The Optimization Problem in Deep Learning

Optimization problems are a crucial aspect of deep learning, and understanding the challenges that arise from optimizing complex functions in high-dimensional spaces is essential for building efficient and effective neural networks. The problem of local optima is a significant concern in optimization problems, where the goal is to find the minimum value of a function, known as the cost function J.

In low-dimensional spaces, it's easy to visualize the shape of the cost function and identify potential local optima. However, when dealing with high-dimensional spaces, such as those encountered in deep learning, these intuitive approaches become less effective. The problem is that in very high-dimensional spaces, most points of zero gradients are not local optima, but rather saddle points. A saddle point is a region where the function has a minimum value along one axis and a maximum value along another axis. This means that if you have twenty thousand parameters, your cost function J is defined over a twenty-thousand dimensional space, and finding a local optimum in this space becomes increasingly unlikely.

In low-dimensional spaces, it's easy to imagine a surface where the gradient is zero, and it looks like a saddle point. However, this intuition doesn't translate well to high-dimensional spaces. In reality, if you have a function defined over a twenty-thousand dimensional space, finding a local optimum requires that all directions are convex or concave, meaning that the curve bends up or down in every direction. This is highly unlikely, and the probability of encountering a saddle point is much higher than encountering a local optimum.

The concept of a saddle point is often visualized as a horse's saddle, where the rider sits in a region with zero derivative. This metaphor helps to illustrate the idea that finding a minimum value in a high-dimensional space requires careful consideration of all possible directions. However, even this analogy is not entirely accurate, and it's essential to understand that finding a local optimum is still a challenging problem in optimization.

Plateaus are another significant challenge in optimization problems. A plateau is a region where the derivative is close to zero for a long time, which can lead to slow learning rates. Gradient descent algorithms, which are commonly used for optimization, may become stuck on these plateaus, taking an extremely long time to find their way off.

To mitigate this issue, more sophisticated optimization algorithms such as momentum or Adam have been developed. These algorithms can help speed up the rate at which the network learns and improves. By using these advanced algorithms, developers can build more efficient neural networks that overcome the challenges of high-dimensional optimization spaces.

In conclusion, understanding the optimization problem in deep learning requires recognizing the limitations of intuitive approaches in low-dimensional spaces. High-dimensional spaces, such as those encountered in deep learning, present significant challenges, including the likelihood of encountering saddle points rather than local optima and the potential for plateaus to slow down learning rates. By acknowledging these challenges and using advanced optimization algorithms, developers can build more effective neural networks that overcome the obstacles of high-dimensional optimization spaces.

The Evolution of Understanding High-Dimensional Spaces

Our understanding of high-dimensional spaces in deep learning is still evolving, and it's essential to recognize that intuition about these spaces is often incomplete or inaccurate. In low-dimensional spaces, such as plotting a figure like this in two dimensions, it's easy to create plots where local optima are prominent. However, when dealing with high-dimensional spaces, these intuitive approaches become less effective.

The concept of saddle points has been widely discussed in the context of optimization problems, and it's essential to understand that these regions can arise in any high-dimensional space. The probability of encountering a saddle point is much higher than encountering a local optimum, making it a significant concern in optimization problems.

The analogy of a horse's saddle helps illustrate the idea that finding a minimum value in a high-dimensional space requires careful consideration of all possible directions. However, even this metaphor has limitations, and it's essential to recognize that finding a local optimum is still a challenging problem in optimization.

The concept of plateaus is another significant challenge in optimization problems. A plateau is a region where the derivative is close to zero for a long time, which can lead to slow learning rates. Gradient descent algorithms may become stuck on these plateaus, taking an extremely long time to find their way off.

By understanding the challenges that arise from high-dimensional spaces and using advanced optimization algorithms, developers can build more efficient neural networks that overcome the obstacles of these complex spaces.

The Role of Optimization Algorithms in Deep Learning

Optimization algorithms play a crucial role in deep learning, and choosing the right algorithm is essential for building effective neural networks. The goal of an optimization algorithm is to find the minimum value of a cost function J, which is defined over a high-dimensional space.

Gradient descent is a widely used optimization algorithm that iteratively updates the model's parameters based on the gradient of the cost function. However, this algorithm can become stuck on plateaus, leading to slow learning rates and inefficient training. To mitigate this issue, more sophisticated optimization algorithms such as momentum or Adam have been developed.

Momentum is an extension of the gradient descent algorithm that adds a term to the update rule to encourage exploration in the direction of the negative gradient. This helps the algorithm escape from plateaus and find better local optima.

Adam is another popular optimization algorithm that combines the benefits of momentum and adaptive learning rate schedules. Adam adjusts the learning rate based on the magnitude of the gradient, which helps it adapt to changing environments and avoid plateaus.

By using these advanced optimization algorithms, developers can build more efficient neural networks that overcome the challenges of high-dimensional optimization spaces. The choice of optimization algorithm is critical in deep learning, and selecting the right one requires careful consideration of the specific requirements of the problem.

"WEBVTTKind: captionsLanguage: enin the early days of deep learning people used to worry a lot about the optimization algorithm getting stuck in bad local optima but as the theory of deep learning has advanced our understanding of local optima is also changing let me show you how we now think about local optima and problems in the optimization problem in deep learning so this was a picture people used to have in mind when they worried about local optima maybe you're trying to optimize some set of parameters and we call them W 1 and W 2 and the height of the surface is the cost function so in this picture it looks like there are a lot of local optima you know in in all those places and it'd be easy for gradient descents or one of the other algorithms to get stuck on a local optimum rather than find this way to a global optimum it turns out that if you are plotting a figure like this in two dimensions then it's easy to create plots like this of a lot of different local optima and these very low dimensional plots used to gather intuition but this intuition isn't actually correct it turns out if you create in your network most points of 0 gradients are not local optima like points like this instead most points of 0 gradients in the cost function are actually saddle points so that's a point with a zero gradient again this is maybe W 1 W 2 and the highest heightens the value of the cost function J but informally a function in a very high dimensional space if the gradient is 0 then in each direction it can either be a convex light function or a concave light function and if you are in say a 20,000 dimensional space then thread to be a local optima all 20,000 directions need to look like this and so the chance of that happening is maybe very small you know maybe 2 to the minus 20000 instead you're much more likely to get some directions where the curve bends up like so as well some directions where the function is bending down rather than have them all Bend upwards so that's why in very high dimensional spaces you're actually much more likely to run into a saddle points like that shown on the right then local optimum oh and as for why the surface is called a saddle point if you can picture maybe this is a sort of shadow you put on a horse right so maybe if this is a horse I guess there's a head of a horse as you I have a horse you know I guess and right well another great drawing of a horse but you get the idea then you the rider will sit here in the saddle so then so that's why this point here where the derivative is zero that point is called a saddle point it's really the point to understand where you're sitting s and that happens to have you know derivative zero and so one of the lessons we learned in history of deep learning is that a lot of our intuitions about low dimensional spaces like what you can plot on the left they really don't transfer to the very high dimensional spaces then our learning algorithms are operating over because if you have twenty thousand parameters then J is V a function over a twenty thousand dimensional vector and you're much more likely to see saddle points than local optimum if local optima aren't a problem then what is a problem it turns out that plateaus can really slow down learning and the plateau is a region where the derivative is close to zero for a long time so if you are here then gradient descent will move down the surface and because the gradient is zero or near zero the surface is quite flat you can actually take a very long time you know to slowly find your way to maybe this point on the plateau and then because of a random perturbation to the left or right maybe then finally I'm gonna switch pen colors for clarity your algorithm can then find this way off the plateau but then to take this very long slope off before it's found this way here and they could get off this plateau so the takeaways from this video are first you actually pretty unlikely to get stuck in bad local optima so long as you're training and reasonably launched new network save a lot of parameters and the cost function J is defined over a relatively high dimensional space but second that plateaus are a problem and they can actually make learning pretty slow and this is where algorithms like momentum or our most proper atom can really help you learning algorithm as well and these are scenarios where more sophisticated optimization algorithms such as atom can actually speed up the rate at which you could move down the plateau and then get off the plateau so because your networks are solving optimization problems over such high dimensional spaces to be honest I don't think anyone has great intuitions about what these spaces really look like and our understanding of them is still evolving but I hope this gives you some better intuition about the challenges that the optimization algorithms may face so that congratulations on coming to the end of this week's content please take a look at this week's quiz as well as the exercise I hope you enjoyed practicing some of these ideas with this week's forum exercise and I look forward to seeing you at the start of next week's videosin the early days of deep learning people used to worry a lot about the optimization algorithm getting stuck in bad local optima but as the theory of deep learning has advanced our understanding of local optima is also changing let me show you how we now think about local optima and problems in the optimization problem in deep learning so this was a picture people used to have in mind when they worried about local optima maybe you're trying to optimize some set of parameters and we call them W 1 and W 2 and the height of the surface is the cost function so in this picture it looks like there are a lot of local optima you know in in all those places and it'd be easy for gradient descents or one of the other algorithms to get stuck on a local optimum rather than find this way to a global optimum it turns out that if you are plotting a figure like this in two dimensions then it's easy to create plots like this of a lot of different local optima and these very low dimensional plots used to gather intuition but this intuition isn't actually correct it turns out if you create in your network most points of 0 gradients are not local optima like points like this instead most points of 0 gradients in the cost function are actually saddle points so that's a point with a zero gradient again this is maybe W 1 W 2 and the highest heightens the value of the cost function J but informally a function in a very high dimensional space if the gradient is 0 then in each direction it can either be a convex light function or a concave light function and if you are in say a 20,000 dimensional space then thread to be a local optima all 20,000 directions need to look like this and so the chance of that happening is maybe very small you know maybe 2 to the minus 20000 instead you're much more likely to get some directions where the curve bends up like so as well some directions where the function is bending down rather than have them all Bend upwards so that's why in very high dimensional spaces you're actually much more likely to run into a saddle points like that shown on the right then local optimum oh and as for why the surface is called a saddle point if you can picture maybe this is a sort of shadow you put on a horse right so maybe if this is a horse I guess there's a head of a horse as you I have a horse you know I guess and right well another great drawing of a horse but you get the idea then you the rider will sit here in the saddle so then so that's why this point here where the derivative is zero that point is called a saddle point it's really the point to understand where you're sitting s and that happens to have you know derivative zero and so one of the lessons we learned in history of deep learning is that a lot of our intuitions about low dimensional spaces like what you can plot on the left they really don't transfer to the very high dimensional spaces then our learning algorithms are operating over because if you have twenty thousand parameters then J is V a function over a twenty thousand dimensional vector and you're much more likely to see saddle points than local optimum if local optima aren't a problem then what is a problem it turns out that plateaus can really slow down learning and the plateau is a region where the derivative is close to zero for a long time so if you are here then gradient descent will move down the surface and because the gradient is zero or near zero the surface is quite flat you can actually take a very long time you know to slowly find your way to maybe this point on the plateau and then because of a random perturbation to the left or right maybe then finally I'm gonna switch pen colors for clarity your algorithm can then find this way off the plateau but then to take this very long slope off before it's found this way here and they could get off this plateau so the takeaways from this video are first you actually pretty unlikely to get stuck in bad local optima so long as you're training and reasonably launched new network save a lot of parameters and the cost function J is defined over a relatively high dimensional space but second that plateaus are a problem and they can actually make learning pretty slow and this is where algorithms like momentum or our most proper atom can really help you learning algorithm as well and these are scenarios where more sophisticated optimization algorithms such as atom can actually speed up the rate at which you could move down the plateau and then get off the plateau so because your networks are solving optimization problems over such high dimensional spaces to be honest I don't think anyone has great intuitions about what these spaces really look like and our understanding of them is still evolving but I hope this gives you some better intuition about the challenges that the optimization algorithms may face so that congratulations on coming to the end of this week's content please take a look at this week's quiz as well as the exercise I hope you enjoyed practicing some of these ideas with this week's forum exercise and I look forward to seeing you at the start of next week's videos\n"