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Previous work has shown that Lyman break galaxies (LBGs) display a range in structures (from single and compact to more clumpy and extended) that is different from typical local star-forming galaxies. Recently, we have introduced a sample of rare, nearby (z < 0.3) starburst galaxies that appear to be good analogs of LBGs. These 'Lyman break analogs' (LBAs) provide an excellent training set for understanding starbursts at different redshifts. We present an application of this by comparing the rest-frame ultraviolet (UV) and optical morphologies of 30 LBAs with those of galaxies at z {approx} 2-4 in the Hubble Ultra Deep Field. We compare LBAs with star-forming sBzK galaxies at z {approx} 2, and LBGs at z {approx} 3-4 at the same intrinsic UV luminosity (L{sub UV} {approx}> 0.3L*{sub z=3}). The UV/optical colors and sizes of LBAs and LBGs are very similar, while the BzK galaxies are somewhat redder and larger. LBAs lie along a mass-metallicity relation that is offset from that of typical local galaxies, but similar to that seen at z {approx} 2. There is significant overlap between the morphologies (G, C, A, and M{sub 20}) of the local and high-redshift samples, although the high-redshift samples are somewhat less concentrated and clumpier than the LBAs. Based on their highly asymmetric morphologies, we find that in the majority of LBAs the starbursts appear to be triggered by interactions/mergers. When the images of the LBAs are degraded to the same sensitivity and linear resolution as the images of LBGs and BzK galaxies, we find that these relatively faint asymmetric features are no longer detectable. This effect is particularly severe in the rest-frame UV. It has been suggested that high-redshift galaxies experience intense bursts unlike anything seen in the local universe, possibly due to cold flows and instabilities. In part, this is based on the fact that the majority ({approx}70%) of LBGs do not show morphological signatures of interactions or mergers. Our results suggest that this evidence is insufficient, since a large fraction of such signatures would likely have been missed in current observations of galaxies at z {approx} 2-4. This leaves open the possibility that clumpy accretion and mergers remain important in driving the evolution of these starbursts, together with rapid gas accretion through other means.