Yi shouldn't win against any other carry top laner in the early game unless they make a serious misplay. Yi's primary source of damage in the early game is auto attacks, while every other carry top laner will usually have some sort of spell or cc damage to boost their early damage up. The extra true damage you get as Yi in the early game is only another ~20 damage per auto attack, while somebody like Riven will use Q to spell weave and deal an extra 20 damage per auto, plus the extra damage from he spell casts, plus the cc she'll get from her spells.
Literally, unless they're an idiot and fight in the middle of the minion wave, or don't make good use of their abilities, or let you engage whenever you have your passive ready without trying to punish you for playing aggressive with it, Yi should never win a fight in lane until items start to factor in. Heck, even with items, it still takes a while until Yi hits critical mass against people like Yas or Riven, and should pretty much never win a fight against a Jax unless his E is down.