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<!doctype html>
<html>
<head>
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1.0, maximum-scale=1.0, user-scalable=no">
<title>reveal.js</title>
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<body>
<div class="reveal">
<div class="slides">
<section data-markdown>
## Modeling real behavior in two-person differential games
John Pearson
[pearsonlab.github.io/modeling-differential-games](https://pearsonlab.github.io/modeling-differential-games)
</section>
<section>
<div class="wrapper" style="">
<img
class="grid-item"
src="https://web.duke.edu/mind/level2/faculty/pearson/assets/images/penaltyshot/kelsey_pk/FinalPaperSVGs/final_move.png"
style="grid-column: 1/2">
<img
class="grid-item"
src="https://web.duke.edu/mind/level2/faculty/pearson/assets/images/penaltyshot/paper_figs/gen_traces.svg"
style="grid-column: 2/3">
<img
class="grid-item"
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style="grid-column: 3/4">
<img
class="grid-item"
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style="grid-column: 4/5">
<img
class="grid-item"
src="https://web.duke.edu/mind/level2/faculty/pearson/assets/images/pacman/pacman_still.svg"
style="grid-column: 1/2">
<img
class="grid-item"
src="https://web.duke.edu/mind/level2/faculty/pearson/assets/images/lab_logo/plab_logo_light.svg"
style="grid-column: 2/4">
<img
class="grid-item"
src="https://web.duke.edu/mind/level2/faculty/pearson/assets/images/mosaic/centers.svg"
style="grid-column: 4/5">
</div>
</section>
<section data-background-image="https://web.duke.edu/mind/level2/faculty/pearson/assets/images/ccn2017/4f.png">
<div style="background-color: rgba(0, 0, 0, 0.75); padding: 10px" class="">
<h3>Social decisions</h3>
<ul>
<li>
Modeling other minds
</li>
<li>
Group foraging
</li>
<li>
Strategic social decisions
</li>
<!-- <section data-markdown> -->
<!-- ## Social decisions
- Modeling other minds
- Group dynamics
- Strategic behavior -->
</section>
<section data-background-image="https://web.duke.edu/mind/level2/faculty/pearson/assets/images/sfn2016/chess.jpg">
<aside class="notes" data-markdown>
### Pros:
- Well-studied, normative solutions
### Cons:
- Highly idealized, limited dynamics
- Biologically aligned?
</aside>
</section>
<section>
<h3>More specifically...</h3>
<div style="width: 50%; margin: auto">
<img src="https://web.duke.edu/mind/level2/faculty/pearson/assets/images/misc/Prisoner's_Dilemma.svg" alt="">
</div>
<p class="ref">Jensen & Riestenberg (<a href="https://commons.wikimedia.org/wiki/File:Prisoner%27s_Dilemma_briefcase_exchange_(colorized).svg">Wikimedia</a>)</p>
</section>
<section data-background-image="https://web.duke.edu/mind/level2/faculty/pearson/assets/images/sfn2016/penalty-shot.jpg">
<aside class="notes" data-markdown>
- Doesn't matter if it's social
- Requires anticipating another agent
- Repeatable, but lots of variation
- Decisions tightly linked to movement
</aside>
</section>
<section data-markdown>
## Differential games
- Continuous states and actions
- Control theory meets game theory
- Classic problem: pursuit and evasion
- Hard enough to prove solutions exist; explicit construction often intractable
</section>
<section>
<h3>Penalty Shot</h3>
<video width="70%" align="center" controls autoplay loop src="https://web.duke.edu/mind/level2/faculty/pearson/assets/videos/penaltyshot/sess130_new.mp4" type="video/mp4">
Your browser does not support the video tag.
</video>
<aside class="notes" data-markdown>
- two monkeys, shooter and goalie (shooter recorded)
- controlled by joysticks
- roles rotated, animals know each other
- repeated sessions
- rapidly learned, rich dynamics
</aside>
</section>
<section>
<div style="width: 50%; float: left; text-align: left">
<video autoplay loop src="https://web.duke.edu/mind/level2/faculty/pearson/assets/videos/penaltyshot/real_trial.mp4" type="video/mp4" id="trace_video">
Your browser does not support the video tag.
</video>
<script>
var vid = document.getElementById("trace_video");
vid.playbackRate = 1.;
</script>
</div>
<div style="width: 50%; float: right; font-size: .9em" >
<h3>Complexity tax</h3>
<ul style="font-size: 0.85">
<li>each trial a different length</li>
<li>how to average, align?</li>
<li>need to "reduce" dynamics</li>
</ul>
</div>
</section>
<section data-markdown>
## Today's plan:
- Model real (not optimal) behavior
- Two approaches
- **Generative model:** focus on dynamics, inferring latent variables
- **Discriminative model:** focus on prediction, sensitivity to inputs
</section>
<section>
<div style="float:left; width:12.5%">
<p></p>
</div>
<div style="float:left; width:25%">
<img src="https://web.duke.edu/mind/level2/faculty/pearson/assets/images/sfn2016/shariq.jpg" >
<p>Shariq Iqbal</p>
</div>
<div style="float:left; width:25.66%">
<img src="https://web.duke.edu/mind/level2/faculty/pearson/assets/images/website/sam.jpg" >
<p>Sam Yin</p>
</div>
<div style="float:left; width:25%; clear:right">
<img src="https://web.duke.edu/mind/level2/faculty/pearson/assets/images/sfn2016/PlattMichael.jpg" >
<p>Michael Platt</p>
</div>
<br>
<br>
<br>
<br>
<br>
<br>
<br>
<p class="ref">Iqbal, Yin et al. (<a href="https://journals.plos.org/ploscompbiol/article?id=10.1371/journal.pcbi.1006895">PLoS Comp Bio, 2019</a>)</p>
</section>
<section>
<h3>Real trials</h3>
<div style="width: 50%; margin: auto">
<img src="https://web.duke.edu/mind/level2/faculty/pearson/assets/images/penaltyshot/paper_figs/real_traces.svg" alt="">
</div>
</section>
<section data-markdown>
### What we want
- Joystick censoring (yes! *but not today*)
- ~~Details of motor execution~~
- Incorporate domain knowledge in model
- Able to generate synthetic data (emulator)
</section>
<section data-markdown>
### Our approach
- Borrow from control theory, time series
- Structured black box models (pieces make sense)
- Neural networks for flexible fitting
</section>
<section>
<h3>Modeling I</h3>
Observed positions at each time ($y_t$):
$$
y_t = \begin{bmatrix}
y_{goalie} &
x_{puck} &
y_{puck}
\end{bmatrix}^\top
$$
<br>
Control inputs ($u_t$) drive changes in observed positions:
$$y_{t + 1} = y_t + v_{max} \sigma(u_t)$$
<br>
<b>Goal</b>: predict control inputs from trial history:
$$u_t = F(y_{1:t})$$
</section>
<section>
<h3>Modeling II</h3>
Assumption: PID control
$$
u_t = u_{t-1} + L * (g_{t} - y_{t}) + \epsilon_t
$$
<br>
<ul>
<li>linear control model: $L$</li>
<li>goal (set point): $g_{t}$</li>
<li>error signal: $e_t \equiv g_{t} - y_{t}$</li>
<li>observation noise: $\epsilon_t$</li>
</ul>
</section>
<section>
<h3>Modeling III</h3>
<h5>Goal model:</h5>
$$
\log p(g) = -\beta E(g|s) - \log Z \\
E(g|s) = \sum_t \left[ \frac{1}{2} \Vert g_t - g_{t-1}\Vert^2 + U(g_t, s_t)\right]
$$
<br>
<h5>How do we interpret this?</h5>
<ul>
<li>Goals minimize an "energy"</li>
<li>"Kinetic" energy favors smoothness</li>
<li>"Potential" $U$ captures player interaction</li>
</ul>
</section>
<section>
<h3>Modeling IV</h3>
<ul>
<li>$U$ is a problem</li>
<li>What if $U$ were just quadratic?</li>
<li>Model $e^U$ as a *mixture* of normals</li>
<li>Use a Gaussian Mixture:
<ul>
<li>$U(g, s) = \sum_k w_k(s)\mathcal{N}(g | \mu_k(s), \lambda_k^{-1}(s))$</li>
<li>i.e., goal mixture depends on current state</li>
</ul>
</li>
</ul>
</section>
</section>
<section>
<h3>Our model</h3>
<img src="https://web.duke.edu/mind/level2/faculty/pearson/assets/images/penaltyshot/paper_figs/model_diagram.svg" style="background:white">
</section>
<section>
<h3>Model fitting</h3>
<h5>Variational Bayes autoencoder</h5>
<ul>
<li>Encoding model:</li>
<ul>
<li>goals: GMM</li>
<li>latent control: PID + Gaussian noise</li>
<li>observed control: soft censoring</li>
</ul>
<br>
<li>Decoding model:</li>
<ul>
<li>state space model <a href="https://arxiv.org/abs/1511.07367">(Archer et al., 2015)</a></li>
<li>= Kalman filter for linear system</li>
</ul>
</ul>
</section>
<section>
<h3>Implementation</h3>
<ul>
<li>Variational Inference:
$$
\max_{\phi, \theta} \mathbb{E}_q[\log p_\theta(x, z)] + \mathbb{H}[q_\phi(z)] \le \log p_\theta(x)
$$
</li>
<li>Stochastic Gradients:
$$
\mathbb{E}_q\left[f(z)\right] \approx f(z^*) \quad z^* \sim q_\phi(z)
$$
</li>
<li>Reparameterization Trick:
$$
Z = h(\epsilon, \phi) \quad \epsilon \sim \mathcal{N}(0, 1)
$$
</li>
<li>Code in <a href="https://www.tensorflow.org/">TensorFlow</a> and <a href="http://edwardlib.org/">Edward</a> </li>
</ul>
</section>
<section>
<h3>It fits!</h3>
<div style="width: 50%; float: left; text-align: left">
<img src="https://web.duke.edu/mind/level2/faculty/pearson/assets/images/penaltyshot/pos_fit.svg" style="border: none; background: none; box-shadow: none">
</img>
</div>
<div style="width: 50%; float: right; text-align: left">
<img src="https://web.duke.edu/mind/level2/faculty/pearson/assets/images/penaltyshot/paper_figs/single_trial_control.svg" style="border: none; background: none; box-shadow: none">
</img>
</div>
</section>
<section>
<h3>Generated Trials</h3>
<video width="70%" align="center" controls autoplay loop src="https://web.duke.edu/mind/level2/faculty/pearson/assets/videos/penaltyshot/gen_data.mp4" type="video/mp4">
Your browser does not support the video tag.
</video>
<aside class="notes" data-markdown>
- Totally new behavior
- Captures some of the richness of original data
- Gives us confidence that our model is plausible
- *Much* stronger test than simple goodness-of-fit
- You can have a model that "fits" but the
observed data are atypical of the model
</aside>
</section>
<section>
<h3>Generated trials</h3>
<div style="width: 50%; float: left">
<img src="https://web.duke.edu/mind/level2/faculty/pearson/assets/images/penaltyshot/paper_figs/real_traces.svg" alt="">
</div>
<div style="width: 50%; float: right">
<img src="https://web.duke.edu/mind/level2/faculty/pearson/assets/images/penaltyshot/paper_figs/gen_traces.svg" alt="">
</div>
</section>
<section>
<h3>A sample trial</h3>
<video width="70%" align="center" controls autoplay loop src="https://web.duke.edu/mind/level2/faculty/pearson/assets/videos/penaltyshot/real_trial.mp4" type="video/mp4">
Your browser does not support the video tag.
</video>
</section>
<section>
<h3>Inferred goals</h3>
<video width="70%" align="center" controls autoplay loop src="https://web.duke.edu/mind/level2/faculty/pearson/assets/videos/penaltyshot/real_trial_goals.mp4" type="video/mp4">
Your browser does not support the video tag.
</video>
</section>
<section>
<h3>Potential energy function</h3>
<video width="70%" align="center" controls autoplay loop src="https://web.duke.edu/mind/level2/faculty/pearson/assets/videos/penaltyshot/real_trial_value.mp4" type="video/mp4">
Your browser does not support the video tag.
</video>
</section>
<section data-markdown>
### What did we do?
- Dynamic control tasks let us leverage motor behavior to study cognitive and social decisions.
- Structured black-box models allow us to carve behavior into interpretable pieces.
- We inferred a value function capable of explaining behavior in terms of goals.
</section>
<section>
<h2>In progress: prey pursuit</h2>
<video width="90%" align="center" controls autoplay loop src="https://web.duke.edu/mind/level2/faculty/pearson/assets/videos/pacman/trial.mp4" type="video/mp4" id="pacman_goals_vid">
Your browser does not support the video tag.
</video>
<p class="ref">Yoo, Yin, Hayden, and Pearson</p>
<script>
var vid = document.getElementById("pacman_goals_vid");
vid.playbackRate = 0.5;
</script>
</section>
<section>
<div style="float:left; width:25%">
<p></p>
</div>
<div style="float:left; width:25%">
<img src="https://web.duke.edu/mind/level2/faculty/pearson/assets/images/website/kelsey.jpg" style="height: 300px; width: auto">
<p>Kelsey McDonald</p>
</div>
<div style="float:left; width:25%">
<img src="https://scholars.duke.edu/file/i1293252/image_1293252.jpg" style="height: 300px; width: auto">
<p>Scott Huettel</p>
</div>
<div style="float:left; width:25%">
</div>
<br>
<br>
<br>
<br>
<br>
<br>
<br>
<br>
<br>
<p class="ref">McDonald et al. (<a href="https://www.biorxiv.org/content/10.1101/385195v1">Nat. Comm., 2019</a>)</p>
</section>
<!-- <section>
<h3>Penalty Shot</h3>
<video width="70%" align="center" controls autoplay loop src="https://web.duke.edu/mind/level2/faculty/pearson/assets/videos/penaltyshot/concat_trials.mp4" type="video/mp4">
Your browser does not support the video tag.
</video>
<p class="ref"><a href="https://doi.org/10.1101/385195">McDonald, Broderick, Huettel, Pearson</a></p>
<aside class="notes" data-markdown>
- two monkeys, shooter and goalie (shooter recorded)
- controlled by joysticks
- roles rotated, animals know each other
- repeated sessions
- rapidly learned, rich dynamics
</aside>
</section> -->
<section>
<div style="width: 50%; float: left; text-align: left">
<video autoplay loop src="https://web.duke.edu/mind/level2/faculty/pearson/assets/videos/penaltyshot/concat_trials.mp4" type="video/mp4" id="trace_video">
Your browser does not support the video tag.
</video>
<script>
var vid = document.getElementById("trace_video");
vid.playbackRate = 1.;
</script>
</div>
<div style="width: 50%; float: right; font-size: .9em" >
<h3>Only this time...</h3>
<ul style="font-size: 0.85">
<li>Human, computer opponents</li>
<li>82 participant shooters, 2 human goalies</li>
<li>Constant x velocity</li>
</ul>
</div>
</section>
<section>
<h3>Highly variable strategies</h3>
<img src="https://web.duke.edu/mind/level2/faculty/pearson/assets/images/penaltyshot/kelsey_pk/trajectories_3.svg" style="border: none; background: none; box-shadow: none; width: 50%; height: auto">
<br>
<img src="https://web.duke.edu/mind/level2/faculty/pearson/assets/images/penaltyshot/kelsey_pk/trajectories_4.svg" style="border: none; background: none; box-shadow: none; width:50%; height: auto">
</section>
<section>
<h3>Modeling I</h3>
<ul>
<li>Observed: most trajectories are piecewise linear</li>
<li>Idea: model velocity changepoints</li>
<li>Want:
<ul>
<li>flexible model</li>
<li>measure of uncertainty</li>
<li>measure of inter-player coupling</li>
</ul>
</li>
</ul>
</section>
<section>
<h3>Modeling II: Gaussian processes</h3>
<p>A Gaussian Process is a distribution over <i>functions</i></p>
<div style="float:left; width:10%">
<p></p>
</div>
<div style="float:left; width:40%">
<img src="https://web.duke.edu/mind/level2/faculty/pearson/assets/images/misc/gp_prior.png", style="height: 300px; width:auto">
<p>Prior</p>
</div>
<div style="float:left; width:40%">
<img src="https://web.duke.edu/mind/level2/faculty/pearson/assets/images/misc/gp_posterior.png" style="height: 300px; width:auto">
<p>Posterior</p>
</div>
<div style="float:right; width:10%"></div>
<p class="ref"><a href="http://www.gaussianprocess.org/gpml/chapters/RW.pdf">Rasmussen & Williams (2006)</a></p>
</section>
<section>
<h3>Modeling III</h3>
<p>Model change point policy as a GP</p>
$$p(\text{change}) = \Phi(f(s(t), \omega(t)))$$
$$f(s(t), \omega(t)) \sim \mathrm{GP}(0, k)$$
$$k(x, x') \propto \prod_i \exp((x_i - x')^2/\lambda_i^2)$$
<br>
<h3>Informally:</h3>
<ul>
<li>strategy depends on game state $s$, opponent $\omega$</li>
<li>log odds of changepoint is a smooth function</li>
</ul>
</section>
<section>
<h3>Inference</h3>
<ul>
<li>Approximate inference via sparse GP methods
<ul>
<li><a href="http://www.jmlr.org/proceedings/papers/v5/titsias09a/titsias09a.pdf">Titsias (2009)</a>
</li>
<li><a href="http://www.jmlr.org/proceedings/papers/v38/hensman15.pdf">Hensman, Matthews, Ghahramani (2015)</a></li>
</ul>
<li>$M$ inducing points define a complete (differentiable) function
<ul>
<li>Takes $\mathcal{O}(NM^2)$ instead of $\mathcal{O}(N^3)$</li>
</ul>
</li>
</li>
<li>Coded in <a href="http://www.jmlr.org/papers/volume18/16-537/16-537.pdf">GPFlow</a> <a href="https://github.com/krm58/PenaltyShot_Behavior">(repo)</a></li>
</ul>
</section>
<section>
<h3>Real and predicted change points</h3>
<div style="float:left; width:50%">
<img src="https://web.duke.edu/mind/level2/faculty/pearson/assets/images/penaltyshot/kelsey_pk/FinalPaperSVGs/real_data_p3.svg" style="height: 300px; width: auto; background:white">
</div>
<div style="float:right; width:50%">
<img src="https://web.duke.edu/mind/level2/faculty/pearson/assets/images/penaltyshot/kelsey_pk/FinalPaperSVGs/pred_data_p3.svg" style="height: 300px; width: auto; background:white">
</div>
</section>
<section>
<h3>Predicting change points</h3>
<video width="70%" align="center" controls autoplay loop src="https://web.duke.edu/mind/level2/faculty/pearson/assets/videos/penaltyshot/concat_switch.mp4" type="video/mp4">
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</video>
</section>
<section>
<h3>Same idea: Predicting wins</h3>
<video width="70%" align="center" controls autoplay loop src="https://web.duke.edu/mind/level2/faculty/pearson/assets/videos/penaltyshot/concat_EV.mp4" type="video/mp4">
Your browser does not support the video tag.
</video>
</section>
<section>
<h3>Modeling III</h3>
<p>How do we measure player coupling?</p>
<ul>
<li>GPs are differentiable! (and $\nabla$GP also a GP)</li>
<li>$\nabla_s f \sim $ sensitivity of policy to current game state</li>
<li>$\varrho \equiv \lvert L^{-1} \nabla f\rvert^2$ (whiten by Cholesky to weight variables equally)</li>
</ul>
</section>
<section>
<h3>How opponent-sensitive are you?</h3>
<video width="70%" align="center" controls autoplay loop src="https://web.duke.edu/mind/level2/faculty/pearson/assets/videos/penaltyshot/concat_oppSens.mp4" type="video/mp4">
Your browser does not support the video tag.
</video>
</section>
<section>
<h3>Which opponent affects you more?</h3>
<img src="https://web.duke.edu/mind/level2/faculty/pearson/assets/images/penaltyshot/kelsey_pk/FinalPaperSVGs/sensitivity_hvsc.svg" style="border: none; background: white; box-shadow: none; width: 50%; height: auto">
</section>
<section>
<h3>But how do you win?</h3>
<img src="https://web.duke.edu/mind/level2/faculty/pearson/assets/images/penaltyshot/kelsey_pk/FinalPaperSVGs/final_move.png" style="border: none; background: white; box-shadow: none; width: 50%; height: auto">
</section>
<section>
<h3>Expected value of final move</h3>
<img src="https://web.duke.edu/mind/level2/faculty/pearson/assets/images/penaltyshot/kelsey_pk/FinalPaperSVGs/KDE_EVsubgame.svg" style="border: none; background: white; box-shadow: none; width: 80%; height: auto">
</section>
<section>
<h3>Could it have been different?</h3>
<img src="https://web.duke.edu/mind/level2/faculty/pearson/assets/images/penaltyshot/kelsey_pk/FinalPaperSVGs/EVsubgame_schematic.svg" style="border: none; background: white; box-shadow: none; width: 80%; height: auto">
</section>
<section>
<h3>Could it have been different?</h3>
<img src="https://web.duke.edu/mind/level2/faculty/pearson/assets/images/penaltyshot/kelsey_pk/FinalPaperSVGs/monkeyEVsubgame.svg" style="border: none; background: white; box-shadow: none; width: 80%; height: auto">
</section>
<section>
<h3>Summary</h3>
<ul>
<li>For simplified dynamics, a discriminative approach can work</li>
<li>GP models give us uncertainty, sensitivity</li>
<li>Can help to dissociate between timing-based and maneuver-based strategies</li>
</ul>
</section>
<section data-markdown>
### Wrapping up:
- Differential games capture key aspects of real-world decisions
- Even simple games give rise to rich dynamics
- Analysis goals should drive modeling approach
</section>
<section>
<h2>Sponsors</h2>
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