High-entropy alloys are solid solutions of multiple principal elements that are capable of reaching composition and property regimes inaccessible for dilute materials. Discovering those with valuable properties, however, too often relies on serendipity, because thermodynamic alloy design rules alone often fail in high-dimensional composition spaces. We propose an active learning strategy to accelerate the design of high-entropy Invar alloys in a practically infinite compositional space based on very sparse data. Our approach works as a closed-loop, integrating machine learning with density-functional theory, thermodynamic calculations, and experiments. After processing and characterizing 17 new alloys out of millions of possible compositions, we identified two high-entropy Invar alloys with extremely low thermal expansion coefficients around 2 × 10-6 per degree kelvin at 300 kelvin. We believe this to be a suitable pathway for the fast and automated discovery of high-entropy alloys with optimal thermal, magnetic, and electrical properties.