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## Can we get some theoretical guarantees of how successful learning can be achieved in a relatively simplified setting?

In this blog, I shall answer the aforementioned question. While third-party materials like this blog might help to introduce some concept, I encourage readers to also go through the sources from which I learned it. This section is primarily taken from a single source – “Understanding Machine Learning: From Theory to Algorithms” by Shai Shalev-Shwartz and Shai Ben-David. More specifically, this blog essentially trims and embarrasingly simplifies Chapter 2.

So let’s get cracking…

### Introduction, Objective, and Motivation

We need to first focus on what do we want to learn and then define what do we mean by the word learning or more specifically a successful learning – i.e., what does it means to have a successful learning.

First let me quote Vladimir N. Vapnik from “The Nature of Statistical Learning Theory”: »we consider the learning problem as a problem of finding a desired dependence using a **limited** number of observations.«

To get a better perspective of what do we want to learn consider this quote (again from Vapnik): »What must one know a priori about an unknown functional dependency in order to estimate it on the basis of observations?«

While I shall provide a more mathematical definition of “learning”, maybe we should spend a bit more time on the philosophy of what exactly we mean by learning.

Now I shall introduce the example that Shalev-Shwartz and Ben-David used in their book – “learn how to predict whether a papaya you see in the market is tasty or not”.

Before diving into, let us first define the learner’s input i.e. the basic statistical learning setting that the learner has access to:

• Domain set: This is a set of objects that we wish to label. Each domain points is referred to as instances in the instance space represented by $$\mathbf{\mathcal{X}}$$.

• Label set: Consider we have only 2 labels i.e. {0,1} or {-1, +1}. We shall denote this set of possible labels with $$\mathbf{\mathcal{Y}}$$.

• Training data: As mentioned, that our Domain set consists of instances we wish to label. But it is not possible to label the entire Domain set. So consider that we label a finite $$\mathbf{m}$$ instances. Hence, $$S = ((x_{1}, y_{1}), (x_{2}, y_{2}), \dots, (x_{m}, y_{m}))$$ is a finite sequence of pairs in $$\mathcal{X}\times\mathcal{Y}$$. It is important to keep in mind that despite the “set” notation of $$S$$, it is a sequence i.e. the same example may appear twice in $$S$$. This finite set or the training data are often called training examples.

• The learner’s output: The objective of the learner is to output a mapping function or a prediction rule $$h: \mathbf{\mathcal{X}} \rightarrow \mathbf{\mathcal{Y}}$$. This function is also called a {predictor, hypothesis, classifier}.

• Data generation Model: We consider that the probability distribution over $$\mathbf{\mathcal{X}}$$ be denoted by $$D$$. It is important to note that we do not assume that the learner knows anything about this distribution.

• Assumption: We assume that there is some “correct” labelling function, $$f:\mathbf{\mathcal{X}} \rightarrow \mathbf{\mathcal{Y}}$$. We shall relax this assumption in our next blog.

• Measure of success: The error of our prediction rule $$h: \mathbf{\mathcal{X}} \rightarrow \mathbf{\mathcal{Y}}$$ is defined to be ($$\stackrel{\text{def}}{=}$$ symbol represent is defined):

$L_{D, f}(h) \stackrel{\text{def}}{=} \mathbb{P}_{x \sim D} [h(x) \neq f(x)] \stackrel{\text{def}}{=} D(\{x \sim D: h(x) \neq f(x)\})$

The error of $$h$$ is the probability of randomly choosing an example $$x \sim D$$ for which $$h(x) \neq f(x)$$. The subscript $$(D, f)$$ indicates that the error is measured with respect to the probability distribution $$D$$ and the correct labelling function $$f$$. $$L_{D, f}$$ is knows as: {generalization error, the risk, true error} of $$h$$.

• Note: Since the learner is blind to the underlying distribution $$D$$ and the correct labelling function $$f$$, it cannot calculate the true error. To overcome this problem, the learner uses what is known as the Empirical Risk for guidance.