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In reinforcement learning, where the state space is discrete and relatively small, a form of learning algorithm commonly used is the Q learning. This involves a look up table $Q(s,a)$ (Or you think of this as a matrix with the states on the row, and actions on the column). The entries are updated as the agent continues to learn.

In the case of continuous state space, or a overly large state space, a look up table is not feasible anymore. And we turn to Q values with linear function approximation. In this case, Q values are now approximated by a linear combination of features.

I was watching a lecture by David Silver on Reinforcement Learning, he mentioned that the look up table is just a special case of the linear function approximation. (The slide I am referring to begins at 28:43.) This never occurred to me, and he showed a little 'proof' that was not so clear to me. Anyone who could give some insights into the matter?

Originally, I just accepted (without proof) that look up table and linear function approximation are just two independent things. It never occurred to me that the two are related.

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In linear approximation, we approximate the value of a state as a linear combination of some feature vector and a vector of weights, i.e. $\hat v(s, a) = \textbf{s}\cdot\textbf{w}$ for a feature vector $\textbf{s}$ and weights $\textbf{w}$

What Silver's getting at in this is that, in the discrete case, we can construct a feature vector $\textbf{s}$ as a list of dummy variables, each an indicator corresponding to a discrete state. This feature vector $\textbf{s}$ will have a zero for all entries except one, which will be one. Each $w_i \in \bf{w}$ will then correspond to a single state $s_i$, and $\hat v$ will just return that weight as the value of $s_i$.

Under this representation, you can view a lookup table value function as a special case of a linear approximation.

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