There is a very famous Yam Shel Shlomo Bava Kama, Perek 7 siman 37, and also after the end of Chapter 7 there is a paragraph, (it begins "garsinan") about siyum being a seudas mitzvah. He makes a new principle, that a feast for completing or publicizing a mitzvah is a seudas mitzvah, as we see from the end of Taanis about Tu B'av. He discusses Bar Mitzvahs, Chanukah, a new house and more; see there if each of these things are seudas mitzvah or not. He says a siyum is definitely a seudas mitzvah, and we see from Shabbos 118b-119a that Abaye would make a yom tov for all the students when a young talmid chacham finished a masechta, yom tov meaning a seuda, that there is an inyan to make a seuda for a siyum. Magen Avraham, O"C 591:33 (commenting on Rama there) cites this.
It is a little surprising that he does not mention an early source - a Midrash - for making a seudah on finishing learning Torah. In Melachim I 3:5-15 Hashem offered Shlomo, in a dream, a choice and Shlomo chose to understand everything, and he did not choose wealth or power over enemies. After he woke up and had so much understanding, he made a big feast. Shir Hashirim Rabbah 1:9 says: R. Elazar said from here we learn to make a seudah on the completion of the Torah.
A few days before we graduated high school, Rav Ahron Solovechik zt"l told us that Chosson Breishis is more important than Chosson Torah. Finishing isn't what's important. The more important thing is to start learning the completed books again, in greater depth, with deeper understanding. (See Reshimos Shiurim of the Rav on Sukkah at the very end, beinyan siyum, similar thoughts are presented there in Rav Ahron's name).
While we're on the topic of this Yam Shel Shlomo, he very strongly supports the minhag of making a seuda for a Bar Mitzvah. He feels "ein mitzvah gedola mizu." However, Igros Moshe O"C I 104 (from the year 1957) says if it were up to him, he would not have anyone make a bar mitzvah the way they make them in the US.
Some points about Bas mitzvah-
Rav Ahron Soloveichik in Perach Mateh Ahron Tefila 7:6 discusses making "barush sheptarani" on one's daughter's bas mitzvah. Basically, he says according to the Levush you would, and according to Magen Avraham peopel claim you would not, but he rejects that and says even to M"A you would.
Rav Moshe Feinstein zt"l was very much against Bas mitzvah seudas (and above we said he wasn't thrilled with Bar Miztvahs either). See Igros Moshe O"C I 104, O"C II 30 & 97 (cites Magen Avraham O"C 225:4 & Dagul Mervavah Y"D 391:2). He syas Bas Mitzvah is no different than a birthday party. It's not a seudas mitzvah, it can't be in a shul, and a reason why Bas Mitzvah is different tyhan Bar Mitzvah is that when a boy turns 13 we see something has changed - before we can't count him in a minyan but now we can. But for a girl there is no recognizable change before 12 and afetr 12, so there is no inyan to have a seuda.