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Programming language researchers have developed a wide array of tools to reason about probabilistic programs, but it remains unclear how to apply these proof techniques to continuous-time stochastic processes. To bridge this gap, we propose a verification method for properties of continuous-time stochastic processes by reasoning about them as standard programs. To demonstrate our approach, we define a continuous semantics for a probabilistic imperative language $\mathrm{pWhile}^{\mathrm{dt}}$ by taking the Kuratowski limit of the standard semantics, which collects limit points of the standard program semantics as the time-step approaches zero. Then, we present a Hoare-style logic that uses loop invariants and decreasing functions to verify properties of continuous programs. To establish soundness, we identify a broad class of assertions that are preserved from the standard semantics to the continuous semantics. To demonstrate our approach, we verify properties of ten continuous-time stochastic processes modeled in $\mathrm{pWhile}^{\mathrm{dt}}$, including variants of Brownian motion and a stochastic hybrid system.

Federated learning (FL) is a machine learning setting where many clients (e.g. mobile devices or whole organizations) collaboratively train a model under the orchestration of a central server (e.g. service provider), while keeping the training data decentralized. FL embodies the principles of focused data collection and minimization, and can mitigate many of the systemic privacy risks and costs resulting from traditional, centralized machine learning and data science approaches. Motivated by the explosive growth in FL research, this paper discusses recent advances and presents an extensive collection of open problems and challenges.