One crucial question in understanding and predicting the learning process, and ultimately the behavior, of modern neural networks, is that of the shape of their loss landscapes. What does this extremely high dimensional landscape look like? Does training generally tend to find minima? Do minima even exist? Is it predictable what type of minima (or regions of lower loss) are found during training? What role does initial randomization play? Are there specific types of basins in the landscape that are qualitatively different from others, that we might care about for safety reasons?
First, let’s just briefly think about very high dimensional spaces. One somewhat obvious observation is that they are absolutely vast. With each added dimension, the volume of the available space increases exponentially. Intuitively we tend to think of 3-dimensional spaces, and often apply this visual/spatial intuition to our understanding of loss landscapes. But this can be extremely misleading. Parameter spaces are utterly incredibly vast to a degree that our brain can hardly fathom. Take GPT3 for instance. It has 175 billion parameters, or dimensions. Let’s assume somewhat arbitrarily that all parameters end up in a range of [-0.5, 0.5], i.e. live in a 175-billion-dimensional unit cube around the origin of that space (as this is not the case, the real parameter space is actually even much, much larger, but bear with me). Even though every single axis only varies by 1 – let’s just for the sake of it interpret this as “1 meter” – even just taking the diagonal from one corner to the opposite one in this high-dimensional cube, you would get a length of ~420km. So if, hypothetically, you were sitting in the middle of this high dimensional unit cube, you could easily touch every single wall with your hand. But nonetheless, all the corners would be more than 200km distant from you.
This may be mind boggling, but is it relevant? I think it is. Take this realization for instance: if you have two minima in this high dimensional space, but one is just a tiny bit “flatter” than the other (meaning the second derivatives overall are a bit closer to 0), then the attractor basin of this flatter minimum is vastly larger than that of the other minimum. This is because the flatness implies a larger radius, and the volume depends exponentially on that radius. So, at 175 billion dimensions, even a microscopically larger radius means an overwhelmingly larger volume. If, for instance, one minimum’s attractor basin has a radius that is just 0.00000001% larger than that of the other minimum, then its volume will be roughly 40 million times larger (if my Javascript code to calculate this is accurate enough, that is). And this is only for GPT3, which is almost 4 years old by now.
The parameter space is just ridiculously large, so it becomes really crucial how the search process through it works and where it lands. It may be that somewhere in this vast space, there are indeed attractor basins that correspond to minima that we find extremely undesirable – certain capable optimizers perhaps, that have situational awareness and deceptive tendencies. If they do exist, what could we possibly tell about them? Maybe these minima have huge attractor basins that are reliably found eventually (maybe once we switch to a different network architecture, or find some adjustment to gradient descent, or reach a certain model size, or whatever), which would of course be bad news. Or maybe these attractor basins are so vanishingly small that we basically don’t have to care about them at all, because all the computer & search capacity of humanity over the next million years would have an almost 0 chance of ever stumbling onto these regions. Maybe they are even so small that they are numerically unstable, and even if your search process through some incredible cosmic coincidence happens to start right in such a basin, the first SGD step would immediately jump out of it due to the limitations of numerical accuracy on the hardware we’re using.
So, what can we actually tell at this point about the nature of high dimensional loss landscapes? While reading up on this topic, one thing that constantly came up is the fact that, the more dimensions you have, the lower the relative number of minima becomes compared to saddle points. Meaning that whenever the training process appears to slow down and it looks like it found some local minimum, it’s actually overwhelmingly likely that what it actually found is a saddle point, hence the training process never halts but keeps moving through parameter space, even if the loss doesn’t change that much. Do local minima exist at all? I guess it depends on the function the neural network is learning to approximate. Maybe some loss landscapes exist where the loss can just get asymptotically closer to some minimum (such as 0), without ever reaching it. And probably other loss landscapes exist where you actually have a global minimum, as well as several local ones.
Some people argue that you probably have no minima at all, because with each added dimension it becomes less and less likely that a given point is a minimum (because not only does the first derivative of a point have to be 0 for it to be a minimum, also all the second derivatives need to be in on it, and all be positive). This sounds compelling, but given that the space itself also grows exponentially with each dimension, we also have overwhelmingly more points to choose from. If you e.g. look at n-dimensional Perlin Noise, its absolute number of local minima within an n-dimensional cube of constant side length actually increases with each added dimension. However, the relative number of local minima compared to the available space still decreases, so it becomes harder and harder to find them.
I’ll keep it at that. This is already not much of a “quick” take. Basically, more research is needed, as my literature review on this subject yielded way more questions than answers, and many of the claims people made in their blog posts, articles and sometimes even papers seemed to be more intuitive / common-sensical or generalized from maybe-not-that-easy-to-validly-generalize-from research.
One thing I’m sure about however is that almost any explanation of how (stochastic) gradient descent works, that uses 3D landscapes for intuitive visualizations, is misleading in many ways. Maybe it is the best we have, but imho all such explainers should come with huge asterisks, explaining that the rules in very high dimensional spaces may look much different than our naive “oh look at that nice valley over there, let’s walk down to its minimum!” understanding, that happens to work well in three dimensions.
Because critical points are non-isolated, there are more important kinds of “flatness” than having small second derivatives. Neural networks have degenerate loss landscapes: their Hessians have zero-valued eigenvalues, which means there are directions you can walk along that don’t change the loss (or that change the loss by a cubic or higher power rather than a quadratic power). The dominant contribution to how volume scales in the loss landscape comes from the behavior of the loss in those degenerate directions. This is much more significant than the behavior of the quadratic directions. The amount of degeneracy is quantified by singular learning theory’s local learning coefficient (LLC).
In the Bayesian setting, the relationship between geometric degeneracy and inductive biases is well understood through Watanabe’s free energy formula. There’s an inductive bias towards more degenerate parts of parameter space that’s especially strong earlier in the learning process.
If, for instance, one minimum’s attractor basin has a radius that is just 0.00000001% larger than that of the other minimum, then its volume will be roughly 40 million times larger (if my Javascript code to calculate this is accurate enough, that is).
Could you share this code? I’d like to take a look.
Maybe I accidentally overpromised here :D this code is just an expression, namely 1.0000000001 ** 175000000000, which, as wolframalpha agrees, yields 3.98e7.
One crucial question in understanding and predicting the learning process, and ultimately the behavior, of modern neural networks, is that of the shape of their loss landscapes. What does this extremely high dimensional landscape look like? Does training generally tend to find minima? Do minima even exist? Is it predictable what type of minima (or regions of lower loss) are found during training? What role does initial randomization play? Are there specific types of basins in the landscape that are qualitatively different from others, that we might care about for safety reasons?
First, let’s just briefly think about very high dimensional spaces. One somewhat obvious observation is that they are absolutely vast. With each added dimension, the volume of the available space increases exponentially. Intuitively we tend to think of 3-dimensional spaces, and often apply this visual/spatial intuition to our understanding of loss landscapes. But this can be extremely misleading. Parameter spaces are utterly incredibly vast to a degree that our brain can hardly fathom. Take GPT3 for instance. It has 175 billion parameters, or dimensions. Let’s assume somewhat arbitrarily that all parameters end up in a range of [-0.5, 0.5], i.e. live in a 175-billion-dimensional unit cube around the origin of that space (as this is not the case, the real parameter space is actually even much, much larger, but bear with me). Even though every single axis only varies by 1 – let’s just for the sake of it interpret this as “1 meter” – even just taking the diagonal from one corner to the opposite one in this high-dimensional cube, you would get a length of ~420km. So if, hypothetically, you were sitting in the middle of this high dimensional unit cube, you could easily touch every single wall with your hand. But nonetheless, all the corners would be more than 200km distant from you.
This may be mind boggling, but is it relevant? I think it is. Take this realization for instance: if you have two minima in this high dimensional space, but one is just a tiny bit “flatter” than the other (meaning the second derivatives overall are a bit closer to 0), then the attractor basin of this flatter minimum is vastly larger than that of the other minimum. This is because the flatness implies a larger radius, and the volume depends exponentially on that radius. So, at 175 billion dimensions, even a microscopically larger radius means an overwhelmingly larger volume. If, for instance, one minimum’s attractor basin has a radius that is just 0.00000001% larger than that of the other minimum, then its volume will be roughly 40 million times larger (if my Javascript code to calculate this is accurate enough, that is). And this is only for GPT3, which is almost 4 years old by now.
The parameter space is just ridiculously large, so it becomes really crucial how the search process through it works and where it lands. It may be that somewhere in this vast space, there are indeed attractor basins that correspond to minima that we find extremely undesirable – certain capable optimizers perhaps, that have situational awareness and deceptive tendencies. If they do exist, what could we possibly tell about them? Maybe these minima have huge attractor basins that are reliably found eventually (maybe once we switch to a different network architecture, or find some adjustment to gradient descent, or reach a certain model size, or whatever), which would of course be bad news. Or maybe these attractor basins are so vanishingly small that we basically don’t have to care about them at all, because all the computer & search capacity of humanity over the next million years would have an almost 0 chance of ever stumbling onto these regions. Maybe they are even so small that they are numerically unstable, and even if your search process through some incredible cosmic coincidence happens to start right in such a basin, the first SGD step would immediately jump out of it due to the limitations of numerical accuracy on the hardware we’re using.
So, what can we actually tell at this point about the nature of high dimensional loss landscapes? While reading up on this topic, one thing that constantly came up is the fact that, the more dimensions you have, the lower the relative number of minima becomes compared to saddle points. Meaning that whenever the training process appears to slow down and it looks like it found some local minimum, it’s actually overwhelmingly likely that what it actually found is a saddle point, hence the training process never halts but keeps moving through parameter space, even if the loss doesn’t change that much. Do local minima exist at all? I guess it depends on the function the neural network is learning to approximate. Maybe some loss landscapes exist where the loss can just get asymptotically closer to some minimum (such as 0), without ever reaching it. And probably other loss landscapes exist where you actually have a global minimum, as well as several local ones.
Some people argue that you probably have no minima at all, because with each added dimension it becomes less and less likely that a given point is a minimum (because not only does the first derivative of a point have to be 0 for it to be a minimum, also all the second derivatives need to be in on it, and all be positive). This sounds compelling, but given that the space itself also grows exponentially with each dimension, we also have overwhelmingly more points to choose from. If you e.g. look at n-dimensional Perlin Noise, its absolute number of local minima within an n-dimensional cube of constant side length actually increases with each added dimension. However, the relative number of local minima compared to the available space still decreases, so it becomes harder and harder to find them.
I’ll keep it at that. This is already not much of a “quick” take. Basically, more research is needed, as my literature review on this subject yielded way more questions than answers, and many of the claims people made in their blog posts, articles and sometimes even papers seemed to be more intuitive / common-sensical or generalized from maybe-not-that-easy-to-validly-generalize-from research.
One thing I’m sure about however is that almost any explanation of how (stochastic) gradient descent works, that uses 3D landscapes for intuitive visualizations, is misleading in many ways. Maybe it is the best we have, but imho all such explainers should come with huge asterisks, explaining that the rules in very high dimensional spaces may look much different than our naive “oh look at that nice valley over there, let’s walk down to its minimum!” understanding, that happens to work well in three dimensions.
I’d like to point out that for neural networks, isolated critical points (whether minima, maxima, or saddle points) basically do not exist. Instead, it’s valleys and ridges all the way down. So the word “basin” (which suggests the geometry is parabolic) is misleading.
Because critical points are non-isolated, there are more important kinds of “flatness” than having small second derivatives. Neural networks have degenerate loss landscapes: their Hessians have zero-valued eigenvalues, which means there are directions you can walk along that don’t change the loss (or that change the loss by a cubic or higher power rather than a quadratic power). The dominant contribution to how volume scales in the loss landscape comes from the behavior of the loss in those degenerate directions. This is much more significant than the behavior of the quadratic directions. The amount of degeneracy is quantified by singular learning theory’s local learning coefficient (LLC).
In the Bayesian setting, the relationship between geometric degeneracy and inductive biases is well understood through Watanabe’s free energy formula. There’s an inductive bias towards more degenerate parts of parameter space that’s especially strong earlier in the learning process.
I heard that there is no local minima in high-dimensional spaces because there will be almost always paths to global minimum.
Could you share this code? I’d like to take a look.
Maybe I accidentally overpromised here :D this code is just an expression, namely
1.0000000001 ** 175000000000, which, as wolframalpha agrees, yields 3.98e7.