Membership in Ris already proven by Abrahamsen, Kleist and Miltzow in [3]. Thealgorithm then needs to verify that the neural network described byΘ fits all data points inD with a total error at mostγ. The goal of this appendix is to build a geometric understanding off(,Θ). We point the interested reader to these articles [6, 26, 49, 66, 92] investigating the set of functions exactly represented by different architecturesofReLUnetworks. To see that this observation is true, consider the following construction.

Decision trees have gained popularity as interpretable surrogate models for learning-based control policies.

Yukai Engineering's latest gadget at CES is a fan for babies Baby FuFu is its name, and baby cooling is its game. Baby FuFu will launch in mid-2026 for around $50 to $60. (Yukai Engineering) Yukai Engineering, maker of the weirdly cute Mirumi robot, has another interesting gadget at CES 2026. Baby FuFu is a portable fan for babies that attaches to strollers. Baby FuFu is modeled on the company's smaller (but otherwise identical-looking) drink-cooling gadget, Nekojita FuFu. Baby FuFu grew out of Nekojita FuFu fans' feedback that their children love not only cooling their food with it but also playing with it, pretending to fan their faces and blow-dry their hair, Yukai Engineering CEO Shunsuke Aoki said. Baby FuFu, positioned on the stroller handle.

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The Public Says'No Thanks' The maker of the eCoffee Energyband says it electrically stimulates your nerves to keep you alert. Researchers are skeptical, and critics see it as a way for China's bosses to keep workers productive. Forget coffee, you can now stay alert by strapping on a wristband that lightly zaps you awake. That's what eCoffee Energyband, a Chinese gadget that sells for just over $100, is claiming to do. First released in late 2023, the product is a lightweight wearable with two electrode pads that sit against the inner wrist.

Many real-world decision-making problems involve optimizing multiple objectives simultaneously, rendering the selection of the most preferred solution a non-trivial problem: All Pareto optimal solutions are viable candidates, and it is typically up to a decision maker to select one for implementation based on their subjective preferences. To reduce the cognitive load on the decision maker, previous work has introduced the Pareto pruning problem, where the goal is to compute a fixed-size subset of Pareto optimal solutions that best represent the full set, as evaluated by a given quality measure. Reframing Pareto pruning as a multiwinner voting problem, we conduct an axiomatic analysis of existing quality measures, uncovering several unintuitive behaviors. Motivated by these findings, we introduce a new measure, directed coverage. We also analyze the computational complexity of optimizing various quality measures, identifying previously unknown boundaries between tractable and intractable cases depending on the number and structure of the objectives. Finally, we present an experimental evaluation, demonstrating that the choice of quality measure has a decisive impact on the characteristics of the selected set of solutions and that our proposed measure performs competitively or even favorably across a range of settings.

We study the problem of learning a $n$-variables $k$-CNF formula $Φ$ from its i.i.d. uniform random solutions, which is equivalent to learning a Boolean Markov random field (MRF) with $k$-wise hard constraints. Revisiting Valiant's algorithm (Commun. ACM'84), we show that it can exactly learn (1) $k$-CNFs with bounded clause intersection size under Lovász local lemma type conditions, from $O(\log n)$ samples; and (2) random $k$-CNFs near the satisfiability threshold, from $\widetilde{O}(n^{\exp(-\sqrt{k})})$ samples. These results significantly improve the previous $O(n^k)$ sample complexity. We further establish new information-theoretic lower bounds on sample complexity for both exact and approximate learning from i.i.d. uniform random solutions.