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171            <h1 class='display-4 my-4'>Contextual Bandits</h1>
172            <div class="d-flex flex-row">
173                <div class=".flex-shrink-1">
174                    <div class="small mb-2" id="date-words"> 19 Nov 2024 - 1623 Words
175                    </div>
176                    
177                    <div class="mb-1">Contextual Bandits and LinUCB algorithm</div>
178                    
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181            <br>
182            <p>This articles provides an overview of <code>Contextual Bandits</code>, specifically focusing on the <code>LinUCB</code> algorithm
183as described in the paper <a href="https://arxiv.org/pdf/1003.0146">A Contextual-Bandit Approach to Personalized News Article Recommendation</a>.</p>
184<hr>
185<h1 id="contextual-bandits">Contextual Bandits</h1>
186<p>In my previous article on <a href="/posts/thompson_sampling/">Thompson sampling</a> I introduced the <code>Multi-arm Bandit</code> problem where
187we have a fixed set of <code>actions</code> (A.K.A <code>arms</code>) and we need to decide which action to take in order to maximise our <code>reward</code>. In the case
188of <code>Thompson Sampling</code> a Bayesian approach is used to continuously update our beliefs about the reward distribution of each
189action.</p>
190<p>The <code>Contextual Multi-arm Bandit</code> problem is an extension of the <code>Multi-arm Bandit</code> problem where we have additional information
191about the <code>context</code> in which the <code>action</code> is taken. This context information can be used to improve the decision making process.</p>
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342
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345
346<p>In the paper linked above the authors describe a <code>Contextual Bandit</code> approach to personalised news article recommendation.
347In this example the <code>action</code> to take is the choice of article to recommend. The <code>context</code> in this kind of problem is
348often represented as both a vector of features describing the user and a vector of features describing the article.</p>
349<p>The user features might include information such as gender, age, location and aggregated previous click history and the
350article features might include information such as the category of the article, the author or the publication date.</p>
351<p>The goal of the <code>Contextual Bandit</code> algorithm is to learn a model that can predict the reward of each action given the context
352so that the algorithm can recommend the best article to the user.</p>
353<p>In the paper the authors describe the contextual bandit formerly as:</p>
354<p>For trials, $t = 1, 2, &hellip; , $ in trial $t$</p>
355<ol>
356<li>For a user $u_t$ there is a set of arms or actions $A_t$ together with their feature vectors $x_{t,a}$. The vector $x_{t,a}$ summarizes information of both the user $u_t$ and arm $a$, and will be referred to as the <code>context</code>.</li>
357<li>Based on observed payoffs in previous trials, the algorithm chooses an arm $a_t \in A_t$, and receives payoff $r_{t,a_t}$ whose expectation depends on both the user $u_t$ and arm $a_t$.</li>
358<li>The algorithm then improves its arm-selection strategy with the new observation, $(x_{t,a_t}, a_t, r_{t,a_t}$). It is important to note that the reward $r_{t,a_t}$ is only observed for the chosen action $a_t$ and not for the other actions in $a\in A_t$.</li>
359</ol>
360<p>The aim is to define a policy or algorithm A that maximises the expected cumulative reward over time $\sum_{t=1}^{T} r_{t,a_t}$. We define the optimal expected payoff as $E[\sum_{t=1}^{T} r_{t,a^{*}_t}]$ where $a^*_t$ is the action with the maximum expected payoff at time $t$.
361Equivalently one can define the <code>regret</code> of the algorithm as the difference between the expected cumulative reward of the algorithm and the expected cumulative reward of the optimal policy. The regret is defined as</p>
362
363<p class="responsive-math" >
364    $$
365\begin{align}
366R(T) = E[\sum_{t=1}^{T} r_{t,a^{*}_t}] - E[\sum_{t=1}^{T} r_{t,a_t}]
367\end{align}
368$$
369</p>
370<h3 id="upper-confidence-bound-algorithms">Upper Confidence Bound algorithms</h3>
371<p>For each trial t, an upper confidence bound (UCB) algorithm estimates both the mean payoff $\hat{\mu}_{t, a}$ as well as an interval $c_{t,a}$ around the mean. The algorithm then chooses the action with the <code>highest upper confidence bound</code>.
372By choosing the action with the highest upper confidence bound the algorithm is able to balance the <code>exploration-exploitation</code> trade-off. The algorithm is able to explore actions with high uncertainty and exploit actions with high expected payoff.
373
374<p class="responsive-math" >
375    $$
376a_t = \underset{a}{\operatorname{argmax}}(\hat{\mu}_{t,a} + c_{t,a})
377$$
378</p></p>
379<h4 id="di
379sjoint-linear-upper-confidence-bound">Disjoint Linear Upper Confidence Bound</h4>
380<p>In the paper the authors propose <code>LinUCB</code> by assuming a <code>linear</code> model for the expected payoff of each action given the context $x_{t,a}$</p>
381
382<p class="responsive-math" >
383    $$
384\begin{align}
385E[r_{t, a} | x_{t,a}] = x_{t,a}^T \theta_a
386\end{align}
387$$
388</p>
389<p>In this first <code>disjoint</code> formulation the parameters $\theta_a$ as well as the standard deviation $c_{t, a}$ are <strong>learned independently for each arm</strong> based on the observed data so far for that arm.
390The algorithm then chooses the action with the highest upper confidence bound as described above.
391In this <code>disjoint</code> formulation with a separate model for each arm the features relate to the user e.g. the users age,
392their interest in different categories of article or their historical behaviour.
393Features about the article wouldn&rsquo;t be helpful in the disjoint problem as the regressions are independent for each arm/action $a$.</p>
394<p>Let $D_a$ be the <code>m x d</code> design matrix whose <code>m</code> rows represent the <code>m</code> previous contexts $x_{t,a}$ where arm <code>a</code> was selected, and let $c_a$ be the <code>m x 1</code> vector of rewards (or clicks) received for each of these contexts. The algorithm then estimates the parameters $\theta_a$ by solving the <code>ridge regression</code> with training data $(D_a, c_a)$</p>
395
396<p class="responsive-math" >
397    $$
398\begin{align}
399\theta_a = (D_a^T D_a + I)^{-1} D_a^T c_a
400\end{align}
401$$
402</p>
403<p>From the paper it can be show that with probability at least $1 - \delta$</p>
404
405<p class="responsive-math" >
406    $$
407\begin{align}
408|x_{t,a}^T\theta_a - E[r_{t, a} | x_{t,a}]| \leq \alpha \sqrt{x_{t,a}^T (D_a^T D_a + I)^{-1} x_{t,a}}
409\end{align}
410$$
411</p>
412<p>where $\alpha = 1 + \sqrt{\log(\frac{2}{\delta})/2}$ and $\delta &gt; 0$.</p>
413<p>From this <code>linUCB</code> selects the action $a_t$ with the highest upper confidence bound</p>
414
415<p class="responsive-math" >
416    $$
417\begin{align}
418a_t &= \underset{a}{\operatorname{argmax}}(x_{t,a}^T \theta_a + \alpha \sqrt{x_{t,a}^T (D_a^T D_a + I)^{-1} x_{t,a}}) \\
419&= \underset{a}{\operatorname{argmax}}(x_{t,a}^T \theta_a + \alpha \sqrt{x_{t,a}^T A_a^{-1} x_{t,a}})
420\end{align}
421$$
422</p>
423<p>where $A_a = D_a^T D_a + I$</p>
424
425
426
427
428<div id="accordion-938762415" class="pb-5">
429    <div class="card">
430      <div class="card-header" id="heading-938762415">
431        <h5 class="mb-0">
432          <button class="btn btn-link" data-toggle="collapse" data-target="#collapse-938762415" aria-expanded="true" aria-controls="collapse-938762415">
433            Click to see the algorithm using iterative parameter updates as described in the paper
434          </button>
435        </h5>
436      </div>
437
438      <div id="collapse-938762415" class="collapse" aria-labelledby="heading-938762415" data-parent="#accordion-938762415">
439        <div class="card-body">
440            <ul>
441<li>For t = 1, 2, &hellip; , T
442<ul>
443<li>Observe context $x_{t,a}$ for each arm $a \in A_t$</li>
444<li>For each arm $a \in A_t$
445<ul>
446<li>If $a$ is new: set
447<ul>
448<li>$A_a = I$ (d x d identity matrix where d is the number of features)</li>
449<li>$b_a = 0$ (d dimensional vector)</li>
450</ul>
451</li>
452<li>Update current mean and variance estimates
453<ul>
454<li>$\theta_a = A_a^{-1} b_a$</li>
455<li>$p_a = x_{t,a}^T \theta_a + \alpha \sqrt{x_{t,a}^T A_a^{-1} x_{t,a}}$</li>
456</ul>
457</li>
458<li>Choose action $a_t = \underset{a}{\operatorname{argmax}} p_a$</li>
459<li>Observe reward $r_{t,a_t}$ and update $A_{a_t}$ and $b_{a_t}$
460<ul>
461<li>$A_{a_t} = A_{a_t} + x_{t,a_t} x_{t,a_t}^T$</li>
462<li>$b_{a_t} = b_{a_t} + x_{t,a_t} r_{t,a_t}$</li>
463</ul>
464</li>
465</ul>
466</li>
467</ul>
468</li>
469</ul>
470
471        </div>
472      </div>
473    </div>
474  </div>
475
476<h4 id="linucb-with-shared-features">LinUCB with Shared Features</h4>
477<p>As described in the paper in many situations it is helpful to use features that are shared across all arms. For instance
478in the news article recommendation it might be useful to have features that interact the users category preferences with
479the categories of the articles. For instance if a user has a preference for sports articles then it might be useful to have a
480feature such as <code>user_likes_sports_and_article_is_sports</code>. This coefficient tieing this feature to the reward is learnt across arms.</p>
481<p>In this case we still have a linear model for the expected payoff but the features are now a combination the <code>shared</code> features $z_{t,a}$ and <code>arm</code> specific features $x_{t,a}$</p>
482
483<p class="responsive-math" >
484    $$
485\begin{align}
486E[r_{t, a} | x_{t,a}] = x_{t,a}^T \theta_a + z_{t,a}^T \beta
487\end{align}
488$$
489</p>
490
491
492
493
494<div id="accordion-361745892" class="pb-5">
495    <div class="card">
496      <div class="card-header" id="heading-361745892">
497        <h5 class="mb-0">
498          <button class="btn btn-link" data-toggle="collapse" data-target="#collapse-361745892" aria-expanded="true" aria-controls="collapse-361745892">
499            Click to see the full algorithm for shared features
500          </button>
501        </h5>
502      </div>
503
504      <div id="collapse-361745892" class="collapse" aria-labelledby="heading-361745892" data-parent="#accordion-361745892">
505        <div class="card-body">
506            <ul>
507<li>Initialise global parameters:
508<ul>
509<li>$A_0 = I_k$ (k x k identity matrix where k is the number of shared features)</li>
510<li>$b_0 = 0$ (k dimensional vector)</li>
511</ul>
512</li>
513<li>For $t=1, 2, 3, &hellip;, T$
514<ul>
515<li>Observe context $x_{t,a}$ for each arm $a \in A_t$</li>
516<li>$\hat{\beta} = A_0^{-1}
516 b_0$</li>
517<li>For all arms $a \in A_t$
518<ul>
519<li>If $a$ is new: set
520<ul>
521<li>$A_a = I$ (d x d identity matrix where d is the number of features)</li>
522<li>$B_a = 0_{d x k} $ (d x k matrix of zeros)</li>
523<li>$b_a = 0$ (d dimensional zero vector)</li>
524</ul>
525</li>
526<li>Update current mean and variance estimates
527<ul>
528<li>$\theta_a = A_a^{-1} (b_a - B_a\hat{\beta})$</li>
529<li>$s_{t, a} = z_{t, a}^T A_0^{-1}z_{t, a} - 2 z_{t, a}^TA_0^{-1}B_a^TA_a^{-1}x_{t, a} +$
530$x_{t, a}^T A_a^{-1}x_{t, a} + x_{t, a}^T A_a^{-1} B_aA_0^{-1}B_a^TA_a^{-1}x_{t, a}$</li>
531<li>$p_a = x_{t, a}^T \theta_a + z_{t, a}^T \hat{\beta} + \alpha \sqrt{s_{t, a}}$</li>
532</ul>
533</li>
534</ul>
535</li>
536<li>Choose arm $a_t = \underset{a}{\operatorname{argmax}} p_a$</li>
537<li>Update shared feature blocks based on current information for chosen action $a_t$
538<ul>
539<li>$A_0 = A_0 + B_{a_t}^TA_{a_t}^{-1}B_{a_t}$</li>
540<li>$b_0 = b_0 + B_{a_t}^TA_{a_t}^{-1}b_{a_t}$</li>
541</ul>
542</li>
543<li>Update the arms specific blocks based on chosen action $a_t$
544<ul>
545<li>$A_{a_t} = A_{a_t} + x_{t, a_t}x_{t, a_t}^T$</li>
546<li>$B_{a_t} = B_{a_t} + x_{t, a_t}z_{t, a_t}^T$</li>
547<li>$b_{a_t} = b_{a_t} + x_{t, a_t}r_{t, a_t}$</li>
548</ul>
549</li>
550<li>Update the share feature blocks based on chosen action $a_t$ and updated information
551<ul>
552<li>$A_0 = A_0 + z_{t, a_t}z_{t, a_t}^T - B_{a_t}^TA_{a_t}^{-1}B_{a_t}$</li>
553<li>$b_0 = b_0 + z_{t, a_t}r_{t, a_t} - B_{a_t}^TA_{a_t}^{-1}b_{a_t}$</li>
554</ul>
555</li>
556</ul>
557</li>
558</ul>
559
560        </div>
561      </div>
562    </div>
563  </div>
564
565<h2 id="simulations-with-disjoint-linucb">Simulations with Disjoint LinUCB</h2>
566<p>In the following simulation we will simulate a disjoint LinUCB algorithm with just one feature in the <code>context</code> $x_{t,a}$ which is just the age of user. To keep the example simple we will consider the case where we have three different articles we can show the users, A - a sports article, B - a politics article and C - a technology article. The <code>reward</code> is simply 1 if the user clicks on the article and 0 if they don&rsquo;t.</p>
567<p>In this simulation we will assume that the user is more likely to click on the sports article if they are younger and more likely to click on the politics article if they are older. The technology article is not dependent on the age of the user. More specifically we will assume that users age is drawn from a uniform distribution between 20 and 60 and the probability of clicking is:</p>
568<ul>
569<li>For sports article is $0.5 - (age - 20)*0.01$</li>
570<li>For politics article is $0.3 + (age - 20)*0.005$</li>
571<li>For technology article is $0.2$</li>
572</ul>
573<p>The visualization shows the true conversion parameter $\theta_k$ for each arm (dotted line) varying by age as well as the predicted linear model and error bounds updated by the LinUCB algorithm. Below, a table displays the number of times each arm is selected and the clicks generated. Finally, a chart shows the cumulative pseudo-regret of the LinUCB algorithm over time, defined as the difference between the optimal reward and the reward generated by the algorithm.
574Note the &ldquo;optimal&rdquo; reward here is defined as selecting the arm with the highest expected reward based on the true conversion parameters.</p>
575<div class="row">
576    <form class="col d-flex pb-3" id="simulation-slider">
577        <button name="play_pause" type="button" style="margin-right: 0.4em; width: 100px" id="play-button">Play</button>
578        <button name="step" type="button" style="margin-right: 0.4em; width: 100px" id="step-button">Step</button>
579        <label style="display: flex; align-items: center; width: 100%">
580            <input name=i type=range min=0 max=1000 value=0 step=1 style="flex-grow:1">
581            <output name=o style="margin-left: 0.4em;">0 Steps</output>
582        </label>
583    </form>
584</div>
585<div class="row">
586    <div class="col-sm-12 pb-3" id="distribution-container"></div>
587    <div class="col-sm-12 pl-5">
588        <table class="table table-bordered table-sm">
589            <thead>
590                <tr>
591                    <th>arm</th>
592                    <th>Observations</th>
593                    <th>Clicks</th>
594                </tr>
595            </thead>
596            <tbody>
597                <tr>
598                    <td>Sports</td>
599                    <td id="arm-A-observations">0</td>
600                    <td id="arm-A-clicks">0</td>
601                </tr>
602                <tr>
603                    <td>Politics</td>
604                    <td id="arm-B-observations">0</td>
605                    <td id="arm-B-clicks">0</td>
606                </tr>
607                <tr>
608                    <td>Technology</td>
609                    <td id="arm-C-observations">0</td>
610                    <td id="arm-C-clicks">0</td>
611                </tr>
612            </tbody>
613        </table>
614    </div>
615    <div class="col-sm-12 pb-3" id="regret"></div>
616
617</div>
618
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629<script>
630    (async () => {
631
632        class disjointLinUCBAgent {
633            constructor(nArms, d, alpha) {
634                console.log(`Creating disjoint LinUCB agent with nArms = ${nArms} and d = ${d}, alpha = ${alpha}`)
635                this.nArms = nArms 
636                this.d = d 
637                this.alpha = alpha
638                this.name = "Disjoint LinUCB"
639                this.A = d3.range(nArms).map(_ => math.identity(d))
640                this.b = d3.range(nArms).map(_ => math.zeros(d).reshape([d, 1]))
641            }
642
643            theta() {
644                return this.A.map((A, i) => math.multiply(math.inv(A), this.b[i]))
645            }
646
647            expectedReward(x) {
648                return this.theta().map((theta, i) => {
649                    let reward = math.multiply(math.transpose(theta), x)
650                    return math.squeeze(reward)
651                })
652            }
653
654            standardError(x) {
655                return this.A.map((A, i) => {
656                    let variance = math.squeeze(math.multiply(
657                        math.transpose(x),
658                        math.multiply(math.inv(A), x)
659                    ))
660                    return math.multiply(this.alpha, math.sqrt(variance))
661                })
662            }
663
664            selectArm(context) {
665                const x = math.matrix(context).reshape([this.d, 1])
666                const standardErrors = this.standardError(x)
667                const expectedRewards = this.expectedReward(x)
668                const ucb = expectedRewards.map((expectedReward, i) => expectedReward + standardErrors[i])
669                const maxUCB = Math.max(...ucb)
670                const bestArms = ucb.map((value, index) => value === maxUCB ? index : -1).filter(index => index !== -1)
671                const arm = bestArms[Math.floor(Math.random() * bestArms.length)]
672                return { expectedRewards, standardErrors, ucb, arm }
673            }
674
675            update(context, arm, reward) {
676                const x = math.matrixFromColumns(context)
677                this.A = this.A.map((A, i) => i === arm ? math.add(A, math.multiply(x, math.transpose(x))) : A)
678                this.b = this.b.map((b, i) => i === arm ? math.add(b, math.multiply(reward, x)) : b)
679            }
680        }
681
682        class Simulator {
683            constructor(agent, simulationData, armNames, permanentData) {
684                this.agent = agent
685                this.simulationData = simulationData
686                this.N = simulationData.length
687                this.t = 0 
688                this.armNames = armNames
689                this.permanentData = permanentData 
690                this.regretData = [[0, 0]]
691                this.frameData = []
692                this.observedAndClicks = Array.from(Array(agent.nArms).keys()).map(_ => [0, 0])
693                this.xs = d3.range(20, 61, 1).map(x => [[1], [x]])
694            }
695
696            run() {
697                console.log(`Running simulation with N = ${this.N}, simulationData = `, this.simulationData)
698                this.simulationData.forEach(data => {
699                    const { context, rewards } = data
700                    const { expectedRewards, standardErrors, ucb, arm } = this.agent.selectArm(context)
701                    const reward = rewards[arm]
702                    const optimalAgentReward = rewards[data.bestArm]
703                    const regret = optimalAgentReward - reward
704                    const newRegretPoint = [this.t + 1, this.regretData[this.t][1] + regret]
705                    console.log(`New regret point = `, newRegretPoint)
706                    this.regretData = [...this.regretData, newRegretPoint]
707                    this.agent.update(context, arm, reward)
708                    this.t += 1
709                    let plotData = this.agent.theta().map((theta, i) => {
710                        const y = this.xs.map(x => math.squeeze(math.multiply(math.transpose(theta), x)))
711                        const s = this.xs.map(x => {
712                            const variance = math.squeeze(math.multiply(
713                                math.transpose(x),
714                                math.multiply(math.inv(this.agent.A[i]), x)
715                            ))
716                            return math.multiply(this.agent.alpha, math.sqrt(variance))
717                        })
718                        const lower = y.map((y, i) => y - s[i])
719                        const upper = y.map((y, i) => y + s[i])
720                        return {
721                            id: this.armNames[i],
722                            values: d3.zip(d3.range(20, 61, 1), y),
723                            confidenceIntervals: d3.zip(d3.range(20, 61, 1), lower, upper),
724                            lineColor: colours[i]
725                        }
726                    })
727                    plotData = plotData.concat(this.permanentData)
728                    this.observedAndClicks = this.observedAndClicks.map((value, i) => {
729                        if (i === arm) {
730                            return [value[0] + 1, value[1] + reward]
731                        } else {
732                            return value
733                        }
734                    })
735                    const observedAndClicks = this.observedAndClicks
736                    this.frameData.push({ plotData, observedAndClicks, regretPlotData: [{"id": "Regret", "values": this.regretData}] })
737                })
738                return this.frameData
739            }
740        }
741
742        function PlayUtility(values, { initial = 0, step_size = 1, steps = null, delay = null, autoplay = true, loop = true, loopDelay = null } = {}) {
743            const form = document.getElementById('simulation-sli
743der')
744            values = Array.from(values)
745            let current_id = initial
746            let frame = null
747            let timer = null
748            let interval = null
749
750            function start() {
751                form.play_pause.textContent = "Pause"
752                if (delay === null) frame = requestAnimationFrame(tick)
753                else interval = setInterval(tick, delay)
754            }
755
756            function stop() {
757                form.play_pause.textContent = "Play"
758                if (frame !== null) cancelAnimationFrame(frame), frame = null
759                if (timer !== null) clearTimeout(timer), timer = null
760                if (interval !== null) clearInterval(interval), interval = null
761            }
762
763            function running() {
764                return frame !== null || timer !== null || interval !== null
765            }
766
767            function tick() {
768                if (form.i.value == values.length - 1) {
769                    if (!loop) {
770                        stop()
771                        return
772                    }
773                    if (loopDelay !== null) {
774                        if (frame !== null) cancelAnimationFrame(frame), frame = null
775                        if (interval !== null) clearInterval(interval), interval = null
776                        timer = setTimeout(() => (step(), start()), loopDelay)
777                        return
778                    }
779                }
780                if (delay === null) frame = requestAnimationFrame(tick)
781                step()
782            }
783
784            function step() {
785                const current_value = form.i.valueAsNumber
786                const next_value = (current_value + step_size + values.length) % values.length
787                form.i.valueAsNumber = next_value
788                form.i.dispatchEvent(new CustomEvent("input", { bubbles: true }))
789            }
790
791            function registerCallback(callback) {
792                d3.select(form.i).on("input", callback)
793            }
794
795            function onInputChange(event) {
796                if (event && event.isTrusted && running()) stop()
797                form.value = values[form.i.valueAsNumber]
798                form.o.value = `${form.i.valueAsNumber} Steps`
799            }
800
801            d3.select(form.i).on("input.a", onInputChange)
802            onInputChange("manual")
803
804            form.play_pause.onclick = () => {
805                if (running()) return stop()
806                form.i.valueAsNumber = (form.i.valueAsNumber + step_size) % values.length
807                form.i.dispatchEvent(new CustomEvent("input", { bubbles: true }))
808                start()
809            }
810            form.step.onclick = () => {
811                if (running()) return stop()
812                step()
813            }
814
815            if (autoplay) start()
816            else stop()
817
818            if (steps !== null) {
819                form.i.min = 0
820                form.i.max = steps - 1
821            }
822
823            return {
824                start,
825                stop,
826                running,
827                step,
828                form,
829                registerCallback,
830                tick,
831            }
832        }
833
834        function updateTable(observedAndClicks) {
835            ['A', 'B', 'C'].forEach((id, i) => {
836                let [observed, clicks] = observedAndClicks[i]
837                d3.select(`#arm-${id}-observations`).text(observed)
838                d3.select(`#arm-${id}-clicks`).text(clicks)
839            })
840        }
841
842        function update(e) {
843            const frameId = playHandle.form.i.valueAsNumber
844            const newPlotData = frameData[frameId].plotData
845            const newRegretData = frameData[frameId].regretPlotData
846            const observedAndClicks = frameData[frameId].observedAndClicks
847            console.log(`[Thompson Sampling] Updating with frameId = ${frameId}`)
848            updateTable(observedAndClicks)
849            chart.update(newPlotData)
850            regretChart.update(newRegretData)
851        }
852
853        function getSimulationData(N) {
854            return d3.range(N).map(d => {
855                let seed = 1000
856                const userAge = d3.randomUniform(20, 60, )()
857                const arms = ["Sports", "Politics", "Technology"]
858                const linearArms = [
859                    age => .5 - (age - 20) * 0.01,
860                    age => .3 + (age - 20) * 0.005,
861                    age => .2
862                ]
863                const trueThetas = linearArms.map(arm => arm(userAge))
864                const maxTheta = Math.max(...trueThetas)
865                const bestArms = trueThetas.map((theta, i) => theta === maxTheta ? i : -1).filter(i => i !== -1)
866                const bestArm = bestArms[Math.floor(Math.random() * bestArms.length)]
867                const rewards = trueThetas.map(theta => d3.randomBernoulli(theta)())
868                return { context: [1, userAge], rewards, bestArm}
869            })
870        }
871
872        const colours = ["#007bff", "#dc3545", "#28a745"]
873        const permanentData = [
874            { id: "Sports-Actual", values: [[20, 0.5], [60, 0.1]], dashArray: true, lineColor: colours[0], legendGroup: "Sports" },
875            { id: "Politics-Actual", values: [[20, 0.3], [60, 0.4]], dashArray: true, lineColor: colours[1], legendGroup: "Politics" },
876            { id: "Technology-Actual", values: [[20, 0.2], [60, 0.2]], dashArray: true, lineColor: colours[2], legendGroup: "Technology" }
877        ]
878        const armNames = ["Sports", "Politics", "Technology"]
879        const chartElement = document.querySelector('#distribution-container')
880        const chart = new LineChart(chartElement, permanentData, { duration: 50, startYAtZero: true, maxY: 1 })
881
882        const regretElement = document.querySelector('#regret')
883        const regretChart = new LineChart(regretElement, [], { duration: 50, startYAtZero: true, curveLine: false })
884
885        window.chart = chart
886        const N = 500 
887        const values = d3.range(N)
888        const settings = {
889            initial: 0,
890            step_size: 1,
891            steps: N,
892            autoplay: false,
893            loop: false,
894            loopDelay: 1000,
895            delay: 50
896        }
897        const playHandle = PlayUtility(values, settings)
898        playHandle.registerCallback(update)
899        console.log("SLEEPING")
900        await new Promise(r => setTimeout(r, 1000))
901        console.log("DONE SLEEPING")
902
903        console.log("Running slow simulations")
904        const simulationData = getSimulationData(N)
905        const alpha = 1 + Math.sqrt(Math.log(2 / 0.99) / 2)
906        const agent = new disjointLinUCBAgent(3, 2, alpha)
907        const simulator = new Simulator(agent, simulationData, armNames, permanentData)
908        const frameData = simulator.run()
909        console.log("Done running simulations")
910
911    })().catch(err => {
912        console.error(err)
913    })
914</script>
914
915
916
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