Reinforcement Learning

Graduate course, Polytechnique, Mathematics and Computer science, 2020

This Reinforcement Learning course focused equally on both the theory and algorithms. Compared to other classes, this course was much more involved from a theoretical standpoint with only a few points left unproved during the class.

Below is a checklist for students to use and revise the concepts that were tackled in class.

Control theory

• Definition of Markov Decision Process
• Markov property
• Discount factor
• Discounted value, Finite time horizon value
• Bellman operator, Bellman optimal operator
• Dynamic Programming principle
• Policy Evaluation: Direct computation, Iteration, Monte-Carlo
• Contraction of Bellman operator, of Bellman optimal operator
• Value Iteration
• Policy Iteration
• Modified Policy Iteration
• Bellman Q-operator.
• Incremental Monte-carlo updates: Temporal Difference, $TD(\lambda)$
• Q-temporal difference, Q-learning
• Function approximation for V: Least-squares TD and Q: Fitted Q-iteration
• Projection vs Contraction

Multi-armed bandits

• The notion of Regret, of optimality gap $\Delta_a$
• What is Exploration? What is Exploitation?
• The optimism in face of uncertainty principle
• Hoeffding inequality for finite samples
• Handling random number of samples with Union bound
• The Upper Confidence bound (UCB) strategy
• The Thompson sampling strategy
• Problem dependent regret lower bound: scaling in $T$, Kullback-Leibler
• Most-confusing instance (e.g. for Bernoulli rewards)
• Problem-free (minimax) regret lower bound: scaling in $T$, $A$
• KL-UCB strategy lower-bound approach
• IMED strategy

Structured Multi-armed bandits

• What is Unimodal structure? Lipschitz structure? Linear structure?
• Graph seen as a linear structure
• Lower-bound for structured bandits: optimization problem
• Most confusing instance for Lipshitz bandits
• IMED for Lipschitz bandits
• Linear regression setup
• Sub-Gaussian noise assumption
• Least-squares estimate
• Optimistic principle for linear bandits
• Information gain

Model-based MDPs

• Average gain criterion
• Poisson equation (gain and bias)
• Diameter of an MDP
• Value Iteration convergence issues
• The span semi-norm
• Intrinsic contraction in span semi-norm
• Stopping criterion for Value Iteration
• Exploration-Exploitation in MDPs
• UCB for MDPs: UCRL
• Building blocks of UCRL: Episode, EVI
• What is an Extended MDP in EVI?
• What is guaranteed when EVI stops?

Planning

• Monte Carlo Tree Search
• What are the 4 main steps of of MCTS strategy?
• UCT rule for the value of each node
• What is a Generative model?
• KL-OLOP combines two main algorithms: which ones?
• What is Best-armed identification (BAI) objective?
• What is Simple regret?
• Fixed-budget objective vs Fixed-confidence objective
• Reduction from cumulative to simple regret
• Sequential Halving
• What do we track in Track-and-stop?
• What is forced exploration?
• UCT rule in max node, versus UCT rule in min node
• Monte-Carlo Graph Search idea
• When to rather use MCGS? When to rather use MCTS?

Deep Reinforcement Learning

• Model-based vs Model-free
• Critic algorithm, Actor algorithm, Actor-critic algorithm
• Example of Critic, Actor, algorithms?
• Q-learning idea
• What is slow/fast network updates? Why was it introduced?
• What is experience replay?
• What is prioritized experience replay?
• Double DQN