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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. 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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, … , $ 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’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 $\delta > 0$.</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, … , 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 $t=1, 2, 3, …, T$ 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’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 “optimal” 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>
914 915 916 917 </div> 918 </div> 919</article> 920 921 922 </div> 923 924 925 926 927 928 929 930 931 932 933 934 935 936 937 938 939 940 941 942 943 944 945 946 947
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Line numbers count LF bytes from the start of the resource, as the search results do. Vendor segments are library code the classifier recognised; they are stored but not indexed. Bytes are shown as Latin1 characters, one per byte.