Agentic AI Mathematics
Tool-using agents as decision processes: how per-step error compounds over a trajectory, the bandit view of tool selection, and what inference-time search buys.
Level 2 — Advanced
Level 3 — Graduate
- Inference-Time Search and Best-of-n11 min
What sampling more candidates actually buys — exponentially decaying failure for linear cost, an unbiased pass@k estimator, and the exact KL price of best-of-n.
Best-of-n with a perfect verifierThe unbiased pass@k estimatorThe KL divergence of best-of-n - Tool Selection and Regret10 min
An agent choosing among tools is running a bandit — and the logarithmic regret of UCB tells you exactly what exploration is worth.
Cumulative regretUCB1 pulls a suboptimal arm O(log T / Δ²) timesLogarithmic regret for UCB1
Problems (11)
Open in the browser| Solved | Problem | Type | Difficulty | Time |
|---|---|---|---|---|
| Trajectory success from per-step reliability | Numerical | 15 min | ||
| Sizing best-of-n | Numerical | 15 min | ||
| The reliability wall | Numerical | 15 min | ||
| Detect-and-retry arithmetic | Proof | 30 min | ||
| Majority voting via Hoeffding | Proof | 30 min | ||
| The pass@k estimator | Proof | 30 min | ||
| What correlation does to voting | Proof | 30 min | ||
| Simulate an agent pipeline end to end run | Coding | 1 day | ||
| The ceiling imposed by a fallible verifier | Proof | 30 min | ||
| The KL divergence of best-of-n abyss | Proof | 1 hour | ||
| The UCB pull bound abyss | Proof | 3 hours |