Specifications from Demonstrations
Prediction, Learning, and Teaching
Marcell Vazquez-Chanlatte
PhD Candidate at UC Berkeley
Committee: Sanjit Seshia, Shankar Sastry, Anca Dragan, Steven Piantadosi
2022-05-09
2022-05-09
What goals & assumptions are implicit in the way we act?
Three types of motivating applications.
Share road with many actors
Detecting Mode Confusion + Unused Features
Nobody reads the manual...
Specializing on the fly
Research Stack
Teaching tasks using demonstrations
Learning tasks from demonstrations
Maximum Entropy Planning
Research Stack
Teaching tasks using demonstrations
Learning tasks from demonstrations
Maximum Entropy Planning
Research Stack
Teaching tasks using demonstrations
Learning tasks from demonstrations
Maximum Entropy Planning
How to infer a task?
Many different signals to use for inference
Decades of research in learning rewards from demonstrations
Decades of research in learning rewards from demonstrations
Inverse Reinforcement Learning
Given a series of demonstrations, what reward, $r(s)$, best explains the behavior? (Abbeel and Ng 2004)
Consider an agent acting in the following stochastic grid world.
Can try to move up, down, left, right
May slip due to wind
What is the agent trying to do?
Probably trying to reach yellow tiles
Although these actions are surprising under that hypothesis
And isn't it easier to go to this yellow tile any way?
A lot of information from a single incomplete demonstration
Communication through actions
-
Essential for interpreting other signals: (Language, Disengagments, Other agents)
Goodman, et al. "Pragmatic language interpretation as probabilistic inference." TiCS `16
McPherson, et al. "Modeling supervisor safe sets for improving collaboration in human-robot teams." IROS `18
Afolabi, et al. "People as sensors: Imputing maps from human actions." IROS `18.
Communication through actions
-
Essential for interpreting other signals: (Language, Disengagments, Other agents)
Goodman, et al. "Pragmatic language interpretation as probabilistic inference." TiCS `16
McPherson, et al. "Modeling supervisor safe sets for improving collaboration in human-robot teams." IROS `18
Afolabi, et al. "People as sensors: Imputing maps from human actions." IROS `18. -
Actions are incredibly diagnostic.
Dragan, et al, "Legibility and predictability of robot motion." HRI `13.
Ho, et al, "Showing versus doing: Teaching by demonstration". NIPS `16
Sadigh, et al. "Planning for autonomous cars that leverage effects on human actions." RSS `16.
How should we represent learned tasks?
Desired Properties
- Decouple task from dynamics.
- Prefer sparse rewards.
- Support describing temporal tasks.
Abel, et al. "On the Expressivity of Markov Reward.", NeurIPS `21.
- Support composition.
Markovian Rewards couple environment with task
Reach yellow. Avoid red.
Taking away top left yellow causes error
Reach yellow. Avoid red.
Solution: Sparse reward + memory
Reach yellow. Avoid red.
Solution: Sparse reward + memory
Reach yellow. Avoid red.
Key question: What memory?
Solution: Sparse reward + memory
Reach yellow. Avoid red.
Key question: What memory?
But first: How to represent task?
Proposal: Tasks as Boolean Specifications
-
A (Boolean) specification,
$\varphi$, is a set of traces.
- We say $\xi$ satisfies $\varphi$, if $\xi \in \varphi$.
- $[\xi \in \varphi]$ is a sparse objective.
Task Specifications derived from Formal Logic, Automata, etc
Will call a collection of task specifications a concept class.
Support incremental learning
Incrementally Learn smaller/simpler rules
Explictly support learning memory
Specifications have our desired properties
Desired Properties
- Decouple task from dynamics.
- Prefer sparse rewards.
- Support describing temporal tasks.
Abel, et al. "On the Expressivity of Markov Reward.", NeurIPS `21.
- Support composition.
Research Stack
Teaching tasks using demonstrations
Learning tasks from demonstrations
Maximum Entropy Planning
Dynamics and Tasks
recipe
frame as inverse reinforcement learning
place holder
frame as inverse reinforcement learning
frame as inverse reinforcement learning
\[
\max_{\pi} \big(\mathbb{E}_{s_{1:\tau}}\big(\sum_{i=1}^\tau r(s_i)~|~\pi\big)\big)
\]
where \[\pi(a~|~s) = \Pr(a~|~s)\]
given a series of demonstrations, what reward, $r(s)$, best explains the behavior? (Abbeel and Ng 2004)
No unique explanatory reward!
\[
\max_{\pi} \Big(\mathbb{E}_{s_{1:\tau}}\big(\sum_{i=1}^\tau r(s_i)~|~\pi\big)\Big)
\]
Maximize Causal Entropy
\[
H(A_{1:\tau}~||~S_{1:\tau}) = \sum_{t=1}^T H\Big(A_{t} \mid S_{1:t}\Big)
\]
subject to feature statistics.
(Ziebart, et al. 2010)Focus on reward matching
Maximize Causal Entropy
\[
H(A_{1:\tau}~||~S_{1:\tau}) = \sum_{t=1}^T H\Big(A_{t} \mid S_{1:t}\Big)
\]
subject to $\mathbb{E}[r]$
- Intuition: Don't over commit to any particular strategy.
\[\Pr(\pi) \propto e^{\lambda\cdot \mathbb{E}[r(\xi)~\mid~ \pi]}\]
- Formally: Minimize worst-case excess bits to encode episode.
What should the reward be?
Satisfaction is a sparse objective.
Proposal: Use indicator.
\[
r(\xi) \triangleq
\begin{cases}
1 & \text{if } \xi \in \varphi\\
0 & \text{otherwise}
\end{cases}
\]
\[\mathbb{E}[r] = \Pr(\xi \in \varphi)\]
Assigns a Demonstration Likelihood
\begin{equation}
\Pr(\pi) \propto e^{\lambda \cdot \mathbb{E}[ \Pr(\xi \in \varphi)~~\mid~\pi~]}
\end{equation}
Proxy policy exponentially favors high value prefixes
\begin{equation}
\ln\pi_\lambda(a~|~s_{1:t}) = Q_\lambda(s_{1:t}, a) - V_\lambda(s_{1:t})
\end{equation}
Will revisit.
Reward Learning offers many ways to rank specifications
- Inverse Optimal Control
- Daniel Kasenberg, Matthias Scheutz. "Interpretable apprenticeship learning with temporal logic specifications." CDC `17
- Glen Chou, Necmiye Ozay, Dmitry Berenson. "Explaining Multi-stage Tasks by Learning Temporal Logic Formulas from Suboptimal Demonstrations." RSS `22
- Bayesian Inference
- Ankit Shah, Pritish Kamath, Julie A. Shah, Shen Li. "Bayesian Inference of Temporal Task Specifications from Demonstrations." NeurIPS `18
- Hansol Yoon, Sriram Sankaranarayanan. "Predictive Runtime Monitoring for Mobile Robots using Logic-Based Bayesian Intent Inference." ICRA `21
- Maximum Entropy Inverse Reinforcement Learning
- Marcell Vazquez-Chanlatte, Susmit Jha, Ashish Tiwari, Mark K. Ho, and Sanjit A. Seshia. "Learning Task Specifications from Demonstrations." NeurIPS. `18.
- Marcell Vazquez-Chanlatte, and Sanjit A. Seshia. "Maximum Causal Entropy Specification Inference from Demonstrations." CAV `20
Key question: What memory?
Problem: No gradient to guide search over discrete structure.
Literature focused on Naïve syntactic search
Literature focused on Naïve syntactic search
Problems with syntatic search
- Conflates inductive bias with search efficiency.
- When teaching even harder to justify.
- Want generic strategy that works with less structured concept classes.
- Ignores the demonstrations!
Solution: Sparse reward + memory
Reach yellow. Avoid red.
Key question: What memory?
Enough memory to seperate good vs bad
Results in walk through labeled example space
Guide search by minimizing surprise
\[
h(\varphi) \triangleq \# \text{ of nats to describe demonstrations assuming } \varphi.
\]
Guide search by minimizing surprise
\[
h(\varphi) \triangleq -\sum_{i=1}^n \ln \Pr(\text{demo}_i~|~\varphi, \text{dynamics})
\]
Will see that $h$ factors through $R^n$
Will try to pull back changes prescribed by gradient.
Key idea 1: Reframe choices along demonstration
Many ways to deviate from demonstration
How valuable is each deviation?
\begin{equation}
\ln\pi_\lambda(a~|~s_{1:t}) = Q_\lambda(s_{1:t}, a) - V_\lambda(s_{1:t})
\end{equation}
Maximum Causal Entropy Policy
\begin{equation}
\ln\pi_{\mathbf{\lambda}}(a~|~s_{1:t}) = Q_{\mathbf{\lambda}}(s_{1:t}, a) - V_{\mathbf{\lambda}}(s_{1:t})
\end{equation}
where
\[
Q_{\mathbf{\lambda}}(s_{1:t}, a) \triangleq \mathbb{E}_{s_{1:t+1}}\left[ V_{\mathbf{\lambda}}(s_{1:t+1})~|~s_{1:t}, a\right]
\]
\[
V_{\mathbf{\lambda}}(s_{1:t}) \triangleq \begin{cases}
\ln \sum_{a} e^{Q_{\mathbf{\lambda}}(s_{1:t}, a)} & \text{if } t \neq \tau,\\
\lambda\cdot [s_{1:\tau} \in \varphi] & \text{otherwise}.
\end{cases}
\]
Find $\lambda$ to match $\Pr(\xi \in \varphi)$.
$\ln\sum_x e^x$ ↦ smax.
How valuable is each deviation?
Each deviation's value summarized by $V_\lambda, Q_\lambda$.
\[
Q_{\mathbf{\lambda}}(s_{1:t}, a) \triangleq \mathbb{E}_{s_{1:t+1}}\left[ V_{\mathbf{\lambda}}(s_{1:t+1})~|~s_{1:t}, a\right]
\]
\[
V_{\mathbf{\lambda}}(s_{1:t}) \triangleq \begin{cases}
\text{smax}_{a} Q(s_{1:t}, a) & \text{if } t \neq \tau,\\
\lambda\cdot [s_{1:\tau} \in \varphi] & \text{otherwise}.
\end{cases}
\]
Reframe as conforming or deviating
$\mathbb{E}$ and $\text{smax}$ are associative and commutative
\[
Q_{\mathbf{\lambda}}(s_{1:t}, a) \triangleq \mathbb{E}_{s_{1:t+1}}\left[ V_{\mathbf{\lambda}}(s_{1:t+1})~|~s_{1:t}, a\right]
\]
\[
V_{\mathbf{\lambda}}(s_{1:t}) \triangleq \begin{cases}
\text{smax}_{a} Q(s_{1:t}, a) & \text{if } t \neq \tau,\\
\lambda\cdot [s_{1:\tau} \in \varphi] & \text{otherwise}.
\end{cases}
\]
Two choices along a demonstration
$\mathbb{E}$ and $\text{smax}$ are associative and commutative
\[
Q_{\mathbf{\lambda}}(s_{1:t}, a) \triangleq \mathbb{E}_{s_{1:t+1}}\left[ V_{\mathbf{\lambda}}(s_{1:t+1})~|~s_{1:t}, a\right]
\]
\[
V_{\mathbf{\lambda}}(s_{1:t}) \triangleq \begin{cases}
\text{smax}_{a} Q(s_{1:t}, a) & \text{if } t \neq \tau,\\
\lambda\cdot [s_{1:\tau} \in \varphi] & \text{otherwise}.
\end{cases}
\]
Two choices along a demonstration
Two choices along a demonstration
- Independent of number of actions and state
- Only need transition probabilities in demonstrations.
Key idea 2: View demonstration as a computation tree
Environment nodes average
Agent nodes smoothmax
Demonstrations map to computation graph
Surprise determined by pivot values.
Surprise factors through a function over pivot values.
\[
\begin{split}
&\widehat{h} : \mathbb{R}^{d} \to \mathbb{R}\\
&h(\varphi) = \widehat{h}(\vec{V}_\varphi)
\end{split}
\]
Gradient of proxy surprise easy to compute.
\[
\frac{\partial \widehat{h}}{\partial V_k}(\vec{V}) =
\sum_{\substack{(i, j) \in E\\i \text{ is ego}}} \#_{(i, j)} \cdot \bigg(\Pr(i \rightsquigarrow k\mid \vec{V}) - \Pr(j \rightsquigarrow k\mid \vec{V})\bigg)
\]
Key idea 3: Toggle trace labels to manipulate pivot values.
Each pivot's subtree summarized by V
Each path given binary label by specification
Flipping value monotonically changes subtree value
$V_3 < V_3'$
Can sample from $\pi$ to generate high weight paths.
Flipping value monotonically changes subtree value
Flipping value monotonically changes subtree value
Surprise Guided Sampler
- Fix a candidate spec and compute proxy gradient.
- Define the pivot distribution, $D$ = softmax $\left (\frac{|\nabla \widehat{h}|}{\beta}\right)$.
- Sample a pivot, $k \sim D$ and a path, $\xi$, using the $\pi_\varphi$ such that:
- $\xi$ pivots at $k$.
- $\nabla_k \widehat{h} > 0 \iff \xi \in \varphi$.
- New label given by $\nabla_k \widehat{h} < 0$.
Candidate sampler can be any exact learner
Lifted in simulated annealing using example buffer
Lifted in simulated annealing using example buffer
Results in walk through labeled example space
Two Experiments
-
Incremental learning using 1 incomplete unlabeled demo
-
Monolithic learning using 2 complete unlabeled demos.
DISS able to quickly find probable task specs
Most probable DFA almost matches ground truth.
-
Incremental learning using 1 incomplete unlabeled demo
-
Monolithic learning using 2 complete unlabeled demos.
Research Stack
Teaching tasks using demonstrations
Learning tasks from demonstrations
Maximum Entropy Planning
Dynamics and Tasks
Can generate pedagogic demonstrations
Helps Teach Conditional Rules
Recipe to generate pedagogic paths
Tasks induce a description length on demonstrations
Find likely paths given ground truth that have:
- Short description length given learned task distribution.
- Long description length given ground truth task.
Research Stack
Teaching tasks using demonstrations
Learning tasks from demonstrations
Maximum Entropy Planning
Summary
Random Bit Model
Idea: Model Markov Decision Process as deterministic transition system with access to $n_c$ coin flips.
Note: Principle of maximum causal entropy + finite horizon together are robust to small dynamics mismatches.
Random Bit Model
Next: Assume \(\#(\text{Actions}) = 2^{n_a} \)
Random Bit Model
Next: Assume \(\#(\text{Actions}) = 2^{n_a} \)
\[
\text{Dynamics} : S \times {\{0, 1\}}^{n_a + n_c} \to S
\]
Random Bit Model
Unrolling \(\tau\) steps and composing with specification results in a predicate.
\[
\psi : {\{0, 1\}}^{\tau\cdot (n_a + n_c)} \to \{0, 1\}
\]
Random Bit Model
\[
\psi : {\{0, 1\}}^{\tau\cdot (n_a + n_c)} \to \{0, 1\}
\]
Proposal: Represent \(\psi\) as a Binary Decision Diagram with bits in causal order.
Random Bit Model
\[
\psi : {\{0, 1\}}^{\tau\cdot (n_a + n_c)} \to \{0, 1\}
\]
Proposal: Represent \(\psi\) as a Binary Decision Diagram with bits in causal order.
Maximum Causal Entropy and BDDs
Q: Can Maximum Entropy Causal Policy be computed on causally ordered BDDs?
Maximum Causal Entropy and BDDs
Q: Can Maximum Entropy Causal Policy be computed on causally ordered BDDs?
A: Yes! Due to:
- Associativity of \(\text{smax}\) and \(\mathbb{E}\).
- \(\text{smax}(\alpha, \alpha) = \alpha + \ln(2)\)
- \(\text{E}(\alpha, \alpha) = \alpha\)
Maximum Causal Entropy and BDDs
- Associativity of \(\text{smax}\) and \(\mathbb{E}\).
- \(\text{smax}(\alpha, \alpha) = \alpha + \ln(2)\)
- \(\text{E}(\alpha, \alpha) = \alpha\)
Maximum Causal Entropy and BDDs
- Associativity of \(\text{smax}\) and \(\mathbb{E}\).
- \(\text{smax}(\alpha, \alpha) = \alpha + \ln(2)\)
- \(\text{E}(\alpha, \alpha) = \alpha\)
Size Bounds
Q: How big can these Causal BDDs be?
\[
|BDD| \leq \overbrace{\underbrace{\tau}_{\text{horizon}} \cdot \big( \log(|A|) + \text{#coins} \big)}^{\text{# inputs}} \cdot \big( \underbrace{|S \times S_\varphi \times A|}_{\text{composed automaton}} \cdot 2^{\text{#coins}} \big)
\]
Size Bounds
\[
|BDD| \leq \overbrace{\underbrace{\tau}_{\text{horizon}} \cdot \big( \log(|A|) + \text{#coins} \big)}^{\text{# inputs}} \cdot \big( \underbrace{|S \times S_\varphi \times A|}_{\text{composed automaton}} \cdot 2^{\text{#coins}} \big)
\]
Linear in horizon!
Note: Using function composition, can build BDD in polynomial time.
Summary
Talk was biased towards Learning Contributions
-
Learn task specifications from (un)labeled and (in)complete demonstrations in a MDP.
- Support incremental and monolithic learning.
- Only needs blackbox access to a MaxEnt planner and Concept Identifer.
Contributions
- Learn task specifications from (un)labeled and (in)complete demonstrations in a MDP.
- Support incremental and monolithic learning.
- Only needs blackbox access to a MaxEnt planner and Concept Identifer.
- Can automatically generate pedagogic demonstrations.
- Efficient MaxEnt planning in symbolic Stochastic Games.
- Marcell Vazquez-Chanlatte, Susmit Jha, Ashish Tiwari, Mark K. Ho, and Sanjit A. Seshia. "Learning Task Specifications from Demonstrations." NeurIPS. `18.
- Marcell Vazquez-Chanlatte, and Sanjit A. Seshia. "Maximum Causal Entropy Specification Inference from Demonstrations." CAV `20
- Marcell Vazquez-Chanlatte, Ameesh Shah, Gil Lederman, and Sanjit A. Seshia. "Demonstration Informed Specification Search" In submission.
- Exact Probablistic Model Checking of Finite Markov Chains
- Sebastian Junges, Steven Holtzen, Marcell Vazquez-Chanlatte, Todd Millstein, Guy Van den Broerk, Sanjit A. Seshia. "Model Checking Finite-Horizon Markov Chains with Probabilistic Inference", CAV `21
By no means solved
Clear path to scaling up
- Need estimate of policy on prefix tree.
- Need a way to sample likely paths from policy.
Plays nicely with approximate methods
- Monte Carlo Tree Search (e.g., Smooth Cruiser [1])
- Function Approximation (e.g., Soft Actor Critic [2])
Promising preliminary results using Graph Neural Networks
Works with any supervised learner
Can in principle do Decision Trees and Symbolic Automata.
Works with any supervised learner
Could use natural language or other signals for priors.
Working on variant for constraints
Fix the reward and infer constraints on behavior.
Multi-agent (inferred) assume guarantee reasoning underexplored
\[ A \implies G\]
Maximum causal entropy correlated equillibria
seem like an interesting model
Ziebart, et al., "Maximum causal entropy correlated equilibria for Markov games." AAAI `10.
Thank you
Thank you
Thank you
Slides: mjvc.me/phd
