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I was reading Elad Hazan's book on Online Convex Optimization(http://ocobook.cs.princeton.edu/OCObook.pdf) and am facing difficulty understanding the proof given for the No regret algorithm for MAB (Pg 102-103). It would be great if someone can provide a clarification on this.

The No-regret (sub-linear) algorithm is given in Algorithm 17 (Pg 102), and proof for no regret is shown in lemma 6.1 . I will make the description as self-contained as possible, but a more detailed presentation can be found in pgs 102-103 in the book mentioned above.

Let $\mathcal{K}=\left[1, n \right]$ denote the set of experts(arms). Let $i_t \in \mathcal{K}$, denote the expert chosen by the algorithm in the $t^{th}$ round. Let $l_t(i_t)$, denote he loss function provided by the adversary in the $t^{th}$ round. Since we are dealing with the bandit setting, we do not know $l_t(i)$ for all $i \in \mathcal{K}\backslash i_t$. Further, the loss functions ($l_t$) are assumed to be bounded between 0 and 1. The way the algorithm works is, at each round we flip a coin, with bias $\delta$. If the outcome of the coin is heads, the algorithm chooses one of the actions i.e. $i_t$, and constructs a estimate of $l_t$ as follows: \begin{equation} \hat{l}_t = \begin{cases} \frac{n}{\delta} l_t(i_t), \text{ if } i = i_t\\ 0, \text{ otherwise} \end{cases} \end{equation}

If however, the outcome of the coin toss was tails, then it simply sets $\hat{l}_t = 0$.

It can be easily shown that by using the above scheme we have $E\left[ \hat{l}_t(i)\right] = l_t(i), \, \forall i \in \mathcal{K}$.

The regret as shown in the book is as follows. \begin{eqnarray} \label{Eq:1} E[regret_T] &=& E\left[\sum_{t=1}^{T} l_t(i_t) - \sum_{t=1}^{T}l_t(i^*)\right] \\ \label{Eq:2} & \leq& E\left[\sum_{t \not \in S_T} \hat{l}_t(i_t) - \sum_{t \not \in S_T} \hat{l}_t(i^*) + \sum_{t \in S_T} 1 \right] \end{eqnarray}

where $S_T \subseteq [1, T]$ denotes the round in which the coin toss was heads, and $i^* = \underset{i \in \mathcal{K}}{\text{arg min}} \sum_{t=1}^{T} l_t(i)$ . I am having a hard time showing why $E[regret_T] \leq E\left[\sum_{t \not \in S_T} \hat{l}(i_t) - \sum_{t \not \in S_t} \hat{l}_t(i^*) + \sum_{t \in S_T} 1 \right] $. The comment in the book is that $i^*$ is independent of $\hat{l}_t$, hence validity of inequality. I did not understand what that was supposed to mean.

My Attempt:

For my attempt, I will be using the some of the notation used in the proof of that book. We know that $\underset{i \in [1, \dotsc, n]}{\text{min}} \sum_{t=1}^T \hat{l}_t (i) \leq \sum_{t=1}^T \hat{l}_t (i), \, \forall i \in [1,n]$. Applying $E$ (expectation) on both sides we get, \begin{eqnarray} E\left[ \underset{i \in [1, \dotsc, n]}{\text{min}} \sum_{t=1}^T \hat{l}_t (i) \right] &\leq & E \left[\sum_{t=1}^T \hat{l}_t (i) \right], \, \forall i \in [1,n] \\ \implies E\left[ \underset{i \in [1, \dotsc, n]}{\text{min}} \sum_{t=1}^T \hat{l}_t (i) \right] &\leq & \underset{i \in [1, \dotsc, n]}{\text{min}} E \left[\sum_{t=1}^T \hat{l}_t (i) \right] \\ &=& \sum_{t=1}^T l_t (i^*) \end{eqnarray} , in the book it is easily shown that $E\left[\hat{l}(i)\right] = l(i)$. In view of the above inequality, we can show

\begin{eqnarray} E[regret_T] &=& E\left[\sum_{t=1}^{T} l(i_t) - \sum_{t=1}^{T}l_t(i^*)\right] \\ & \leq& E\left[ \sum_{t=1}^{T} \hat{l}(i_t) - \underset{i \in [1, \dotsc, n]}{\text{min}} \sum_{t=1}^T \hat{l}_t (i) \right] \end{eqnarray}

Clearly, I am making some mistakes in the attempt above. I will be grateful if someone can point me to those and help clarify the reasoning in the book. I say my attempt is incorrect because, in the way I have shown, I completely disregarded the set $S_T$. Without this set, the book shows the regret to be $\mathcal{O}(\sqrt{T})$, whereas with the $S_T$ set the regret is shown to be $\mathcal{O}(T^{\frac{3}{4}})$

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I don't think you wrote anything incorrect, it's just that you didn't finish your proof yet. How are you going to go from there to an actual bound on the regret? You're going to have to use the regret guarantee of the gradient descent algorithm. But we didn't run Gradient Descent on all of the time steps. We only ran it on the time steps not in $S_T$. So we get a regret bound for the steps not in $S_T$ from gradient descent, which is $O\left(\frac{n}{\delta}\sqrt{T}\right)$. And we have to use a separate bound for the time steps in $S_T$. The proof uses the easiest one: at each such step, at worst, the algorithm has loss $1$ while $i^*$ has loss zero, so the contribution to regret from that step is at most $1$, for a total of $|S_T|$, which in expectation is $\delta T$.

As a sidenote, I am not sure that independence of $i^*$ and $\hat{\ell}_t$ is really being used, maybe it is just linearity of expectation (?).

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  • $\begingroup$ The algo given in pg 102 seems to apply GD at all times (lines 7 and 10). However during steps not in $S_T$, it sets $\hat{f}_t, \hat{l}_t = 0$. So we could have disregarded all those estimates (functions) when computing $i_{t+1}$ while performing GD. In view of this, I am still confused as to why you say that the Algo only applies GD to time steps in not in $S_T$. Could you please expand your answer addressing this, or may be just post a comment clarifying it. - Thanks $\endgroup$ – csTheoryBeginner Nov 26 '18 at 21:27
  • $\begingroup$ @csTheoryBeginner, good question, you are right about the updates (lines 7 and 10), but it is not true for the actions taken. Line 9 of the alg is SGD but line 5 is to draw an action uniformly at random. $\endgroup$ – usul Dec 1 '18 at 16:54

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