Functions are the very essence of our existence, the
mathematical tapestry that weaves through the fabric of our reality. They
encapsulate everything, from the mellifluous notes that caress your eardrums to
the incandescent photons currently dancing upon your retinas. They permeate
through diverse domains of mathematics, manifesting as high school algebra's
humble polynomials or elegant calculus's smooth, one-variable functions.
Functions are the lexicon of our world. They
are the keys to understanding, modeling, and predicting the universe that
envelopes us. In the realm of artificial intelligence, the goal is to create
programs that possess the cognitive prowess to comprehend, model, and
prophesize the world autonomously. To achieve this, these programs must craft
their own functions, a task entrusted to the domain of function approximation.
In essence, functions are systems that convert
inputs into outputs, a numerical transformation. Visualize them as pathways
connecting 'x' inputs to 'y' outputs, tracing a trajectory on a graph.
Now, imagine a scenario
where we possess only partial knowledge of this function, knowing some 'x' and
'y' values but oblivious to the underlying formula. Can we reverse-engineer
this enigmatic function? Surprisingly, yes. We can construct a function
approximation, and that's where neural networks come into play.
Enter neural networks, the artisans of function construction. At its core, a function is a marvelous contraption that ingests a set of inputs and gracefully yields a corresponding set of outputs, establishing a profound relationship between them. The neural networks solve a rather intriguing problem, they endeavor to approximate functions whose exact definitions elude us. Instead, we possess a meager smattering of data points, our dataset, comprising inputs and their corresponding outputs. Our task? Craft a function that not only accommodates these data points but also gracefully extrapolates to predict outputs for inputs not found within our dataset. This process, akin to the art of fitting curves, is the heart of function approximation. This neural network, in itself, is a function, poised to approximate some enigmatic target function. Its inputs and outputs, often referred to as features and predictions, come in the guise of vectors, arrays of numerical wonders.
This overarching function is an intricate tapestry
of simple functions called neurons. These neurons, voracious consumers of
inputs, multiply each by its own weight, summing them up along with an extra
weight known as the bias. Allow me to cast this equation in the grandeur of
linear algebra: inputs into one vector, weights into another, culminating in
the glorious dot product. We add the product of each input and weight, summing
them to obtain this numeric gem, which subsequently waltzes into a quaint
activation function. In our case, it's a relu, gracefully curving our neuron's
mathematical path.
As these weights shift and dance, our neural network
unfolds, ushering the original inputs into layers of neurons, each armed with
their unique weights and individual output values. These outputs harmoniously
coalesce into a vector, serenading the next layer, and the dance continues. The
ultimate aim? Minimize the network's error or loss, a yardstick quantifying the
divergence between predicted and actual outputs. The magic underpinning this
alchemy goes by the name of backpropagation.
Remember, the activation function defines our neuron's mathematical persona, while the weights mold and shape its trajectory. We furnish our network's core with inputs, each leading to a troupe of neurons adorned with their distinctive weights. Their harmonious chorus yields an output vector, a masterpiece in the making. The mission: to diminish the network's error or loss over time.
It's imperative to recognize that while our neural
networks are theoretically universal function approximators, practicality often
tells a different tale. Empirical measurement, the litmus test of any
scientific endeavor, invites us to calculate and compare the error rates of our
neural networks. Science, you see, thrives on experimentation, on the
validation of our approximations.
Our neural network
endeavors to mold the surface around the sphere, taming the data points with a
mathematical embrace.
