ALMA observations of the long wavelength dust continuum are used to estimate the interstellar medium (ISM) masses in a sample of 708 galaxies at z=0.3 to 4.5 in the COSMOS field. The galaxy sample has known farinfrared luminosities and, hence, star formation rates (SFRs) and stellar masses ( * M ) from the optical-infrared spectrum fitting. The galaxies sample SFRs from the main sequence (MS) to 50 times above the MS. The derived ISM masses are used to determine the dependence of gas mass on redshift, * M , and specific SFR (sSFR) relative to the MS. The ISM masses increase approximately with the 0.63 power of the rate of increase in SFRs with redshift and the 0.32 power of the sSFR/sSFR MS . The SF efficiencies also increase as the 0.36 power of the SFR redshift evolution and the 0.7 power of the elevation above the MS; thus the increased activities at early epochs are driven by both increased ISM masses and SF efficiency. Using the derived ISM mass function, we estimate the accretion rates of gas required to maintain continuity of the MS evolution (>  M 100 yr −1 at z > 2.5). Simple power-law dependencies are similarly derived for the gas accretion rates. We argue that the overall evolution of galaxies is driven by the rates of gas accretion. The cosmic evolution of total ISM mass is estimated and linked to the evolution of SF and active galactic nucleus activity at early epochs.