Answers.Body: <p>The original arXiv DQN <a href="https://arxiv.org/abs/1312.5602" rel="nofollow noreferrer">paper</a> used the Mean-Squared Error (MSE) loss function. The loss of a sample experience tuple <span class="math-container">$(s, a, r, s')$</span> is as follows:</p>
  <p><span class="math-container">$$ \left([r + \gamma\max_{a'}Q(s', a'; \theta_{i-1})] - Q(s, a;\theta_i)\right)^2.$$</span></p>
  <p>For completeness sake, the target values (expression in brackets) are generated with the previous weights of the network <span class="math-container">$\theta_{i-1}$</span> when determining the current weights <span class="math-container">$\theta_i$</span> via gradient descent.</p>
  <p>In contrast to using MSE as a loss function, the Nature DQN <a href="https://www.nature.com/articles/nature14236" rel="nofollow noreferrer">paper</a> instead clipped the error (expression inside outer parentheses) to be between <span class="math-container">$-1$</span> and <span class="math-container">$1$</span> (see Methods, Training algorithm for deep Q-networks, last paragraph). The authors state that clipping the error corresponds to using an absolute value loss function outside of the interval <span class="math-container">$(-1, 1)$</span> during gradient descent. In practice, this generally produces smaller gradients than those of MSE loss, and the authors state that this form of error clipping <strong>improved the stability</strong> of the DQN algorithm. Some deep learning libraries implement this loss function, or a similar variant, as the <strong>Huber loss function</strong>.  Also for completeness sake, the Nature paper replaced parameters <span class="math-container">$\theta_{i-1}$</span> with those of a separate target network, denoted as <span class="math-container">$\theta_i^{-}$</span>, for further stability of the algorithm.</p>
  <p>Without more details, it's difficult to determine if the loss function is the culprit of poor performance for your problem, but this answer is at least a starting point regarding better loss functions in practice.</p>
  
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Answers.Text: you could also mention sparsity of rewards being an issue if OP is using them
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Answers.Text: Good point. It's kind of hard to know the extent of OP's issues with the problem. At least my answer provides guidance for the question in the title and body.
  
  Just so I'm on the same page as you @DavidIreland, how would you incorporate sparse rewards as an issue into this answer? Does it relate to the loss functions, or is it a more general issue that I should discuss?
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Answers.Text: When I read that OP’s problem was a maze I assumed that it would be some kind of sparse reward (0 every step, 1 to reach the goal, for instance). In this case I don’t think it’s a problem with the loss function, so just something you could point out as to why it might not be working, though I think in terms of answering the actual question related to the loss you’ve provided a great answer.
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Body: <p>I am a newbie in reinforcement learning, and I was doing a project on solving an agent maze solver using deep Q Learning.  Currently, I am using the MSE loss function, but the agent is very slow or not even reaching the target.  Is there any better loss function to improve the agent performance?</p>
  
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Comments.Text: We would really need to know more about your problem and setup to answer this more specifically. You didn't tell us   e.g. for how the agent is being trained, what the reward function is, etc.
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Tags: <reinforcement-learning><deep-learning><q-learning><dqn><intelligent-agent>
Title: Are there better loss functions than MSE for maze solver using deep learning?
UserDisplayName: Lim
ThreadText: # Are there better loss functions than MSE for maze solver using deep learning?
  
  (Asked by: username_0 on 2022-11-14)
  
  I am a newbie in reinforcement learning, and I was doing a project on solving an agent maze solver using deep Q Learning. Currently, I am using the MSE loss function, but the agent is very slow or not even reaching the target. Is there any better loss function to improve the agent performance?
  
  **Tags:** reinforcement-learning, deep-learning, q-learning, dqn, intelligent-agent
  
  > **Comments:**
  > **username_1**: We would really need to know more about your problem and setup to answer this more specifically. You didn't tell us   e.g. for how the agent is being trained, what the reward function is, etc.  
  
  ---
  
  ## 1 answers
  
  ### Answer by username_2 on 2022-11-14. Score: 2
  
  The original arXiv DQN [paper](https://arxiv.org/abs/1312.5602) used the Mean-Squared Error (MSE) loss function. The loss of a sample experience tuple $(s, a, r, s')$ is as follows:
  
  $$ \left([r + \gamma\max\_{a'}Q(s', a'; \theta\_{i-1})] - Q(s, a;\theta\_i)\right)^2.$$
  
  For completeness sake, the target values (expression in brackets) are generated with the previous weights of the network $\theta\_{i-1}$ when determining the current weights $\theta\_i$ via gradient descent.
  
  In contrast to using MSE as a loss function, the Nature DQN [paper](https://www.nature.com/articles/nature14236) instead clipped the error (expression inside outer parentheses) to be between $-1$ and $1$ (see Methods, Training algorithm for deep Q-networks, last paragraph). The authors state that clipping the error corresponds to using an absolute value loss function outside of the interval $(-1, 1)$ during gradient descent. In practice, this generally produces smaller gradients than those of MSE loss, and the authors state that this form of error clipping **improved the stability** of the DQN algorithm. Some deep learning libraries implement this loss function, or a similar variant, as the **Huber loss function**. Also for completeness sake, the Nature paper replaced parameters $\theta\_{i-1}$ with those of a separate target network, denoted as $\theta\_i^{-}$, for further stability of the algorithm.
  
  Without more details, it's difficult to determine if the loss function is the culprit of poor performance for your problem, but this answer is at least a starting point regarding better loss functions in practice.
  
  > **Comments:**
  > **username_3**: you could also mention sparsity of rewards being an issue if OP is using them  
  > **username_2**: Good point. It's kind of hard to know the extent of OP's issues with the problem. At least my answer provides guidance for the question in the title and body.  Just so I'm on the same page as you @DavidIreland, how would you incorporate sparse rewards as an issue into this answer? Does it relate to the loss functions, or is it a more general issue that I should discuss?  
  > **username_3**: When I read that OP’s problem was a maze I assumed that it would be some kind of sparse reward (0 every step, 1 to reach the goal, for instance). In this case I don’t think it’s a problem with the loss function, so just something you could point out as to why it might not be working, though I think in terms of answering the actual question related to the loss you’ve provided a great answer.  
  
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