Datasets:

sci-modeling-bench
/

design-bench

Reinforcement learning studies an agent that repeatedly observes an environment, selects actions through a policy, and receives rewards. In an episodic problem, a trajectory ends after a finite number of transitions, and its return is the cumulative reward collected during that episode. A policy's expected return is an expectation over all sources of randomness, not the outcome of a single episode.

Scope

Covered

Agent–environment interaction, observations, actions, rewards, and policies.
Trajectories, episodes, undiscounted and discounted return.
Policy value and expected episodic performance.
Environment termination and external time-limit truncation.

Not covered

A particular controller architecture or parameter serialization.
A particular environment's reward constants or episode limit.
A training recipe, model-selection procedure, or candidate-ranking method.

Key concepts and notation

Term or symbol	Definition
(S_t)	Environment state at time (t)
(O_t)	Observation available to the agent at time (t)
(A_t)	Action selected by the agent
(R_{t+1})	Reward produced after action (A_t)
(\pi(a\mid o))	Policy distribution over actions conditional on an observation
(\tau)	Trajectory of observations, actions, and rewards
(T)	Final transition index of an episode
(G_t)	Return accumulated after time (t)
(\gamma)	Discount factor, conventionally in ([0,1])

Core knowledge

Agent–environment interaction

At each time step, an agent receives an observation, uses a policy to select an action, and the environment produces a reward and a new state [1]. The environment includes everything outside the policy's immediate action selection, including physical dynamics, contacts, process noise, and the reward rule.

The environment state and the agent's observation need not be identical. An observation can omit variables, transform them, or contain noisy measurements. A policy defined on observations therefore does not necessarily have direct access to the complete physical state.

Policies can be deterministic or stochastic

A deterministic policy maps an observation to one action. A stochastic policy defines a conditional distribution:

[ A_t \sim \pi(\cdot\mid O_t). ]

Even in deterministic dynamics, a stochastic policy can produce different trajectories from the same initial state. Conversely, a deterministic policy can have variable outcomes when initial conditions, disturbances, or the environment are random.

Episodic trajectories and return

An episodic task divides interaction into finite trajectories. For an undiscounted episode, the return after time (t) is

[ G_t=\sum_{k=t}^{T-1}R_{k+1}. ]

For discounted return,

[ G_t=\sum_{k=t}^{T-1}\gamma^{k-t}R_{k+1}. ]

Discounting changes the relative contribution of rewards at different times. It is not implied merely by reporting an episodic return; the return convention must be stated [1].

Expected policy performance

The value of a policy from an initial-state distribution is an expectation:

[ J(\pi)=\mathbb{E}_{\tau\sim\pi}[G_0]. ]

The expectation includes randomness in initialization, transitions, policy actions, and any other stochastic component. One observed return is a sample from this distribution. Repeated episodes estimate (J(\pi)), but the estimate remains uncertain for any finite sample.

Termination and truncation

Gymnasium distinguishes termination from truncation [2]. Termination indicates that an environment-defined terminal condition has been reached. Truncation indicates that an external boundary, commonly a time limit, ended the trajectory even though the underlying process need not be terminal.

This distinction matters scientifically: a short terminated episode may represent physical failure, while a full-length truncated episode may represent survival until the experimental horizon. The numerical return alone does not always identify which occurred.

Conditions, limitations, and uncertainty

Return is meaningful only together with the reward definition and episode convention.
High return does not identify which physical behavior or reward component caused it.
Equal expected returns can arise from different return distributions and different failure modes.
A finite episode limit can change both the observed return and the meaning of successful survival.
An observation can omit state variables needed for fully Markovian control.

Related knowledge resources

continuous_control_and_gaussian_policies: stochastic policies for continuous actions.
stochastic_rollout_evaluation_and_uncertainty: repeated evaluation of policy returns.
proximal_policy_optimization_and_policy_checkpoints: one method for learning policies.

References

Sutton RS, Barto AG. Reinforcement Learning: An Introduction. 2nd ed. MIT Press; 2018. http://incompleteideas.net/book/the-book-2nd.html. [Textbook]
Farama Foundation. Gymnasium core API: Env.step termination and truncation semantics. https://gymnasium.farama.org/api/env/. Accessed 2026-07-23. [Official documentation]