Как самому зарегистрировать SIM-карту на Госуслугах? - коротко
Чтобы зарегистрировать SIM-карту через Госуслуги, авторизуйтесь в личном кабинете, выберите услугу регистрации номера и следуйте инструкциям. Потребуется подтверждение личности и ввод данных SIM-карты.
Как самому зарегистрировать SIM-карту на Госуслугах? - развернуто
Ре# Value Functions
Value functions are functions of states (or state-action pairs) that estimate how good it is for the agent to be in a given state (or how good it is to perform a given action in a given state). The notion of "how good" is defined in terms of future rewards that can be expected, or, to be precise, in terms of expected return.
There are two main types of value functions in reinforcement learning: state-value functions and action-value functions.
State-Value Function
The state-value function for a policy $\pi$, denoted $V^{\pi}(s)$, is the expected return when starting from state $s$ and following policy $\pi$ thereafter.
Mathematically, it is defined as:
$$ V^{\pi}(s) = \mathbb{E}_{\pi}\left[G_t \mid St = s\right] = \mathbb{E}{\pi}\left[\sum{k=0}^{\infty} \gamma^k R{t+k+1} \mid S_t = s\right] $$
where $\mathbb{E}_{\pi}[\cdot]$ denotes the expected value given that the agent follows policy $\pi$, and $\gamma$ is the discount factor.
Action-Value Function
The action-value function for a policy $\pi$, denoted $Q^{\pi}(s, a)$, is the expected return when starting from state $s$, taking action $a$, and following policy $\pi$ thereafter.
Mathematically, it is defined as:
$$ Q^{\pi}(s, a) = \mathbb{E}_{\pi}\left[G_t \mid S_t = s, At = a\right] = \mathbb{E}{\pi}\left[\sum{k=0}^{\infty} \gamma^k R{t+k+1} \mid S_t = s, A_t = a\right] $$
Optimal Value Functions
The optimal state-value function, denoted $V^*(s)$, is the maximum value function over all policies:
$$ V^*(s) = \max_{\pi} V^{\pi}(s) $$
The optimal action-value function, denoted $Q^*(s, a)$, is the maximum action-value function over all policies:
$$ Q^*(s, a) = \max_{\pi} Q^{\pi}(s, a) $$
For any Markov decision process (MDP), there exists at least one optimal policy $\pi^*$ that achieves the optimal value functions.
Bellman Equations
Value functions satisfy recursive relationships known as the Bellman equations.
The Bellman equation for $V^{\pi}$ is:
$$ V^{\pi}(s) = \sum{a} \pi(a \mid s) \sum{s', r} p(s', r \mid s, a) \left[r + \gamma V^{\pi}(s')\right] $$
The Bellman equation for $Q^{\pi}$ is:
$$ Q^{\pi}(s, a) = \sum{s', r} p(s', r \mid s, a) \left[r + \gamma \sum{a'} \pi(a' \mid s') Q^{\pi}(s', a')\right] $$
The Bellman optimality equation for $V^*$ is:
$$ V^(s) = \max{a} \sum{s', r} p(s', r \mid s, a) \left[r + \gamma V^(s')\right] $$
The Bellman optimality equation for $Q^*$ is:
$$ Q^(s, a) = \sum{s', r} p(s', r \mid s, a) \left[r + \gamma \max{a'} Q^(s', a')\right] $$
Relationship Between Value Functions
The value functions are related as follows:
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The state-value function can be expressed in terms of the action-value function:
$$ V^{\pi}(s) = \sum_{a} \pi(a \mid s) Q^{\pi}(s, a) $$
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The action-value function can be expressed in terms of the state-value function:
$$ Q^{\pi}(s, a) = \sum_{s', r} p(s', r \mid s, a) \left[r + \gamma V^{\pi}(s')\right] $$
Estimating Value Functions
In practice, value functions are often estimated using methods such as:
- Dynamic programming (for known MDPs)
- Monte Carlo methods (for episodic tasks)
- Temporal difference learning (for both episodic and continuing tasks)
These methods are used to approximate the value functions when exact computation is infeasible due to the size of the state or action spaces.