A large sample of spectroscopically confirmed galaxies at 1.4<z<3.7, with complementary imaging in the near- and mid-IR from the ground and from Hubble and Spitzer, is used to infer the average star formation histories (SFHs) of typical galaxies from z~7 to 2. For a subset of 302 galaxies at 1.5<z<2.6, we perform a comparison of star formation rates (SFRs) determined from SED modeling (SFRs[SED]) and those calculated from deep Keck UV and Spitzer/MIPS 24 micron imaging (SFRs[IR+UV]). Exponentially declining SFHs yield SFRs[SED] that are 5-10x lower on average than SFRs[IR+UV], indicating that declining SFHs may not be accurate for typical galaxies at z>2. The SFRs of z~2-3 galaxies are directly proportional to their stellar masses M*, with unity slope---a result that is confirmed with Spitzer/IRAC stacks of 1179 UV-faint (R>25.5) galaxies---for M*>5e8 Msun and SFRs >2 Msun/yr. We interpret this result in the context of several systematic biases that can affect determinations of the SFR-M* relation. The average specific SFRs at z~2-3 are similar within a factor of two to those measured at z>4, implying an average SFH where SFRs increase with time. A consequence of these rising SFHs is that (a) a substantial fraction of UV-bright z~2-3 galaxies had faint sub-L* progenitors at z>4; and (b) gas masses must increase with time from z=7 to 2, over which time the net cold gas accretion rate---as inferred from the specific SFR and the Kennicutt-Schmidt relation---is ~2-3x larger than the SFR . However, if we evolve to higher redshift the SFHs and masses of the halos that are expected to host L* galaxies at z~2, we find that <10% of the baryons accreted onto typical halos at z>4 actually contribute to star formation at those epochs. These results highlight the relative inefficiency of star formation even at early cosmic times when galaxies were first assembling. [Abridged]
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