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Is it correct that for SARSA to converge to the optimal value function (and policy)

  1. The learning rate parameter $\alpha$ must satisfy the conditions: $$\sum \alpha_{n^k(s,a)} =\infty \quad \text{and}\quad \sum \alpha_{n^k(s,a)}^{2} <\infty \quad \forall s \in \mathcal{S}$$ where $n_k(s,a)$ denotes the $k^\text{th}$ time $(s,a)$ is visited

  2. $\epsilon$ (of the $\epsilon$-greedy policy) must be decayed so that the policy converges to a greedy policy.

  3. Every state-action pair is visited infinitely many times.

Are any of these conditions redundant?

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    $\begingroup$ There is probably some relation between the fact that the TD(0) function converge to the true Value function with probability 1 when the learning rate parameter is decayed like above. There also exists some interval of alphas $\alpha \in (0,p)$ such that the TD estimate converges in expected value to the true value function $\endgroup$ – KaneM Feb 27 at 13:17
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The paper Convergence Results for Single-Step On-Policy Reinforcement-Learning Algorithms by Satinder Singh et al. proves that SARSA(0), in the case of a tabular representation of the value functions, converges to the optimal value function, provided certain assumptions are met

  1. Infinite visits to every state-action pair
  2. The learning policy becomes greedy in the limit

The properties are more formally stated in lemma 1 (page 7 of the pdf) and theorem 1 (page 8). The Robbins–Monro conditions should ensure that each state-action pair is visited infinitely often.

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  • $\begingroup$ I was already writing this answer before the other answer was published, but these answers are equivalent. I am only citing the paper that originally proved this. $\endgroup$ – nbro Feb 27 at 13:40
  • $\begingroup$ Thanks for the answer. Sorry about my poor formatting of posts here. Just learning about markdown. $\endgroup$ – KaneM Feb 27 at 13:45
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    $\begingroup$ @KaneM No problem! I have a look at source code of my and your answer (now after my edits) to understand better how markdown works :) Anyway, I am not completely sure that the Robbins-Monro conditions ensure that each state is visited infinitely often. Maybe it's the other way around: if each state-action pair is visited infinitely often, then the Robbins-Monro conditions are satisfied. $\endgroup$ – nbro Feb 27 at 13:48
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    $\begingroup$ @KaneM These conditions are very related. I'm linking to a paper that seems to relate the two in a more precise way, but I haven't yet fully read this paper. $\endgroup$ – nbro Feb 27 at 14:03
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    $\begingroup$ I wonder if the Robbins-Monro conditions are not present, does there exist some range of $\alpha$ such that the policy converges in some notion of expectation to the optimal policy. $\endgroup$ – KaneM Feb 27 at 14:32
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I have the conditions for convergence in these notes SARSA convergence by Nahum Shimkin.

  1. The Robbins-Monro conditions above hold for $α_t$.

  2. Every state-action pair is visited infinitely often

  3. The policy is greedy with respect to the policy derived from $Q$ in the limit

  4. The controlled Markov chain is communicating: every state can be reached from any other with positive probability (under some policy).

  5. $\operatorname{Var}{R(s, a)} < \infty$, where $R$ is the reward function

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