Breaking the Curse of Dimensionality in Decision-Making for Autonomous Systems

Assistant Professor Zachary Sunberg

University of Colorado Boulder

September 6th, 2024

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

Example 1: Autonomous Driving

Example 2: Tornados

Video: Eric Frew

Example 2: Tornados

Example 3: Search and Rescue

What do they have in common?

Driving: what are the other road users going to do?

Tornado Forecasting: what is going on in the storm?

Search and Rescue: where is the lost person?

All are sequential decision-making problems with uncertainty!

All can be modeled as a POMDP (with a very large state and observation spaces).

Outline

The Promise and Curse of POMDPs
Breaking the Curse
Applications
Multiple Agents

Part I: The Promise and Curse of POMDPs

Types of Uncertainty

Aleatory

Epistemic (Static)

Epistemic (Dynamic)

Interaction

MDP

POMDP

Game

Markov Decision Process (MDP)

\(\mathcal{S}\) - 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

Aleatory

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

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

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

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

Reinforcement Learning

\(\mathcal{S}\) - 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

Aleatory

Epistemic (Static)

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

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

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

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)

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

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

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

\begin{aligned} & \mathcal{S} = \mathbb{Z} \quad \quad \quad ~~ \mathcal{O} = \mathbb{R} \\ & s' = s+a \quad \quad o \sim \mathcal{N}(s, |s-10|) \\ & \mathcal{A} = \{-10, -1, 0, 1, 10\} \\ & R(s, a) = \begin{cases} 100 & \text{ if } a = 0, s = 0 \\ -100 & \text{ if } a = 0, s \neq 0 \\ -1 & \text{ otherwise} \end{cases} & \\ \end{aligned}

State

Timestep

Accurate Observations

Goal: \(a=0\) at \(s=0\)

Optimal Policy

Localize

\(a=0\)

POMDP Example: Light-Dark

Solving a POMDP

Environment

Belief Updater

Planner

\(a = +10\)

True State

\(s = 7\)

Observation \(o = -0.21\)

\(b\)

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

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

\(Q(b, a)\)

\(O(|\mathcal{S}|^2)\)

Why are POMDPs difficult?

Curse of History
Curse of dimensionality
1. State space
2. Observation space
3. Action space

Tree size: \(O\left(\left(|A||O|\right)^D\right)\)

Curse of Dimensionality

\(d\) dimensions, \(k\) segments \(\,\rightarrow \, |S| = k^d\)

1 dimension

e.g. \(s = x \in S = \{1,2,3,4,5\}\)

\(|S| = 5\)

2 dimensions

e.g. \(s = (x,y) \in S = \{1,2,3,4,5\}^2\)

\(|S| = 25\)

3 dimensions

e.g. \(s = (x,y,x_h) \in S = \{1,2,3,4,5\}^3\)

\(|S| = 125\)

(Discretize each dimension into 5 segments)

\(x\)

\(y\)

\(x_h\)

Part II: Breaking the Curse

Integration

Find \(\underset{s\sim b}{E}[f(s)]\)

\[=\sum_{s \in S} f(s) b(s)\]

Monte Carlo Integration

\(Q_N \equiv \frac{1}{N} \sum_{i=1}^N f(s_i)\)

\(s_i \sim b\) i.i.d.

\(\text{Var}(Q_N) = \text{Var}\left(\frac{1}{N} \sum_{i=1}^N f(s_i)\right)\)

\(= \frac{1}{N^2} \sum_{i=1}^N\text{Var}\left(f(s_i)\right)\)

\(= \frac{1}{N} \text{Var}\left(f(s_i)\right)\)

\[P(|Q_N - E[f(s_i)]| \geq \epsilon) \leq \frac{\text{Var}(f(s_i))}{N \epsilon^2}\]

(Bienayme)

(Chebyshev)

Curse of dimensionality!

Particle Filter POMDP Approximation

\[b(s) \approx \sum_{i=1}^N \delta_{s}(s_i)\; w_i\]

[Sunberg and Kochenderfer, ICAPS 2018, T-ITS 2022]

How do we prove convergence?

Key 1: Self Normalized Infinite Renyi Divergence Concentation

\(\mathcal{P}\): state distribution conditioned on observations (belief)

\(\mathcal{Q}\): marginal state distribution (proposal)

Key 2: Sparse Sampling

Expand for all actions (\(\left|\mathcal{A}\right| = 2\) in this case)

...

Expand for all \(\left|\mathcal{S}\right|\) states

\(C=3\) states

SS-\(\omega\) is close to Belief MDP

SS-\(\omega\) close to Particle Belief MDP (in terms of Q)

PF Approximation Accuracy

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

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

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

No direct dependence on \(|\mathcal{S}|\) or \(|\mathcal{O}|\)!

Particle belief planning suboptimality

\(C\) is too large for any direct safety guarantees. But, in practice, works extremely well for improving efficiency.

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

Why are POMDPs difficult?

Curse of History
Curse of dimensionality
1. State space
2. Observation space
3. Action space

Tree size: \(O\left(\left(|A|C\right)^D\right)\)

Solve simplified surrogate problem for policy deep in the tree

[Lim, Tomlin, and Sunberg, 2021]

Part III: Applications

Navigation among Pedestrians

[Gupta, Hayes, & Sunberg, AAMAS 2022]

Previous solution: 1-D POMDP (92s avg)

Our solution (65s avg)

State:

Vehicle physical state
Human physical state
Human intention

Meteorology

State: (physical state of aircraft, which forecast is the truth)
Action: (flight direction, drifter deploy)
Reward: Terminal reward for correct weather prediction

Example 2: Tornado Prediction

Drone Search and Rescue

State:

Location of Drone
Location of Human

Baseline

Our POMDP Planner

[Ray, Laouar, Sunberg, & Ahmed, ICRA 2023]

Drone Search and Rescue

[Ray, Laouar, Sunberg, & Ahmed, ICRA 2023]

Space Domain Awareness

(Result for simplified dynamical system)

State:

Position, velocity of object-of-interest
Anomalies: navigation failure, suspicious maneuver, thruster failure, etc.

Innovation: Large language models allow analysts to quickly specify anomaly hypotheses

Catalog Maintenance Plan

Practical Safety Guarantees

Three Contributions

Recursive constraints (solves "stochastic self-destruction")
Undiscounted POMDP solutions for estimating probability
Much faster motion planning with Gaussian uncertainty

State:

Position of rover
Environment state: e.g. traversibility
Internal status: e.g. battery, component health

[Ho et al., UAI 24], [Ho, Feather, Rossi, Sunberg, & Lahijanian, UAI 24], [Ho, Sunberg, & Lahijanian, ICRA 22]

Explainability: Reward Reconciliation

Calculate Outcomes

Calculate Weight Update

\( \frac{\epsilon - \alpha_a \cdot \Delta \mu_{h-a}}{\Delta\mu_{h-j} \cdot \Delta \mu_{h-a}} \Delta\mu_{h-j}\)

Estimate Weight with Update

\( \alpha[2]\)

\( \alpha[1]\)

\( a_{h}\) - optimal

\( a_{a}\) - optimal

\( \alpha_{h}\)

\( \alpha_{a}\)

\( R(s,a) = \alpha \cdot \boldsymbol{\phi}(s,a)\)

\(a_a\) outcomes: \(\mu_a\)

\(a_h\) outcomes: \(\mu_h\)

\(a_a\)

\(a_h\)

\( \hat{\alpha}_{h}\)

\(a_h\)

\(a_a\)

[Kraske, Saksena, Buczak, & Sunberg, ICAA 2024]

Part IV: Multiple Agents

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

POSG Example: Missile Defense

POMDP Solution:

Assume a distribution for the missile's actions
Update belief according to this distribution
Use a POMDP planner to find the best defensive action

Nash equilibrium: All players play a best response to the other players

Fundamentally impossible for POMDP solvers to compute.

May include stochastic behavior (bluffing)

A shrewd missile operator will use different actions, invalidating our belief

Simplified SDA Game

...

\(N\)

[Becker & Sunberg, AMOS 2022]

Counterfactual Regret Minimization Training

[Becker & Sunberg, AMOS 2022]

Finding a Nash Equilibrium: Poker

Image: Russel & Norvig, AI, a modern approach

P1: A

P1: K

P2: A

P2: K

Tree Search Algorithms for POSGs

[Becker & Sunberg, NeurIPS 2024 (Under Review)]

Tree Search Algorithms for POSGs

[Becker & Sunberg, NeurIPS 2024 (Under Review)]

Thank You!

www.cu-adcl.org

Funding orgs: (all opinions are my own)

VADeR

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

Part V: Open Source Research Software

Good Examples

Open AI Gym interface
OMPL
ROS

Challenges for POMDP Software

There is a huge variety of
- Problems
  - Continuous/Discrete
  - Fully/Partially Observable
  - Generative/Explicit
  - Simple/Complex
- Solvers
  - Online/Offline
  - Alpha Vector/Graph/Tree
  - Exact/Approximate
- Domain-specific heuristics
POMDPs are computationally difficult.

Explicit

Black Box

("Generative" in POMDP lit.)

\(s,a\)

\(s', o, r\)

Previous C++ framework: APPL

"At the moment, the three packages are independent. Maybe one day they will be merged in a single coherent framework."

Open Source Research Software

Performant
Flexible and Composable
Free and Open
Easy for a wide range of people to use (for homework)
Easy for a wide range of people to understand

C++

Python, C++

Python, Matlab

Python, C++

2013

We love [Matlab, Lisp, Python, Ruby, Perl, Mathematica, and C]; they are wonderful and powerful. For the work we do — scientific computing, machine learning, data mining, large-scale linear algebra, distributed and parallel computing — each one is perfect for some aspects of the work and terrible for others. Each one is a trade-off.

We are greedy: we want more.

2012

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

Mountain Car

partially_observable_mountaincar = QuickPOMDP(
    actions = [-1., 0., 1.],
    obstype = Float64,
    discount = 0.95,
    initialstate = ImplicitDistribution(rng -> (-0.2*rand(rng), 0.0)),
    isterminal = s -> s[1] > 0.5,

    gen = function (s, a, rng)        
        x, v = s
        vp = clamp(v + a*0.001 + cos(3*x)*-0.0025, -0.07, 0.07)
        xp = x + vp
        if xp > 0.5
            r = 100.0
        else
            r = -1.0
        end
        return (sp=(xp, vp), r=r)
    end,

    observation = (a, sp) -> Normal(sp[1], 0.15)
)

using POMDPs
using QuickPOMDPs
using POMDPPolicies
using Compose
import Cairo
using POMDPGifs
import POMDPModelTools: Deterministic

mountaincar = QuickMDP(
    function (s, a, rng)        
        x, v = s
        vp = clamp(v + a*0.001 + cos(3*x)*-0.0025, -0.07, 0.07)
        xp = x + vp
        if xp > 0.5
            r = 100.0
        else
            r = -1.0
        end
        return (sp=(xp, vp), r=r)
    end,
    actions = [-1., 0., 1.],
    initialstate = Deterministic((-0.5, 0.0)),
    discount = 0.95,
    isterminal = s -> s[1] > 0.5,

    render = function (step)
        cx = step.s[1]
        cy = 0.45*sin(3*cx)+0.5
        car = (context(), circle(cx, cy+0.035, 0.035), fill("blue"))
        track = (context(), line([(x, 0.45*sin(3*x)+0.5) for x in -1.2:0.01:0.6]), stroke("black"))
        goal = (context(), star(0.5, 1.0, -0.035, 5), fill("gold"), stroke("black"))
        bg = (context(), rectangle(), fill("white"))
        ctx = context(0.7, 0.05, 0.6, 0.9, mirror=Mirror(0, 0, 0.5))
        return compose(context(), (ctx, car, track, goal), bg)
    end
)

energize = FunctionPolicy(s->s[2] < 0.0 ? -1.0 : 1.0)
makegif(mountaincar, energize; filename="out.gif", fps=20)