episode stringlengths 45 100 | text stringlengths 1 528 | timestamp_link stringlengths 56 56 |
|---|---|---|
Tomaso Poggio: Brains, Minds, and Machines | Lex Fridman Podcast #13 | was you should have at least twice the number of data | https://karpathy.ai/lexicap/0013-large.html#00:42:04.520 |
Tomaso Poggio: Brains, Minds, and Machines | Lex Fridman Podcast #13 | than the number of parameters. | https://karpathy.ai/lexicap/0013-large.html#00:42:09.720 |
Tomaso Poggio: Brains, Minds, and Machines | Lex Fridman Podcast #13 | Maybe 10 times is better. | https://karpathy.ai/lexicap/0013-large.html#00:42:12.880 |
Tomaso Poggio: Brains, Minds, and Machines | Lex Fridman Podcast #13 | Now, the way you train neural networks these days | https://karpathy.ai/lexicap/0013-large.html#00:42:15.480 |
Tomaso Poggio: Brains, Minds, and Machines | Lex Fridman Podcast #13 | is that they have 10 or 100 times more parameters | https://karpathy.ai/lexicap/0013-large.html#00:42:19.560 |
Tomaso Poggio: Brains, Minds, and Machines | Lex Fridman Podcast #13 | than data, exactly the opposite. | https://karpathy.ai/lexicap/0013-large.html#00:42:23.480 |
Tomaso Poggio: Brains, Minds, and Machines | Lex Fridman Podcast #13 | And it has been one of the puzzles about neural networks. | https://karpathy.ai/lexicap/0013-large.html#00:42:26.760 |
Tomaso Poggio: Brains, Minds, and Machines | Lex Fridman Podcast #13 | How can you get something that really works | https://karpathy.ai/lexicap/0013-large.html#00:42:34.080 |
Tomaso Poggio: Brains, Minds, and Machines | Lex Fridman Podcast #13 | when you have so much freedom? | https://karpathy.ai/lexicap/0013-large.html#00:42:37.120 |
Tomaso Poggio: Brains, Minds, and Machines | Lex Fridman Podcast #13 | From that little data, it can generalize somehow. | https://karpathy.ai/lexicap/0013-large.html#00:42:40.640 |
Tomaso Poggio: Brains, Minds, and Machines | Lex Fridman Podcast #13 | Right, exactly. | https://karpathy.ai/lexicap/0013-large.html#00:42:43.000 |
Tomaso Poggio: Brains, Minds, and Machines | Lex Fridman Podcast #13 | Do you think the stochastic nature of it | https://karpathy.ai/lexicap/0013-large.html#00:42:44.200 |
Tomaso Poggio: Brains, Minds, and Machines | Lex Fridman Podcast #13 | is essential, the randomness? | https://karpathy.ai/lexicap/0013-large.html#00:42:46.400 |
Tomaso Poggio: Brains, Minds, and Machines | Lex Fridman Podcast #13 | So I think we have some initial understanding | https://karpathy.ai/lexicap/0013-large.html#00:42:48.160 |
Tomaso Poggio: Brains, Minds, and Machines | Lex Fridman Podcast #13 | why this happens. | https://karpathy.ai/lexicap/0013-large.html#00:42:50.640 |
Tomaso Poggio: Brains, Minds, and Machines | Lex Fridman Podcast #13 | But one nice side effect of having | https://karpathy.ai/lexicap/0013-large.html#00:42:52.240 |
Tomaso Poggio: Brains, Minds, and Machines | Lex Fridman Podcast #13 | this overparameterization, more parameters than data, | https://karpathy.ai/lexicap/0013-large.html#00:42:56.480 |
Tomaso Poggio: Brains, Minds, and Machines | Lex Fridman Podcast #13 | is that when you look for the minima of a loss function, | https://karpathy.ai/lexicap/0013-large.html#00:43:00.920 |
Tomaso Poggio: Brains, Minds, and Machines | Lex Fridman Podcast #13 | like stochastic gradient descent is doing, | https://karpathy.ai/lexicap/0013-large.html#00:43:04.720 |
Tomaso Poggio: Brains, Minds, and Machines | Lex Fridman Podcast #13 | you find I made some calculations based | https://karpathy.ai/lexicap/0013-large.html#00:43:08.240 |
Tomaso Poggio: Brains, Minds, and Machines | Lex Fridman Podcast #13 | on some old basic theorem of algebra called the Bezu | https://karpathy.ai/lexicap/0013-large.html#00:43:12.120 |
Tomaso Poggio: Brains, Minds, and Machines | Lex Fridman Podcast #13 | theorem that gives you an estimate of the number | https://karpathy.ai/lexicap/0013-large.html#00:43:19.040 |
Tomaso Poggio: Brains, Minds, and Machines | Lex Fridman Podcast #13 | of solution of a system of polynomial equation. | https://karpathy.ai/lexicap/0013-large.html#00:43:23.240 |
Tomaso Poggio: Brains, Minds, and Machines | Lex Fridman Podcast #13 | Anyway, the bottom line is that there are probably | https://karpathy.ai/lexicap/0013-large.html#00:43:25.960 |
Tomaso Poggio: Brains, Minds, and Machines | Lex Fridman Podcast #13 | more minima for a typical deep networks | https://karpathy.ai/lexicap/0013-large.html#00:43:30.520 |
Tomaso Poggio: Brains, Minds, and Machines | Lex Fridman Podcast #13 | than atoms in the universe. | https://karpathy.ai/lexicap/0013-large.html#00:43:36.080 |
Tomaso Poggio: Brains, Minds, and Machines | Lex Fridman Podcast #13 | Just to say, there are a lot because | https://karpathy.ai/lexicap/0013-large.html#00:43:39.480 |
Tomaso Poggio: Brains, Minds, and Machines | Lex Fridman Podcast #13 | of the overparameterization. | https://karpathy.ai/lexicap/0013-large.html#00:43:42.120 |
Tomaso Poggio: Brains, Minds, and Machines | Lex Fridman Podcast #13 | A more global minimum, zero minimum, good minimum. | https://karpathy.ai/lexicap/0013-large.html#00:43:44.760 |
Tomaso Poggio: Brains, Minds, and Machines | Lex Fridman Podcast #13 | A more global minima. | https://karpathy.ai/lexicap/0013-large.html#00:43:50.280 |
Tomaso Poggio: Brains, Minds, and Machines | Lex Fridman Podcast #13 | Yeah, a lot of them. | https://karpathy.ai/lexicap/0013-large.html#00:43:51.560 |
Tomaso Poggio: Brains, Minds, and Machines | Lex Fridman Podcast #13 | So you have a lot of solutions. | https://karpathy.ai/lexicap/0013-large.html#00:43:53.200 |
Tomaso Poggio: Brains, Minds, and Machines | Lex Fridman Podcast #13 | So it's not so surprising that you can find them | https://karpathy.ai/lexicap/0013-large.html#00:43:54.560 |
Tomaso Poggio: Brains, Minds, and Machines | Lex Fridman Podcast #13 | relatively easily. | https://karpathy.ai/lexicap/0013-large.html#00:43:57.920 |
Tomaso Poggio: Brains, Minds, and Machines | Lex Fridman Podcast #13 | And this is because of the overparameterization. | https://karpathy.ai/lexicap/0013-large.html#00:44:00.400 |
Tomaso Poggio: Brains, Minds, and Machines | Lex Fridman Podcast #13 | The overparameterization sprinkles that entire space | https://karpathy.ai/lexicap/0013-large.html#00:44:04.200 |
Tomaso Poggio: Brains, Minds, and Machines | Lex Fridman Podcast #13 | with solutions that are pretty good. | https://karpathy.ai/lexicap/0013-large.html#00:44:07.920 |
Tomaso Poggio: Brains, Minds, and Machines | Lex Fridman Podcast #13 | It's not so surprising, right? | https://karpathy.ai/lexicap/0013-large.html#00:44:09.720 |
Tomaso Poggio: Brains, Minds, and Machines | Lex Fridman Podcast #13 | It's like if you have a system of linear equation | https://karpathy.ai/lexicap/0013-large.html#00:44:11.240 |
Tomaso Poggio: Brains, Minds, and Machines | Lex Fridman Podcast #13 | and you have more unknowns than equations, then you have, | https://karpathy.ai/lexicap/0013-large.html#00:44:14.400 |
Tomaso Poggio: Brains, Minds, and Machines | Lex Fridman Podcast #13 | we know, you have an infinite number of solutions. | https://karpathy.ai/lexicap/0013-large.html#00:44:18.520 |
Tomaso Poggio: Brains, Minds, and Machines | Lex Fridman Podcast #13 | And the question is to pick one. | https://karpathy.ai/lexicap/0013-large.html#00:44:22.040 |
Tomaso Poggio: Brains, Minds, and Machines | Lex Fridman Podcast #13 | That's another story. | https://karpathy.ai/lexicap/0013-large.html#00:44:24.480 |
Tomaso Poggio: Brains, Minds, and Machines | Lex Fridman Podcast #13 | But you have an infinite number of solutions. | https://karpathy.ai/lexicap/0013-large.html#00:44:25.440 |
Tomaso Poggio: Brains, Minds, and Machines | Lex Fridman Podcast #13 | So there are a lot of value of your unknowns | https://karpathy.ai/lexicap/0013-large.html#00:44:27.520 |
Tomaso Poggio: Brains, Minds, and Machines | Lex Fridman Podcast #13 | that satisfy the equations. | https://karpathy.ai/lexicap/0013-large.html#00:44:31.040 |
Tomaso Poggio: Brains, Minds, and Machines | Lex Fridman Podcast #13 | But it's possible that there's a lot of those solutions that | https://karpathy.ai/lexicap/0013-large.html#00:44:33.160 |
Tomaso Poggio: Brains, Minds, and Machines | Lex Fridman Podcast #13 | aren't very good. | https://karpathy.ai/lexicap/0013-large.html#00:44:36.360 |
Tomaso Poggio: Brains, Minds, and Machines | Lex Fridman Podcast #13 | What's surprising is that they're pretty good. | https://karpathy.ai/lexicap/0013-large.html#00:44:37.560 |
Tomaso Poggio: Brains, Minds, and Machines | Lex Fridman Podcast #13 | So that's a good question. | https://karpathy.ai/lexicap/0013-large.html#00:44:39.160 |
Tomaso Poggio: Brains, Minds, and Machines | Lex Fridman Podcast #13 | Why can you pick one that generalizes well? | https://karpathy.ai/lexicap/0013-large.html#00:44:40.160 |
Tomaso Poggio: Brains, Minds, and Machines | Lex Fridman Podcast #13 | Yeah. | https://karpathy.ai/lexicap/0013-large.html#00:44:42.840 |
Tomaso Poggio: Brains, Minds, and Machines | Lex Fridman Podcast #13 | That's a separate question with separate answers. | https://karpathy.ai/lexicap/0013-large.html#00:44:44.120 |
Tomaso Poggio: Brains, Minds, and Machines | Lex Fridman Podcast #13 | One theorem that people like to talk about that kind of | https://karpathy.ai/lexicap/0013-large.html#00:44:47.120 |
Tomaso Poggio: Brains, Minds, and Machines | Lex Fridman Podcast #13 | inspires imagination of the power of neural networks | https://karpathy.ai/lexicap/0013-large.html#00:44:51.160 |
Tomaso Poggio: Brains, Minds, and Machines | Lex Fridman Podcast #13 | is the universality, universal approximation theorem, | https://karpathy.ai/lexicap/0013-large.html#00:44:53.800 |
Tomaso Poggio: Brains, Minds, and Machines | Lex Fridman Podcast #13 | that you can approximate any computable function | https://karpathy.ai/lexicap/0013-large.html#00:44:57.840 |
Tomaso Poggio: Brains, Minds, and Machines | Lex Fridman Podcast #13 | with just a finite number of neurons | https://karpathy.ai/lexicap/0013-large.html#00:45:00.960 |
Tomaso Poggio: Brains, Minds, and Machines | Lex Fridman Podcast #13 | in a single hidden layer. | https://karpathy.ai/lexicap/0013-large.html#00:45:02.840 |
Tomaso Poggio: Brains, Minds, and Machines | Lex Fridman Podcast #13 | Do you find this theorem one surprising? | https://karpathy.ai/lexicap/0013-large.html#00:45:04.360 |
Tomaso Poggio: Brains, Minds, and Machines | Lex Fridman Podcast #13 | Do you find it useful, interesting, inspiring? | https://karpathy.ai/lexicap/0013-large.html#00:45:07.680 |
Tomaso Poggio: Brains, Minds, and Machines | Lex Fridman Podcast #13 | No, this one, I never found it very surprising. | https://karpathy.ai/lexicap/0013-large.html#00:45:12.600 |
Tomaso Poggio: Brains, Minds, and Machines | Lex Fridman Podcast #13 | It was known since the 80s, since I entered the field, | https://karpathy.ai/lexicap/0013-large.html#00:45:16.440 |
Tomaso Poggio: Brains, Minds, and Machines | Lex Fridman Podcast #13 | because it's basically the same as Weierstrass theorem, which | https://karpathy.ai/lexicap/0013-large.html#00:45:22.640 |
Tomaso Poggio: Brains, Minds, and Machines | Lex Fridman Podcast #13 | says that I can approximate any continuous function | https://karpathy.ai/lexicap/0013-large.html#00:45:27.560 |
Tomaso Poggio: Brains, Minds, and Machines | Lex Fridman Podcast #13 | with a polynomial of sufficiently, | https://karpathy.ai/lexicap/0013-large.html#00:45:32.000 |
Tomaso Poggio: Brains, Minds, and Machines | Lex Fridman Podcast #13 | with a sufficient number of terms, monomials. | https://karpathy.ai/lexicap/0013-large.html#00:45:34.560 |
Tomaso Poggio: Brains, Minds, and Machines | Lex Fridman Podcast #13 | So basically the same. | https://karpathy.ai/lexicap/0013-large.html#00:45:38.120 |
Tomaso Poggio: Brains, Minds, and Machines | Lex Fridman Podcast #13 | And the proofs are very similar. | https://karpathy.ai/lexicap/0013-large.html#00:45:39.360 |
Tomaso Poggio: Brains, Minds, and Machines | Lex Fridman Podcast #13 | So your intuition was there was never | https://karpathy.ai/lexicap/0013-large.html#00:45:41.680 |
Tomaso Poggio: Brains, Minds, and Machines | Lex Fridman Podcast #13 | any doubt that neural networks in theory | https://karpathy.ai/lexicap/0013-large.html#00:45:43.520 |
Tomaso Poggio: Brains, Minds, and Machines | Lex Fridman Podcast #13 | could be very strong approximators. | https://karpathy.ai/lexicap/0013-large.html#00:45:45.680 |
Tomaso Poggio: Brains, Minds, and Machines | Lex Fridman Podcast #13 | Right. | https://karpathy.ai/lexicap/0013-large.html#00:45:48.000 |
Tomaso Poggio: Brains, Minds, and Machines | Lex Fridman Podcast #13 | The question, the interesting question, | https://karpathy.ai/lexicap/0013-large.html#00:45:48.800 |
Tomaso Poggio: Brains, Minds, and Machines | Lex Fridman Podcast #13 | is that if this theorem says you can approximate, fine. | https://karpathy.ai/lexicap/0013-large.html#00:45:50.760 |
Tomaso Poggio: Brains, Minds, and Machines | Lex Fridman Podcast #13 | But when you ask how many neurons, for instance, | https://karpathy.ai/lexicap/0013-large.html#00:45:58.520 |
Tomaso Poggio: Brains, Minds, and Machines | Lex Fridman Podcast #13 | or in the case of polynomial, how many monomials, | https://karpathy.ai/lexicap/0013-large.html#00:46:03.200 |
Tomaso Poggio: Brains, Minds, and Machines | Lex Fridman Podcast #13 | I need to get a good approximation. | https://karpathy.ai/lexicap/0013-large.html#00:46:06.400 |
Tomaso Poggio: Brains, Minds, and Machines | Lex Fridman Podcast #13 | Then it turns out that that depends | https://karpathy.ai/lexicap/0013-large.html#00:46:11.360 |
Tomaso Poggio: Brains, Minds, and Machines | Lex Fridman Podcast #13 | on the dimensionality of your function, | https://karpathy.ai/lexicap/0013-large.html#00:46:15.960 |
Tomaso Poggio: Brains, Minds, and Machines | Lex Fridman Podcast #13 | how many variables you have. | https://karpathy.ai/lexicap/0013-large.html#00:46:18.080 |
Tomaso Poggio: Brains, Minds, and Machines | Lex Fridman Podcast #13 | But it depends on the dimensionality | https://karpathy.ai/lexicap/0013-large.html#00:46:20.520 |
Tomaso Poggio: Brains, Minds, and Machines | Lex Fridman Podcast #13 | of your function in a bad way. | https://karpathy.ai/lexicap/0013-large.html#00:46:22.120 |
Tomaso Poggio: Brains, Minds, and Machines | Lex Fridman Podcast #13 | It's, for instance, suppose you want | https://karpathy.ai/lexicap/0013-large.html#00:46:25.080 |
Tomaso Poggio: Brains, Minds, and Machines | Lex Fridman Podcast #13 | an error which is no worse than 10% in your approximation. | https://karpathy.ai/lexicap/0013-large.html#00:46:28.000 |
Tomaso Poggio: Brains, Minds, and Machines | Lex Fridman Podcast #13 | You come up with a network that approximate your function | https://karpathy.ai/lexicap/0013-large.html#00:46:35.040 |
Tomaso Poggio: Brains, Minds, and Machines | Lex Fridman Podcast #13 | within 10%. | https://karpathy.ai/lexicap/0013-large.html#00:46:38.120 |
Tomaso Poggio: Brains, Minds, and Machines | Lex Fridman Podcast #13 | Then it turns out that the number of units you need | https://karpathy.ai/lexicap/0013-large.html#00:46:40.440 |
Tomaso Poggio: Brains, Minds, and Machines | Lex Fridman Podcast #13 | are in the order of 10 to the dimensionality, d, | https://karpathy.ai/lexicap/0013-large.html#00:46:44.520 |
Tomaso Poggio: Brains, Minds, and Machines | Lex Fridman Podcast #13 | how many variables. | https://karpathy.ai/lexicap/0013-large.html#00:46:48.360 |
Tomaso Poggio: Brains, Minds, and Machines | Lex Fridman Podcast #13 | So if you have two variables, these two words, | https://karpathy.ai/lexicap/0013-large.html#00:46:50.080 |
Tomaso Poggio: Brains, Minds, and Machines | Lex Fridman Podcast #13 | you have 100 units and OK. | https://karpathy.ai/lexicap/0013-large.html#00:46:54.840 |
Tomaso Poggio: Brains, Minds, and Machines | Lex Fridman Podcast #13 | But if you have, say, 200 by 200 pixel images, | https://karpathy.ai/lexicap/0013-large.html#00:46:57.240 |
Tomaso Poggio: Brains, Minds, and Machines | Lex Fridman Podcast #13 | now this is 40,000, whatever. | https://karpathy.ai/lexicap/0013-large.html#00:47:02.920 |
Tomaso Poggio: Brains, Minds, and Machines | Lex Fridman Podcast #13 | We again go to the size of the universe pretty quickly. | https://karpathy.ai/lexicap/0013-large.html#00:47:06.840 |
Tomaso Poggio: Brains, Minds, and Machines | Lex Fridman Podcast #13 | Exactly, 10 to the 40,000 or something. | https://karpathy.ai/lexicap/0013-large.html#00:47:09.800 |
Tomaso Poggio: Brains, Minds, and Machines | Lex Fridman Podcast #13 | And so this is called the curse of dimensionality, | https://karpathy.ai/lexicap/0013-large.html#00:47:14.120 |
Tomaso Poggio: Brains, Minds, and Machines | Lex Fridman Podcast #13 | not quite appropriately. | https://karpathy.ai/lexicap/0013-large.html#00:47:18.680 |
Tomaso Poggio: Brains, Minds, and Machines | Lex Fridman Podcast #13 | And the hope is with the extra layers, | https://karpathy.ai/lexicap/0013-large.html#00:47:22.280 |
Tomaso Poggio: Brains, Minds, and Machines | Lex Fridman Podcast #13 | you can remove the curse. | https://karpathy.ai/lexicap/0013-large.html#00:47:24.200 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.