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Safe Learning

An educational book with interactive chapters, worked examples and graded practice

Start here, even if some earlier knowledge feels rusty

Begin with Primer 0, then use the primer refreshers and section practice to rebuild the skills needed later. Exercises progress from Easy calculations through Medium applications to Hard proofs and assumption checks, with separate hints and worked answers.

Open the study guide to check your starting point, find practice by topic, and choose a reading route.

Six math primers (0, A–E) take you from a first course in linear algebra to every tool the section uses: the language of proofs, limits and compactness; norms, positive definiteness and the SVD; calculus, convexity and optimization; probability and concentration; dynamical systems and feedback; MDPs, reinforcement learning and neural networks. Then fifteen modules build up the mathematics of learning with safety guarantees: from Lagrangian duality, LMIs and Gaussian-process confidence bounds, through safe Bayesian optimization, viability, constrained policy optimization, barrier functions and safety filters, to Lipschitz-certified networks and neural controllers in closed loop.

Two research lines get extra depth: the Trimpe group (RWTH Aachen, DSME — safe exploration, safe BO, viability) and Patricia Pauli (TU/e, Control Systems Technology — certified neural networks via robust control). Every page has step-by-step derivations, an interactive explorer with real computations, exercises with worked solutions, a verified reading list and flashcards, and opens with a “Before you start” box that links to exactly the primer sections and earlier modules it relies on.

Read this as a book

Follow the complete reading path through six foundation chapters and fifteen learning/control chapters. Each now includes a practical application lab, chapter review and new transfer exercises. Finish with three connected projects; use the glossary when a term needs rebuilding.

Topic Map

Select a topic to open its page. On a small screen, scroll the map sideways to keep the labels readable.

Math Primers (start here) Foundations Safe Exploration (Trimpe lens) Constrained Deep RL Control-Theoretic Safety Certified Neural Networks (Pauli lens) Reference Prerequisite Shared concepts

Math Primers (start here)

0. Mathematical Language, Proofs & Limits — Sets and quantifiers, functions and fixed points, proof techniques, limits and geometric series, sup/inf, compactness, O-notation A. Linear Algebra II: Norms, Positive Definiteness & the SVD — Norms, spectral theorem, PSD matrices, SVD, block matrices, log det, function spaces B. Calculus, Convexity & Optimization — Gradients, matrix calculus, Lipschitz continuity, convexity, descent, constraints, LP/QP C. Probability, Concentration & Information — Gaussians and conditioning, Hoeffding, high-probability bounds, martingales, KL, CVaR, tests D. Dynamical Systems, Stability & Feedback Control — State space, Lyapunov, comparison lemma, LQR/PID, frequency response, Lur'e systems, MPC E. MDPs, Reinforcement Learning & Neural Networks — Bellman equations, policy gradients, trust regions, modern RL, layers, margins, adversarial examples

Foundations

1. The Safe Learning Landscape — What "safe" means, three traditions, who's who, timeline, notation 2. Math Toolkit I: Duality, LMIs & the S-Procedure — KKT, SDPs, Schur complement, S-lemma, quadratic constraints, dissipativity 3. Math Toolkit II: Kernels, GPs & Uncertainty Bounds — RKHS, GP regression, information gain, frequentist confidence bounds, the βt problem

Safe Exploration (Trimpe lens)

4. Safe Bayesian Optimization: SafeOpt & Controller Tuning — Safe sets, expanders, maximisers, guarantees, BO for controller tuning 5. Is Safe BO Actually Safe? Real-β-SafeOpt and LoSBO — The β-heuristic gap, Lipschitz-only safety, LoS-GP-UCB, event-triggered safe BO 6. Global Safe Exploration of Dynamical Systems: GoSafe & GoSafeOpt — Disconnected safe regions, backup policies, boundary conditions, safe MDP exploration 7. Viability & Safe Value Functions — Viability kernels, penalty threshold theorem, entropy regularisation, uncertainty-aware safe RL

Constrained Deep RL

8. CMDPs, Duality & Lagrangian Methods — Occupancy measures, zero duality gap, primal-dual, PID Lagrangians, CVaR, state augmentation 9. Trust Regions, CPO & Modern Safe Policy Optimization — Performance difference, CPO closed form, FOCOPS/CUP/P3O, model-based, offline, benchmarks

Control-Theoretic Safety

10. Barrier Functions, Reachability & Safety Filters — Nagumo, CBF-QP, HOCBF, robust CBFs, HJ reachability, predictive safety filters, shields 11. Lyapunov Certificates, Safe Model-Based RL & Learning-Based MPC — Certified regions of attraction, neural certificates, learning-based and certified approximate MPC

Certified Neural Networks (Pauli lens)

12. Lipschitz Bounds via SDP: LipSDP and Beyond — Incremental QCs, the LipSDP LMI, training under LMI constraints, CNNs as dynamical systems 13. Lipschitz-by-Design Networks & Direct Parameterizations — Cayley, SLL, Sandwich layers, LipKernel, RENs, certified robustness SOTA 14. Neural Networks in the Loop: QCs, IQCs & Dissipativity — Stability LMIs, Zames–Falb multipliers, offset-free tracking, dissipative RNNs, synthesis 15. Verification & Distribution-Free Guarantees — IBP, CROWN, α,β-CROWN, randomized smoothing, conformal prediction, scenario approach

Reference

Study Guide & Practice Routes — Start with gaps in your knowledge, find graded assignments, and follow the prerequisites Formula Sheet — All key definitions, theorems and update rules on one printable page Paper Atlas — The verified bibliography (240+ papers), filterable by topic, year and importance Open Problems & Research Gaps — 30 open problems and research directions, each linked to the modules that teach its background

Suggested reading paths

Coming straight from a first linear algebra course? Start with Primers 0 and A, then read B–E as far as your track needs. The lists below include the numbered prerequisite modules for each track; use each page’s “Before you start” box to find the exact sections to review.