Conformal prediction constructs prediction sets with finite-sample coverage guarantees, but its calibration stage is structurally constrained to a scalar score function and a single threshold variable -- forcing shapes of prediction sets to be fixed before calibration, typically through data splitting. We introduce multi-variable conformal prediction (MCP), a framework that extends conformal prediction to vector-valued score functions with multiple simultaneous calibration variables. Building on scenario theory as a principled framework for certifying data-driven decisions, MCP unifies prediction set design and calibration into a single optimization problem, eliminating data splitting without sacrificing coverage guarantees. We propose two computationally efficient variants: RemMCP, grounded in constrained optimization with constraint removal, which admits a clean generalization of split conformal prediction; and RelMCP, based on iterative optimization with constraint relaxation, which supports non-convex score functions at the cost of possibly greater conservatism. Through numerical experiments on ellipsoidal and multi-modal prediction sets, we demonstrate that RemMCP and RelMCP consistently meet the target coverage with prediction set sizes smaller than or comparable to those of baselines with data split, while considerably reducing variance across calibration runs -- a direct consequence of using all available data for shape optimization and calibration simultaneously.

We propose and analyze a model-based bootstrap for transition kernels in finite controlled Markov chains (CMCs) with possibly nonstationary or history-dependent control policies, a setting that arises naturally in offline reinforcement learning (RL) when the behavior policy generating the data is unknown. We establish distributional consistency of the bootstrap transition estimator in both a single long-chain regime and the episodic offline RL regime. The key technical tools are a novel bootstrap law of large numbers (LLN) for the visitation counts and a novel use of the martingale central limit theorem (CLT) for the bootstrap transition increments. We extend bootstrap distributional consistency to the downstream targets of offline policy evaluation (OPE) and optimal policy recovery (OPR) via the delta method by verifying Hadamard differentiability of the Bellman operators, yielding asymptotically valid confidence intervals for value and $Q$-functions. Experiments on the RiverSwim problem show that the proposed bootstrap confidence intervals (CIs), especially the percentile CIs, outperform the episodic bootstrap and plug-in CLT CIs, and are often close to nominal ($50\%$, $90\%$, $95\%$) coverage, while the baselines are poorly calibrated at small sample sizes and short episode lengths.

We propose an algorithm to sample from composite log-concave distributions over $\mathbb{R}^d$, i.e., densities of the form $π\propto e^{-f-g}$, assuming access to gradient evaluations of $f$ and a restricted Gaussian oracle (RGO) for $g$. The latter requirement means that we can easily sample from the density $\text{RGO}_{g,h,y}(x) \propto \exp(-g(x) -\frac{1}{2h}||y-x||^2)$, which is the sampling analogue of the proximal operator for $g$. If $f + g$ is $α$-strongly convex and $f$ is $β$-smooth, our sampler achieves $\varepsilon$ error in total variation distance in $\widetilde{\mathcal O}(κ\sqrt d \log^4(1/\varepsilon))$ iterations where $κ:= β/α$, which matches prior state-of-the-art results for the case $g=0$. We further extend our results to cases where (1) $π$ is non-log-concave but satisfies a Poincaré or log-Sobolev inequality, and (2) $f$ is non-smooth but Lipschitz.

We introduce Pion, a spectrum-preserving optimizer for large language model (LLM) training based on orthogonal equivalence transformation. Unlike additive optimizers such as Adam and Muon, Pion updates each weight matrix through left and right orthogonal transformations, preserving its singular values throughout training. This yields an optimization mechanism that modulates the geometry of weight matrices while keeping their spectral norm fixed. We derive the Pion update rule, systematically examine its design choices, and analyze its convergence behavior along with several key properties. Empirical results show that Pion offers a stable and competitive alternative to standard optimizers for both LLM pretraining and finetuning.

Chinese firm unveils'transformer' style manned robot NewsFeed Chinese firm unveils'transformer' style manned robot Unitree Robotics has released footage of its CEO piloting the GD01, a 2.7-metre transformable mecha that smashes through walls with its mechanical arms. The machine is priced from $650,000. Starmer at risk because he pushed Labour to be'new Conservative Party' Trump skirts question on US'red lines' for Iran ceasefire How one Palestinian teen's life changed forever after Israeli gunfire

Lenovo's latest ThinkPad X13 Gen 7 helps launch the AMD Ryzen AI 400, a chip we haven't seen anything of since its January launch. The laptop has also earned itself a top score from iFixit for its accessible, replaceable components.

If You Have $650,000 and Don't Buy This Giant Mecha Robot You're a Fool China's Unitree, famous for making low-cost dancing robots, will now sell you a giant, wall-smashing mecha. Unitree is a Chinese company known for making adorable, relatively affordable robots that dance and shuffle and such. Last night, it revealed its latest creation, which is something of a departure: a giant, walking, crawling, transforming, wall-smashing "mecha" called the GD01. An introductory video for the GD01--set to a thundering rock guitar soundtrack--shows the company's founder and CEO, Xingxing Wang, holding hands with the robot before climbing into its prodigious, open-air belly. A disclaimer added to Unitree's social media post reads: "Please everyone be sure to use the robot in a Friendly and Safe manner."

In a United States court, OpenAI chief executive Sam Altman has rejected claims from fellow tech mogul Elon Musk that he betrayed the artificial intelligence company's original vision. Tuesday marked the start of Altman's testimony in a contentious trial unfolding in Oakland, California, between some of tech's richest and most powerful titans. He alleged that OpenAI's leader persuaded him to invest $38bn, based on a goal of improving humanity, only to see the company pivot to a for-profit venture in 2019. On the witness stand on Tuesday, Altman instead framed Musk as a competitor obsessed with exercising control over OpenAI. "It does not fit with my conception of the words'stealing a charity' to look at what has actually happened here," Altman told the court.

When China's leader Xi Jinping hosts his American counterpart in Beijing this week, Donald Trump will be reminded of his last visit in 2017 - he was wooed hard, complete with dinner inside the Forbidden City, an honour no US president before him had received. This week's reception promises to be just as grand, including a stop inside Zhongnanhai, the rarefied compound where China's top leadership lives and works. The agenda too will be just as thorny, with Iran being a new source of tension, alongside trade, technology and Taiwan. But a lot has changed as Trump returns to a stronger and far more assertive China. Now well into an unprecedented third term, an ambitious Xi has been pushing forward with plans for new productive forces with heavy investments in renewable energy, robotics and artificial intelligence.

The new camera options will join the other features Apple will reportedly highlight at WWDC 2026: performance improvements and AI . The biggest change Apple is making to the Camera app is to make it more customizable. Rather than being stuck with the company's predetermined interface for shooting photos and capturing videos, you'll reportedly be able to tweak it to your liking. The app will reportedly also include more advanced options like controls for depth-of-field, exposure and the company's photo styles feature. Apple offers a theoretically easy way to tweak these settings on the iPhone by using the Camera Control button, but changing things from the touchscreen should be even easier.