Recent years have witnessed substantial progress in understanding the behavior of EM for mixture models that are correctly specified. Given that model misspecification is common in practice, it is important to understand EM in this more general setting. We provide non-asymptotic guarantees for the population and sample-based EM algorithms when used to estimate parameters of certain misspecified Gaussian mixture models.

Could Iran be using China's BeiDou system? Footage shows a fire burning near Dubai International Airport after a drone ignited a fuel tank, according to authorities in the UAE. Civil defence crews say the blaze is under control. What is force majeure and why are some Gulf countries invoking it?

We consider semidefinite programs (SDPs) of size nwith equality constraints. In order to overcome scalability issues, Burer and Monteiro proposed a factorized approach based on optimizing over a matrix Y of size n ksuch that X = YY is the SDP variable. The advantages of such formulation are twofold: the dimension of the optimization variable is reduced, and positive semidefiniteness is naturally enforced. However, optimization in Y is non-convex. In prior work, it has been shown that, when the constraints on the factorized variable regularly define a smooth manifold, provided k is large enough, for almost all cost matrices, all second-order stationary points (SOSPs) are optimal. Importantly, in practice, one can only compute points which approximately satisfy necessary optimality conditions, leading to the question: are such points also approximately optimal? To answer it, under similar assumptions, we use smoothed analysis to show that approximate SOSPs for a randomly perturbed objective function are approximate global optima, with k scaling like the square root of the number of constraints (up to log factors). Moreover, we bound the optimality gap at the approximate solution of the perturbed problem with respect to the original problem.

We develop a novel computationally efficient and general framework for robust hypothesis testing. The new framework features a new way to construct uncertainty sets under the null and the alternative distributions, which are sets centered around the empirical distribution defined via Wasserstein metric, thus our approach is data-driven and free of distributional assumptions. We develop a convex safe approximation of the minimax formulation and show that such approximation renders a nearly-optimal detector among the family of all possible tests. By exploiting the structure of the least favorable distribution, we also develop a tractable reformulation of such approximation, with complexity independent of the dimension of observation space and can be nearly sample-size-independent in general. Real-data example using human activity data demonstrated the excellent performance of the new robust detector.

Race on to establish globally recognised'AI-free' logo Organisations worldwide are racing to develop a universally recognised label for human-made products and services as part of the growing backlash against AI use. Declarations like Proudly Human, Human-made, 'No A.I and AI-free are appearing across films, marketing, books and websites. It is in response to fears that jobs or entire professions are being swept away in a wave of AI-powered automation. BBC News has counted at least eight different initiatives trying to come up with a label that could get the kind of global recognition that the Fair Trade logo has for ethically made products. But with so many competing labels - as well as confusion over the definition of AI-free - experts say consumers are in danger of being left confused unless a single standard can be agreed on.

'We will go wherever they hide': Rooting out IS in Somalia A figure appears in the picture, moving through a valley. He has been to fetch water for his friends, says the drone operator. He is running and carrying something on his back, adds another soldier. The man on the screen is near a cave, which the army believes is a hideout for 50 to 60 IS fighters. The Puntland Defence Forces have about 500 soldiers stationed at this base in the north-east of Somalia. Ten years ago the barren and inhospitable landscape was home to only a few nomadic communities, but that changed when IS established a foothold here, shifting its focus to Africa as its fighters were driven out of their strongholds in Syria and Iraq.

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The first reports appeared on foreign screens, beyond the reach of most Iranians. On 28 February Prime Minister Benjamin Netanyahu said there were signs that the tyrant is no more, suggesting Supreme Leader Ayatollah Ali Khamenei had been killed in a joint US-Israeli strike. Iranians watching state television, however, encountered silence. Government officials would neither confirm nor deny Khamenei's death. On one of the state broadcaster's channels, IRTV3, one news presenter urged viewers to trust him and the latest information the government had.

Offline reinforcement learning (RL) aims to learn decision policies from a fixed batch of logged transitions, without additional environment interaction. Despite remarkable empirical progress, offline RL remains fragile under distribution shifts: value-based methods can overestimate the value of unseen actions, yielding policies that exploit model errors rather than genuine long-term rewards. We propose \emph{Bayesian Conservative Policy Optimization (BCPO)}, a unified framework that converts epistemic uncertainty into \emph{provably conservative} policy improvement. BCPO maintains a hierarchical Bayesian posterior over environment/value models, constructs a \emph{credible lower bound} (LCB) on action values, and performs policy updates under explicit KL regularization toward the behavior distribution. This yields an uncertainty-calibrated analogue of conservative policy iteration in the offline regime. We provide a finite-MDP theory showing that the pessimistic fixed point lower-bounds the true value function with high probability and that KL-controlled updates improve a computable return lower bound. Empirically, we verify the methodology on a real offline replay dataset for the CartPole benchmark obtained via the \texttt{d3rlpy} ecosystem, and report diagnostics that link uncertainty growth and policy drift to offline instability, motivating principled early stopping and calibration