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The reward function formally describes the feedback an agent receives from the environment, often denoted as $$R$$. Specifically, $$r(s, a, s')$$ represents the reward for taking action $$a$$ in state $$s$$ and transitioning to the next state $$s'$$. For a sequence of state-action pairs, the reward at a specific time step $$t$$ is written as $$r_t = r(s_t, a_t, s_{t+1})$$. In deterministic decision-making processes, where the next state $$s_{t+1}$$ is entirely determined by the current state $$s_t$$ and action $$a_t$$, the notation simplifies to $$r(s_t, a_t)$$.

Reward Function in Reinforcement Learning

An agent is being trained to navigate a maze and reach a specific goal location. After extensive training, the agent is observed to be taking an unnecessarily long, winding path to the goal, often revisiting the same locations multiple times before finally reaching the destination. Analyze the agent's reward structure provided below and explain the logical flaw that is most likely causing this inefficient behavior.

Diagnosing Flawed Agent Behavior

A reinforcement learning agent controls a robot vacuum cleaner. The primary goal is for the robot to collect all pieces of trash in a room as quickly as possible. Which of the following reward function designs would be most effective at encouraging the desired behavior without leading to unintended negative consequences?

Consider an agent navigating a 10x10 grid. The agent's goal is to move from a starting square to a designated goal square. Certain squares are marked as 'hazards'. The agent receives a reward after each move. Your task is to design a reward function that encourages the agent to reach the goal efficiently while avoiding the hazards. Describe the reward values you would assign for the following three outcomes:
1. Reaching the goal square.
2. Entering a hazard square.
3. Making any other move (i.e., moving to a normal, non-goal, non-hazard square).

Provide a specific numerical value for each outcome and briefly justify your choices.

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