Sure, here’s a concise definition and formulation of online reinforcement learning (online RL), with context using \(D\) as the current data batch or sample.

Online Reinforcement Learning: Definition and Formulation

Definition:

Online reinforcement learning refers to the paradigm where an agent interacts with an environment in a sequential, step-by-step fashion, updating its policy continually based on each newly observed data point or experience, rather than learning from a fixed, static dataset. The policy and/or value function is adapted immediately or frequently as new data \(D\) arrives.

Formulation:

Let the environment be modeled as a Markov Decision Process (MDP) defined by the tuple \((\mathcal{S}, \mathcal{A}, P, r, \gamma)\), where:

• \(\mathcal{S}\): state space
• \(\mathcal{A}\): action space
• \(P(s'|s, a)\): transition probability
• \(r(s, a)\): reward function
• \(\gamma\): discount factor

At each timestep \(t\):

1. The agent observes the current state \(s_t\).
2. It receives the current data \(D_t\) from the environment (e.g., observation, transition tuple).
3. It selects an action \(a_t \sim \pi_\theta(a|s_t)\), where \(\pi_\theta\) is the policy parameterized by \(\theta\).
4. The environment returns reward \(r_t\) and the next state \(s_{t+1}\).
5. The tuple \(D_t = (s_t, a_t, r_t, s_{t+1})\) is immediately used to update the policy (and/or value function).

Online RL policy update:

Where:

• \(J(\theta; D_t)\): Policy objective estimated using the current data \(D_t\)
• \(\alpha\): learning rate

For example, with policy gradient (REINFORCE):

where \(\hat{A}_t\) is an advantage estimate from the current data \(D_t\).

Contextual Example (with \(D\) as current data)

• At time \(t\), the agent interacts with the environment and observes new data \(D_t\).
• The agent immediately updates its policy parameters using \(D_t\).
• The updated policy is used in the next timestep, making the learning process truly online, as opposed to offline RL where all data is collected first and then learning occurs.

Key Point:
In online RL, the policy is updated sequentially as each new data point \(D_t\) arrives, and each update can influence the agent's immediate future behavior.

If you want a pseudocode format for this context, just let me know!

Enviroment unchanged

Yes, in the context of online reinforcement learning (RL), the environment is typically assumed to be relatively unchanged or stationary over short periods. This means that while the agent interacts with the environment and learns from its experiences (data \(D_t\)), the fundamental dynamics of the environment—such as transition probabilities \(P\) and reward structure \(r\)—are assumed to remain consistent during the learning process.

Here’s why this assumption is crucial:

1. Stationarity Assumption: Online RL assumes that the environment's basic characteristics (state transitions, rewards) do not change drastically or unpredictably from one timestep to the next. This allows the agent to use recent data \(D_t\) to update its policy effectively without frequent recalibration due to environmental shifts.

2. Learning Efficiency: By assuming stationarity, the agent can focus on learning optimal policies given the current environment state, rather than constantly adapting to new, unpredictable conditions. This improves learning efficiency and stability.

3. Real-World Applications: Many real-world applications of online RL, such as autonomous driving or financial trading, operate under the assumption that the underlying environment changes slowly enough to justify the stationarity assumption over short timeframes.

However, it's important to note that while the environment is assumed to be stationary in online RL, the agent's policy can adapt and evolve based on new data \(D_t\), enabling it to respond effectively to long-term changes or trends that may emerge over time.

If you have more specific questions or need further clarification on how online RL applies to your context, feel free to ask!

修改理論

Your understanding is absolutely correct—the way this article currently defines the environment and policy introduces some fundamental theoretical problems for RL and FL (Federated Learning) theory. Let’s clarify the core issues and propose a more theoretically sound design, specifically for the RL component in an online federated setting like ATFL.

Key Theoretical Issues

1. Environment Should Be Stationary

• Current Issue: In the current framework, the encoder (which changes during training) is used as part of the environment for the RL agent.
• Why Problematic: In RL, the environment is assumed to be stationary during policy learning (at least within each episode/round). If you let the encoder (part of the agent/model) change and treat it as the environment, you violate this assumption.
• What to Fix: The environment must be the external data-generating process (e.g., the real-world trajectories and their distributions), not any part of the model that is updated during learning.

2. Policy Definition Should Be Explicit and Decoupled

• Current Issue: The article’s framework does not clearly separate the policy (i.e., the RL agent that selects the tile size) from the encoders/decoders (which are part of the learning system). Also, the concept of “policy” is not visible to the client.

• What to Fix:

  • Policy: The RL policy should only refer to the logic (parameters) that chooses actions (tile sizes), not the encoders/decoders themselves.
  • Client: The client should receive the current policy (for tile size selection) from the server and keep it fixed within an hour, updating only at the start of the next hour.

3. Aggregation Logic Should Be on the Policy, Not Encoders

• Current Issue: Aggregating encoder parameters across clients mixes up the definition of “policy” and “environment,” and undermines any convergence guarantee you want for RL.

• What to Fix:

  • The aggregation (at the server) should be for the policy parameters (or Q-values/actor-critic heads), not the encoder weights.

4. Theoretical Guarantee Is Tied to This Separation

• If the encoder is changing (and part of the “environment”), you cannot guarantee the convergence of PPO or any RL algorithm, because the Markov property is broken.

• You should have:

  • A stationary environment (the city, its trajectories, and their statistics).
  • A client policy that is updated only at the end of each period (e.g., each hour), not within.

How to Redesign the Framework

1. Proper Environment Definition

• Environment: Defined as the real-world data distribution and trajectory generation process for each client (city).
• Stationary: The environment is assumed fixed within each RL episode/hour.

2. Policy Handling

• Policy: The RL agent's policy \(\pi_\theta\) selects tile sizes based on observed data statistics (the “state” is a summary of recent data, e.g., tile embedding variance, not model parameters).

• Where:

  • The policy is maintained and updated on the server.
  • The client receives a tile size decision at the start of each hour and keeps it unchanged during the hour (no policy change within the hour).

• Policy Update Timing: At the end of each hour (or RL episode), server gathers state transitions, rewards, and updates \(\pi_\theta\) (not encoders).

3. Encoder Handling

• Encoders: Only updated by standard local SGD during the hour, as in vanilla FL. Never aggregated as “policies.”
• Aggregation: Server may aggregate encoder parameters for federated averaging, but this is distinct from the RL policy.

4. Client-Server Interaction

• Client: Receives tile size (from server policy) at start of hour, uses it for all encoding in that hour, and trains its own model locally.
• Server: Aggregates policy transitions (state, action, reward, next state), updates the policy \(\pi_\theta\), and may also aggregate encoder parameters via FedAvg or personalized FL.

Can You Treat the Client Policy as the Encoder?

No—you cannot treat the encoder as the policy.

• The encoder is part of the model (agent) that processes data, not the logic that selects actions in the RL sense.
• The policy in this context is specifically the decision function that selects tile sizes based on data statistics.

Should the server have a policy?

• Yes, the server maintains the global policy (the PPO agent or equivalent), because only the server sees transitions from all clients and can optimize the global policy.
• Clients do not need to maintain policies locally; they simply apply the server’s selected action (tile size) for each round/hour.

Summary Table: Correct Design

Sample Reworded Paragraph (for Paper Revision)

In our revised framework, we define the environment as the underlying trajectory data distribution for each client (city), which remains stationary during each federated round (hour). The policy is implemented as a global RL agent (maintained by the server) that selects optimal tile sizes for each client based on aggregated client statistics. Clients receive the selected tile size at the start of each hour and do not alter it during local updates. Model encoders and decoders are updated locally via self-supervised learning and are periodically aggregated by the server, distinct from the RL policy. This separation ensures theoretical soundness and enables convergence guarantees for the RL-based adaptation of spatial representation.

If you want, I can help you rephrase your methods section or provide more technical diagrams or pseudocode to clarify this logic for a reviewer. Just let me know!