For deriving upper bounds both in the stochastic online learning setting and the multi-armed bandit problems we will use two classical tools: Hoeffding's inequality in combination with union bounds.

In this post we will prove a corollary that allows us to control the deviation of the empirical mean from the true mean for all $$t \leq n$$, given $$k \leq K$$ sources.

That is, we will look at $$X_{k,1}, \ldots, X_{k,n}$$ i.i.d. random variables and bound the probability of the following event

$$\bigg \{ \bigcap\limits_{k \leq K} \bigcap\limits_{t \leq n}|\frac{1}{t} \sum_\limits{i\leq t} X_{k,i} - \mu_k | \leq c \bigg \} \tag{1}\label{intro}$$

for some constant $$c$$, which we will specify at the end of this article.

## Theorem 1 (Hoeffding's inequality)

Let $$X_1, \ldots, X_n$$ be i.i.d. random variables such that their distribution has support in $$[0,1]$$, and mean $$\mu$$. It holds that

$$\mathbb{ P }\bigg(|\frac{1}{n} \sum_\limits{i=1}^n X_i - \mu| > \epsilon \bigg) \leq 2\exp\left(-2n\epsilon^2\right)$$

Equivalently it holds with probability larger than $$1-\delta$$ that

$$|\frac{1}{n} \sum_\limits{i=1}^n X_i - \mu| \leq \sqrt{\frac{\log(2/\delta)}{2n}}.$$

## Theorem 2 (Union bound)

Let $$\xi_1, \ldots, \xi_n$$ be events of probability $$1 - \delta_1, \ldots, 1 - \delta_n$$. It holds that

$$\mathbb{ P }\left(\bigcap\limits_{i}\xi_i \right) \geq 1 - \sum\limits_i \delta_i.$$

A proof for Hoeffding can be seen e.g. on Wikipedia. Union bound can be proven by looking at the complement of the event and using the sub-additivity of the probability measure.

The Hoeffding inequality gives us an upper bound on the probability that the empirical mean deviates from the expected value by more than a certain amount. Note that this holds for an arbitrary but fixed $$n$$. The following corollary provides us an upper bound for all $$t \leq n$$.

## Corollary (Hoeffding and union bound)

Let for any $$k \leq K, X_{k,1}, \ldots, X_{k,n}$$ be i.i.d random variables such that their distribution $$\nu_k$$ has support in $$[0,1]$$, and mean $$\mu_k$$. It holds that

$$\mathbb{ P }(\exists k \leq K, \exists t\leq n, |\frac{1}{t} \sum\limits_{i\leq t} X_{k,i} - \mu_k | > \frac{\epsilon}{\sqrt{t}}) \leq 2nK \exp\left(-2\epsilon^2\right)$$

### Proof

By Hoeffding we know that for some $$t \leq n$$

$$\mathbb{ P }(|\frac{1}{t} \sum\limits_{i\leq t} X_{k,i} - \mu_k | > \frac{\epsilon}{\sqrt{t}} ) \leq 2 exp\left(-2\epsilon^2\right) \tag{2}\label{bound}$$

With the sub-additivity of the probability measure we get

\begin{align*} &\mathbb{ P }\bigg(\exists k \leq K, \exists t\leq n, |\frac{1}{t} \sum\limits_{i\leq t} X_{k,i} - \mu_k | > \frac{\epsilon}{\sqrt{t}} \bigg) \\ &= \mathbb{ P }\bigg(\bigcup\limits_{k \leq K} \bigcup\limits_{t \leq n} \bigg\{ |\frac{1}{t} \sum\limits_{i\leq t} X_{k,i} - \mu_k | > \frac{\epsilon}{\sqrt{t}} \bigg\} \bigg) \\ &\leq \sum\limits_{k\leq K}\sum\limits_{t\leq n} \mathbb{ P }\left( |\frac{1}{t} \sum\limits_{i\leq t} X_{k,i} - \mu_k | > \frac{\epsilon}{\sqrt{t}} \right) \\ &\overset{\eqref{bound}}{\leq} \sum\limits_{k\leq K}\sum\limits_{t\leq n} 2 \exp\left(-2 \epsilon^2 \right)\\ &\leq Kn 2 \exp\left(-2 \epsilon^2 \right) \tag*{$\blacksquare$} \end{align*}

## Implications

Let $$\epsilon = \sqrt{\frac{\log(2nK/\delta)}{2}}$$. It follows

$$\mathbb{ P }\bigg(\exists k \leq K, \exists t\leq n, |\frac{1}{t} \sum\limits_{i\leq t} X_{k,i} - \mu_k | > \sqrt{\frac{\log(2nK/\delta)}{2t}} \bigg) \leq \delta.$$

So for the complement we get

$$\mathbb{ P }\bigg(\color{royalblue}{\forall k \leq K}, \color{royalblue}{\forall t\leq n}, |\frac{1}{t} \sum\limits_{i\leq t} X_{k,i} - \mu_k | \color{royalblue}{\leq} \sqrt{\frac{\log(2nK/\delta)}{2t}} \bigg) \color{royalblue}{> 1- \delta} .$$

Using the set notation once more, we get

$$\mathbb{ P }\bigg(\color{royalblue}{\bigcap\limits_{k \leq K}} \color{royalblue}{\bigcap\limits_{t \leq n}} |\frac{1}{t} \sum\limits_{i\leq t} X_{k,i} - \mu_k | \leq \frac{\epsilon}{\sqrt{t}} \bigg) > 1- \delta$$

and we finally have the bound on the event $$\eqref{intro}$$ from the beginning of this post.

Over the next few posts we will use the above technique for proving upper bounds of the Follow the Leader and UCB algorithm.

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