6 ms·

We understand how they work mechanically, but not why they work from a theoretical stand-point (which is I assume what you're trying to say by "abstractly"). W

We understand how they work mechanically, but not why they work from a theoretical stand-point (which is I assume what you're trying to say by "abstractly"). Why it is that ASGD and backprop converges on a non-convex optimization problem, and what kinds of model topologies make it do better / worse? That's all basically art right now.

Houshalter 11y ago
No that's not what he's trying to say. We know why gradient descent and back-propagation works. But no one understands what an actual trained neural net is doing. You can look at the weights, and you can watch the inputs and outputs, but it is very difficult to understand why it does what it does. It just fit a model to data, but there is no explanation why that model is best. The weights are not interprettable by humans. There have been some attempts at making models which humans can interpret. One program named Eureqa fit the simplest possible mathematical expression possible to a set of samples. A biologist tried it on his data and found that it created an expression which actually fit the data really well. But he couldn't publish it because he couldn't explain why it worked. It just did. But there was no understanding, no explanation.
- dhj 11y ago
  Do you have a citation or blog post for this application of Eureqa that gave a good result but wasn't publishable from a biological study perspective? I have a background in BME and CS and am very interested in learning about this anecdote.
  - Houshalter 11y ago
    It was in an interview on an episode of radiolab: http://www.radiolab.org/story/91712-limits-of-science/ http://www.radiolab.org/story/91712-limits-of-science/ There aren't many details. I believe the scientist who said it is Dr. Gurol Suel, from that radiolab page.
- Someone 11y ago
  "We know why gradient descent and back-propagation works." I would phrase that differently: we know when gradient descent and back-propagation work, not why they work for so many real world problems. For the when, there are zillions of published mathematical results stating "if a problem has property X, this and this method will find a solution with property Y in time T", in zillions of variations (Y can be the true optimum, a value within x% of the real optimum, the true optimum 'most of the time', etc. T can be 'eventually', 'after O(n^3) iterations', 'always', etc) However, for most real-world problems we do not know whether they have property X, or even how to go about arguing that it is likely they have property X, other than the somewhat circular "algorithm A seems to work well for it, and we know it works for problems of type X"
  - Leszek 11y ago
    Well, we do know why gradient descent works (for smooth data), at least for finding a local minimum, because finding the minimum is basically what it does by construction. Similarly, we certainly know how back-propagation works, because it's simple calculus, backwards application of the chain rule. Perhaps what you're trying to say is we don't know why finding local minima of these problems is good at solving the problem?
- argonaut 11y ago
  If that is what he was trying to say, then he's wrong. Hiddencost is right. We do not have a solid theoretical understanding of neural nets and why backprop, and all the other tricks (e.g. relu, dropout) works. There is basic intuition on why they work, but no rigorous theory. Even worse, we have no theory on which architectures work better or worse and why. So whenever you read about some 50-layer net trained with this architecture with this padding/stride/normalization, and you wonder how they came up with that, the answer is: some grad student sat there, thought about his past experiences and the papers he's read of architectures that have worked well, and then spent months trying a bunch of things.
  - Houshalter 11y ago
    I don't think that's quite right. There is a lot of understanding about neural nets and why different method work. Maybe not to the level that satisfies purist mathematicians, but nothing really satisfies them. Yes it's true that hyperparameters seem arbitrary, but that's just a consequence of the no-free-lunch theorem. Some models will always fit some problems better than others. There is no such thing as a perfect model. NNs turn out to be a really good prior for real world problems. I wrote about why that might be here: http://houshalter.tumblr.com/post/120134087595/approximating-solomonoff-induction http://houshalter.tumblr.com/post/120134087595/approximating... But in principle, that doesn't apply to things like layer numbers. In theory the best neural net has infinite layers of infinite size and infinite convolution, with a stride of 1. Because you can fit every other model into that, and as long as the parameters are properly regularized it can also avoid overfitting too. In the real world of course, we are limited to merely 50 layers, and need to cut corners with convolution sizes and such. Likewise for training. The best algorithm for training is bayesian inference on the parameters. Since that is usually very impractical, we use approximations like maximum-likelihood or dropout.
    - argonaut 11y ago
      No, that's not even what the theory says. In theory, all you need is a single hidden layer, and all you need to do is keep increasing the number of hidden units until things work, because that can model any function (universal approximation theorem). What you said makes no sense theoretically (an infinite number of layers and convolutions makes no sense - I think you meant to use the word arbitrary). In the real world, we need to stack layers, for some reason that is only vaguely understood. It goes without saying that the reason we have 50 (vs. deeper) layer nets and strides, among other things, is not solely to reduce computational cost.
- nl 11y ago
  Sorry, hiddencost is correct here. While we understand the math of backpropegation, we don't really understand why it lets neural networks converge as well as they do. If you asked a mathematician they'd point out that yes, backpropegation can work, but there are no guarantees even around the probability that it will work. And yet, it does work, and often enough and in enough different cases for it to be useful. Yes, interpretability can be an issue, but that's not what is being discussed here. There have been some attempts at making models which humans can interpret. One program named Eureqa fit the simplest possible mathematical expression possible to a set of samples. A biologist tried it on his data and found that it created an expression which actually fit the data really well. But he couldn't publish it because he couldn't explain why it worked. It just did. But there was no understanding, no explanation. This is a pretty well understood issue in the ML community. Different tools have different strengths: Regression (sometimes), Tree based classifiers and Bayesian nets are nice for explaining your data, but others methods (SVMs, Neural Networks etc) can be better for prediction in some circumstances.
  - Houshalter 11y ago
    Gradient descent is guaranteed to move the parameters closer to a more fit solution. That's almost the definition of GD. I guess you are saying that we don't know why it doesn't just get stuck in local optimas? I'm not certain, but for one, stochastic gradient descent is usually used, which helps break out of local optimas. And second, they don't seem to be as much of a problem as you add more hidden units. I was under the impression there was a mathematical explanation for both of those, though I haven't done much research. Off the top of my head, I think as you add infinite hidden units, a subset of them will fit the function exactly just by chance, and gradient descent will only increase their parameters. In fact as long as they are in some large possibility space of the correct solution, GD can move the parameters downhill to the optimal solution. Not that we even want the optimal solution, just a good enough one.
    - argonaut 11y ago
      Neural nets virtually always get stuck in local optima. I have no idea what you're talking about.

Houshalter 11y ago

No that's not what he's trying to say. We know why gradient descent and back-propagation works. But no one understands what an actual trained neural net is doing. You can look at the weights, and you can watch the inputs and outputs, but it is very difficult to understand why it does what it does. It just fit a model to data, but there is no explanation why that model is best. The weights are not interprettable by humans. There have been some attempts at making models which humans can interpret. One program named Eureqa fit the simplest possible mathematical expression possible to a set of samples. A biologist tried it on his data and found that it created an expression which actually fit the data really well. But he couldn't publish it because he couldn't explain why it worked. It just did. But there was no understanding, no explanation.

dhj 11y ago
Do you have a citation or blog post for this application of Eureqa that gave a good result but wasn't publishable from a biological study perspective? I have a background in BME and CS and am very interested in learning about this anecdote.
- Houshalter 11y ago
  It was in an interview on an episode of radiolab: http://www.radiolab.org/story/91712-limits-of-science/ http://www.radiolab.org/story/91712-limits-of-science/ There aren't many details. I believe the scientist who said it is Dr. Gurol Suel, from that radiolab page.
Someone 11y ago
"We know why gradient descent and back-propagation works." I would phrase that differently: we know when gradient descent and back-propagation work, not why they work for so many real world problems. For the when, there are zillions of published mathematical results stating "if a problem has property X, this and this method will find a solution with property Y in time T", in zillions of variations (Y can be the true optimum, a value within x% of the real optimum, the true optimum 'most of the time', etc. T can be 'eventually', 'after O(n^3) iterations', 'always', etc) However, for most real-world problems we do not know whether they have property X, or even how to go about arguing that it is likely they have property X, other than the somewhat circular "algorithm A seems to work well for it, and we know it works for problems of type X"
- Leszek 11y ago
  Well, we do know why gradient descent works (for smooth data), at least for finding a local minimum, because finding the minimum is basically what it does by construction. Similarly, we certainly know how back-propagation works, because it's simple calculus, backwards application of the chain rule. Perhaps what you're trying to say is we don't know why finding local minima of these problems is good at solving the problem?
argonaut 11y ago
If that is what he was trying to say, then he's wrong. Hiddencost is right. We do not have a solid theoretical understanding of neural nets and why backprop, and all the other tricks (e.g. relu, dropout) works. There is basic intuition on why they work, but no rigorous theory. Even worse, we have no theory on which architectures work better or worse and why. So whenever you read about some 50-layer net trained with this architecture with this padding/stride/normalization, and you wonder how they came up with that, the answer is: some grad student sat there, thought about his past experiences and the papers he's read of architectures that have worked well, and then spent months trying a bunch of things.
- Houshalter 11y ago
  I don't think that's quite right. There is a lot of understanding about neural nets and why different method work. Maybe not to the level that satisfies purist mathematicians, but nothing really satisfies them. Yes it's true that hyperparameters seem arbitrary, but that's just a consequence of the no-free-lunch theorem. Some models will always fit some problems better than others. There is no such thing as a perfect model. NNs turn out to be a really good prior for real world problems. I wrote about why that might be here: http://houshalter.tumblr.com/post/120134087595/approximating-solomonoff-induction http://houshalter.tumblr.com/post/120134087595/approximating... But in principle, that doesn't apply to things like layer numbers. In theory the best neural net has infinite layers of infinite size and infinite convolution, with a stride of 1. Because you can fit every other model into that, and as long as the parameters are properly regularized it can also avoid overfitting too. In the real world of course, we are limited to merely 50 layers, and need to cut corners with convolution sizes and such. Likewise for training. The best algorithm for training is bayesian inference on the parameters. Since that is usually very impractical, we use approximations like maximum-likelihood or dropout.
  - argonaut 11y ago
    No, that's not even what the theory says. In theory, all you need is a single hidden layer, and all you need to do is keep increasing the number of hidden units until things work, because that can model any function (universal approximation theorem). What you said makes no sense theoretically (an infinite number of layers and convolutions makes no sense - I think you meant to use the word arbitrary). In the real world, we need to stack layers, for some reason that is only vaguely understood. It goes without saying that the reason we have 50 (vs. deeper) layer nets and strides, among other things, is not solely to reduce computational cost.
nl 11y ago
Sorry, hiddencost is correct here. While we understand the math of backpropegation, we don't really understand why it lets neural networks converge as well as they do. If you asked a mathematician they'd point out that yes, backpropegation can work, but there are no guarantees even around the probability that it will work. And yet, it does work, and often enough and in enough different cases for it to be useful. Yes, interpretability can be an issue, but that's not what is being discussed here. There have been some attempts at making models which humans can interpret. One program named Eureqa fit the simplest possible mathematical expression possible to a set of samples. A biologist tried it on his data and found that it created an expression which actually fit the data really well. But he couldn't publish it because he couldn't explain why it worked. It just did. But there was no understanding, no explanation. This is a pretty well understood issue in the ML community. Different tools have different strengths: Regression (sometimes), Tree based classifiers and Bayesian nets are nice for explaining your data, but others methods (SVMs, Neural Networks etc) can be better for prediction in some circumstances.
- Houshalter 11y ago
  Gradient descent is guaranteed to move the parameters closer to a more fit solution. That's almost the definition of GD. I guess you are saying that we don't know why it doesn't just get stuck in local optimas? I'm not certain, but for one, stochastic gradient descent is usually used, which helps break out of local optimas. And second, they don't seem to be as much of a problem as you add more hidden units. I was under the impression there was a mathematical explanation for both of those, though I haven't done much research. Off the top of my head, I think as you add infinite hidden units, a subset of them will fit the function exactly just by chance, and gradient descent will only increase their parameters. In fact as long as they are in some large possibility space of the correct solution, GD can move the parameters downhill to the optimal solution. Not that we even want the optimal solution, just a good enough one.
  - argonaut 11y ago
    Neural nets virtually always get stuck in local optima. I have no idea what you're talking about.