are the same for any batch size scaling $α\in [0,1)$, and that under a commensurate scaling of the learning rate, dynamics of SGD, SME, and gradient flow are mutually distinct, with those of SGD and SME coinciding in the special case of a linear model. We recover a known dynamical mean-field characterization of gradient flow in a limit of small learning rate, and of one-pass/online SGD in a limit of increasing sample size $n/d \to \infty$.

Adaptive sampling schemes are well known to create complex dependence that may invalidate conventional inference methods. A recent line of work shows that this need not be the case for UCB-type algorithms in multi-armed bandits. A central emerging theme is a `stability' property with asymptotically deterministic arm-pull counts in these algorithms, making inference as easy as in the i.i.d. setting. In this paper, we study the precise arm-pull dynamics in another canonical class of Thompson-sampling type algorithms. We show that the phenomenology is qualitatively different: the arm-pull count is asymptotically deterministic if and only if the arm is suboptimal or is the unique optimal arm; otherwise it converges in distribution to the unique invariant law of an SDE. This dichotomy uncovers a unifying principle behind many existing (in)stability results: an arm is stable if and only if its interaction with statistical noise is asymptotically negligible. As an application, we show that normalized arm means obey the same dichotomy, with Gaussian limits for stable arms and a semi-universal, non-Gaussian limit for unstable arms. This not only enables the construction of confidence intervals for the unknown mean rewards despite non-normality, but also reveals the potential of developing tractable inference procedures beyond the stable regime. The proofs rely on two new approaches. For suboptimal arms, we develop an `inverse process' approach that characterizes the inverse of the arm-pull count process via a Stieltjes integral. For optimal arms, we adopt a reparametrization of the arm-pull and noise processes that reduces the singularity in the natural SDE to proving the uniqueness of the invariant law of another SDE. We prove the latter by a set of analytic tools, including the parabolic Hörmander condition and the Stroock-Varadhan support theorem.

The achievement of the 2030 Sustainable Development Goals (SDGs) is dependent upon strategic resource distribution. We propose a causal discovery framework using Panel Vector Autoregression, along with both country-specific fixed effects and PCMCI+ conditional independence testing on 168 countries (2000-2025) to develop the first complete causal architecture of SDG dependencies. Utilizing 8 strategically chosen SDGs, we identify a distributed causal network (i.e., no single 'hub' SDG), with 10 statistically significant Granger-causal relationships identified as 11 unique direct effects. Education to Inequality is identified as the most statistically significant direct relationship (r = -0.599; p < 0.05), while effect magnitude significantly varies depending on income levels (e.g., high-income: r = -0.65; lower-middle-income: r = -0.06; non-significant). We also reject the idea that there exists a single 'keystone' SDG. Additionally, we offer a proposed tiered priority framework for the SDGs namely, identifying upstream drivers (Education, Growth), enabling goals (Institutions, Energy), and downstream outcomes (Poverty, Health). Therefore, we conclude that effective SDG acceleration can be accomplished through coordinated multi-dimensional intervention(s), and that single-goal sequential strategies are insufficient.

Learning a stationary diffusion amounts to estimating the parameters of a stochastic differential equation whose stationary distribution matches a target distribution. We build on the recently introduced kernel deviation from stationarity (KDS), which enforces stationarity by evaluating expectations of the diffusion's generator in a reproducing kernel Hilbert space. Leveraging the connection between KDS and Stein discrepancies, we introduce the Stein-type KDS (SKDS) as an alternative formulation. We prove that a vanishing SKDS guarantees alignment of the learned diffusion's stationary distribution with the target. Furthermore, under broad parametrizations, SKDS is convex with an empirical version that is $ε$-quasiconvex with high probability. Empirically, learning with SKDS attains comparable accuracy to KDS while substantially reducing computational cost and yields improvements over the majority of competitive baselines.

The world's first ultra-realistic robot artist Ai-Da Robot can create architectural designs using camera eyes and AI algorithms. Her Space Pod concept envisions modular housing.

In a number of instances, investigation showed that while one Telegram channel had been shut down, another with a near-identical name remained active. In a number of instances, investigation showed that while one Telegram channel had been shut down, another with a near-identical name remained active. Millions of people around the world are creating and sharing deepfake nudes on the secure messaging app Telegram, a Guardian analysis has shown, as the spread of advanced AI tools industrialises the online abuse of women. The Guardian has identified at least 150 Telegram channels - large encrypted group chats popular for their secure communication - that appear to have users in many countries, from the UK to Brazil, China to Nigeria, Russia to India. Some of them offer "nudified" photos or videos for a fee: users can upload a photo of any woman, and AI will produce a video of that woman performing sexual acts.

Damning new video shows Alex Pretti running at ICE agents and screaming in their faces before smashing feds' tail light Costco faces lawsuit over beloved $4.99 rotisserie chicken amid claims customers were misled What bread's really doing to your body: ROSAMUND DEAN gave it up for a month and tested her health before and after. From blood pressure, to cholesterol and her weight, the results are truly shocking... Man who Beckhams have gone to war with: Why David and Victoria fear'smiling crocodile' Nelson Peltz is behind Brooklyn's outburst... as ALISON BOSHOFF reveals 14-year legal row that proves his'bully billionaire' reputation Kim Kardashian says Harry and Meghan's team asked for removal of party photos when they'realised it was Remembrance Day' 'I can kill you at any time:' Surgeon's fatal threats to ex-wife before she was found dead with her new partner at home, new documents show Jealous killer's cold last words as he's executed for brutal murders of his ex and her new boyfriend Ilhan Omar accuses Trump of being'obsessed' with her as police identify potential liquid sprayed at her during town hall attack Weekend's monster storm to bring freezing temperatures to MIAMI as cold snap ices over millions of Americans Megyn Kelly blasts Alex Pretti for'stalking, harassing and terrorizing' ICE after video shows him kicking SUV's tail light and spitting at agents Telling detail stitched into Melania's Dior dress that hints at her true ambitions, as the First Lady rings the New York Stock Exchange bell: JANE TIPPETT The Kristi Noem call that left Tom Homan seething. The freeze-out no one saw... and why insiders say his Minneapolis mission is do or die Origins of Egypt's Great Pyramid upended as new clues point to lost civilization from 20,000 years ago Margot Robbie stuns in Elizabeth Taylor's iconic necklace as she is joined by Jacob Elordi at Wuthering Heights premiere in LA Hilarious live gaffe on David Muir's World News Tonight that'triggered behind the scenes meltdown' Alex Pretti spits at ICE agents and smashes federal SUV's tail light in shocking footage taken 11 days before he was shot dead Extraordinary transformation of beloved child star who has'self-canceled' and ditched Hollywood to live off grid in POVERTY as'Catholic extremist' NASA gives a glimpse inside Orion's cramped quarters where four astronauts will live for 10 days as they whizz around the moon - 'the smell would be intolerable!' With the first launch window for Artemis II now just days away, NASA has shared a glimpse inside the cramped quarters of the Orion spacecraft. Four astronauts - Reid Wiseman, Victor Glover, Christina Koch, and Jeremy Hansen - will spend 10 days living inside the capsule as they whizz around the moon.

Dozens killed in RSF drone attack in war-torn Sudan's South Kordofan Dozens of people have been killed in a drone attack by the paramilitary Rapid Support Forces (RSF) on a key town in war-torn Sudan's South Kordofan state, according to local media reports. Multiple areas of Dilling, including the headquarters of the Sudanese army's 54th Brigade and the central market, were struck by suicide drones during Wednesday's attack, the Sudan Tribune reported, citing local sources and medical groups. Dilling lies halfway between Kadugli - the besieged state capital - and el-Obeid, the capital of neighbouring North Kordofan province, which the RSF has sought to encircle. The RSF and the SAF have been waging a brutal civil war for control of Sudan since April 2023, which has killed thousands of people and displaced millions. Since the siege was lifted, Dilling has endured a wave of drone attacks that have destroyed service facilities and caused several casualties.