constituents

\(m\) data points \(\qty(x_i,y_{i})\)

requirements

we desire \(c_{j}\) such that:

\begin{equation} y = c_1 + c_{2} x + c_3 x^{2} + \dots \end{equation}

Given our set of basis functions \(\phi_{j}(x)\) for input \(x\), our goal is:

\begin{equation} y = c_1 \phi_{1} + c_2 \phi_{2} + \dots + c_{n}\phi_{n} \end{equation}

the \(\phi\) are the model function which determines our neural networks.

additional information

Monomial basis and vandermonde Matrix

to do this, we put stuff in matrix form following forms, called the matrix of monomial basis:

Previous monte-carlo tree search methods which are not competitive to PBVI, SARSOP, etc., but those are affected by close-up history.

key point: monte-cargo roll outs best-first tree search + unweighted particle filter (instead of categorical beliefs)

Background

• History: a trajectory of some \(h = \{a_1, o_1, …\}\)
• generative model: we perform a random sample of possible next state (weighted by the action you took, meaning an instantiation of \(s’ \sim T(\cdot | s,a)\)) and reward \(R(s,a)\) from current state
• Rollout: keep sampling at each point, rolling out and calculating future reward

monte-carlo tree search

1. loop:
   1. sample \(s\) from the belief distribution \(B(h)\) for each node and call that the node state
   2. loop until we reach a leaf:
      1. sample exploratino using UCB 1 via the belief
      2. get observation, reward, next state
   3. add leaf node, add node for each available action
   4. Rollout
   5. backpropegate the obtained value with discounts backwards via POMDP Bellman Backup

During runtime, we choose the action with the best action, prune the tree given what you observed, and do this again in a different.

POMDPs with continuous actions are hard. So POMCP or (belief update + MCTS).

So instead, let’s try improving that. Unlike just POMCP, not only do we have \(B(h)\), we also have \(W(h)\), which is the weight of a specific state sampled. Naively applying POMCP on continuous states will give a wide-ass tree because each sampled state will not be the same as before.

double progressive widening

We want to use sampling to sample from observation. This will eventually lead to a suboptimal QMDP policy—this is because there are no state uncertainty?

Upper bounds of alpha vectors

QMDP and FIB represents an upper bound of the true optimal alpha vector values.

• QMDP
• Fast Informed Bound

FIB is a generally lower bound than QMDP.

Lower bounds of alpha vectors

BAWS and blind lower bound represents

Faster:

• best-action worst-state
• blind lower bound

Slower:

• Point-Based Value Iteration
  • “Perseus”: Randomized PBVI
• HSVI
• SARSOP

point selection

see point selection

What if our initial state never change or is deterministically changing? For instance, say, for localization. This should make solving a POMDP easier.

POMDP-lite

• \(X\) fully observable states
• \(\theta\) hidden parameter: finite amount of values \(\theta_{1 \dots N}\)
• where \(S = X \times \theta\)

we then assume conditional independence between \(x\) and \(\theta\). So: \(T = P(x’|\theta, x, a)\), where \(P(\theta’|\theta,x,a) = 1\) (“our hidden parameter is known or deterministically changing”)