The transfer of volatiles from the Earth's interior to the atmosphere occurs through degassing of magma, the dynamics of which assert a significant control on volcanic eruptions. The first and most critical step in degassing is the nucleation of gas bubbles, which requires that a sufficient number of volatile molecules cluster together to overcome the free energy associated with the formation of a new interface between nucleus and surrounding melt. This free energy is a function of surface tension, typically assumed to equate to the macroscopically measurable value. Surface tension estimates inferred from bubble nucleation experiments in silicate melts are, however, lower than direct macroscopic measurements, making it difficult to accurately predict magma ascent and decompression rates from measured bubble number densities in pyroclasts. We provide a potential resolution to this problem through an integrated study of bubble nucleation experiments and modeling thereof, based on nonclassical nucleation theory. We find that surface tension between critical bubble nuclei and the surrounding melt depends on the degree of supersaturation and is lower than the macroscopically measured value. This is consistent with the view that far from equilibrium the interface between a nucleus and surrounding metastable bulk phase is diffuse instead of sharp. As a consequence, the increase in nucleation rate with supersaturation is significantly larger at high supersaturations than predicted by classical nucleation theory.