Jethro's Braindump

Reinforcement Learning ⭐

tags
Machine Learning

Reinforcement Learning is the field of learning and decision-making under uncertainty. An agent acts with a learnable behaviour policy in an environment with initially unknown dynamics and reward. The agent observes the environment’s state (sometimes partially: POMDPs) and chooses an action.

behaviour
\(\Pi(a | s)\)
action
\(a_t \in A\)
reward
\(r_{t+1} \in R\)
state
\(s_{t+1} \in S\)
dynamics
\(T(s_{t+1} | s_t, a_t)\)
reward
\(R(r_{t+1} | s_t, a_t)\)

The dynamics and rewards here reflect the markovian assumption. States can be represented in many ways: in a tabular fashion (state 1 \(\rightarrow\) state 2) or as features.

Key Challenges

  1. Exploration vs Exploitation
  2. Function approximation
  3. Credit Assignment

Making Complex Decisions

Earlier, we were concerned with environments with one-shot, episodic decision problems. Sequential decision problems incorporate utilities, uncertainty and sensing. These include searching and planning problems as special cases.

Utilities over time

A finite horizon for decision making means that there is a fixed time \(N\) after which nothing matters. In these scenarios, the optimal action in a given state could change over time, i.e. the optimal policy is non-stationary.

It turns out that under stationarity, there are only 2 coherent ways to assign utilities to sequences:

additive rewards
\(U_h([s_0, s_1, \dots, s_n]) = R(s_0) + R(s_1) + \dots + R(s_n)\)
discounted rewards
\(U_h([s_0, s_1, \dots, s_n]) = R(s_0) + \gamma \cdot R(s_1) + \dots + \gamma^2 \cdot R(s_n)\)

This discount factor \(\gamma\) is a number between 0 and 1. Assuming stationarity has several problems. First, if the environment does not contain a terminal state, then utilities of undiscounted rewards go to infinite, and comparing two infinitely state sequences would be impossible. With discounted rewards, the utility of an infinite sequence can be made finite.

However, if the environment contains a terminal state, and the agent is guaranteed to reach a terminal state eventually, then this policy is called a proper policy, and the above issue goes away. Infinite sequences can be compared in terms of the average reward obtained per time step.

Optimal policies and the utilities of states

First, we can derive the expected utility of executing a policy \(\pi\) in \(s\):

\begin{equation} U^\pi (s) = \mathbb{E} \left[ \sum_{t=0}^\infty \gamma^t R(S_t) \right] \end{equation}

where the expectation is with respect to the probability distribution over state sequences. determined by \(s\) and \(\pi\). Then $π^*(s) = argmax_π U^π (s)*.

A consequence of using discounted utilities with infinite horizons is that the optimal policy is independent of the starting state. This allows us to compute the true utility of the state as \(U^{\pi^*} (s)\). The utility function allows the agent to select actions by using the principle of maximum expected utility from the earlier chapter: \(\pi^*(s) = argmax_{a \in A(s) } \sum_{s^{i}} P(s’ |s, a)U(s’)\).

Summary

Problem Bellman Equation Algorithm
Prediction Bellman Expectation Equation Iterative Policy Evaluation
Control Bellman Expectation Equation + Greedy Policy Improvement Policy Iteration
Control Bellman Optimality Equation Value Iteration

Passive Reinforcement Learning

We start with a passive learning agent using a state-based representation in a fully observable environment.

In passive learning, the agent’s policy \(\pi\) is fixed: in state \(s\), it always executes the action \(\pi(s)\). Its goal is simply to learn how good the policy is – the utility function \(U^\pi (s)\).

The passive learning task is similar to policy evaluation, but the agent does not know the transition model \(P(s'|s, a)\) and the reward function \(R(s)\).

The agent executes a number of trials using the policy \(\pi\), and experiences a sequence of state transitions. At each state its percepts receives the current state and the reward of the state.

We write the utility as:

\begin{equation} U^\pi (s) = E\left[\sum_{t=0}^\infty \gamma^t R(S_t) \right] \end{equation}

Direct Utility Estimation (MC Learning)

The main idea of direct utility estimation is that the utility of a state is the expected total reward from that state onward, and each trial provides a sample of this quantity for each state visited.

Direct utility estimation reduces the reinforcement learning problem to a supervised inductive learning problem, where each example has the state as input, and the observed reward-to-go as output.

However, it misses an important source of information: that the utility of states are not independent. This means it misses many opportunities for learning. For example, if a state has high expected utility, then neighbouring states should also have high expected utility.

The utility of each state equals its own reward plus the expected utility of its successor states: i.e. it obeys the Bellman Equation for a fixed policy.

We can view directed utility estimation as searching for \(U\) in a hypothesis space that is much larger than it needs to be, since it includes many functions that violate the Bellman equations.

Adaptive Dynamic Programming

An ADP agent takes advantage of the constraints among the utilities of states by learning the transition model that connects them and solving the corresponding MDP using a dynamic programming method.

For a passive learning agent, the task is as simply as plugging in the learnt transition model and the rewards into the Bellman equations to calculate the utility of each state.

The task of learning the model is easy, because the environment is fully observable. This means we have a supervised learning task where the input is a state-action pair, and the output is the resulting state. We keep track of how often each action outcome occurs and estimate the transition probability \(P(s’ | s, a)\) from the frequency with which \(s’\) is reached when executing \(a\) in \(s\).

function PASSIVE-ADP_AGENT(percept) returns an action
  inputs: percept, indicating state s' and reward signal r'
  persistent: \pi, a fixed policy
    mdp: MDP with model P, rewards R, and discount \gamma
    U: a table of utilities, initially empty
    N_{sa}: a table of frequencies for each state-action pair
    N_{s'|s,a}: a table of outcome frequencies
    s, a: the previous state and action
  if s' is new then $U[s'] <- r'; R[s'] <- r'
  if s is not null then
    increment N_{sa}[s, a] and N_{s'|s,a}[s', s, a]
    for each t such that N_{s'|s, a}[t,s,a] is nonzero do
      P(t|s, a) <- N_{s'|s, a}[t,s,a] / N_{sa}[s, a]
    U <- POLICY-EVALUATION(\pi, U, mdp)
  if s'.TERMINAL? then s,a <- null else s,a <- s', \pi[s']
  return a
Code Snippet 1: A passive RL agent based on ADP.

This approach is computationally intractable for large state spaces. In addition, it uses the maximum-likelihood estimation for learning the transition model.

A more nuanced approach would be Bayesian reinforcement learning, which assumes a prior probability \(P(h)\) for each hypothesis \(h\) about what the true model is. The posterior probability \(P(h|e)\) is obtained via Bayes’ rule. Then \(\pi^* = argmax_\pi \sum_h P(h|e) u_h^\pi\).

Another approach, derived from robust control theory, allows for a set of possible models \(H\) and defines an optimal robust policy as one that gives the best outcome in the worst case over \(H\): \(\pi^* = argmax_\pi min_h u_h^\pi\).

Temporal-difference Learning

TD learning involves using the observed transitions to adjust the utilities such that the constraint equations are met.

When a transition occurs from state \(s\) to state \(s’\), we apply the update rule:

\begin{equation} U^\pi(s’) \leftarrow U^\pi(s) + \alpha (R(s) + \gamma U^\pi(s’) -U^\pi(s)) \end{equation}

Where \(\alpha\) is the learning rate. The difference in utilities gives rise to the name temporal-difference.

function PASSIVE-TD-AGENT(percept) returns an action
  inputs: percept, with current state s' and reward r'
  persistent: \pi, a fixed policy
    U, a table of utilities, initially empty
    N_s, a table of frequencies
    s, a, r, the previous state, action and reward

  if s' is new then U[s'] <- r'
  if s is not null then
    increment N_s[s]
    U[s] <- U[s] + \alpha N_s[s] (r + \gamma U[s'] - U[s])
  if s'.TERMINAL? then s, a r <- null else s,a,r <- s', \pi[s'], r'
  return a

TD learning learns slower than ADP and shows much higher variability, but is simpler and requires less computation. TD learning does not need a transition model to perform updates.

ADP and TD are closely related. Both try to make local adjustments to the utility estimates in order to make each state “agree” with its successors. However, TD adjusts a state to agree with its observed successor, while ADP adjusts the state to agree with all of the successors that might occur, weighted by their probabilities.

ADP can be made more efficient by approximating the algorithms for value or policy iteration. For example, the prioritized sweeping heuristic prefers adjustments to states that have undergone a large adjustment in their own utility schemes. This enables them to handle state spaces that are far too large for a full ADP. An approximation algorithm can use a minimum adjustment size that decreases as the environment model becomes more accurate, eliminating very long value iterations that occur early in learning due to large changes in the model.

Active Reinforcement Learning

A passive learning agent has a fixed policy that determines its behaviour. An active agent must learn what actions to take.

First, the agent will need to learn a complete model with outcome probabilities for all actions, rather than the model for the fixed policy. The learning mechanism for the passive ADP agent will work for this

Next, the agent has a choice of actions. The utilities it learns are defined by the optimal policy, governed by the Bellman Equations. Having obtained a utility function for the given model, the agent can extract an optimal action by one-step look-ahead to maximise the expected utility.

Potential Pitfalls

A greedy agent, that picks the best action given the learned model, very seldom converges to the optimal policy for the environment and sometimes converges to horrible policies.

This is because the learned model is not the same as the true environment. What is optimal in the learned model might not be optimal in the true environment.

An agent therefore has to make a tradeoff between exploitation to maximise its reward, and exploration to maximise its long-term well-being. The question on whether there is an optimal exploration policy is a subfield of statistical decision theory called the bandit problem.

An agent has to be greedy in the limit of infinite exploration, or GLIE. This is the scenario where the learned model is the true model. There are several GLIE schemes, one of the simplest is to have the agent choose a random action a fraction \(\frac{1}{t}\) of the time and to follow the greedy policy otherwise. This can be extremely slow to converge.

A more sensible approach is to assign some eight to actions that the agent has not tried very often,while tending to avoid actions that are believed to be of low utility. This can be achieved by altering the constraint equation to assign higher utility estimates to unexplored state-action pairs.

\begin{equation} U^+(s) \leftarrow R(s) + \gamma max_{a} f\left( \sum_{s’} P(s’ | s, a) U^+(s), N(s, a) \right) \end{equation}

\(f(u, n)\) is called the exploration function. It determines how greed is traded off against curiosity. The function should be increasing in \(u\) and decreasing in \(n\).

Learning an action-utility function

An active TD agent is no longer equipped with a fixed policy, so if it learns a utility function \(U\), it will need to learn a model in order to be able choose an action based on \(U\) via one-step look-ahead. The update rule for TD remains unchanged. IT can be shown that the TD algorithm will converge to the same values as ADP as the number of training sequences tends to infinity.

RL Applications

Personalizer
ICML 2017 Tutorial on Real World Interactive Learning
Robotics
Deep Reinforcement Learning, Decision Making, and Control
Tutoring Systems
Reinforcement Learning with People - NIPS 2017 - YouTube

Resources

<biblio/rl.bib>

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