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In addition to "We need a cluster for deep learning", the second most popular mostly untrue thing I hear is "We have no idea how neural networks learn". Howeve

In addition to "We need a cluster for deep learning", the second most popular mostly untrue thing I hear is "We have no idea how neural networks learn". However, there are many papers that explore various ways to make a network learn, and they keep improving on performance, suggesting they're on to something. There are also many papers that discuss possible theoretical implications of experimental results. But what does Knowing Why The Network Works mean exactly? "It works because universal approximation and gradient descent", but that's not a very satisfactory answer. "It works because it starts at a general solution and, over the course of many iterations, takes many small steps in an ever changing direction defined by a gradient approximation generated by looking at the difference between an average error and a target output (which should trend towards 0)". What would a satisfactory "why" even look like exactly? As in, what form might it take compared to some other scientific discipline where we do know what's going on? Personally, I think the whole thing is a red herring -- people in the field have some idea of how neural nets work, and there are many disciplines considered by many to be mature sciences that are far from settled on a grand theoretical scale. That said, the theory I'm most interested in is recent attempts to connect a memory module to neural networks so they can "learn" to store important/complex/distributed information that can be recalled with high accuracy later. That will make it easier to do things like ask a neural network to remember your name, or where you left your keys, or whatever.

andrewflnr 11y ago
Heck, I want that for the neural net in my head.
TTPrograms 11y ago
The why is largely why does gradient descent converge to a good answer instead of getting stuck in a local minima.
- shmel 11y ago
  Mainly because in high-dimensional space saddle points are much more common and many local minima have very close fitness. Look for example http://arxiv.org/abs/1405.4604 http://arxiv.org/abs/1405.4604 and following papers.
- robotresearcher 11y ago
  Because the solution space is convex if you've chosen your representation well.
  - gipp 11y ago
    This is definitely not the case for general neural networks, though.
    - robotresearcher 11y ago
      If gradient descent is working reliably, the problem is convex. See the sibling comments for the intuition for large dimensional spaces.
      - gipp 11y ago
        "The problem is convex" and "the algorithm is unlikely to get stuck in a local minimum on realistic problems" are very different things.
        robotresearcher 11y ago
        Right. Hence the word 'reliably'.
- modeless 11y ago
  Because our intuition about local minima is wrong in extremely high dimensional spaces. In two and three dimensions, local minima are common. In a million dimensions, local minima are rare. The intuitive explanation is that for a local minimum to exist, the function must be curving up (first derivative = 0, second derivative >= 0) simultaneously in every dimension. It makes sense that as you add more dimensions this becomes less and less likely, and for a million dimensions it's vanishingly unlikely. What you get instead of local minima are saddle points, where some dimensions are curving up and some are curving down. Saddle points can also be problematic for optimization but they can be dealt with using fancy optimization techniques. For a more rigorous explanation, see http://arxiv.org/abs/1406.2572 http://arxiv.org/abs/1406.2572
  - bendbro 11y ago
    Wouldn't it still be likely to settle on a local minima when the deciding factors for the existence of a local minima are limited to those that contribute to the likelihood of a single output category, and not whether all functions are curving up? An example I can think of would be an absurd million input neural network, where one of the inputs only has a pronounced effect on one of the outputs. It seems like it would be possible for the path of the input to output to be dragged downhill in the context of all outputs, but uphill in the context of the single output it affects. Is what I've described not likely, or am I just completely off base?
    - modeless 11y ago
      An interesting question. A million input neural network isn't necessarily absurd. Images are very high dimensional. One could easily imagine a one megapixel input to a neural net. But in natural images no single pixel is indicative of any single image characteristic by itself. I think this isn't a coincidence but a common characteristic of "natural" high dimensional data, on which neural nets tend to work well. So yes, I'd say what you've described is not likely for a large category of "natural" high dimensional data which probably includes most of the data we care about in the real world.
  - sunstone 11y ago
j2kun 11y ago
> What would a satisfactory "why" even look like exactly? As in, what form might it take compared to some other scientific discipline where we do know what's going on? To me this is obvious: a proof is an adequate answer. All of the explanations given in this area are heuristic reasons. They're nice, but we can't know if they're correct or if we're being fooled by our intuition without a proof.
gone35 11y ago
But what does Knowing Why The Network Works mean exactly? I would say it means: Why do these algorithms seem to be doing "well" despite their well-known theoretical intractability? Is it something about the instances? Or is it simply a matter of scale (so the local optima they are getting stuck in are not as obvious)? Or is there something "deep" we do not understand yet about these algorithms (or the theory)? Either way, there is something going on here for sure.
david_ar 11y ago
> What would a satisfactory "why" even look like exactly? Before deep learning happened, neural networks used to be popular for regression and interpolation problems. For a long time, our understanding of how they worked wasn't much better than our current understanding of how deep learning works. In 1994, Radford Neal showed [1] that weight-decay neural networks were in fact an approximation to Bayesian inference on a Gaussian process prior over the space of possible functions. Amongst other benefits, this allowed better approximations to be utilised, and essentially (along with the advent of SVMs) marked the end the first neural network era. Something like that is what I would consider a satisfactory "why" [1] http://www.cs.toronto.edu/~radford/pin.abstract.html http://www.cs.toronto.edu/~radford/pin.abstract.html

andrewflnr 11y ago

Heck, I want that for the neural net in my head.

TTPrograms 11y ago

The why is largely why does gradient descent converge to a good answer instead of getting stuck in a local minima.

shmel 11y ago
Mainly because in high-dimensional space saddle points are much more common and many local minima have very close fitness. Look for example http://arxiv.org/abs/1405.4604 http://arxiv.org/abs/1405.4604 and following papers.
robotresearcher 11y ago
Because the solution space is convex if you've chosen your representation well.
- gipp 11y ago
  This is definitely not the case for general neural networks, though.
  - robotresearcher 11y ago
    If gradient descent is working reliably, the problem is convex. See the sibling comments for the intuition for large dimensional spaces.
    - gipp 11y ago
      "The problem is convex" and "the algorithm is unlikely to get stuck in a local minimum on realistic problems" are very different things.
      - robotresearcher 11y ago
        Right. Hence the word 'reliably'.
modeless 11y ago
Because our intuition about local minima is wrong in extremely high dimensional spaces. In two and three dimensions, local minima are common. In a million dimensions, local minima are rare. The intuitive explanation is that for a local minimum to exist, the function must be curving up (first derivative = 0, second derivative >= 0) simultaneously in every dimension. It makes sense that as you add more dimensions this becomes less and less likely, and for a million dimensions it's vanishingly unlikely. What you get instead of local minima are saddle points, where some dimensions are curving up and some are curving down. Saddle points can also be problematic for optimization but they can be dealt with using fancy optimization techniques. For a more rigorous explanation, see http://arxiv.org/abs/1406.2572 http://arxiv.org/abs/1406.2572
- bendbro 11y ago
  Wouldn't it still be likely to settle on a local minima when the deciding factors for the existence of a local minima are limited to those that contribute to the likelihood of a single output category, and not whether all functions are curving up? An example I can think of would be an absurd million input neural network, where one of the inputs only has a pronounced effect on one of the outputs. It seems like it would be possible for the path of the input to output to be dragged downhill in the context of all outputs, but uphill in the context of the single output it affects. Is what I've described not likely, or am I just completely off base?
  - modeless 11y ago
    An interesting question. A million input neural network isn't necessarily absurd. Images are very high dimensional. One could easily imagine a one megapixel input to a neural net. But in natural images no single pixel is indicative of any single image characteristic by itself. I think this isn't a coincidence but a common characteristic of "natural" high dimensional data, on which neural nets tend to work well. So yes, I'd say what you've described is not likely for a large category of "natural" high dimensional data which probably includes most of the data we care about in the real world.
- sunstone 11y ago

j2kun 11y ago

> What would a satisfactory "why" even look like exactly? As in, what form might it take compared to some other scientific discipline where we do know what's going on? To me this is obvious: a proof is an adequate answer. All of the explanations given in this area are heuristic reasons. They're nice, but we can't know if they're correct or if we're being fooled by our intuition without a proof.

gone35 11y ago

But what does Knowing Why The Network Works mean exactly? I would say it means: Why do these algorithms seem to be doing "well" despite their well-known theoretical intractability? Is it something about the instances? Or is it simply a matter of scale (so the local optima they are getting stuck in are not as obvious)? Or is there something "deep" we do not understand yet about these algorithms (or the theory)? Either way, there is something going on here for sure.

david_ar 11y ago

> What would a satisfactory "why" even look like exactly? Before deep learning happened, neural networks used to be popular for regression and interpolation problems. For a long time, our understanding of how they worked wasn't much better than our current understanding of how deep learning works. In 1994, Radford Neal showed [1] that weight-decay neural networks were in fact an approximation to Bayesian inference on a Gaussian process prior over the space of possible functions. Amongst other benefits, this allowed better approximations to be utilised, and essentially (along with the advent of SVMs) marked the end the first neural network era. Something like that is what I would consider a satisfactory "why" [1] http://www.cs.toronto.edu/~radford/pin.abstract.html http://www.cs.toronto.edu/~radford/pin.abstract.html