# Consistent Value Function Definitions

Mathematically defining value function and policies, as well as some of the emergent properties of these definitions. In this post, I will define the standard γ-discounted value functions used in reinforcement learning.

In this post, I will define the standard γ-discounted value functions used in reinforcement learning. From these definitions, I will discuss two important emergent properties of value functions that prove the self-consistency of the definitions. I will build up these concepts mathematically, focusing on writing out every step in the derivations and discussing the implications of each step. These equations are the foundation of many important mathematical proofs in RL and understanding them completely is important to building a theoretical understanding of RL.

## Value Functions

Value functions are at the core of reinforcement learning. For any given state, an agent can query a value function to determine the “value” associated with being in that state. We traditionally define “value” as being the sum of rewards obtained into the future. Because of its dependence on what rewards the agent will see in the future, a value function must be defined for a given strategy of behavior; a policy. That is, the value of a state depends on how the agent behaves after visiting that state; a discussion of “value” that is independent of behavior is meaningless.

We denote a policy as a function which maps a state to a probability distribution over actions. Formally,

where 𝒮 denotes the set of all possible states that the agent can visit (often called the “state space”), 𝒜 denotes the set of all possible actions (often called the “action space”), and Δ(𝒜) denotes the standard simplex over the set of actions. The standard simplex is simply a formal way of writing a probability distribution over the action space. Simply put, a policy takes a state and returns a weighting over which actions the agent should take in that state. A large weighting leads to a frequency of selecting that action, a small weighting leads towards a low frequency.

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