AIXI

AIXI is a theoretical mathematical formalism for artificial general intelligence.It combines Solomonoff induction with sequential decision theory.AIXI was first proposed by Marcus Hutter in 2000 and several results regarding AIXI are proved in Hutter's 2005 book Universal Artificial Intelligence. AIXI is a theoretical mathematical formalism for artificial general intelligence.It combines Solomonoff induction with sequential decision theory.AIXI was first proposed by Marcus Hutter in 2000 and several results regarding AIXI are proved in Hutter's 2005 book Universal Artificial Intelligence. AIXI is a reinforcement learning agent. It maximizes the expected total rewards received from the environment. Intuitively, it simultaneously considers every computable hypothesis (or environment). In each time step, it looks at every possible program and evaluates how many rewards that program generates depending on the next action taken. The promised rewards are then weighted by the subjective belief that this program constitutes the true environment. This belief is computed from the length of the program: longer programs are considered less likely, in line with Occam's razor. AIXI then selects the action that has the highest expected total reward in the weighted sum of all these programs. AIXI is a reinforcement learning agent that interacts with some stochastic and unknown but computable environment μ {displaystyle mu } . The interaction proceeds in time steps, from t = 1 {displaystyle t=1} to t = m {displaystyle t=m} , where m ∈ N {displaystyle min mathbb {N} } is the lifespan of the AIXI agent. At time step t, the agent chooses an action a t ∈ A {displaystyle a_{t}in {mathcal {A}}} (e.g. a limb movement) and executes it in the environment, and the environment responds with a 'percept' e t ∈ E = O × R {displaystyle e_{t}in {mathcal {E}}={mathcal {O}} imes mathbb {R} } , which consists of an 'observation' o t ∈ O {displaystyle o_{t}in {mathcal {O}}} (e.g., a camera image) and a reward r t ∈ R {displaystyle r_{t}in mathbb {R} } , distributed according to the conditional probability μ ( o t r t | a 1 o 1 r 1 . . . a t − 1 o t − 1 r t − 1 a t ) {displaystyle mu (o_{t}r_{t}|a_{1}o_{1}r_{1}...a_{t-1}o_{t-1}r_{t-1}a_{t})} , where a 1 o 1 r 1 . . . a t − 1 o t − 1 r t − 1 a t {displaystyle a_{1}o_{1}r_{1}...a_{t-1}o_{t-1}r_{t-1}a_{t}} is the 'history' of actions, observations and rewards. The environment μ {displaystyle mu } is thus mathematically represented as a probability distribution over 'percepts' (observations and rewards) which depend on the full history, so there is no Markov assumption (as opposed to other RL algorithms). Note again that this probability distribution is unknown to the AIXI agent. Furthermore, note again that μ {displaystyle mu } is computable, that is, the observations and rewards received by the agent from the environment μ {displaystyle mu } can be computed by some program (which runs on a Turing machine), given the past actions of the AIXI agent . The only goal of the AIXI agent is to maximise ∑ t = 1 m r t {displaystyle sum _{t=1}^{m}r_{t}} , that is, the sum of rewards from time step 1 to m. The AIXI agent is associated with a stochastic policy π : ( A × E ) ∗ → A {displaystyle pi :({mathcal {A}} imes {mathcal {E}})^{*} ightarrow {mathcal {A}}} , which is the function it uses to choose actions at every time step, where A {displaystyle {mathcal {A}}} is the space of all possible actions that AIXI can take and E {displaystyle {mathcal {E}}} is the space of all possible 'percepts' that can be produced by the environment. The environment (or probability distribution) μ {displaystyle mu } can also be thought of as a stochastic policy (which is a function): μ : ( A × E ) ∗ × A → E {displaystyle mu :({mathcal {A}} imes {mathcal {E}})^{*} imes {mathcal {A}} ightarrow {mathcal {E}}} , where the ∗ {displaystyle *} is the Kleene star operation. In general, at time step t {displaystyle t} (which ranges from 1 to m), AIXI, having previously executed actions a 1 … a t − 1 {displaystyle a_{1}dots a_{t-1}} (which is often abbreviated in the literature as a < t {displaystyle a_{<t}} ) and having observed the history of percepts o 1 r 1 . . . o t − 1 r t − 1 {displaystyle o_{1}r_{1}...o_{t-1}r_{t-1}} (which can be abbreviated as e < t {displaystyle e_{<t}} ), chooses and executes in the environment the action, a t {displaystyle a_{t}} , defined as follows