Why Neural Networks can learn (almost) anything
Summary
TLDRThis script introduces the concept of neural networks as universal function approximators. It begins by explaining functions as a system of inputs and outputs, then explores how neural networks can be trained to reverse-engineer and approximate functions from data. The video demonstrates the mechanics of a neural network, showing how simple building blocks like neurons can be combined to construct complex functions. It emphasizes the importance of non-linearities in allowing neural networks to learn and highlights their ability to approximate any function given enough data and neurons. The script discusses the potential of neural networks to learn and emulate intelligent behavior, while acknowledging practical limitations and the necessity of sufficient training data. It concludes by emphasizing the transformative impact of neural networks in fields like computer vision and natural language processing.
Takeaways
- 🤖 Neural networks are a form of function approximators that can learn to represent complex patterns and relationships in data.
- 🧩 Neural networks are composed of interconnected neurons, which are simple linear functions that can be combined to create more complex non-linear functions.
- 🚀 Neural networks can be trained using algorithms like backpropagation to automatically adjust their parameters and improve their approximation of a target function.
- 🌀 Neural networks can learn to approximate any function to any desired degree of precision, making them universal function approximators.
- 🖼️ Neural networks can learn to approximate functions that represent various tasks, such as image classification, language translation, and more, by encoding inputs and outputs as numerical data.
- 💻 Neural networks are theoretically Turing-complete, meaning they can solve any computable problem, given enough data and resources.
- 🔮 The success of neural networks in approximating functions depends on the availability of sufficient data that accurately represents the underlying function.
- 🚧 Neural networks have practical limitations, such as finite resources and challenges in the learning process, that constrain their ability to approximate certain functions.
- 🤯 Neural networks have revolutionized fields like computer vision and natural language processing by providing a way to solve problems that require intuition and fuzzy logic, which are difficult to manually program.
- 🚀 The humble function is a powerful concept that allows neural networks to construct complex representations and approximate a wide range of intelligent behaviors.
Q & A
What is a neural network learning in this video?
-The neural network is learning the shape of the infinitely complex fractal known as the Mandelbrot set.
What is the fundamental mathematical concept that needs to be understood in order to grasp how a neural network can learn?
-The fundamental mathematical concept that needs to be understood is the concept of a function, which is informally defined as a system of inputs and outputs.
How can a function be approximated if the actual function itself is unknown?
-A function approximator can be used to construct a function that captures the overall pattern of the data, even if there is some noise or randomness present.
What is a neural network in the context of function approximation?
-A neural network is a function approximator that can learn to approximate any function by combining simple computations.
What is the basic building block of a neural network?
-The basic building block of a neural network is a neuron, which is a simple linear function that takes in inputs, multiplies them by weights, adds a bias, and produces an output.
Why is a non-linearity needed in a neural network?
-A non-linearity, such as the rectified linear unit (ReLU), is needed to prevent the neural network from simplifying down to a single linear function, which would limit its ability to learn more complex patterns.
What algorithm is commonly used to automatically tune the parameters of a neural network?
-The most common algorithm for automatically tuning the parameters of a neural network is called backpropagation.
Can neural networks learn any function?
-Neural networks have been rigorously proven to be universal function approximators, meaning they can approximate any function to any degree of precision, as long as there is enough data to describe the function.
What are some practical limitations of neural networks?
-Practical limitations of neural networks include finite network size, constraints introduced by the learning process, and the requirement for sufficient data to accurately approximate the target function.
What are some areas where neural networks have been particularly successful?
-Neural networks have been indispensable in fields like computer vision, natural language processing, and other areas of machine learning, where they have been able to learn intuitions and fuzzy logic that are difficult for humans to manually program.
Outlines
🧠 Neural Network Learning the Mandelbrot Set
This paragraph introduces the video's topic, which is about an artificial neural network learning the shape of the infinitely complex Mandelbrot fractal set. It provides context by explaining what the Mandelbrot set is and emphasizing its complexity. The paragraph then transitions into discussing the fundamental mathematical concept of a function, which is a system that takes inputs and produces outputs. It poses the question of whether it's possible to reverse engineer a function that produced a given data set of inputs and outputs, even if there is some noise or randomness. The concept of a function approximator, which is what a neural network is, is introduced.
🔄 How Neural Networks Approximate Functions
This paragraph delves deeper into how neural networks work as function approximators. It introduces a visualization tool that demonstrates a neural network taking two inputs (x1 and x2) and producing one output. Through this visual representation, the paragraph explains how the network constructs a shape that accurately distinguishes between different data points, effectively learning and approximating the underlying function that describes the data. The concept of neurons, the building blocks of neural networks, is introduced, with each neuron being a simple linear function that takes inputs, multiplies them by weights, and produces an output. The paragraph then explores the limitations of linear functions and the need for non-linearities, such as the rectified linear unit (ReLU), to overcome these limitations and enable more complex function approximations.
🔀 Neural Networks as Universal Function Approximators
This paragraph discusses the power of neural networks as universal function approximators, capable of approximating any function to any desired degree of precision. It emphasizes that by adding more neurons and layers, neural networks can piece together even the most complicated approximations, capturing intricate patterns like spirals. The paragraph highlights that neural networks can learn anything that can be expressed as a function, including the infinitely complex Mandelbrot set. It then expands on the concept of encoding various inputs (images, text, audio) as numbers and using neural networks to process them, as they can simulate any processing that can be written as a function. The paragraph also touches on the Turing completeness of neural networks, implying their ability to solve the same kinds of problems that computers can. It concludes by acknowledging some practical limitations and considerations, while emphasizing the transformative impact of neural networks on fields like computer vision and natural language processing.
Mindmap
Keywords
💡Function
💡Neural Network
💡Neuron
💡Activation Function
💡Back Propagation
💡Function Approximation
💡Universal Function Approximator
💡Turing Completeness
💡Machine Learning
💡Fractal
Highlights
A neural network is learning the shape of the infinitely complex fractal known as the Mandelbrot set.
A function is a system of inputs and outputs - numbers in, numbers out.
If we know some of a function's inputs and outputs, but not the function itself, we can reverse engineer the function that produced the data.
A neural network is a function approximator that can construct a function to describe a data set, even with some noise or randomness.
A neuron is a function that takes inputs, multiplies them by weights, adds a bias, and produces a single output.
A neural network is a network of neurons, where each neuron's output becomes the input for the next layer of neurons.
Neurons are the building blocks of a neural network, and they can be combined to construct more complicated functions.
Linear functions can only combine to make a single linear function, which is not sufficient for approximating complex data.
Adding a non-linearity, such as the rectified linear unit (ReLU) activation function, allows neurons to overcome their individual limitations and build more complex functions.
Backpropagation is the most common algorithm for automatically adjusting the weights and biases of a neural network to improve its function approximation.
Neural networks can be proven to be universal function approximators, capable of approximating any function to any desired degree of precision.
Neural networks can learn any intelligent behavior, process, or task that can be expressed as a function, including computer vision, natural language processing, and other machine learning problems.
Under certain assumptions, neural networks are Turing complete, meaning they can solve the same types of problems as any computer and can learn to simulate any algorithm.
Practical limitations, such as network size, learning process constraints, and availability of sufficient training data, can restrict what neural networks can learn.
Neural networks have transformed fields like computer vision and natural language processing by providing a way to solve problems that require intuition and fuzzy logic, which are difficult to manually program.
Transcripts
you are currently watching an artificial
neural network learn
in particular it's learning the shape of
an infinitely complex fractal known as
the mandelbrot set
this is what that set looks like
complexity all the way down
now in order to understand how a neural
network can learn the mandelbrot set
really how it can learn anything at all
we will need to start with a fundamental
mathematical concept
what is a function
informally a function is just a system
of inputs and outputs numbers in numbers
out
in this case you input an x and it
outputs a y
you can plot all of a function's x and y
values in a graph where it draws out a
line what is important is that if you
know the function you can always
calculate the correct output y given any
input x
but say we don't know the function and
instead only know some of its x and y
values we know the inputs and outputs
but we don't know the function used to
produce them
is there a way to reverse engineer that
function that produced this data
if we could construct such a function we
could use it to calculate a y value
given an x value that is not in our
original data set this would work even
if there was a little bit of noise in
our data a little randomness we can
still capture the overall pattern of the
data and continue producing y values
that aren't perfect but close enough to
be useful what we need is a function
approximation and more generally a
function approximator
that is what a neural network is
this is an online tool for visualizing
neural networks and i'll link it in the
description below this particular
network takes two inputs x1 and x2 and
produces one output technically this
function would create a
three-dimensional surface but it's
easier to visualize in two dimensions
this image is rendered by passing the x
y coordinate of each pixel into the
network which then produces a value
between negative one and one that is
used as the pixel value these points are
our data set and are used to train the
network when we begin training it
quickly constructs a shape that
accurately distinguishes between blue
and orange points building a decision
boundary that separates them it is
approximating the function that
describes the data it's learning and is
capable of learning the different data
sets that we throw at it
so what is this middle section then well
as the name implies this is the network
of neurons each one of these nodes is a
neuron which takes in all the inputs
from the previous layer of neurons and
produces one output which is then fed to
the next layer
inputs and outputs sounds like we're
dealing with a function
indeed a neuron itself is just a
function one that can take any number of
inputs and has one output each input is
multiplied by a weight and all are added
together along with bias the weights and
bias make up the parameters of this
neuron values that can change as the
network learns
to keep it easy to visualize we'll
simplify this down to a two-dimensional
function with only one input and one
output
now neurons are our building blocks of
the larger network building blocks that
can be stretched and squeezed and
shifted around and ultimately work with
other blocks to construct something
larger than themselves the neuron as
we've defined it here works like a
building block it is actually an
extremely simple linear function one
which forms a flat line or plane when
there's more than one input
with the two parameters the weight and
bias we can stretch and squeeze and move
our function up and down and left and
right
as such we should be able to combine it
with other neurons to form a more
complicated function one built from lots
of linear functions
so let's start with a target function
one we want to approximate i've
hard-coded a bunch of neurons whose
parameters were found manually and if we
weight each one and add them up as would
happen in the final neuron of the
network we should get a function that
looks like the target function
well that didn't work at all what
happened
well if we simplify our equation
distributing weights and combining like
terms we end up with a single linear
function
turns out linear functions can only
combine to make one linear function this
is a big problem because we need to make
something more complicated than just a
line we need something that is not
linear a non-linearity
in our case we will be using a relu a
rectified linear unit we use it as our
activation function meaning we simply
apply it to our previous naive neuron
this is about as close as you can get to
a linear function without actually being
one and we can tune it with the same
parameters as before
however you may notice that we can't
actually lift the function off of the
x-axis which seems like a pretty big
limitation
well let's give it a shot anyway and see
if it performs any better than our
previous attempt
we're still trying to approximate the
same function and we're using the same
weights and biases as before but this
time we're using a value as our
activation function
and just like that the approximation
looks way better unlike before our
function cannot simplify down to a flat
linear function if we add the neurons
one by one we can see the simple value
functions building on one another and
the inability for one neuron to lift
itself off the x-axis doesn't seem to be
a problem many neurons working together
overcome the limitations of individual
neurons
now i manually found these weights and
biases but how would you find them
automatically the most common algorithm
for this is called back propagation and
is in fact what we're watching when we
run this program
it essentially tweaks and tunes the
parameters of the network bit by bit to
improve the approximation and the
intricacies of this algorithm are really
beyond the scope of this video i'll link
some better explanations in the
description
now we can see how this shape is formed
and why it looks like it's made up of
sort of sharp linear edges it's the
nature of the activation function we're
using
we can also see why if we use no
activation function at all the network
utterly fails to learn we need those
non-linearities
so what if we try learning a more
complicated data set like this spiral
let's give it a go
seems to be struggling a little bit to
capture the pattern no problem if we
need a more complicated function we can
add more building blocks more neurons
and layers of neurons and the network
should be able to piece together a
better approximation something that
really captures the spiral
it seems to be working
in fact no matter what the data set is
we can learn it that is because neural
networks can be rigorously proven to be
universal function approximators they
can approximate any function to any
degree of precision you could ever want
you can always add more neurons
this is essentially the whole point of
deep learning because it means that
neural networks can approximate anything
that can be expressed as a function a
system of inputs and outputs
this is an extremely general way of
thinking about the world the mandelbrot
set for instance can be written as a
function and learned all the same this
is just a scaled-up version of the
experiment we were just looking at but
with an infinitely complex data set
we don't even really need to know what
the manual brought set is the network
learns it for us and that's kind of the
point if you can express any intelligent
behavior any process any task as a
function then a network can learn it for
instance your input could be an image
and your output a label as to whether
it's a cat or a dog or your input could
be text in english and your output a
translation to spanish you just need to
be able to encode your inputs and
outputs as numbers but computers do this
all the time images video text audio
they can all be represented as numbers
and any processing you may want to do
with this data so long as you can write
it as a function can be emulated with a
neural network
it goes deeper than this though under a
few more assumptions neural networks are
provably turing complete meaning they
can solve all of the same kinds of
problems that any computer can solve
an implication of this is that any
algorithm written in any programming
language can be simulated on a neural
network but rather than being manually
written by a human it can be learned
automatically with a function
approximator
neural networks can learn anything
okay that is not true first off you
can't have an infinite number of neurons
there are practical limitations on
network size and what can be modeled in
the real world
i've also ignored the learning process
in this video and just assumed that you
can find the optimal parameters
magically how you realistically do this
introduces its own constraints on what
can be learned
additionally in order for neural
networks to approximate a function you
need the data that actually describes
that function if you don't have enough
data your approximation will be all
wrong it doesn't matter how many neurons
you have or how sophisticated your
network is you just have no idea what
your actual function should look like
it also doesn't make a lot of sense to
use a function approximator when you
already know the function you wouldn't
build a huge neural network to say learn
the mandelbrot set when you can just
write three lines of code to generate it
unless of course you want to make a cool
background visual for a youtube video
there are countless other issues that
have to be considered but for all these
complications neural networks have
proven themselves to be indispensable
for a number of really rather famously
difficult problems for computers
usually these problems require a certain
level of intuition and fuzzy logic that
computers generally lack and are very
difficult for us to manually write
programs to solve
fields like computer vision natural
language processing and other areas of
machine learning have been utterly
transformed by neural networks
and this is all because of the humble
function a simple yet powerful way to
think about the world and by combining
simple computations we can get computers
to construct any function we could ever
want
neural networks can learn almost
anything
[Music]
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