Effective horizon in Reinforcement Learning

The horizon is fundamentally a property of the environment, not of the agent. Here’s why:

1. Environment-Defined Horizon

• The environment specifies whether episodes end after a fixed number of steps (finite horizon) or continue indefinitely (infinite horizon).

• For example:

  • A chess game has a natural finite horizon (the game ends in a finite number of moves).
  • CartPole can run forever (until termination), so it’s modeled as an infinite-horizon MDP.

Thus, the MDP definition itself includes the horizon — either explicitly (finite episodes) or implicitly (continuing process).

2. Agent’s Perspective

• While the environment dictates the horizon, the agent’s design interacts with it:

  • In finite-horizon settings, optimal policies can be non-stationary (they depend on how many steps remain).
  • In infinite-horizon discounted settings, the optimal policy is stationary.

• The agent can choose to ignore long-term rewards (e.g., by setting a smaller discount factor), but that doesn’t change the environment’s horizon — it just changes how the agent values rewards.

3. Example

• Environment property: An episode ends after 200 steps → horizon = 200.
• Agent choice: Uses $\gamma = 0.9$, so it effectively only cares about the next ~10 steps (effective horizon).

So:

• True horizon = defined by the environment (termination structure).
• Effective horizon = shaped by the agent (via discounting or modeling choices).

In summary, the horizon is primarily a property of the environment, but the agent’s discount factor or planning depth determines its effective horizon of concern.

Continuing on examples, let’s think about CartPole which is a continuing control task but is usually implemented with a max episode length (finite horizon). Let’s break down how different values of $\gamma$ affect the learning dynamics:

1. Low Discount Factor ($\gamma \ll 1$, e.g., 0.5–0.8)

• Effective horizon is short ($\frac{1}{1-\gamma}$ ≈ 2–5 steps).
• The agent strongly prioritizes immediate reward.

• In CartPole, this means:

  • The agent learns quick, reflex-like corrections to prevent imminent pole falls.
  • It often ignores strategies that require short-term sacrifice for longer stability (e.g., moving the cart away to counteract pole drift).
• Learning dynamics: Fast convergence, but to myopic policies that may fail to generalize well.

2. Moderate Discount Factor ($\gamma \approx 0.9–0.99$)

• Effective horizon is longer ($\approx 10–100$ steps).
• The agent balances immediate and long-term stability.

• In CartPole:

  • The agent learns to not just react to the pole angle but also anticipate drift, moving the cart preemptively.
  • Policies are smoother and more robust, keeping the pole balanced for much longer.
• Learning dynamics: Slightly slower, but better long-term performance and higher returns.

3. Near-Undiscounted ($\gamma \to 1.0$)

• Effective horizon approaches infinity.
• The agent tries to optimize stability over the entire episode length.

• In CartPole:

  • The optimal strategy is to keep the pole upright indefinitely.
  • However, exploration becomes harder: since rewards far in the future matter almost as much as immediate ones, the credit assignment problem becomes severe.
  • Algorithms like Q-learning may converge more slowly or become unstable.
• Learning dynamics: High variance in updates, need for more careful exploration and function approximation.

4. Interaction with Episode Termination

• CartPole usually ends at 200 or 500 steps. Even with $\gamma=1$, the true horizon is bounded by the environment.

• So:

  • With $\gamma=1$, the agent tries to maximize survival up to that cap.
  • With lower $\gamma$, it acts as though the episode is much shorter, ignoring long-term risks.

Takeaways

• $\gamma$ controls the agent’s effective foresight.
• Small $\gamma$ → greedy, reactive behavior, fast but suboptimal.
• Large $\gamma$ → farsighted, anticipatory behavior, slower but higher return.

• In CartPole, $\gamma \approx 0.95–0.99$ usually works best: it’s enough foresight to learn balancing, but not so large that learning becomes unstable.

Policy Gradient Methods and $\gamma$

Policy gradient methods optimize the expected return:

The gradient estimate (REINFORCE-style) is:

where

is the discounted return from time $t$.

Thus, $\gamma$ directly influences what signal is used to weight policy updates.

1. Low $\gamma$ (myopic, e.g., 0.7–0.9)

• Effect on returns: $G_t$ mostly reflects rewards in the next few steps.
• Impact on gradient updates: Policy updates are driven by short-term outcomes.

• Learning dynamics:

  • Faster convergence because variance in $G_t$ is smaller (fewer far-off rewards).
  • But the learned policy is often short-sighted (e.g., in CartPole: it may learn to twitch reactively rather than balancing smoothly).
• Bias/Variance trade-off: Lower variance, higher bias (since long-term credit assignment is ignored).

2. Moderate $\gamma$ (0.95–0.99, common default)

• Effect on returns: $G_t$ incorporates a meaningful portion of the episode.
• Impact on gradient updates: Policy updates balance short-term corrective actions and long-term planning.

• Learning dynamics:

  • More stable policies (e.g., in CartPole: anticipatory movements, not just reactive corrections).
  • Slightly noisier gradients than with low $\gamma$, but usually the sweet spot in practice.

3. High $\gamma$ (→ 1.0, farsighted)

• Effect on returns: Almost the entire episode influences $G_t$.
• Impact on gradient updates: Credit assignment becomes very difficult — an action at time $t$ gets credit (or blame) for rewards that might happen hundreds of steps later.

• Learning dynamics:

  • High variance in gradient estimates → slower or unstable learning.
  • Exploration plays a bigger role: bad initial actions may spoil the whole return.
  • In CartPole, if you set $\gamma=1.0$, REINFORCE gradients will be very noisy because most episodes end with zero reward until the agent stumbles into balance.

4. Connection to Variance Reduction (Baselines, Advantage Functions)

• Policy gradient methods often use an advantage estimate:

  \[A_t = G_t - V(s_t)\]

  where $V(s_t)$ is a learned baseline.

• With larger $\gamma$:

  • $G_t$ has higher variance → baselines (or GAE in PPO/A2C) become critical.
  • Generalized Advantage Estimation (GAE) introduces an extra $\lambda$ parameter to trade off bias/variance, often mitigating issues caused by high $\gamma$.

5. CartPole Example

• $\gamma=0.9$: The agent learns quick reflexes but might fail to keep the pole up for 200 steps consistently.
• $\gamma=0.99$: The agent learns smooth cart movements to stabilize the pole for long durations.
• $\gamma=1.0$: In vanilla REINFORCE, training often diverges due to huge gradient variance. With PPO+GAE, it can still work, but training is slower.

Key Insights

• $\gamma$ in policy gradient methods controls the time horizon of credit assignment.
• Low $\gamma$: lower variance, but myopic.
• High $\gamma$: farsighted, but very noisy gradients.
• In practice: $\gamma \approx 0.95–0.99$ is a good trade-off.
• Techniques like baselines, GAE, and actor-critic architectures are especially important at high $\gamma$.

What about GRPO?

GRPO is a modern variant of policy gradient methods tailored for large language models. It mirrors PPO’s structure but eliminates the value function by using group-based advantage estimates:

• You sample a group of completions (outputs) from the current policy for the same prompt.

• The advantage for each completion is normalized relative to the group’s mean and standard deviation of rewards, i.e.,

  \[A_i = \frac{r_i - \text{mean}(r)}{\text{std}(r)}\]

  ([DigitalOcean][1], [RLHF Book][2]).

• The policy update uses a clipped surrogate objective (like PPO) plus a KL divergence penalty to avoid policy drift ([DeepWiki][3], [DigitalOcean][1], [RLHF Book][2]).

Notably, GRPO replaces a learned baseline (value function) with a statistical, group-based normalization, bringing computational efficiency and stability ([Medium][4], [RLHF Book][2]).

Does $\gamma$ Play a Role in GRPO?

No—GRPO does not involve a discount factor. Unlike conventional RL methods (REINFORCE, actor-critic, PPO), GRPO treats each completion’s reward as a flat score—no discounting is applied over time or tokens.

• It computes advantages using raw, normalized rewards (e.g., completion-level rewards), with no reference to $\gamma$ ([DigitalOcean][1], [RLHF Book][2]).
• The algorithm is designed for tasks where the outcome (e.g., correctness of generated text) is evaluated globally, not stepwise—with no per-token discounted return.

Why Doesn’t GRPO Use $\gamma$?

In environments like language model fine-tuning, each action (token generation) doesn’t have an incremental reward. Instead:

• The model generates a full completion and is assessed as a whole (e.g., correctness, helpfulness).
• There’s no sequential reward structure that needs discounting—hence no need for $\gamma$.

GRPO’s normalization over groups addresses variance and baseline without needing value estimation per token.

Summary Table

TL;DR

• GRPO does not involve a discount factor $\gamma$.
• It relies on group-based normalization of final rewards and a clipped PPO-like objective with KL regularization.
• As a result, $\gamma$ has no influence on GRPO’s behavior or updates.

The broader role of $\gamma$ remains vital in sequential RL algorithms—but for GRPO’s domain (LLM fine-tuning with final-output rewards), it simply doesn’t come into play.

Where $\gamma$ belongs when applying GRPO to sequential RL

An interesting research direction is to adapt GRPO (group-relative policy optimization) from LLM-style, trajectory-level rewards to classic sequential RL (e.g., CartPole, MuJoCo, Atari).

1. design choices (where $\gamma$ enters),
2. concrete algorithmic variants (with pseudocode),
3. how $\gamma$ changes learning dynamics and what to do about it (variance, credit assignment), and
4. practical hyperparameter recommendations.

In sequential RL you have per-step rewards $r_t$. The discount factor $\gamma$ defines the discounted return used for credit assignment:

GRPO originally uses trajectory-level scalar rewards (no discounting). You must decide how to map the sequential rewards into the scalar signals that GRPO normalizes over a group. Typical choices:

A. Trajectory-level discounted return (simple): compute a single $G=\sum_{t=0}^{T}\gamma^{t}r_t$ per trajectory and treat each trajectory as one sample in a group. Group-normalize these trajectory returns.

B. Per-timestep discounted returns (fine-grained): compute $G_t$ for each timestep and perform group-normalization of advantages at the timestep-level (either across the whole minibatch or within same time index / episode length bucket).

C. Advantage + group-normalize: compute advantage $A_t = \hat{G}t - V\phi(s_t)$ (or use GAE), then group-normalize the $A_t$ values across the sampled batch.

D. Hybrid: use trajectory-level normalization for episodic return signal and per-step baselines/advantage for policy updates.

Which to choose depends on task sparsity and horizon — below I explain tradeoffs.

Algorithmic variants (pseudocode + explanations)

Variant A — Trajectory-level GRPO (straightforward)

Use when you want the closest analogue to GRPO for episodic tasks.

1. Sample a group $G$ of $N$ trajectories ${\tau_i}$ under current policy.

2. For each trajectory compute discounted return

   \[R_i = \sum_{t=0}^{T_i} \gamma^t r_{i,t}.\]

3. Compute group mean $\mu$ and std $\sigma$ of ${R_i}$. Form normalized reward:

   \[\tilde{R}_i = \frac{R_i - \mu}{\sigma + \epsilon}.\]

4. For each trajectory, compute policy surrogate loss (PPO-like):

   \[L(\theta) = \mathbb{E}_{i,t}\left[\mathrm{clip}\Big(\frac{\pi_\theta(a_{i,t}|s_{i,t})}{\pi_{\text{old}}(a_{i,t}|s_{i,t})}, 1-\epsilon,1+\epsilon\Big)\cdot \tilde{R}_i \right] - \beta\,\mathrm{KL}(\pi_{\text{old}}\|\pi_\theta).\]

   Note: use the same normalized trajectory return $\tilde{R}_i$ as the scalar weight for all timesteps in that trajectory.

5. Update policy with standard PPO/GD.

Comments: simple, low memory (only one scalar per trajectory). But using the same scalar for all timesteps reduces temporal credit resolution — appropriate when trajectory-level reward is the natural signal (sparse episodic tasks).

Variant B — Per-timestep GRPO with discounted returns

Better when fine-grained credit assignment matters.

1. Sample transitions (or full trajectories). For each transition compute $G_{i,t}$ (discounted return).
2. Compute group mean/std over the set ${G_{i,t}}$ (optionally normalize within time buckets to avoid mixing different time depths).

3. Use normalized $\tilde{G}_{i,t}$ as the weight in the surrogate loss:

   \[L(\theta)=\mathbb{E}_{i,t}\left[\mathrm{clip}\big(\rho_{i,t},1-\epsilon,1+\epsilon\big)\cdot \tilde{G}_{i,t}\right] - \beta\,\mathrm{KL}.\]

   where $\rho_{i,t}$ is the importance ratio.

Comments: higher variance (many $G_{i,t}$ samples), but better credit assignment. If you use this, pairing with a learned baseline (value) or GAE is recommended.

Variant C — Advantage GRPO (recommended for policy-gradient stability)

Combine GAE/value baseline with group normalization.

1. Collect batch of trajectories. Fit/compute value $V_\phi$.

2. Compute advantages (GAE):

   \[\hat{A}_{i,t} = \text{GAE}_\lambda(s_{i,t}).\]

3. Compute group statistics over ${\hat{A}_{i,t}}$ and normalize:

   \[\tilde{A}_{i,t} = \frac{\hat{A}_{i,t}-\mu}{\sigma+\epsilon}.\]
4. Use $\tilde{A}_{i,t}$ in PPO-style surrogate loss.

Comments: retains per-step credit assignment while inheriting GRPO’s variance-stabilization via group normalization. This is the closest to modern actor-critic practice and typically the most robust.

How does $\gamma$ affect dynamics in these variants?

General principles:

• $\gamma$ controls effective horizon and credit assignment.

  • Small $\gamma$ ⇒ short effective horizon ⇒ $G$ (or $G_t$) dominated by immediate rewards → lower variance, more bias toward myopic policies.
  • Large $\gamma$ (close to 1) ⇒ long effective horizon ⇒ returns depend on many future steps → higher variance in $G$ or $A$.

• Group-normalization reduces scale/variance across the group but does not remove long-horizon credit assignment difficulty.

  • Normalizing returns/advantages makes gradients numerically stable and comparable across trajectories, but if $\gamma$ is large and the reward is sparse, individual $G$ or $\hat{A}$ values will still have huge variability because of delayed outcomes — you still need larger sample sizes or temporal-credit solutions.

• Specific effects per variant

  • Variant A (trajectory-level): with large $\gamma$ you emphasize long-term survival; group-normalization helps training stability but you lose which actions in the episode caused success/failure. Long gamma + trajectory-level scalar → you may need longer trajectories and bigger group sizes to get reliable signal.
  • Variant B (per-timestep): large $\gamma$ dramatically increases variance of $G_{t}$; normalization helps, but you’ll see noisy policy updates unless you also use baselines or GAE.
  • Variant C (advantage+GAE): you mitigate variance via baseline and the $\lambda$ parameter. For large $\gamma$, set $\lambda$ < 1 to reduce variance (GAE is essentially another bias/variance knob complementary to $\gamma$).

• Exploration / optimization

  • Larger $\gamma$ makes early exploratory actions more consequential — may require stronger exploration (entropy, larger policy noise) and larger groups/batches so that the group normalization reliably captures performance differences.
  • Smaller $\gamma$ reduces the dependency on long strategic behavior, making exploration less costly but risking suboptimal long-run behavior.

Thought Experiments

1. Start with conventional RL defaults: for many control tasks $\gamma \in [0.98,0.995]$ is common; for episodic tasks with short horizons $\gamma=0.99$ is a good starting point. For CartPole, $\gamma=0.99$ is typical.

2. Prefer Variant C (Advantage + GAE + group-normalize) as your first implementation. It gives the best bias/variance control.

3. If rewards are sparse or horizon is long (high $\gamma$):

   • Increase batch / group size (GRPO benefits from larger groups to estimate group mean/std reliably).
   • Use GAE with $\lambda \in [0.9,0.97]$ (lower $\lambda$ to reduce variance if needed).
   • Consider normalizing advantages per-trajectory length buckets (to avoid mixing early- and late-timestep advantages).
   • Use stronger entropy regularization or explicit exploration schedules.

4. If rewards are dense or you want fast convergence:

   • You can reduce $\gamma$ slightly (0.95–0.99), which reduces variance and speeds learning — but check asymptotic performance.

5. Group size & normalization

   • Small groups (e.g., groups of 4–8 trajectories) work for LLMs but in sequential RL prefer groups = minibatch of many transitions or several full episodes (e.g., 32–256 trajectories/transitions) so the mean/std estimates are stable.
   • Add clipping / robust statistics (e.g., clip normalized values to a reasonable range) to avoid a few outliers dominating.

6. Time-dependent policies / finite-horizon tasks

   • If the environment is finite-horizon and optimal policy is time-dependent, include the time index in the policy input or maintain a time-conditioned policy.

Practical pseudocode for the recommended approach (Variant C)

for iteration=1..N:
    collect K trajectories (or M transitions) using π_old
    compute value estimates Vφ(s) (fit value network if needed)
    compute GAE advantages A_hat_{i,t} with gamma, lambda
    compute group mean μ and std σ over {A_hat_{i,t}} in batch
    normalized_A = (A_hat - μ) / (σ + eps)
    compute PPO clipped surrogate loss using normalized_A as advantage weights
    add KL penalty / entropy regularizer
    update policy θ (and value φ)
    optionally anneal gamma?  (usually do not)

Notes:

• Do not change $\gamma$ on-policy within the same batch — choose $\gamma$ at experiment start.
• You can run ablations where you sweep $\gamma$ ∈ {0.95, 0.98, 0.99, 0.995, 1.0} and also sweep group size and $\lambda$.

Some open questions

• Normalization bias: group normalization transforms the objective: when you normalize advantages, you’re changing the relative scale and possibly introducing bias in gradient direction (it’s empirical but often helpful). Be mindful when comparing to baselines.
• Non-i.i.d. returns across trajectories: if trajectories are correlated (e.g., same start state), group statistics may underestimate variance — consider stratified grouping.
• Gamma vs episode cap: if environment has a hard episode cap $T$, $\gamma$ interacts with that cap; effective horizon ≈ min(T, 1/(1-γ)). When T is small relative to 1/(1-γ), raising gamma past that has diminishing returns.
• Theoretic sample complexity: increasing $\gamma$ generally worsens sample complexity due to larger variance; group-normalization reduces variance but does not change the fundamental dependence on horizon.

Quick actionable checklist to implement & tune

1. Implement GRPO with advantage normalization (Variant C). Use GAE.
2. Choose initial $\gamma=0.99$ (CartPole/MuJoCo) or $\gamma=0.998$ (very long-horizon control) — pick based on environment horizon.
3. Group/minibatch size: start with 64–256 transitions; increase if normalized stats are noisy.
4. If training is unstable with high $\gamma$: reduce $\lambda$ in GAE, or reduce $\gamma$ slightly, increase group size, add value function regularization.
5. Report ablations: (γ, λ, group size, clipping ε, KL β). Track both learning speed and final performance.

Final takeaway (one-paragraph)

You must retain $\gamma$ as the discounted-return horizon when moving GRPO into sequential RL. Use GRPO’s group-normalization as a variance-stabilizing wrapper around standard advantage estimation (GAE + baseline). For stability and good credit assignment, prefer per-timestep advantages (Variant C) normalized across the group — tune $\gamma$ to reflect the environment’s true horizon (typical defaults 0.98–0.995 for continuous control), and counteract the higher variance induced by large $\gamma$ with larger groups, GAE $\lambda<1$, and stronger baselines/regularizers.