What else am I doing?

Adversarial Policies in POMDPs

 

Professor Zachary Sunberg

University of Colorado Boulder

CAAMS Winter '25 IAB Meeting

Autonomous Decision and Control Laboratory

cu-adcl.org

  • Algorithmic Contributions
    • Scalable algorithms for partially observable Markov decision processes (POMDPs)
    • Motion planning with safety guarantees
    • Game theoretic algorithms
  • Theoretical Contributions
    • Particle POMDP approximation bounds
  • Applications
    • Space Domain Awareness
    • Autonomous Driving
    • Autonomous Aerial Scientific Missions
    • Search and Rescue
    • Space Exploration
    • Ecology
  • Open Source Software
    • POMDPs.jl Julia ecosystem

PI: Prof. Zachary Sunberg

PhD Students

Postdoc

Markov Decision Process (MDP)

  • \(\mathcal{S}\) - State space
  • \(T(s' \mid s, a)\) - Transition probability distribution
  • \(\mathcal{A}\) - Action space
  • \(R(s, a)\) - Reward

Aleatory

\([x, y, z,\;\; \phi, \theta, \psi,\;\; u, v, w,\;\; p,q,r]\)

\(\mathcal{S} = \mathbb{R}^{12}\)

\(\mathcal{S} = \mathbb{R}^{12} \times \mathbb{R}^\infty\)

\[\underset{\pi:\, \mathcal{S} \to \mathcal{A}}{\text{maximize}} \quad \text{E}\left[ \sum_{t=0}^\infty R(s_t, a_t) \right]\]

Partially Observable Markov Decision Process (POMDP)

  • \(\mathcal{S}\) - State space
  • \(T(s' \mid s, a)\) - Transition probability distribution
  • \(\mathcal{A}\) - Action space
  • \(R(s, a)\) - Reward
  • \(\mathcal{O}\) - Observation space
  • \(Z(o \mid a, s')\) - Observation probability distribution

Aleatory

Epistemic (Static)

Epistemic (Dynamic)

\([x, y, z,\;\; \phi, \theta, \psi,\;\; u, v, w,\;\; p,q,r]\)

\(\mathcal{S} = \mathbb{R}^{12}\)

\(\mathcal{S} = \mathbb{R}^{12} \times \mathbb{R}^\infty\)

Examples of POMDPs

How Our POMDP Algorithms Work

1. Low-sample particle filtering

2. Sparse Sampling

[Lim, Becker, Kochenderfer, Tomlin, & Sunberg, JAIR 2023; Sunberg & Kochenderfer, ICAPS 2018; Others]

Solving a POMDP

Environment

Belief Updater

Planner

\(a = +10\)

\(b\)

\[b_t(s) = P\left(s_t = s \mid b_0, a_0, o_1 \ldots a_{t-1}, o_{t}\right)\]

\[ = P\left(s_t = s \mid b_{t-1}, a_{t-1}, o_{t}\right)\]

\(Q(b, a)\)

True State

\(s = 7\)

Observation \(o = -0.21\)

Key Theoretical Breakthrough

  • Solving POMDPs exactly is intractable (PSPACE-Complete / NP-hard) [Papadimitriou, 1987]
  • However, our online algorithms can find \(\epsilon\)-suboptimal policies with no dependence on the size of the state or observation spaces.

[Lim, Becker, Kochenderfer, Tomlin, & Sunberg, JAIR 2023]

\[|Q_{\mathbf{P}}^*(b,a) - Q_{\mathbf{M}_{\mathbf{P}}}^*(\bar{b},a)| \leq \epsilon \quad \text{w.p. } 1-\delta\]

For any \(\epsilon>0\) and \(\delta>0\), if \(C\) (number of particles) is high enough,

No direct relationship between \(C\) and \(|\mathcal{S}|\) or \(|\mathcal{O}|\)

Adversarial Policies in POMDPs

What is the worst case probabilistic model for the orange evader?

This cannot be calculated by solving a (PO)MDP!

(Every (PO)MDP has an optimal deterministic policy)

The POMDP is a good model for information gathering, but it is incomplete:

  • Cannot reason about an adversary that can change their policy in response
  • Cannot synthesize deceptive stochastic policies

Partially Observable Stochastic Game (POSG)

Image: Russel & Norvig, AI, a modern approach

P1: A

P1: K

P2: A

P2: A

P2: K

Planning in POSGs

Partially Observable Markov Decision Process (POMDP)

  • \(\mathcal{S}\) - State space
  • \(T:\mathcal{S}\times \mathcal{A} \times\mathcal{S} \to \mathbb{R}\) - Transition probability distribution
  • \(\mathcal{A}\) - Action space
  • \(R:\mathcal{S}\times \mathcal{A} \to \mathbb{R}\) - Reward
  • \(\mathcal{O}\) - Observation space
  • \(Z:\mathcal{S} \times \mathcal{A}\times \mathcal{S} \times \mathcal{O} \to \mathbb{R}\) - Observation probability distribution

Aleatory

Epistemic (Static)

Epistemic (Dynamic)

Partially Observable Stochastic Game (POSG)

Aleatory

Epistemic (Static)

Epistemic (Dynamic)

Interaction

  • \(\mathcal{S}\) - State space
  • \(T(s' \mid s, \bm{a})\) - Transition probability distribution
  • \(\mathcal{A}^i, \, i \in 1..k\) - Action spaces
  • \(R^i(s, \bm{a})\) - Reward function (cooperative, opposing, or somewhere in between)
  • \(\mathcal{O}^i, \, i \in 1..k\) - Observation spaces
  • \(Z(o^i \mid \bm{a}, s')\) - Observation probability distributions

[Becker & Sunberg, AAMAS 2025]

Planning in POSGs

Our approach: combine particle filtering and information sets

Joint Belief

Joint Action

[Becker & Sunberg, AAMAS 2025]

Open (related) questions:

  1. Is this useful for continuous observations?
  2. What information needs to be carried step to step?
  3. Is this useful for games where both players know a partition of the state space exactly?

Planning in POSGs

Autonomous Decision and Control Laboratory

cu-adcl.org

  • Algorithmic Contributions
    • Scalable algorithms for partially observable Markov decision processes (POMDPs)
    • Motion planning with safety guarantees
    • Game theoretic algorithms
  • Theoretical Contributions
    • Particle POMDP approximation bounds
  • Applications
    • Space Domain Awareness
    • Autonomous Driving
    • Autonomous Aerial Scientific Missions
    • Search and Rescue
    • Space Exploration
    • Ecology
  • Open Source Software
    • POMDPs.jl Julia ecosystem

PI: Prof. Zachary Sunberg

PhD Students

Postdoc

Interaction Uncertainty

[Peters, Tomlin, and Sunberg 2020]

Space Domain Awareness Games

Open question: are there \(\mathcal{S}\)- and \(\mathcal{O}\)-independent algorithms for POMGs?

Incomplete Information Extensive form Game

Our new algorithm for POMGs

c_I = 100.0 \\ c_T = 1.0 \\ c_{TR} = 10.0

COVID POMDPs

Planning Rebuilding Ecosystems

POMDPs.jl - An interface for defining and solving MDPs and POMDPs in Julia

Open Source Software

https://github.com/JuliaPOMDP/POMDPs.jl

Autonomous Decision and Control Laboratory

cu-adcl.org

  • Algorithmic Contributions
    • Scalable algorithms for partially observable Markov decision processes (POMDPs)
    • Motion planning with safety guarantees
    • Game theoretic algorithms
  • Theoretical Contributions
    • Particle POMDP approximation bounds
  • Applications
    • Space Domain Awareness
    • Autonomous Driving
    • Autonomous Aerial Scientific Missions
    • Search and Rescue
    • Space Exploration
    • Ecology
  • Open Source Software
    • POMDPs.jl Julia ecosystem

PI: Prof. Zachary Sunberg

PhD Students

Postdoc

Thank You!

zachary.sunberg.net

ADCL Students

BOMCP

[Mern, Sunberg, et al. AAAI 2021]

MPC for Intermittent Rotor Failures

UAV Component Failures

Reward Decomposition for Adaptive Stress Testing

Voronoi Progressive Widening

[Lim, Tomlin, & Sunberg CDC 2021]

Active Information Gathering for Safety

Sparse PFT

I(t) = \int_0^\infty I(t-\tau)\beta(\tau)d\tau

COVID POMDP

Individual Infectiousness

Infection Age

Incident Infections

\beta(\tau)
\tau
I
I(t) = \int_0^\infty I(t-\tau)\beta(\tau)d\tau
\beta(\tau)

Need

Test sensitivity is secondary to frequency and turnaround time for COVID-19 surveillance

Larremore et al.

\beta(\tau) \propto \log_{10}(\text{viral load})

Viral load represented by piecewise-linear hinge function

(t_0, 3)
(t_{\text{peak}}, V_{\text{peak}})
(t_f,6)
t_0 \sim \mathcal{U}[2.5,3.5]
t_\text{peak} - t_0 \sim 0.2 + \text{Gamma}(1.8)
V_\text{peak} \sim \mathcal{U}[7,11]
t_f - t_\text{peak} \sim \mathcal{U}[5,10]

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