We address the problem of nonparametric estimation of the spectral density for a centered stationary Gaussian time series under local differential privacy constraints. Specifically, we propose new interactive privacy mechanisms for three tasks: estimating a single covariance coefficient, estimating the spectral density at a fixed frequency, and estimating the entire spectral density function. Our approach achieves faster rates through a two-stage process: we apply first the Laplace mechanism to the truncated value and then use the former privatized sample to gain knowledge on the dependence mechanism in the time series. For spectral densities belonging to H\"older and Sobolev smoothness classes, we demonstrate that our estimators improve upon the non-interactive mechanism of Kroll (2024) for small privacy parameter $\alpha$, since the pointwise rates depend on $n\alpha^2$ instead of $n\alpha^4$. Moreover, we show that the rate $(n\alpha^4)^{-1}$ is optimal for estimating a covariance coefficient with non-interactive mechanisms. However, the $L_2$ rate of our interactive estimator is slower than the pointwise rate. We show how to use these estimators to provide a bona-fide locally differentially private covariance matrix estimator.

Ask the search engine Ecosia about "Paris to Prague" and flight booking websites dominate the results. Ecosia's CEO Christian Kroll would prefer to present more train options, which he considers better for the environment. But because its results are licensed from Google and Microsoft's Bing, Ecosia has little control over what's shown. Kroll is ready for that to change. Berlin-based Ecosia, which donates its profits to tree planting, and its Paris-based competitor Qwant are announcing Tuesday that they will team up to develop an index of the web.

In the era of search wars fought between giants, it's tough to be small. Berlin-based Ecosia offers a search engine for the climate-conscious, promising to be carbon-negative by investing all of its profits into planting trees--more than 180 million of them since it launched in 2009. It's not likely to topple Google, but it has won a stable clientele of around 20 million users with that green branding and by repackaging search results from Microsoft's Bing. But after a decade of little change in the search business, everything is now in flux, thanks to generative AI. "I've never seen so much change in the market as in the last six months," says Christian Kroll, Ecosia's CEO. The tumult has forced Ecosia to rethink its business plan in order to compete with new chatbot-like search engines built on large language models.

Ecosia, the search engine that uses its profits to plant trees, is launching a new shopping feature. The company, which was founded in 2009, donates its expendable funds to tree-planting organizations. It claims to have planted 130 million trees across 30 countries around the world. Ecosia Shopping recommends products on Amazon, Kelkoo and Idealo, and other shopping partners that have been sustainably made, are reused, or have been refurbished. The feature is available now in the UK, Germany and France.

At first glance, the Berlin startup doesn't seem so different from others: a factory floor in the rear courtyard of a building in the city's Neukölln district, stacked preserving jars filled with muesli in the kitchen, a discarded ping-pong surface repurposed as a conference table. The employees are young, relaxed and very international. The company's head and founder, Christian Kroll, is 35 years old, the same age as Mark Zuckerberg. The two men also share a quirk: To avoid wasting time in the mornings choosing an outfit, he always wears the same thing -- in his case, blank white T-shirts made from organic cotton. Zuckerberg's favorite color, by contrast, is gray.

As we feel our way through the haze that is a combination of corporate party and holiday-season prep, Monday kicks off with stories on Alexa's new security talents, a car coming in 2021 that we already drove, and expect Year In Review reports to start hitting Engadget later this week. Amazon's framework can arm your system with just your voice. Amazon has upgraded its voice assistant to work with security systems. You can arm or disarm them, specify certain modes (home, away and night) and simply check in. The functionality is available now in the US, with companies like Abode, ADT, Honeywell, Ring and Scout Alarm already using it.