Index of /besson/publis/slides/2019_06__About_Bernoulli_GLRTest__Seminar_at_PANAMA_IRISA_Rennes
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Publications associated with this talk
Joint work with my advisor Emilie Kaufmann:
- [*"Analyse non asymptotique d'un test séquentiel de détection de
ruptures et application aux bandits non
stationnaires"*](https://hal.inria.fr/hal-02006471/document)\
by **L. Besson** & [E.
Kaufmann](http://chercheurs.lille.inria.fr/ekaufman/research.html)
$\hookrightarrow$ presented at **GRETSI**, in Lille (France), next
August 2019
\vspace*{30pt}
- [*"The Generalized Likelihood Ratio Test meets klUCB: an Improved
Algorithm for Piece-Wise Non-Stationary
Bandits"*](https://hal.inria.fr/hal-02006471/document)\
by **L. Besson** & [E.
Kaufmann](http://chercheurs.lille.inria.fr/ekaufman/research.html)\
February 2019, pre-print on
[[HAL-02006471]{style="color: blue"}](https://hal.inria.fr/hal-02006471)
and
[[arXiv:1902.01575]{style="color: blue"}](https://arxiv.org/abs/1902.01575)
Outline of the talk
===================
Outline of the talk
1. (Stationary) Multi-armed bandits problems
2. Piece-wise stationary multi-armed bandits problems
3. The B-GLR test and its finite time properties
4. The BGLR-T + klUCB algorithm
5. Regret analysis
6. Numerical simulations
1. (Stationary) Multi-armed bandits problems
============================================
1\. (Stationary) Multi-armed bandits problems
1. [ **(Stationary) Multi-armed bandits problems** ]{.alert}
2. [ Piece-wise stationary multi-armed bandits problems
]{style="color: gray"}
3. [ The B-GLR test and its finite time properties
]{style="color: gray"}
4. [ The BGLR-T + klUCB algorithm ]{style="color: gray"}
5. [ Regret analysis ]{style="color: gray"}
6. [ Numerical simulations ]{style="color: gray"}
What is a bandit problem?
-------------------------
Multi-armed bandits
$=$ Sequential decision making problems in uncertain environments :
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$\hookrightarrow$ Interactive demo
[[`perso.crans.org/besson/phd/MAB_interactive_demo/`]{style="color: blue"}](https://perso.crans.org/besson/phd/MAB_interactive_demo/)\
Ref: \[Bandits Algorithms, Lattimore & Szepesvári, 2019\], on
[[`tor-lattimore.com/downloads/book/book.pdf`]{style="color: blue"}](https://tor-lattimore.com/downloads/book/book.pdf)
Mathematical model
------------------
Mathematical model
- Discrete time steps $t = 1, \dots, T$\
The *horizon* $T$ is fixed and usually unknown
- At time $t$, an *agent plays the arm* $A(t)\in\{1,\dots,K\}$,\
then she observes the *iid random reward* $r(t) \sim \nu_k$,
$r(t)\in\mathbb{R}$
\pause
- Usually, we focus on Bernoulli arms
$\nu_k = \mathrm{Bernoulli}(\mu_k)$, of mean $\mu_k\in[0,1]$, giving
binary rewards $r(t) \in\{0,1\}$.
\pause
- **Goal** : maximize the sum of rewards $\sum\limits_{t=1}^T r(t)$
- or [maximize the sum of expected rewards
$\mathbb{E}\left[ \sum\limits_{t=1}^T r(t) \right]$]{.alert}
\pause
- Any efficient policy must balance [between exploration and
exploitation]{.alert}: it must explore all arms to discover the best
one, while exploiting the arms known to be good so far.
Naive solutions
---------------
Two examples of bad solutions
$i)$ Pure exploration
- Play arm $A(t) \sim \mathcal{U}(\{1,\dots,K\})$ uniformly at random
- $\implies$ Mean expected rewards
$\frac{1}{T} \mathbb{E}\left[ \sum\limits_{t=1}^T r(t) \right] = \frac{1}{K} \sum\limits_{k=1}^K \mu_k \ll \max_k \mu_k$
\pause
$ii)$ Pure exploitation
- Count the number of samples and the sum of rewards of each arm
$N_k(t) = \sum\limits_{s < t} \mathbbm{1}(A(s)=k)$ and
$X_k(t) = \sum\limits_{s < t} r(s) \mathbbm{1}(A(s)=k)$
- Estimate the [unknown]{.alert} mean $\mu_k$ with
$\widehat{\mu_k}(t) = X_k(t) / N_k(t)$
- Play the arm of maximum empirical mean :
$A(t) = \arg\max_k \widehat{\mu_k}(t)$
- Performance depends on the first draws, and can be very poor!
$\hookrightarrow$ Interactive demo
[[`perso.crans.org/besson/phd/MAB_interactive_demo/`]{style="color: blue"}](https://perso.crans.org/besson/phd/MAB_interactive_demo/)
The *"Upper Confidence Bound"* algorithm
----------------------------------------
A first solution: *"Upper Confidence Bound"* algorithm
- Compute
$\mathrm{UCB}_k(t) = X_k(t) / N_k(t) + \sqrt{\alpha \log(t) / N_k(t)}$\
$=$ an [upper confidence bound]{.alert} on the [unknown]{.alert}
mean $\mu_k$
- Play the arm of maximal UCB : $A(t) = \arg\max_k \mathrm{UCB}_k(t)$\
$\hookrightarrow$ Principle of "optimism under uncertainty"
- $\alpha$ balances between *exploitation* ($\alpha\to0$) and
*exploration* ($\alpha\to\infty$)
\pause
- [UCB is efficient]{.alert}: the best arm is identified correctly
(with high probability) if there are enough samples (for $T$ large
enough)
- $\implies$ Expected rewards attains the maximum
($\forall \alpha>1/2$)
$$\text{For~} T\to\infty, \;\;\; \frac{1}{T} \mathbb{E}\left[ \sum\limits_{t=1}^T r(t) \right] \to \max_k \mu_k$$
Elements of the proof for UCB algorithm
Elements of proof of convergence (for $K$ Bernoulli arms)
- Suppose the first arm is the best:
$\textcolor{deeppurple}{\mu^*} = \textcolor{deeppurple}{\mu_1} > \mu_2 \geq \ldots \geq \mu_K$
- $\mathrm{UCB}_k(t) = X_k(t) / N_k(t) + \sqrt{\alpha \log(t) / N_k(t)}$
- Hoeffding's inequality gives
$\mathbb{P}(\mathrm{UCB}_k(t) < \mu_k(t)) \leq \mathcal{O}(\frac{1}{t^{2 \alpha}})$\
$\implies$ the different $\mathrm{UCB}_k(t)$ are true "Upper
Confidence Bounds" on the (unknown) $\mu_k$ (most of the times)
- And if a suboptimal arm $k>\textcolor{deeppurple}{1}$ is sampled, it
implies
$\mathrm{UCB}_k(t) > \mathrm{UCB}_{\textcolor{deeppurple}{1}}(t)$,
but $\mu_k < \textcolor{deeppurple}{\mu_1}$: Hoeffding's inequality
also proves that any "wrong ordering" of the $\mathrm{UCB}_k(t)$ is
unlikely
- We can prove that suboptimal arms $k$ are sampled about $o(T)$
times\
$\implies \mathbb{E}\left[ \sum\limits_{t=1}^T r(t) \right] \underset{T\to\infty}{\to} \textcolor{deeppurple}{\mu^*} \times \mathcal{O}(T) + \sum\limits_{k: \Delta_k>0} \mu_k \times o(T)$
[But... at which speed do we have this convergence?]{.alert}
Regret of a bandit algorithm
----------------------------
Measure the performance of an algorithm $\mathcal{A}$ with its mean
regret $R_{\mathcal{A}}(T)$
- Difference in the accumulated rewards between an "oracle" and
$\mathcal{A}$
- The "oracle" algorithm always plays [the (unknown) best arm
$k^* = \arg\max \mu_k$]{style="color: deeppurple"} (we note the best
mean [$\mu_{k^*} = \mu^*$]{style="color: deeppurple"})
- Maximize the sum of expected rewards $\Longleftrightarrow$ [minimize
the regret]{.alert}
$$\alert{ R_{\mathcal{A}}(T) } = \mathbb{E}\left[ \sum\limits_{t=1}^T \textcolor{deeppurple}{r_{k^*}}(t) \right] - \sum\limits_{t=1}^T \mathbb{E}\left[ r(t) \right] = T \textcolor{deeppurple}{\mu^*} - \sum\limits_{t=1}^T \mathbb{E}\left[ r(t) \right].$$
\pause
\vspace*{10pt}
Typical regime for stationary bandits (lower & upper bounds)
- No algorithm $\mathcal{A}$ can obtain a regret better than
$R_{\mathcal{A}}(T) \geq \Omega(\log(T))$
- And an efficient algorithm $\mathcal{A}$ obtains
$R_{\mathcal{A}}(T) \leq \mathcal{O}(\log(T))$
Regret of two UCB algorithms
----------------------------
Regret of the UCB algorithm and another algorithm
For any problem with $K$ arms following Bernoulli distributions, of
means $\mu_1,\dots,\mu_K \in[0,1]$, and [optimal mean
$\mu^*$]{style="color: deeppurple"}, then
For the UCB algorithm
$$R_T^{\mathrm{UCB}} \leq ( \sum_{\substack{k=1,\dots,K \\ \mu_k < \textcolor{deeppurple}{\mu^*}}} \frac{8}{(\mu_k - \textcolor{deeppurple}{\mu^*})} ) \log(T) + o(\log(T)) = \mathcal{O}\left( \alert{C(\mu_1,\dots,\mu_K)} \log(T) \right).$$
\<2-\>For the kl-UCB algorithm: a smaller regret upper-bound
$$R_T^{\mathrm{kl}\text{-}\mathrm{UCB}} \leq ( \sum_{\substack{k=1,\dots,K \\ \mu_k < \textcolor{deeppurple}{\mu^*}}} \frac{(\mu_k - \textcolor{deeppurple}{\mu^*})}{\mathrm{kl}(\textcolor{deeppurple}{\mu^*}, \mu_k)} ) \log(T) + o(\log(T)).$$
If $\mathrm{kl}(x, y) = x \log(x/y) + (1-x) \log((1-x)/(1-y))$ is the
binary relative entropy (*ie*, Kullback-Leibler divergence of two
Bernoulli of means $x$ and $y$)
2. Piece-wise stationary multi-armed bandits problems
=====================================================
2\. Piece-wise stationary MAB problems
1. [ (Stationary) Multi-armed bandits problems ]{style="color: gray"}
2. [ **Piece-wise stationary multi-armed bandits problems** ]{.alert}
3. [ The B-GLR test and its finite time properties
]{style="color: gray"}
4. [ The BGLR-T + klUCB algorithm ]{style="color: gray"}
5. [ Regret analysis ]{style="color: gray"}
6. [ Numerical simulations ]{style="color: gray"}
Non stationary MAB problems
Stationary MAB problems Arm $k$ gives rewards sampled from [the same
distribution]{style="color: blue"} for any time step:
$\forall t, r_k(t) \overset{\text{iid}}{\sim} \nu_k = \mathrm{Bernoulli}(\mu_k)$.
\pause
Non stationary MAB problems? Arm $k$ gives rewards sampled a [(possibly)
different distributions]{.alert} for any time step:
$\forall t, r_k(t) \overset{\text{iid}}{\sim} \nu_k\alert{(t)} = \mathrm{Bernoulli}(\mu_k\alert{(t)})$.
$\implies$ harder problem! And very hard if $\mu_k(t)$ can change at any
step!
\pause
**Piece-wise stationary** problems! $\hookrightarrow$ we focus on the
easier case when there are at most $o(\sqrt{T})$ intervals on which the
means are all stationary (= **sequence**)
Definitions
-----------
Break-points and stationary sequences
Define
- The number of break-points\
$\Upsilon_T = \sum\limits_{t=1}^{T-1} \mathbbm{1}(\exists k\in \{1,\dots,K\}$
$:$ $\mu_k(t) \neq \mu_k(t+1) )$
- The $i$-th break-point\
$\tau^{i} = \inf\{t > \tau^{i-1} : \exists k : \mu_k(t) \neq \mu_k(t+1)\}$
(with $\tau^0=0$)
\<2-\>**Hypotheses** on piece-wise stationary problems
- The rewards $r_k(t)$ generated by each arm $k$ are [*iid* on each
interval]{.alert} $[ \tau^{i} + 1, \tau^{i+1} ]$ (the $i$-th
sequence)
- There are $\Upsilon_T = o(\sqrt{T})$ break-points
- And [$\Upsilon_T$ can be known before-hand]{.alert}
- All sequences are "long enough"
Example of a piece-wise stationary MAB problem We plots the means
[$\mu_1(t)$]{style="color: red"}, [$\mu_2(t)$]{style="color: green"},
[$\mu_3(t)$]{style="color: blue"} of $K=3$ arms. There are
$\Upsilon_T=4$ break-points and $5$ sequences between $t=1$ and
$t=T=5000$:
{width="100%"}
Extending the definition of regret
----------------------------------
Regret for piece-wise stationary bandits?
The "oracle" algorithm plays the (unknown) best arm
$k^*(t) = \arg\max \mu_k(t)$ (which changes between stationary
sequences)
$$\alert{ R_{\mathcal{A}}(T) } = \mathbb{E}\left[ \sum\limits_{t=1}^T r_{k^*(t)}(t) \right] - \sum\limits_{t=1}^T \mathbb{E}\left[ r(t) \right] = \left(\alert{\sum_{t=1}^T \max_k \mu_k(t)} \right) - \sum\limits_{t=1}^T \mathbb{E}\left[ r(t) \right].$$
\pause
\vspace*{10pt}
Typical regime for piece-wise stationary bandits
- The conjectured lower-bound is
$R_{\mathcal{A}}(T) \geq \Omega(\sqrt{K \Upsilon_T T})$
- Currently, state-of-the-art algorithms $\mathcal{A}$ obtain
$$R_{\mathcal{A}}(T) \leq \mathcal{O}(K \sqrt{\Upsilon_T T \log(T)})$$
3. The B-GLR test and its finite time properties
================================================
3\. The B-GLR test and its finite time properties
1. [ (Stationary) Multi-armed bandits problems ]{style="color: gray"}
2. [ Piece-wise stationary multi-armed bandits problems
]{style="color: gray"}
3. [ **The B-GLR test and its finite time properties** ]{.alert}
4. [ The BGLR-T + klUCB algorithm ]{style="color: gray"}
5. [ Regret analysis ]{style="color: gray"}
6. [ Numerical simulations ]{style="color: gray"}
Break-point detection
---------------------
The break-point detection problem
Imagine the following game...
- You observe data $X_1,X_2,\ldots,X_t,\ldots \in[0,1]$
- You know $X_t$ is generated by a certain (unknown) distribution
- [Your goal]{.alert} is to distinguish between two hypotheses:
- [The distributions have the same mean ("no break-point")\
$\exists \mu_0, \mathbb{E}[X_1] = \mathbb{E}[X_2] = \ldots = \mathbb{E}[X_t] = \mu_0$]{style="color: deeppurple"}
- [The distributions have changed mean at a break-point at time
$\tau$\
$\exists \mu_0, \mu_1, \tau, \mathbb{E}[X_1] = \ldots = \mathbb{E}[X_{\tau}] = \mu_0, \mathbb{E}[X_{\tau+1}] = \mathbb{E}[X_{\tau+2}] = \ldots = \mu_1$]{style="color: meca"}
\pause
A [sequential break-point detection]{.alert} is a stopping time,
measurable under $\mathcal{F}_t = \sigma(X_1,\dots,X_t)$, which rejects
the hypothesis [$\mathcal{H}_0$]{style="color: deeppurple"} when
$\widehat{\tau} < +\infty$.
Likelihood ratio test for Bernoulli observations
------------------------------------------------
Bernoulli likelihood ratio test
**Hypothesis**: all distributions are Bernoulli
The problem boils down to distinguishing
- [$(\exists \mu_0 : \forall i\in\mathbb{N}^*, X_i \overset{\text{i.i.d.}}{\sim} \cB(\mu_0))$]{style="color: deeppurple"},
against the alternative
- [$(\exists \mu_0 \neq \mu_1, \tau > 1 : X_1, \ldots, X_\tau \overset{\text{i.i.d.}}{\sim} \cB(\mu_0) \text{~et~} X_{\tau+1}, \ldots \overset{\text{i.i.d.}}{\sim} \cB(\mu_1))$]{style="color: meca"}.
\pause
The [Likelihood Ratio statistic]{.alert} for this hypothesis test, after
observing $X_1,\dots,X_n$, is given by
$$\mathcal{L}(n) = \frac{\sup\limits_{\textcolor{meca}{\mu_0,\mu_1,\tau < n}}\ell(X_1,\dots,X_n ; \textcolor{meca}{\mu_0,\mu_1,\tau})}{\sup\limits_{\textcolor{deeppurple}{\mu_0}}\ell(X_1,\dots,X_n ;\textcolor{deeppurple}{\mu_0})},$$
where $\ell(X_1,\dots,X_n ; \textcolor{deeppurple}{\mu_0})$ (resp.
$\ell(X_1,\dots,X_n ; \textcolor{meca}{\mu_0,\mu_1,\tau})$) is the
likelihood of the observations under a model in
[$\mathcal{H}_0$]{style="color: deeppurple"} (resp.
[$\mathcal{H}_1$]{style="color: meca"}).
\pause
[$\hookrightarrow$ High values of this statistic tends to reject
[$\mathcal{H}_0$]{style="color: deeppurple"} over
[$\mathcal{H}_1$]{style="color: meca"}.]{.alert}
Expression of the Bernoulli Likelihood ratio
We can rewrite this statistic
$\mathcal{L}(n) = \frac{\sup\limits_{\textcolor{meca}{\mu_0,\mu_1,\tau < n}}\ell(X_1,\dots,X_n ; \textcolor{meca}{\mu_0,\mu_1,\tau})}{\sup\limits_{\textcolor{deeppurple}{\mu_0}}\ell(X_1,\dots,X_n ;\textcolor{deeppurple}{\mu_0})}$,
by using Bernoulli likelihood, and shifting means
$\widehat{\mu}_{k:k'} = \frac{1}{k'-k+1} \sum\limits_{s=k}^{k'} X_s$ :
$$\begin{aligned}
\log \mathcal{L}(n) = \sup_{s \in \{2,\dots,n-1\}} \bigl[
& s \times \mathrm{kl} (\underbrace{\widehat{\mu}_{1:s}}_{\text{before change}},\underbrace{\widehat{\mu}_{1:n}}_{\text{all data}} ) \\
+ & (n-s) \times \mathrm{kl} (\underbrace{\widehat{\mu}_{s+1:n}}_{\text{after change}},\underbrace{\widehat{\mu}_{1:n}}_{\text{all data}} ) \bigr].
\end{aligned}$$
Where
$\mathrm{kl}(x,y) =x \ln\bigl(x/y\bigr) + (1-x)\ln\bigl((1-x)/(1-y)\bigr)$
is the binary relative entropy
The BGLR-T
----------
The Bernoulli Generalized likelihood ratio test (BGLR)
- We can extend the Bernoulli likelihood ratio test if the
observations are [sub-Bernoulli]{.alert}.
- And any bounded distributions on $[0,1]$ is sub-Bernoulli
- $\implies$ the BGLR test can be applied for any bounded
observations!
\pause
The BGRL-T sequential break-point detection test The [BGLR-T]{.alert} is
the stopping time
$$\widehat{\tau}_{\delta} = \inf \bigl\{ n \in \mathbb{N}^* : \max_{s \in \{2,\dots,n-1\}} \bigl[s \, \mathrm{kl}\left(\widehat{\mu}_{1:s},\widehat{\mu}_{1:n}\right) + (n-s) \, \mathrm{kl}\left(\widehat{\mu}_{s+1:n},\widehat{\mu}_{1:n}\right)\bigr] \geq \beta(n,\delta) \bigr\}$$
with a [threshold function]{.alert} $\beta(n,\delta)$ specified later.
False alarm
-----------
Probability of false alarm
A good test should not detect any break-point if there is no break-point
to detect...
False alarm The stopping time is $\widehat{\tau}_\delta$, and a
break-point is detected if $\widehat{\tau}_\delta < \infty$.
Let $\mathbb{P}_{\textcolor{deeppurple}{\mu_0}}$ be a probability model
under which the observations are $X_t \in[0,1]$ and
[$\mathbb{E}[X_t] = \mu_0$ for all $t$]{style="color: deeppurple"}.
The [false alarm probability]{.alert} is
$\mathbb{P}_{\textcolor{deeppurple}{\mu_0}}(\widehat{\tau}_\delta < \infty)$.
[$\implies$ Goal: controlling the false alarm event!]{.alert} (in high
probability)
First result for the BGLR test
Controlling the false alarm probability For any [confidence
level]{.alert} $0<\delta<1$, the BGLR test satisfies
$$\mathbb{P}_{\textcolor{deeppurple}{\mu_0}}(\widehat{\tau}_\delta < \infty) \leq \delta$$
with the threshold function
$$\beta(n,\delta)= 2\,\mathcal{T}\left(\frac{\ln(3n\sqrt{n}/\delta)}{2}\right) + 6\ln(1+\ln(n)) \simeq \ln(3n \sqrt{n}/\delta).$$
Where $\mathcal{T}(x)$ verifies $\mathcal{T}(x)\simeq x + \ln(x)$ for
$x$ large enough
\<2\>Proof ? Hard to explain in a short time...\
$\hookrightarrow$ pre-print on
[[HAL-02006471]{style="color: blue"}](https://hal.inria.fr/hal-02006471)
and
[[arXiv:1902.01575]{style="color: blue"}](https://arxiv.org/abs/1902.01575)
Delay of detection
------------------
Delay of detection
A good test should detect a break-point "fast enough" if there is a
break-point to detect, with enough samples before the break-point
Delay of detection Let $\mathbb{P}_{\textcolor{meca}{\mu_0,\mu_1,\tau}}$
be a probability model under which $X_t \in[0,1]$ and [$X_t$ a mean
$\mu_0$ for all $t \leq \tau$, and $\mu_1$ for all $t > \tau$, with
$\mu_0 \neq \mu_1$]{style="color: meca"}.
The [gap]{.alert} of this break-point is $\Delta = |\mu_0 - \mu_1|$.
The [delay of detection]{.alert} is
$u = \widehat{\tau}_{\delta} - \tau$.
[$\implies$ Goal: controlling the delay of detection!]{.alert} (in high
probability)
Second result for the BGLR test
Controlling the delay of detection On a break-point of amplitude
$\Delta = |\mu_1 - \mu_0|$, the B-GLRT test satisfies
$$\mathbb{P}_{\textcolor{meca}{\mu_0,\mu_1,\tau}} (\widehat{\tau}_{\delta} \geq \textcolor{meca}{\tau} + u) \leq \exp\left( -\frac{2\textcolor{meca}{\tau} u}{\textcolor{meca}{\tau} + u}\left(\max\left[ 0, \Delta - \sqrt{\frac{\textcolor{meca}{\tau} + u}{2\textcolor{meca}{\tau} u} \beta(\textcolor{meca}{\tau} + u,\delta)} \right]\right)^2 \right).$$
with the same threshold function
$\beta(n,\delta) \simeq \ln(3n \sqrt{n}/\delta)$.
Consequence In high probability, the delay $\widehat{\tau}_\delta$ of
BGLR is $\mathcal{O}(\Delta^{-2} \ln(1/\delta))$ if enough samples are
observed before the break-point at time $\tau$.
Summary of results for BGLR-T
-----------------------------
BGLR is an efficient break-point detection test!
- We just saw that by choosing
- a confidence level $\delta$,
- and a good threshold function
$\beta(n,\delta) \simeq \ln(3n^{3/2}/\delta)$
- we can control the two properties of the BGLR test:
- its [false alarm probability]{.alert}:
$\mathbb{P}_{\textcolor{darkpurple}{\mu_0}}(\widehat{\tau}_\delta < \infty) \leq \delta$
- its [detection delay]{.alert}:
$\mathbb{P}_{\textcolor{meca}{\mu_0,\mu_1,\tau}} (\widehat{\tau}_{\delta} \geq \tau + u)$
decreases exponentially fast wrt $u$ (if there are enough
samples before and after the break-point).
- $\implies$ it is an efficient break-point detection test
\pause
Finite time guarantees The two guarantees are [finite time]{.alert} and
not asymptotic, this kind of results are quite recent!
4. The BGLR-T + klUCB algorithm
===============================
4\. The BGLR-T + klUCB algorithm
1. [ (Stationary) Multi-armed bandits problems ]{style="color: gray"}
2. [ Piece-wise stationary multi-armed bandits problems
]{style="color: gray"}
3. [ The B-GLR test and its finite time properties
]{style="color: gray"}
4. [ **The BGLR-T + klUCB algorithm** ]{.alert}
5. [ Regret analysis ]{style="color: gray"}
6. [ Numerical simulations ]{style="color: gray"}
BGRL test + kl-UCB index
------------------------
Our algorithm: BGRL test + kl-UCB index (1/2)
Main ideas
- We compute a UCB index on each arm
- Most of the times, we select the arm with highest index
$A_t = \arg\max\limits_{k\in \{1,\dots,K\}} \mbox{kl-UCB}_k(t)$
- We use a BGLR test to detect changes on the played arm
- If a break-point is detected, we reset the memories of *all arms*
\pause
The kl-UCB indexes
- $\tau_k(t)$ is the time of last restart of arm $k$ before time $t$,
- $n_k(t)$ counts the the selections, and $\widehat{\mu}_k(t)$ is the
empirical means of observations since the last restart of arm $k$,
- Let
$\mbox{kl-UCB}_k(t) = \max \bigl\{ q\in[0,1] : n_k(t) \times \mathrm{kl}\left(\widehat{\mu}_k(t),q\right) \leq f(t - \tau_k(t)) \bigr\}$
- $f(t) = \ln(t) + 3 \ln(\ln(t))$ controls the width of the UCB.
Our algorithm: BGRL test + kl-UCB index (2/2)
How do we use the BGLR test? (parameter $\delta$) From observations
$Z_1,\cdots,Z_n$ we detect a break-point with confidence level $\delta$
when
$$\sup_{1 < s < n} \left[s \times \mathrm{kl} \left( \widehat{Z}_{1:s}, \widehat{Z}_{1:n}\right) + (n-s) \times \mathrm{kl} \left( \widehat{Z}_{s+1:n}, \widehat{Z}_{1:n} \right) \right] \geq \beta(n,\delta)$$
\pause
Forced exploration (parameter $\alpha$)
- We use a forced exploration uniformly on all arms...\
*ie*, in average, arm $k$ is forced to be sampled at least
$T \times \alpha / K$ times
- $\implies$ so we can detect break-points on all the arms
- and not only on the arm played by the kl-UCB indexes
The BGLR + kl-UCB algorithm
0.1
0.9
\Donnees{\emph{Parameters of the problem} : $T\in\mathbb{N}^*$, $K\in\mathbb{N}^*$\;}
\Donnees{\emph{Parameters of the algorithm} : $\alpha \in (0, 1)$, $\delta>0$\;}
**Initialisation :** $\forall k \in \{1,\dots,K\}$, $\tau_k = 0$ and
$n_k = 0$\
5. Regret analysis
==================
5\. Regret analysis
1. [ (Stationary) Multi-armed bandits problems ]{style="color: gray"}
2. [ Piece-wise stationary multi-armed bandits problems
]{style="color: gray"}
3. [ The B-GLR test and its finite time properties
]{style="color: gray"}
4. [ The BGLR-T + klUCB algorithm ]{style="color: gray"}
5. [ **Regret analysis** ]{.alert}
6. [ Numerical simulations ]{style="color: gray"}
Hypotheses
----------
Hypotheses of our theoretical analysis
- Denote $\tau^{i}$ the position of $i$-th break-point
- and $\mu_k^{i}$ the mean of arm $k$ on segment
$[\tau^i, \tau^{i+1}]$.
- Let $b(i) \in \arg\max_k \mu_k^{i}$
- and the largest gap at break-point $i$ is
$\Delta^{i} = \max\limits_{k=1,\dots,K} |\mu_k^{i} - \mu_k^{i-1}| >0$.
\pause
Assumption Let
$d^{i} = d^{i}(\alpha,\delta) = \lceil \frac{4K}{\alpha(\Delta^{i})^2}\beta(T,\delta) + \frac{K}{\alpha} \rceil$.
Assume that all sequences are long enough:
$$\forall i \in \{1,\dots,\Upsilon_T\}, \;\;\;\; \tau^{i} - \tau^{i-1} \geq 2\max( d^{i}, d^{i-1} ).$$
The minimum length of the $i$-th sequence depends on the amplitude of
the changes at [the beginning]{style="color: darkred"} and [the
end]{style="color: darkblue"} of the sequence
([$\Delta^i$]{style="color: darkred"} and
[$\Delta^{i+1}$]{style="color: darkblue"})
Regret upper-bound
------------------
Theoretical result
Under this hypothesis, we obtained a *finite time* upper-bound on the
regret $R_T$, with explicit dependency from the problem difficulty.
The exact bound uses:
- the divergences $\mathrm{kl}(\mu_{k}^{i},\mu_{b(i)}^{i})$ to account
for the difficulty of the stationary problem on any sequence,
- $\Delta^{i}$ express the difficulty of detecting the $i$-th
break-point,
as well as
- the parameter $\alpha$ probability of forced exploration,
- and the parameter $\delta$ the confidence level of the break-point
detection algorithm.
Simplified form of the regret upper-bound
Regret upper bound for BGLR + kl-UCB
- Let $\alpha = \sqrt{\Upsilon_T \ln(T) / T}$ and
$\delta = 1 / \sqrt{\Upsilon_T T}$
- And for a problem satisfying our assumption
- Then the regret of our algorithm, with parameters $\alpha$ and
$\delta$, satisfies
$$R_T =\mathcal{O}\left( \frac{K}{\left(\textcolor{deeppurple}{\Delta^{\text{change}}}\right)^2}\sqrt{T \Upsilon_T \ln(T)} + \frac{(K-1)}{\textcolor{meca}{\Delta^{\text{opt}}}} \Upsilon_T\ln(T) \right),$$
- with
$\textcolor{deeppurple}{\Delta^{\text{change}}} = \min_{i} \Delta^{i}$
[the smallest detection gap between two stationary
segments]{style="color: deeppurple"},
- $\textcolor{meca}{\Delta^{\text{opt}}}$ [the smallest value of
sub-optimality gap on a stationary segment]{style="color: meca"}.
\pause
$\implies$ $R_T = \mathcal{O}(K \sqrt{T \Upsilon_T \log(T)})$ if we hide
the dependency on the gaps.
Comparison with other algorithms
--------------------------------
Comparison with other state-of-the-art approaches
Our algorithm (BGLR + kl-UCB)
- Hypotheses: bounded rewards, known $T$ and
$\Upsilon_T = o(\sqrt{T})$, and long enough stationary sequences
- We obtain $R_T = \mathcal{O}(K \sqrt{\Upsilon_T T \log(T)})$
\pause
And two recent competitors:
CUSUM-UCB [\[Liu & Lee & Shroff, AAAI 2018\]]{style="color: gray"}
- Also require [prior knowledge of a lower-bound on the gaps]{.alert}
- They obtained
$R_T = \mathcal{O}(K \sqrt{\Upsilon_T T \log(T / \Upsilon_T)})$
M-UCB [\[Cao & Zhen & Kveton & Xie, AISTATS 2019\]]{style="color: gray"}
- Also require [prior knowledge of a lower-bound on the gaps]{.alert}
- They obtained $R_T = \mathcal{O}(K \sqrt{\Upsilon_T T \log(T)})$
6. Numerical simulations
========================
6\. Numerical simulations
1. [ (Stationary) Multi-armed bandits problems ]{style="color: gray"}
2. [ Piece-wise stationary multi-armed bandits problems
]{style="color: gray"}
3. [ The B-GLR test and its finite time properties
]{style="color: gray"}
4. [ The BGLR-T + klUCB algorithm ]{style="color: gray"}
5. [ Regret analysis ]{style="color: gray"}
6. [ **Numerical simulations** ]{.alert}
Setup of the experiments
------------------------
Numerical simulations
We consider three problems with
- $K=3$ arms, Bernoulli distributed
- $T=5000$ time steps (fixed horizon)
- $\Upsilon_T=4$ break-points ($=5$ stationary sequences)
- Algorithms can use this prior knowledge of $T$ and $\Upsilon_T$
- $1000$ independent runs, we plot the average regret
\pause
Reference
- We used my open-source Python library for simulations of multi-armed
bandits problems, **SMPyBandits**\
$\hookrightarrow$ Published online at
[[`SMPyBandits.GitHub.io`]{style="color: blue"}](https://SMPyBandits.GitHub.io)
- More experiments are included in the long version of the paper!\
$\hookrightarrow$ pre-print on
[[HAL-02006471]{style="color: blue"}](https://hal.inria.fr/hal-02006471)
and
[[arXiv:1902.01575]{style="color: blue"}](https://arxiv.org/abs/1902.01575)
Problem 1: only local changes
{width="100%"}
We plots the means: [$\mu_1(t)$]{style="color: red"},
[$\mu_2(t)$]{style="color: green"}, [$\mu_3(t)$]{style="color: blue"}.
Results on problem 1 {width="115%"}
$\implies$ BGLR achieves the best performance among non-oracle
algorithms!
Problem 2: only global changes
{width="100%"}
Results on problem 2 {width="115%"}
$\implies$ BGLR again achieves the best performance!
Pb 3: non-uniform lenghts of stationary sequences
{width="100%"}
Results on problem 3 {width="115%"}
$\implies$ BGLR achieves the best performance among non-oracle
algorithms!
Conclusions from the simulations
--------------------------------
Interpretation of the simulations (1/2)
Conclusions in terms of regret
- Empirically we can check that the [BGLR test is efficient]{.alert}:
- it has a [low false alarm probability]{.alert}
- it has a [small delay]{.alert} if the stationary sequences are
long enough
And this is true even if the hypotheses of our analysis are not
satisfied!
- Using the kl-UCB indexes policy gives good performance
$\implies$ Our algorithm (BGLR test + kl-UCB) is efficient
$\implies$ We verified that it obtains state-of-the-art performance!
Interpretation of the simulations (2/2)
What about the efficiency in terms of time and memory?
Memory: efficient Our algorithm is as efficient as other
state-of-the-art strategies!\
Memory $= \mathcal{O}(K d_{\max})$ for $K$ arms and horizon $T$.
\<2-\>Time: slow! But it is too slow! Time $= \mathcal{O}(K T d_{\max})$
for $K$ arms and horizon $T$.
$\hookrightarrow$ we proposed two numerical tweaks to speed it up
$\implies$ BGLR test + kl-UCB can be as fast as M-UCB or CUSUM-UCB
$\hookrightarrow$ see the long version on
[[HAL-02006471]{style="color: blue"}](https://hal.inria.fr/hal-02006471)
and
[[arXiv:1902.01575]{style="color: blue"}](https://arxiv.org/abs/1902.01575)
($d_{\max}$ $=$ duration of the longer stationary sequence,
$T \leq (1+\Upsilon_T) d_{\max}$)
Conclusion
==========
Summary
-------
Summary
What we just presented...
- The Multi-Armed Bandits problem (MAB)
- Stationary, then [piece-wise stationary]{.alert}
- The BGLR test is efficient
- to detect break-points with [no false alarm]{.alert} and [low
delay]{.alert}
- for Bernoulli (or sub-Bernoulli) data,
- and does not need to know the amplitude of the break-point
- We can combine it with an efficient MAB policy: [BGLR +
kl-UCB]{.alert}
- Its regret bound is
$R_T = \mathcal{O}(K \sqrt{T \Upsilon_T \log(T)})$
(state-of-the-art)
- On numerical simulations, our algorithm outperforms other efficient
policies, and can be as fast as its best competitors
Thanks
------
Conclusion
Thanks for your attention .
\vspace*{20pt}
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