6.9. decisions

Some notes on decision theory based on Berkeley’s CS 188 course and “Artificial Intelligence” Russel & Norvig 3rd Edition

6.9.1. game trees - R&N 5.2-5.5

  • like search (adversarial search)

  • minimax algorithm

    • ply - half a move in a tree

    • for multiplayer, the backed-up value of a node n is the vector of the successor state with the highest value for the player choosing at n

    • time complexity - \(O(b^m)\)

    • space complexity - \(O(bm)\) or even \(O(m)\)

  • alpha-beta pruning cuts in half the exponential depth

    • once we have found out enough about n, we can prune it

    • depends on move-ordering

      • might want to explore best moves = killer moves first

    • transposition table can hash different movesets that are just transpositions of each other

  • imperfect real-time decisions

    • can evaluate nodes with a heuristic and cutoff before reaching goal

    • heuristic uses features

    • want quiescent search - consider if something dramatic will happen in the next ply

      • horizon effect - a position is bad but isn’t apparent for a few moves

      • singular extension - allow searching for certain specific moves that are always good at deeper depths

    • forward pruning - ignore some branches

      • beam search - consider only n best moves

      • PROBCUT prunes some more

  • search vs lookup

    • often just use lookup in the beginning

    • program can solve and just lookup endgames

  • stochastic games

    • include chance nodes

    • change minimax to expectiminimax

    • \(O(b^m numRolls^m)\)

    • cutoff evaluation function is sensitive to scaling - evaluation function must be a positive linear transformation of the probability of winning from a position

    • can do alpha-beta pruning analog if we assume evaluation function is bounded in some range

    • alternatively, could simulate games with Monte Carlo simulation

6.9.1.1. utilities / decision theory – R&N 16.1-16.3, mazzonni quant finance book

  • lottery - any function of a random variable

  • utility function - lottery that satisfiers certain properties (e.g. transitivity)

    • expected utility = von Neumann-Morgenstern utility

  • goal: maximize utility by taking actions (focus on single actions)

    • utility function U(s) gives utility of a state

    • actions are probabilistic: \(P[RESULT(a)=s' \vert a,e]\)

      • s - state, e - observations, a - action

  • soln: pick action with maximum expected utility

    • expected utility \(EU(a\vert e) = \sum_{s'} P(RESULT(a)=s' \vert a,e) U(s')\)

  • notation

    • A>B - agent prefers A over B

    • A~B - agent is indifferent between A and B

  • preference relation has 6 axioms of utility theory

    1. orderability - A>B, A~B, or A<B

    2. transitivity

    3. continuity

    4. substitutability - can do algebra with preference eqns

    5. monotonicity - if A>B then must prefer higher probability of A than B

    6. decomposability - 2 consecutive lotteries can be compressed into single equivalent lottery

  • these axioms yield a utility function

    • isn’t unique (ex. affine transformation yields new utility function)

    • value function = ordinal utility function - sometimes ranking, numbers not needed

    • agent might not be explicitly maximizing the utility function

6.9.1.2. utility functions

  • preference elicitation - finds utility function

    • normalized utility to have min and max value

    • assess utility of s by asking agent to choose between s and \((p: \min, (1-p): \max)\)

  • people have complicated utility functions

    • ex. micromort - one in a million chance of death

    • ex. QALY - quality-adjusted life year

  • risk

    • agents exhibits monotonic preference for more money

    • gambling has expected monetary value = EMV

    • risk averse = when utility of money is sublinear

      • risk premium = value agent will accept in lieu of lottery = certainty equivalent= insurance premium

    • risk-neutral = linear

    • risk-seeking = supralinear

    • absolute risk aversion \(ARA(x) = - \frac{u''(x)}{u'(x)} \) : higher is more risk averse

    • relative risk aversion \(ARA(x) = - \frac{x \cdot u''(x)}{u'(x)} \)

  • optimizer’s curse - tendency for E[utility] to be too high because we keep picking high utility randomness

  • normative theory - how idealized agents work

  • descriptive theory - how actual agents work

    • certainty effect - people are drawn to things that are certain

    • ambiguity aversion

    • framing effect - wording can influence people’s judgements

    • anchoring effect - buy middle-tier wine because expensive is there

6.9.2. decision theory / VPI – R&N 16.5 & 16.6

  • note: here we are just making 1 decision

  • decision network (sometimes called influence diagram)

    1. chance nodes - represent RVs (like BN)

    2. decision nodes - points where decision maker has a choice of actions

    3. utility nodes - represent agent’s utility function

  • can ignore chance nodes

    • then action-utility function = Q-function maps directly from actions to utility

decision_nets

  • evaluation

    1. set evidence

    2. for each possible value of decision node

    • set decision node to that value

    • calculate probabilities of parents of utility node

    • calculate resulting utility

    1. return action with highest utility

6.9.2.1. the value of information

  • information value theory - enables agent to choose what info to acquire

    • observations only affect agent’s belief state

    • value of info = difference in best expected value with/without info

    • maximum \(EU(\alpha|e) = \underset{a}{\max} \sum_{s'} P(Result(a)=s'|a, e) U(s')\)

  • value of perfect information VPI - assume we can obtain exact evidence for a variable (ex. variable \(T=t\))

    • \(VPI(T) = \mathbb{E}_{T}\left[ EU(\alpha|e, T) \right] - \underbrace{EU(\alpha \vert e)}_{\text{original EU}}\)

    • first term expands to \(\sum_t P(T=t \vert e) \cdot EU(\alpha \vert e, T=t) \)

      • within each of these EU, we take a max over actions

    • VPI not linearly additive, but is order-independent

    • intuition

      • info is more valuable when it is likely to cause a change of plan

      • info is more valuable when the new plan will be much better than the old plan

  • information-gathering agent

    • myopic - greedily obtain evidence which yields highest VPI until some threshold

    • conditional plan - considers more things

6.9.3. mdps and rl - R&N 17.1-17.4

  • sequences of actions

  • fully observable - agent knows its state

  • markov decision process - all these things are given

    • set of states s

    • set of actions a

    • stochastic transition model \(P(s' \vert s,a)\)

    • reward function R(s)

      • utility aggregates rewards, for models more complex than mdps reward can be a function of past sequences of actions / observations

  • want policy \(\pi (s)\) - what action to do in state s

    • optimal policy yields highest expected utlity

  • optimizing MDP - multiattribute utility theory

    • could sum rewards, but results are infinite

    • instead define objective function (maps infinite sequences of rewards to single real numbers)

      • ex. discounting to prefer earlier rewards (most common)

        • discount reward n steps away by \(\gamma^n, 0<\gamma<1\)

      • ex. set a finite horizon and sum rewards

        • optimal action in a given state could change over time = nonstationary

      • ex. average reward rate per time step

      • ex. agent is guaranteed to get to terminal state eventually - proper policy

  • expected utility executing \(\pi\): \(U^\pi (s) = \mathbb E_{s_1,...,s_t}\left[\sum_t \gamma^t R(s_t)\right]\)

    • when we use discounted utilities, \(\pi\) is independent of starting state

    • \(\pi^*(s) = \underset{\pi}{argmax} \: U^\pi (s) = \underset{a}{argmax} \sum_{s'} P(s' \vert s,a) U'(s)\)

  • experience replay: instead of learning from samples one by one, want to reduce correlation between subsequent samples

    • take a large batch of samples and sample randomly from it, rather than going sequentially

6.9.3.1. value iteration

  • value iteration - calculates utility of each state and uses utilities to find optimal policy

    • bellman eqn: \(U(s) = R(s) + \gamma \: \underset{a}{\max} \sum_{s'} P(s' \vert s, a) U(s')\)

    • start with arbitrary utilities

    • recalculate several times with Bellman update to approximate solns to bellman eqn

  • value iteration eventually converges

    • contraction - function that brings variables together

      • contraction only has 1 fixed point

      • Bellman update is a contraction on the space of utility vectors and therefore converges

      • error is reduced by factor of \(\gamma\) each iteration

    • also, terminating condition - if \( \vert \vert U_{i+1}-U_i \vert \vert < \epsilon (1-\gamma) / \gamma\) then \( \vert \vert U_{i+1}-U \vert \vert <\epsilon\)

    • what actually matters is policy loss \( \vert \vert U^{\pi_i}-U \vert \vert \) - the most the agent can lose by executing \(\pi_i\) instead of the optimal policy \(\pi^*\)

      • if \( \vert \vert U_i -U \vert \vert < \epsilon\) then \( \vert \vert U^{\pi_i} - U \vert \vert < 2\epsilon \gamma / (1-\gamma)\)

6.9.3.2. policy iteration

  • another way to find optimal policies

    1. policy evaluation - given a policy \(\pi_i\), calculate \(U_i=U^{\pi_i}\), the utility of each state if \(\pi_i\) were to be executed

    • like value iteration, but with a set policy so there’s no max

      • \(U_i(s) = R(s) + \gamma \: \sum_{s'} P(s' \vert s, \pi_i(s)) U_i(s')\)

      • can solve exactly for small spaces, or approximate (set of lin. eqs.)

    1. policy improvement - calculate a new MEU policy \(\pi_{i+1}\) using \(U_i\)

    • same as above, just \(\pi^*(s) = \underset{\pi}{argmax} \: U^\pi (s) = \underset{a}{argmax} \sum_{s'} P(s' \vert s,a) U'(s)\)

  • asynchronous policy iteration - don’t have to update all states at once

6.9.3.3. partially observable markov decision processes (POMDP)

  • agent is not sure what state it’s in

  • same elements but add sensor model \(P(e \vert s)\)

  • have distr \(b(s)\) for belief states

    • updates like the HMM: \(b'(s') = \alpha P(e \vert s') \sum_s P(s' \vert s, a) b(s)\)

    • changes based on observations

  • optimal action depends only on the agent’s current belief state

    • use belief states as the states of an MDP and solve as before

    • changes because state space is now continuous

  • value iteration

    1. expected utility of executing p in belief state is just \(b \cdot \alpha_p\) (dot product)

    2. \(U(b) = U^{\pi^*}(b)=\underset{p}{\max} \: b \cdot \alpha_p\)

    • belief space is continuous [0, 1] so we represent it as piecewise linear, and store these discrete lines in memory

      • do this by iterating and keeping any values that are optimal at some point

        • remove dominated plans

    • generally this is far too inefficient

  • dynamic decision network - online agent

6.9.4. reinforcement learning – R&N 21.1-21.6

  • reinforcement learning - use observed rewards to learn optimal policy for the environment

    • in ch 17, agent had model of environment (P(s’|s, a) and R(s))

  • 2 problems

    • passive - given \(\pi\), learn \(U^\pi (s)\)

    • active - explore states to find utilities and exploit to get highest reward

  • 2 model types, 3 agent designs

    • model-based: can predict next state/reward before taking action (for MDP, requires learning \(P(s'|s,a)\))

      • utility-based agent - learns utility function on states

        • requires model of the environment

    • model-free

      • Q-learning agent: learns action-utility function = Q-function maps actions \(\to\) utility

      • reflex agent: learns policy that maps directly from states to actions

6.9.4.1. passive reinforcement learning

  • given policy \(\pi\), learn \(U^\pi (s) = \mathbb E\left[ \sum_{t=0}^{\infty} \gamma^t R(S_t)\right]\)

    • like policy evaluation, but transition model / reward function are unknown

  • direct utility estimation: treat states independently

    • run trials to sample utility

    • average to get expected total reward for each state = expected total reward from each state

  • adaptive dynamic programming (ADP) - sample to estimate transition model \(P(s'|s, a)\) and rewards \(R(s)\), then plug into Bellman eqn to find \(U^\pi(s)\) (plug in at each step)

    • we might want to enforce a prior on the model (two ways)

      1. Bayesian reinforcement learning - assume a prior \(P(h)\) on transition model h

      • use prior to calculate \(P(h \vert e)\)

      • use \(P(h|e)\) to calculate optimal policy: \(\pi^* = \underset{\pi}{argmax} \sum_h P(h \vert e) u_h^\pi\)

        • \(u_h^\pi\)= expected utility over all possible start states, obtained by executing policy \(\pi\) in model h

      1. give best outcome in the worst case over H (from robust control theory)

      • \(\pi^* = \underset{\pi}{argmax}\: \underset{h}{\min} \: u_h^\pi\)

  • temporal-difference learning - adjust utility estimates towards local equilibrium for correct utilities

    • like an approximation of ADP

    • when we transition \(s \to s'\), update \(U^\pi(s) = U^\pi (s) + \alpha \left[R(s) - U^\pi (s) + \gamma \:U^\pi (s') \right]\)

      • \(\alpha\) should decrease over time to converge

    • prioritized sweeping - prefers to make adjustments to states whose likely successors have just undergone a large adjustment in their own utility estimates

      • speeds things up

6.9.4.2. active reinforcement learning

  • no longer following set policy

    • explore states to find their utilities and exploit model to get highest reward

    • must explore all actions, not just those in the policy

  • bandit problems - determining exploration policy

    • n-armed bandit - pulling n levelers on a slot machine, each with different distr.

    • Gittins index - function of number of pulls / payoff

  • coorect schemes should be GLIE - greedy in the limit of infinite exploration - visits all states infinitely, but eventually become greedy

6.9.4.2.1. agent examples

  • ex. choose random action \(1/t\) of the time

  • ex. active adp agent

    • give optimistic utility to relatively unexplored states

    • uses exploration function f(u, numTimesVisited) around the sum in the bellman eqn

      • high utilities will propagate

  • ex. active TD agent

    • now must learn transitions (same as adp)

    • update rule same as passive TD

6.9.4.2.2. learning action-utility function

  • \(U(s) = \underset{a}{\max} \: Q(s,a)\)

    • ADP version: \(Q(s, a) = R(s) + \gamma \sum_{s'} P(s'|s, a) \underset{a'}{\max} Q(s', a')\)

    • TD version: \(Q(s,a) = Q(s,a) + \alpha [R(s) - Q(s,a) + \gamma \: \underset{a'}{\max} Q(s', a')]\) - this is what is usually referred to as Q-learning

  • SARSA (state-action-reward-state-action) is related: \(Q(s,a) = Q(s,a) + \alpha [R(s) + \gamma \: Q(s', a') - Q(s,a) ]\)

    • here, a’ is action actually taken

  • Q-learning is off-policy (only uses best Q-value)

    • more flexible

  • SARSA is on-policy (pays attention to actual policy being followed)

  • can approximate Q-function with something other than a lookup table

    • ex. linear function of parameters \(\hat{U}_\theta(s) = \theta_1f_1(s) + ... + \theta_n f_n(s)\)

      • can learn params online with delta rule = wildrow-hoff rule: \(\theta_i = \theta - \alpha \: \frac{\partial Loss}{\partial \theta_i}\)