Extending active inference to other intrinsic motivations
Martin Biehl
- Background to intrinsic motivations
- Formal framework for intrinsic motivations
- Perception-action loop
- Generative model
- Prediction / complete posterior
- Action selection
- Some intrinsic motivations:
- Free Energy Minimization
- Predictive information maximization
- Knowledge seeking
- Empowerment maximization
- Curiosity
- Active inference
Overview
Motivation is something that generates behavior for an agent (robot, living organism)
- similar to reward function in reinforcement learning (RL)
Background on intrinsic motivations
Originally from psychology e.g. (Ryan and Deci, 2000):
activity for its inherent satisfaction rather than separable consequence
for the fun or challenge entailed rather than because of external products, pressures or reward
Examples (Oudeyer, 2008):
- infants grasping, throwing, biting new objects,
- adults playing crosswords, painting, gardening, reading novels...
Background on intrinsic motivations
But can always argue:
- these things possibly increase the probability of survival in some way
- "learned" by evolution
- cannot be sure they have no purpose
Background on intrinsic motivations
Background on intrinsic motivations
Developmental robotics:
- study developmental processes of infants
- motor skill acquisition
- language acquisition
- implement similar processes in robots
Working definition compatible with Oudeyer (2008):
Motivation is intrinsic if it:
- embodiment independent,
- semantic free / information theoretic,
This includes the approach by Schmidhuber (2010):
Motivation is intrinsic if it
- rewards improvement of some model quality measure.
Background on intrinsic motivations
Embodiment independent means it should work (without changes) for any form of agent:
Background on intrinsic motivations
and produce "worthwhile" behavior
Embodiment independent means it should work for any form of agent:
Background on intrinsic motivations
- any number or kind of
- sensors
- actuators
- rewired
- sensors
- actuators
this implies
- cannot assume there is a special sensor whose value is to be maximized
- so reward functions of MDPs, POMDPs, and standard RL are not available
Background on intrinsic motivations
Semantic free, information theoretic:
- relations between sensors, actuators, and internal variables count
- specific values don't
- information theory quantifies relations
- if \(f\) and \(g\) are bijective functions then $$\text{I}(X:Y)=\text{I}(f(X):g(Y))$$
- so the values of \(X\) or \(Y\) can play no role in mutual information.
Background on intrinsic motivations
Background on intrinsic motivations
Another important but not defining feature is usually known from evolution:
Background on intrinsic motivations
Another important but not defining feature is usually known from evolution:
open endedness
The motivation should not vanish until the capacities of the agent are exhausted.
Background on intrinsic motivations
Other applications of intrinsic motivations:
- sparse reward reinforcement learning problems
- Human level and artificial general AI
Background on intrinsic motivations
Sparse reward reinforcement learning:
- Add additional term rewarding model improvement / curiosity / control
- when not obtaining reward this lets the agent find useful behaviour (hopefully)
Background on intrinsic motivations
AGI:
- implement predictive model that continually improves through experience
- implement action selection / optimization that chooses according to the prediction
- drive learning with intrinsic motivation
- no limit?
- possibly hardware limit
- ignore this here.
Background on intrinsic motivations
Advantages of intrinsic motivations
- scalability :
- no need to redesign reward function for different environments / agents
- environment kind and size does not change reward function
- agent complexity does not change reward function
Disadvantage:
- often (very) hard to compute
- too general, faster if available:
- specifically designed (dense) reward
- imitation learning
Examples:
- hunger is not an intrinsic motivation
- not embodiment (digestive system) independent
- eating more doesn't improve our model of the world
Background on intrinsic motivations
Examples:
- maximizing stored energy is closer to an intrinsic motivation
- real world agents need energy but not virtual ones
- doesn't directly improve the world model
- but maybe indirectly
- open ended?
Background on intrinsic motivations
Examples:
- maximizing money is also close to an intrinsic motivation
- but it only exists in some societies
- may also indirectly improve our model
- open ended?
Background on intrinsic motivations
Examples:
- minimizing prediction error of the model is an intrinsic motivation
- any agent that remembers its predictions can calculate the prediction error
- reducing it improves the model (at least locally)
Background on intrinsic motivations
Background on intrinsic motivations
dark room problem
Examples:
- minimizing prediction error of the model is an intrinsic motivation
- any agent that remembers its predictions can calculate the prediction error
- reducing it improves the model (at least locally)
- not open ended
Solution for dark room problem
-
maximizing the decrease of the prediction error (prediction progress) is an intrinsic motivation
- improves the predictions of the model in one area until more progress can be made in another
- may be open ended
Background on intrinsic motivations
- Background to intrinsic motivations
- Formal framework for intrinsic motivations
- Perception-action loop
- Generative model
- Prediction / complete posterior
- Action selection
- Some intrinsic motivations:
- Free Energy Minimization
- Predictive information maximization
- Knowledge seeking
- Empowerment maximization
- Curiosity
- Active inference
Overview
2. Formal framework for intrinsic motivations
- Perception-action loop
Similar to reinforcement learning for POMDPs :
- partially observable environment
- unknown environment transition dynamics
But we assume no extrinsic reward
2. Formal framework for intrinsic motivations
- Perception-action loop
\(E\) : Environment state
\(S\) : Sensor state
\(A\) : Action
\(M\) : Agent memory state
\(\newcommand{\p}{\text{p}} \p(e_0)\) : initial distribution
\(\newcommand{\p}{\text{p}}\p(s|e)\) : sensor dynamics
\(\newcommand{\p}{\text{p}}\p(m'|s,a,m)\) : memory dynamics
\(\newcommand{\p}{\text{p}}\p(a|m)\) : action generation
\(\newcommand{\p}{\text{p}}\p(e'|a',e)\) : environment dynamics
2. Formal framework for intrinsic motivations
- Perception-action loop
\(E\) : Environment state
\(S\) : Sensor state
\(A\) : Action
\(M\) : Agent memory state
\(\newcommand{\p}{\text{p}} \p(e_0)\): initial distribution
\(\newcommand{\p}{\text{p}}\p(s|e)\) : sensor dynamics
\(\newcommand{\p}{\text{p}}\p(m'|s,a,m)\) : memory dynamics
\(\newcommand{\p}{\text{p}}\p(a|m)\) : action generation
\(\newcommand{\p}{\text{p}}\p(e'|a',e)\) : environment dynamics
2. Formal framework for intrinsic motivations
- Perception-action loop
\(E\) : Environment state
\(S\) : Sensor state
\(A\) : Action
\(M\) : Agent memory state
\(\newcommand{\p}{\text{p}} \p(e_0)\): initial distribution
\(\newcommand{\p}{\text{p}}\p(s|e)\) : sensor dynamics
\(\newcommand{\p}{\text{p}}\p(m'|s,a,m)\) : memory dynamics
\(\newcommand{\p}{\text{p}}\p(a|m)\) : action generation
\(\newcommand{\p}{\text{p}}\p(e'|a',e)\) : environment dynamics
2. Formal framework for intrinsic motivations
- Perception-action loop
\(E\) : Environment state
\(S\) : Sensor state
\(A\) : Action
\(M\) : Agent memory state
\(\newcommand{\p}{\text{p}} \p(e_0)\): initial distribution
\(\newcommand{\p}{\text{p}}\p(s|e)\) : sensor dynamics
\(\newcommand{\p}{\text{p}}\p(m'|s,a,m)\) : memory dynamics
\(\newcommand{\p}{\text{p}}\p(a|m)\) : action generation
\(\newcommand{\p}{\text{p}}\p(e'|a',e)\) : environment dynamics
2. Formal framework for intrinsic motivations
- Perception-action loop
\(E\) : Environment state
\(S\) : Sensor state
\(A\) : Action
\(M\) : Agent memory state
\(\newcommand{\p}{\text{p}} \p(e_0)\): initial distribution
\(\newcommand{\p}{\text{p}}\p(s|e)\) : sensor dynamics
\(\newcommand{\p}{\text{p}}\p(m'|s,a,m)\) : memory dynamics
\(\newcommand{\p}{\text{p}}\p(a|m)\) : action generation
\(\newcommand{\p}{\text{p}}\p(e'|a',e)\) : environment dynamics
2. Formal framework for intrinsic motivations
- Perception-action loop
\(E\) : Environment state
\(S\) : Sensor state
\(A\) : Action
\(M\) : Agent memory state
\(\newcommand{\p}{\text{p}} \p(e_0)\): initial distribution
\(\newcommand{\p}{\text{p}}\p(s|e)\) : sensor dynamics
\(\newcommand{\p}{\text{p}}\p(m'|s,a,m)\) : memory dynamics
\(\newcommand{\p}{\text{p}}\p(a|m)\) : action generation
\(\newcommand{\p}{\text{p}}\p(e'|a',e)\) : environment dynamics
2. Formal framework for intrinsic motivations
- Perception-action loop
\(E\) : Environment state
\(S\) : Sensor state
\(A\) : Action
\(M\) : Agent memory state
\(\newcommand{\p}{\text{p}} \p(e_0)\): initial distribution
\(\newcommand{\p}{\text{p}}\p(s|e)\) : sensor dynamics
\(\newcommand{\p}{\text{p}}\p(m'|s,a,m)\) : memory dynamics
\(\newcommand{\p}{\text{p}}\p(a|m)\) : action generation
\(\newcommand{\p}{\text{p}}\p(e'|a',e)\) : environment dynamics
2. Formal framework for intrinsic motivations
- Perception-action loop
\(E\) : Environment state
\(S\) : Sensor state
\(A\) : Action
\(M\) : Agent memory state
\(\newcommand{\p}{\text{p}} \p(e_0)\): initial distribution
\(\newcommand{\p}{\text{p}}\p(s|e)\) : sensor dynamics
\(\newcommand{\p}{\text{p}}\p(m'|s,a,m)\) : memory dynamics
\(\newcommand{\p}{\text{p}}\p(a|m)\) : action generation
\(\newcommand{\p}{\text{p}}\p(e'|a',e)\) : environment dynamics
2. Formal framework for intrinsic motivations
- Perception-action loop
\(E\) : Environment state
\(S\) : Sensor state
\(A\) : Action
\(M\) : Agent memory state
\(\newcommand{\p}{\text{p}} \p(e_0)\): initial distribution
\(\newcommand{\p}{\text{p}}\p(s|e)\) : sensor dynamics
\(\newcommand{\p}{\text{p}}\p(m'|s,a,m)\) : memory dynamics
\(\newcommand{\p}{\text{p}}\p(a|m)\) : action generation
\(\newcommand{\p}{\text{p}}\p(e'|a',e)\) : environment dynamics
2. Formal framework for intrinsic motivations
- Perception-action loop
\(E\) : Environment state
\(S\) : Sensor state
\(A\) : Action
\(M\) : Agent memory state
\(\newcommand{\p}{\text{p}} \p(e_0)\): initial distribution
\(\newcommand{\p}{\text{p}}\p(s|e)\) : sensor dynamics
\(\newcommand{\p}{\text{p}}\p(m'|s,a,m)\) : memory dynamics
\(\newcommand{\p}{\text{p}}\p(a|m)\) : action generation
\(\newcommand{\p}{\text{p}}\p(e'|a',e)\) : environment dynamics
2. Formal framework for intrinsic motivations
- Perception-action loop
\newcommand{\p}{\text{p}} \p(e_{0:T},s_{0:T},a_{1:T},m_{1:T})
= \left( \prod_{t=1}^T \p(a_t|m_t) \p(m_t|s_{t-1},a_{t-1},m_{t-1}) \p(s_t|e_t) \p(e_t|a_t,e_{t-1}) \right) \p(s_0|e_0) \p(e_0)
Joint distribution until final time \(t=T\) :
2. Formal framework for intrinsic motivations
- Perception-action loop
Assumptions :
- constant environment and sensor dynamics given $$\newcommand{\p}{\text{p}}\p(e_{t_1}|a_{t_1},e_{t_1-1})=\p(e_{t_2}|a_{t_2},e_{t_2-1})$$ $$\newcommand{\p}{\text{p}}\p(s_{t_1}|e_{t_1})=\p(s_{t_2}|e_{t_2})$$
- perfect agent memory : $$\newcommand{\pt}{{\prec t}}m_t := (s_\pt,a_\pt) := sa_\pt \Rightarrow \newcommand{\p}{\text{p}}\p(m'|s,a,m) $$
2. Formal framework for intrinsic motivations
- Perception-action loop
Assumptions :
- constant environment and sensor dynamics given $$\newcommand{\p}{\text{p}}\p(e_{t_1}|a_{t_1},e_{t_1-1})=\p(e_{t_2}|a_{t_2},e_{t_2-1})$$ $$\newcommand{\p}{\text{p}}\p(s_{t_1}|e_{t_1})=\p(s_{t_2}|e_{t_2})$$
- perfect agent memory : $$\newcommand{\pt}{{\prec t}}m_t := (s_\pt,a_\pt) := sa_\pt \Rightarrow \newcommand{\p}{\text{p}}\p(m'|s,a,m) $$
2. Formal framework for intrinsic motivations
- Perception-action loop
Only missing:
- action generation mechanism \(\newcommand{\p}{\text{p}} \p(a|m)\)
- takes current sensor-action history \(sa_{\prec t}\) to generate new action \(a_t\)
- reuse of computational results on previous history \(sa_{\prec t-1}\) not reflected here but possible
2. Formal framework for intrinsic motivations
Remarks:
- Intrinsic motivations quantify statistical relations between sensor values, actions, and beliefs/internal variables.
- Taking actions according to such measures requires to predict them.
- Possible by using parameterized / generative model.
- Encodes beliefs and predictions in probability distributions over parameters and latent variables.
- Easy to rigorously express many intrinsic motivations.
- Naive computation is intractable.
- Making it tractable is not discussed.
2. Formal framework for intrinsic motivations
1. Generative model
- Internal to the agent
- For parameters write \(\Theta=(\Theta^1,\Theta^2,\Theta^3)\)
- \(\xi=(\xi^1,\xi^2,\xi^3)\) are fixed hyperparameters that encode priors over the parameters
2. Formal framework for intrinsic motivations
2. Generative model
Model split up into three parts:
- sensor dynamics model \(\newcommand{\q}{\text{q}}\newcommand{\hs}{\hat{s}}\q(\hat{s}|\hat{e},\theta)\)
- environment dynamics model \(\newcommand{\q}{\text{q}}\newcommand{\ha}{\hat{a}}\q(\hat{e}'|\ha,\hat{e},\theta)\)
- initial environment distribution \(\newcommand{\q}{\text{q}}\newcommand{\hs}{\hat{s}}\q(\hat{e}|\theta)\)
2. Formal framework for intrinsic motivations
2. Generative model
Model split up into three parts:
- sensor dynamics model \(\newcommand{\q}{\text{q}}\newcommand{\hs}{\hat{s}}\q(\hat{s}|\hat{e},\theta)\)
- environment dynamics model \(\newcommand{\q}{\text{q}}\newcommand{\ha}{\hat{a}}\q(\hat{e}'|\ha,\hat{e},\theta)\)
- initial environment distribution \(\newcommand{\q}{\text{q}}\newcommand{\hs}{\hat{s}}\q(\hat{e}|\theta)\)
2. Formal framework for intrinsic motivations
2. Generative model
Model split up into three parts:
- sensor dynamics model \(\newcommand{\q}{\text{q}}\newcommand{\hs}{\hat{s}}\q(\hat{s}|\hat{e},\theta)\)
- environment dynamics model \(\newcommand{\q}{\text{q}}\newcommand{\ha}{\hat{a}}\q(\hat{e}'|\ha,\hat{e},\theta)\)
- initial environment distribution \(\newcommand{\q}{\text{q}}\newcommand{\hs}{\hat{s}}\q(\hat{e}|\theta)\)
2. Formal framework for intrinsic motivations
3. Inference / prediction
As time passes in the perception action loop
- new sensor values \(S_t=s_t\) and actions \(A_t=a_t\) are generated
- get stored in \(M_t=m_t=sa_{\prec t}\)
As time passes in the perception action loop
- new sensor values \(S_t=s_t\) and actions \(A_t=a_t\) are generated
- get stored in \(M_t=m_t=sa_{\prec t}\)
2. Formal framework for intrinsic motivations
3. Inference / prediction
So at \(t\) agent can plug \(m_t=sa_{\prec t}\) into model
- updates the probability distribution to a posterior:
\newcommand{\tT}{{t:T}}
\newcommand{\thT}{{t:\hT}}
\newcommand{\hs}{\hat{s}}
\newcommand{\pt}{{\prec t}}
\newcommand{\pet}{{\preceq t}}
\newcommand{\ha}{\hat{a}}
\newcommand{\he}{\hat{e}}
\newcommand{\q}{{q}}
\newcommand{\hT}{{\hat{T}}}
\newcommand{\diff}{\text{d}}
\q(\hs_\thT,\he_{0:\hT},\ha_\thT,\theta|sa_\pt,\xi)
2. Formal framework for intrinsic motivations
3. Inference / prediction
So at \(t\) agent can plug \(m_t=sa_{\prec t}\) into model
- updates the probability distribution to a posterior:
\newcommand{\tT}{{t:T}}
\newcommand{\thT}{{t:\hT}}
\newcommand{\hs}{\hat{s}}
\newcommand{\pt}{{\prec t}}
\newcommand{\pet}{{\preceq t}}
\newcommand{\ha}{\hat{a}}
\newcommand{\he}{\hat{e}}
\newcommand{\q}{{q}}
\newcommand{\hT}{{\hat{T}}}
\newcommand{\diff}{\text{d}}
\q(\hs_\thT,\he_{0:\hT},\ha_\thT,\theta|sa_\pt,\xi)
2. Formal framework for intrinsic motivations
3. Inference / prediction
predicts consequences of assumed actions \(\blue{\hat{a}_{t:\hat{T}}}\) for relations between:
\newcommand{\tT}{{t:T}}
\newcommand{\thT}{{t:\hT}}
\newcommand{\hs}{\hat{s}}
\newcommand{\pt}{{\prec t}}
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\newcommand{\ha}{\hat{a}}
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\newcommand{\q}{\text{q}}
\newcommand{\hT}{{\hat{T}}}
\newcommand{\diff}{\text{d}}
\q(\hs_\thT,\he_{0:\hT},\theta|\ha_\thT,sa_\pt,\xi)
- parameters \(\red{\Theta}\)
- latent variables \(\red{\hat{E}_{0:\hat{T}}}\)
- future sensor values \(\red{\hat{S}_{t:\hat{T}}}\)
2. Formal framework for intrinsic motivations
3. Inference / prediction
Call \(\text{q}(\hat{s}_{t:\hat{T}},\hat{e}_{0:\hat{T}},\theta|\hat{a}_{t:\hat{T}},sa_{\prec t},\xi)\) the complete posteriors.
2. Formal framework for intrinsic motivations
3. Inference / prediction
2. Formal framework for intrinsic motivations
3. Prediction
Note: can use same model to predict consequences of closed loop policies \(\blue{\hat{\Pi}}\)
- here \(\hat{\Pi}\) is a parameterization of a general policy $$\newcommand{\q}{\text{q}}\pi(\hat{a}_t|sa_{\prec t}) = \q(\hat{a}_t|sa_{\prec t},\pi)$$
2. Formal framework for intrinsic motivations
3. Prediction
Note: can use same model to predict consequences of closed loop policies \(\blue{\hat{\Pi}}\)
- simpler policies are also possible $$\newcommand{\q}{\text{q}}\pi(\hat{a}_t|s_{t-1}) = \q(\hat{a}_t|s_{t-1},\pi)$$
\newcommand{\tT}{{t:T}}
\newcommand{\thT}{{t:\hT}}
\newcommand{\hs}{\hat{s}}
\newcommand{\pt}{{\prec t}}
\newcommand{\pet}{{\preceq t}}
\newcommand{\ha}{\hat{a}}
\newcommand{\he}{\hat{e}}
\newcommand{\q}{\text{q}}
\newcommand{\hT}{{\hat{T}}}
\newcommand{\diff}{\text{d}}
\q(\hs_\thT,\he_{0:\hT},\theta|\ha_\thT,sa_\pt,\xi) = \q(\hs_\thT,\he_\thT|\ha_\thT,\he_{t-1},\theta) \q(\he_{\prec t},\theta|sa_\pt,\xi)
2. Formal framework for intrinsic motivations
3. Inference / prediction
Complete posteriors factorizes:
\newcommand{\tT}{{t:T}}
\newcommand{\thT}{{t:\hT}}
\newcommand{\hs}{\hat{s}}
\newcommand{\pt}{{\prec t}}
\newcommand{\pet}{{\preceq t}}
\newcommand{\ha}{\hat{a}}
\newcommand{\he}{\hat{e}}
\newcommand{\q}{\text{q}}
\newcommand{\hT}{{\hat{T}}}
\newcommand{\diff}{\text{d}}
\q(\hs_\thT,\he_{0:\hT},\theta|\ha_\thT,sa_\pt,\xi) = \q(\hs_\thT,\he_\thT|\ha_\thT,\he_{t-1},\theta) \q(\he_{\prec t},\theta|sa_\pt,\xi)
2. Formal framework for intrinsic motivations
3. Inference / prediction
into posterior and predictive factor:
2. Formal framework for intrinsic motivations
4. Action selection
- define intrinsic motivations as functions \(\mathfrak{M}\) of the complete posteriors and a given sequence \(\hat{a}_{t:\hat{T}}\) of future actions:
\newcommand{\tT}{{t:T}}
\newcommand{\hT}{\hat{T}}
\newcommand{\thT}{{t:\hT}}
\newcommand{\hs}{\hat{s}}
\newcommand{\pt}{{\prec t}}
\newcommand{\pet}{{\preceq t}}
\newcommand{\set}{{\succeq t}}
\newcommand{\ha}{\hat{a}}
\newcommand{\he}{\hat{e}}
\newcommand{\q}{\text{q}}
\newcommand{\diff}{\text{d}}
\newcommand{\ptau}{{\prec \tau}}
\newcommand{\petau}{{\preceq \tau}}
\newcommand{\stau}{{\succ \tau}}
\newcommand{\setau}{{\succeq \tau}}
\newcommand{\mot}{\mathfrak{M}}
\mot(\q(.,.,.|.,sa_\pt,\xi),\ha_\thT)
- To act find best sequence:
\newcommand{\hT}{\hat{T}}
\newcommand{\thT}{{t:\hT}}
\newcommand{\hs}{\hat{s}}
\newcommand{\pt}{{\prec t}}
\newcommand{\pet}{{\preceq t}}
\newcommand{\set}{{\succeq t}}
\newcommand{\ha}{\hat{a}}
\newcommand{\he}{\hat{e}}
\newcommand{\q}{\text{q}}
\newcommand{\diff}{\text{d}}
\newcommand{\ptau}{{\prec \tau}}
\newcommand{\petau}{{\preceq \tau}}
\newcommand{\stau}{{\succ \tau}}
\newcommand{\setau}{{\succeq \tau}}
\newcommand{\argmax}{\text{argmax}}
\newcommand{\mot}{\mathfrak{M}}
\ha^*_\thT(sa_\pt):=\argmax_{\ha_\thT} \mot(\q(.,.,.|.,sa_\pt,\xi),\ha_\thT)
2. Formal framework for intrinsic motivations
4. Action selection
General reinforcement learning (RL) evaluates actions by expected cumulative reward \(Q(\hat{a}_{t:\hat{T}},sa_{\prec t}):=\mathbb{E}[R|\hat{a}_{t:\hat{T}},sa_{\prec t}]\):
\newcommand{\tT}{{t:T}}
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\newcommand{\ptau}{{\prec \tau}}
\newcommand{\petau}{{\preceq \tau}}
\newcommand{\stau}{{\succ \tau}}
\newcommand{\setau}{{\succeq \tau}}
Q(\hat{a}_{t:\hat{T}},sa_{\prec t}) := \sum_{\hs_\thT} \q(\hs_\thT|\ha_\thT,sa_\pt,\xi) \sum_{\tau=t}^{\hat{T}} r(\hs\ha_{t:\tau}, sa_\pt)
\newcommand{\hT}{\hat{T}}
\newcommand{\thT}{{t:\hT}}
\newcommand{\hs}{\hat{s}}
\newcommand{\pt}{{\prec t}}
\newcommand{\pet}{{\preceq t}}
\newcommand{\set}{{\succeq t}}
\newcommand{\ha}{\hat{a}}
\newcommand{\he}{\hat{e}}
\newcommand{\q}{\text{q}}
\newcommand{\diff}{\text{d}}
\newcommand{\ptau}{{\prec \tau}}
\newcommand{\petau}{{\preceq \tau}}
\newcommand{\stau}{{\succ \tau}}
\newcommand{\setau}{{\succeq \tau}}
Q(\pi,sa_{\prec t}) := \sum_{\hs_\thT} \q(\hs\ha_\thT|sa_\pt,\pi,\xi) \sum_{\tau=t}^{\hat{T}} r(\hs\ha_{t:\tau}, sa_\pt)
Standard RL : reward \(r_t\) is one of the sensor values so $$r(\hat{s}\hat{a}_{t:\tau},sa_{\prec t})=r(s_\tau)=r_\tau$$
For the case of evaluating policies \(Q(\pi,sa_{\prec t}):=\mathbb{E}[R|\pi,sa_{\prec t}]\):
2. Formal framework for intrinsic motivations
4. Action selection
From best sequence:
\newcommand{\tT}{{t:T}}
\newcommand{\hT}{\hat{T}}
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\newcommand{\hs}{\hat{s}}
\newcommand{\pt}{{\prec t}}
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\newcommand{\ptau}{{\prec \tau}}
\newcommand{\petau}{{\preceq \tau}}
\newcommand{\stau}{{\succ \tau}}
\newcommand{\setau}{{\succeq \tau}}
\newcommand{\argmax}{\text{argmax}}
\p(a_t|sa_\pt):=\delta_{\ha^*_t(sa_\pt)}(a_t)
Select / perform the first action:
\newcommand{\hT}{\hat{T}}
\newcommand{\thT}{{t:\hT}}
\newcommand{\hs}{\hat{s}}
\newcommand{\pt}{{\prec t}}
\newcommand{\pet}{{\preceq t}}
\newcommand{\set}{{\succeq t}}
\newcommand{\ha}{\hat{a}}
\newcommand{\he}{\hat{e}}
\newcommand{\q}{\text{q}}
\newcommand{\diff}{\text{d}}
\newcommand{\ptau}{{\prec \tau}}
\newcommand{\petau}{{\preceq \tau}}
\newcommand{\stau}{{\succ \tau}}
\newcommand{\setau}{{\succeq \tau}}
\newcommand{\argmax}{\text{argmax}}
\newcommand{\mot}{\mathfrak{M}}
\ha^*_\thT(sa_\pt):=\argmax_{\ha_\thT} \mot(\q(.,.,.|.,sa_\pt,\xi),\ha_\thT)
2. Formal framework for intrinsic motivations
4. Action selection
From best sequence:
\newcommand{\tT}{{t:T}}
\newcommand{\hT}{\hat{T}}
\newcommand{\thT}{{t:\hT}}
\newcommand{\hs}{\hat{s}}
\newcommand{\pt}{{\prec t}}
\newcommand{\pet}{{\preceq t}}
\newcommand{\set}{{\succeq t}}
\newcommand{\ha}{\hat{a}}
\newcommand{\he}{\hat{e}}
\newcommand{\q}{\text{q}}
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\newcommand{\ptau}{{\prec \tau}}
\newcommand{\petau}{{\preceq \tau}}
\newcommand{\stau}{{\succ \tau}}
\newcommand{\setau}{{\succeq \tau}}
\newcommand{\argmax}{\text{argmax}}
\p(a_t|sa_\pt):=\delta_{\ha^*_t(sa_\pt)}(a_t)
Select / perform the first action:
\newcommand{\hT}{\hat{T}}
\newcommand{\thT}{{t:\hT}}
\newcommand{\hs}{\hat{s}}
\newcommand{\pt}{{\prec t}}
\newcommand{\pet}{{\preceq t}}
\newcommand{\set}{{\succeq t}}
\newcommand{\ha}{\hat{a}}
\newcommand{\he}{\hat{e}}
\newcommand{\q}{\text{q}}
\newcommand{\diff}{\text{d}}
\newcommand{\ptau}{{\prec \tau}}
\newcommand{\petau}{{\preceq \tau}}
\newcommand{\stau}{{\succ \tau}}
\newcommand{\setau}{{\succeq \tau}}
\newcommand{\argmax}{\text{argmax}}
\newcommand{\mot}{\mathfrak{M}}
\ha^*_\thT(sa_\pt):=\argmax_{\ha_\thT} \mot(\q(.,.,.|.,sa_\pt,\xi),\ha_\thT)
- General requirement is conditional probability \(\text{q}(\hat{s}_{t:\hat{T}},\hat{e}_{0:\hat{T}},\theta|\hat{a}_{t:\hat{T}})\) how this is obtained by agent does not matter.
- drop \((sa_{\prec t},\xi)\) in following
- Background to intrinsic motivations
- Formal framework for intrinsic motivations
- Perception-action loop
- Generative model
- Prediction / complete posterior
- Action selection
- Some intrinsic motivations:
- Free Energy Minimization
- Predictive information maximization
- Knowledge seeking
- Empowerment maximization
- Curiosity
- Active inference
Overview
3. Some intrinsic motivations
1. Free energy minimization
Actions should lead to environment states expected to have precise sensor values (e.g. Friston, Parr et al., 2017):
\newcommand{\hT}{\hat{T}}
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\newcommand{\mot}{\mathfrak{M}}
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\q(\he_\thT|\ha_\thT)=
\int \sum_{\hs_\thT,\he_\pt} \q(\hs_\thT,\he_{0:\hT},\theta|\ha_\thT) \diff \theta
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\newcommand{\hS}{\hat{S}}
\newcommand{\hE}{\hat{E}}
\begin{aligned}
\mot(\q(.,.,.|.,\xi),\ha_\thT) :&=-\HS_{\q}(\hS_\thT|\hE_\thT,\ha_\thT)\\
&= \sum_{\he_\thT} \q(\he_\thT|\ha_\thT) \sum_{\hs_\thT} \q(\hs_\thT|\he_\thT) \log \q(\hs_\thT|\he_\thT)\\
\end{aligned}
Get \(\text{q}(\hat{e}_{t:\hat{T}}|\hat{a}_{t:\hat{T}})\) frome the complete posterior:
3. Some intrinsic motivations
1. Free energy minimization
- random noise source are avoided
- will get stuck in known "dark room traps"
- we know $$\text{H}_{\text{q}}(\hat{S}_{t:\hat{T}}|\hat{a}_{t:\hat{T}})=0\Rightarrow\text{H}_{\text{q}}(\hat{S}_{t:\hat{T}}|\hat{E}_{t:\hat{t}},\hat{a}_{t:\hat{T}})=0$$
- such an optimal action sequence \(\hat{a}_{t:\hat{T}}\) exists e.g. if there is a "dark room" in the environment
- even if room cannot be escaped once entered and the agent knows it!
- solved by adding KL divergence to constructed desired sensory experience
- breaks purpose of intrinsic motivations (not scalable)
- Free energy is not suitable for AGI
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\begin{aligned}
\mot^{PI}(\d(.,.,.|.),\ha_\thT) :&= \I_{\q}(\hS_{t:t+k-1}:\hS_{t+k:t+2k-1}|\ha_\thT)\\
&=\sum_{\hs_{t:t+2k-1}} \q(\hs_{t:t+2k-1}|\ha_\thT) \log \frac{\q(\hs_{t+k:t+2k-1}|\hs_{t:t+k-1},\ha_\thT)}{\q(\hs_{t:t+k-1}|\ha_\thT)}\\
\end{aligned}
3. Some intrinsic motivations
2. Predictive information maximization
Actions should lead to the most complex sensor stream:
- Next \(k\) sensor values should have max mutual information with the subsequent \(k\).
- Can get needed distributions from complete posterior.
3. Some intrinsic motivations
2. Predictive information maximization
- random noise source are avoided as they produce no mutual information
- will not get stuck in known "dark room traps"
- from $$\text{H}_{\text{q}}(\hat{S}_{t+k:t+2k-1}|\hat{a}_{t:\hat{T}})=0\Rightarrow\text{I}_{\text{q}}(\hat{S}_{t:t+k-1},\hat{S}_{t+k:t+2k-1}|\hat{a}_{t:\hat{T}})=0$$
- possible long term behavior:
- ergodic sensor process
- finds a subset of environment states that allows this ergodicity
3. Some intrinsic motivations
2. Predictive information maximization
Georg Martius, Ralf Der
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\begin{aligned}
\mot^{KSA}(\q(.,.,.|.),\ha_\thT) :&= \I_{\q}(\hS_\thT:\hE_{0:\hT},\Theta|\ha_\thT)\\
&=\sum_{\hs_\thT} \int \q(\hs_\thT,\he_{0:\hT},\theta|\ha_\thT) \log \frac{\q(\he_{0:\hT},\theta|\hs_\thT,\ha_\thT)}{\q(\theta)} \diff \theta
\end{aligned}
3. Some intrinsic motivations
3. Knowledge seeking
Actions should lead to sensor values that tell the most about hidden (environment) variables \(\hat{E}_{0:\hat{T}}\) and model parameters \(\Theta\):
- Also known as information gain maximization
- Can get needed distributions from complete posterior.
3. Some intrinsic motivations
3. Knowledge seeking
- avoids random noise sources once they are known
- similar to prediction progress
- can rewrite as $$\text{H}_{\text{q}}(\hat{E}_{0:\hat{T}},\Theta|\hat{a}_{t:\hat{T}})-\text{H}_{\text{q}}(\hat{E}_{0:\hat{T}},\Theta|\hat{S}_{t:\hat{T}},\hat{a}_{t:\hat{T}})$$
- will not get stuck in known "dark room traps"
- from $$\text{H}_{\text{q}}(\hat{S}_{t:\hat{T}}|\hat{a}_{t:\hat{T}})=0\Rightarrow\text{I}_{\text{q}}(\hat{S}_{t:\hat{T}}:\hat{E}_{0:\hat{T}},\Theta|\hat{a}_{t:\hat{T}})=0$$
- technical results exist by Orseau et al. (2013) (off-policy prediction!)
- possible long term behavior:
- when model is known does nothing / random walk
3. Some intrinsic motivations
3. Knowledge seeking
Bellemare et al. (2016)
Bellemare, M. G., Srinivasan, S., Ostrovski, G., Schaul, T., Saxton, D., and Munos, R. (2016). Unifying Count-Based Exploration and Intrinsic Motivation. arXiv:1606.01868 [cs]. arXiv: 1606.01868.
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\begin{aligned}
\mot^{EM}(\d(.,.,.|.),\ha_\thTa) :&= \max_{\d(\ha_{\hTa+1:\hT})} \; \I_{\d}(\hA_{\hTa+1:\hT}:\hS_\hT|\ha_\thTa) \\
&=\max_{\d(\ha_{\hTa+1:\hT})} \; \sum_{\ha_{\hTa+1:\hT},\hs_\hT} \d(\ha_{\hTa+1:\hT}) \d(\hs_\hT|\ha_\thT) \log \frac{\d(\hs_\hT|\ha_\thT)}{\d(\hs_\hT|\ha_\thTa)}.
\end{aligned}
3. Some intrinsic motivations
4. Empowerment maximization
Actions should lead to control over as many future experiences as possible:
- Actions \(\hat{a}_{t:\hat{T}_a}\) are taken such that subsequent actions \(\hat{a}_{\hat{T}_a+1:\hat{T}}\) have control
- Can get needed distributions from complete posterior.
3. Some intrinsic motivations
4. Empowerment maximization
- avoids random noise sources because they cannot be controlled
- will not get stuck in known "dark room traps"
- from $$\text{H}_{\text{q}}(\hat{S}_{\hat{T}}|\hat{a}_{t:\hat{T}_a})=0\Rightarrow\text{I}_{\text{q}}(\hat{A}_{\hat{T}_a+1:\hat{T}}\hat{S}_{\hat{T}}:|\hat{a}_{t:\hat{T}_a})=0$$
- similar to energy and money maximization but more general
- possible long term behavior:
- remains in (or maintains) the situation where it expects the most control over future experience
- exploration behavior not fully understood
- Belief empowerment may solve it...
3. Some intrinsic motivations
4. Empowerment
Guckelsberger et al. (2016)
Guckelsberger, C., Salge, C., & Colton, S. (2016). Intrinsically Motivated General Companion NPCs via Coupled Empowerment Maximisation. 2016 IEEE Conf. Computational Intelligence in Games (CIG’16), 150–157
3. Some intrinsic motivations
5a. Curiosity
Actions should lead to surprising sensor values.
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\begin{aligned}
\mot(\q(.,.,.|.,\xi),\ha_\thT) :&=+\HS_{\q}(\hS_\thT|\ha_\thT)\\
&= \sum_{\hs_\thT} \q(\hs_\thT|\ha_\thT)(- \log \q(\hs_\thT|\ha_\thT))\\
\end{aligned}
- Also called Shannon knowledge seeking (Orseau)
- maximize expected surprise (=entropy)
- Get density from the complete posterior.
3. Some intrinsic motivations
5a. Curiosity
- will not get stuck in known "dark room traps"
- it directly pursues the opposite situation
- will get stuck at random noise sources
- in deterministic environments not a problem
- proven to asymptotically drive an agent to the behavior of an agent that knows the environment (under some technical conditions, see Orseau, 2011)
3. Some intrinsic motivations
5b. Curiosity
Actions should lead to surprising environment states.
- I have never seen this explicitly
- Get density from the complete posterior.
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\begin{aligned}
\mot(\q(.,.,.|.,\xi),\ha_\thT) :&=+\HS_{\q}(\hE_\thT|\ha_\thT)\\
&= \sum_{\hs_\thT} \q(\he_\thT|\ha_\thT)(- \log \q(\he_\thT|\ha_\thT))\\
\end{aligned}
3. Some intrinsic motivations
5b. Curiosity
Actions should lead to surprising embedding of sensor values:
- Something between the last two
- Get density from the complete posterior.
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\begin{aligned}
\mot(\q(.,.,.|.,\xi),\ha_\thT) :&=+\HS_{\q}(f(\hS_\thT)|\ha_\thT)\\
&= \sum_{\hs_\thT} \q(f(\hs_\thT)|\ha_\thT)(- \log \q(f(\hs_\thT)|\ha_\thT))\\
\end{aligned}
3. Some intrinsic motivations
5. Curiosity
Burda et al. (2018) with permission.
Burda, Y., Edwards, H., Pathak, D., Storkey, A., Darrell, T., and Efros, A. A. (2018). Large-Scale Study of Curiosity-Driven Learning. arXiv:1808.04355 [cs, stat]. arXiv: 1808.04355.
4.Active inference
1. Variational complete posteriors
- Complete posteriors are computationally intractable
- approximation usually necessary
- introduce variational complete posterior
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\re(\hs_\thT,\he_{0:\hT},\theta|\ha_\thT,\phi)
- the parameter \(\phi\) ranges over a (usually simpler) class of conditional distributions than the true posterior
- obtaining \(\phi^*\) that parameterizes the best approximation to the true posterior is called variational inference
4.Active inference
1. Variational complete posteriors
- Active inference promises to do inference and action selection in a single optimization step
- before we used direct non-variational updating of the complete posteriors and then chose the best actions
- now construct a way to do both (possibly approximately) at once.
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\begin{aligned}
\phi^*_{sa_\pt,\xi}:&=\text{argmin}_\phi \;\F[\phi,sa_\pt,\xi]\\
&=\text{argmin}_\phi \;\sum_{\he_\pt} \int \re(\he_\pt,\theta|\phi) \log \frac{\re(\he_\pt,\theta|\phi)} {\q(s_\pet,\he_\pt,\theta|a_\pt,\xi)} \diff \theta\\
&=\text{argmin}_\phi \;- \log \q(s_\pt|a_\pt,\xi) + \KL[\re(\hE_\pt,\Theta|\phi)||\q(\hE_\pt,\Theta|sa_\pt,\xi)]\\
&=\text{argmin}_\phi\; \KL[\re(\hE_\pt,\Theta|\phi)||\q(\hE_\pt,\Theta|sa_\pt,\xi)]\\
\end{aligned}
4.Active inference
2. Variational inference
- Each parameter value \(\phi\) specifies a "complete posterior-like" object
- not only \(\phi^*_{sa_{\prec t},\xi}\) (the optimized one)
- could also select action according to the predictions of the consequences of actions specified by such an arbitrary \(\phi\)
- define an according policy
4.Active inference
2. Variational policy
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\re(\hs_\thT,\he_{0:\hT},\theta|\ha_\thT,\phi)
- optimal action sequence according to \(\phi\):
4.Active inference
2. Variational policy
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\ha^*_\thT(\phi):=\argmax_{\ha_\thT} \mot(\re(.,.,.|.,\phi),\ha_\thT)
- turn this into a stochastic policy using softmax:
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\q(\ha_\thT|\phi):=\frac{1}{Z} e^{\mot(\re(.,.,.|.,\phi),\ha_\thT)}
- introduce another policy \(\text{r}(\hat{a}_{t:\hat{T}}|\pi)\) parameterized by \(\pi\)
- minimize at the same time
- the difference of \(\text{r}(\hat{a}_{t:\hat{T}}|\pi)\) to \(\text{q}(\hat{a}_{t:\hat{T}}|\phi)\)
- the difference of the approximate posterior to the true one:
4.Active inference
3. Active inference
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\begin{aligned}
\phi^*_{sa_\pt,\xi},\pi^*_{sa_\pt,\xi} :
&=\text{argmin}_{\phi,\pi} \left(\F[\phi,sa_\pt,\xi]+\KL[\re(\hA_\thT|\pi)||\q(\hA_\thT|\phi)]\right)
\end{aligned}
- act according to \(\text{r}(\hat{a}_{t:\hat{T}}|\pi^*_{sa_{\prec t},\xi})\)
- can be used with any intrinsic motivation \(\mathfrak{M}\) at least in principle
- performance probably worse than first optimizing \(\phi\) and then \(\pi\):
4.Active inference
3. Active inference
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\begin{aligned}
\phi^*_{sa_\pt,\xi}:
&=\text{argmin}_{\phi} \; \F[\phi,sa_\pt,\xi] \\
\pi^*_{sa_\pt,\xi}:&=\text{argmin}_{\pi}\; \KL[\re(\hA_\thT|\pi)||\q(\hA_\thT|\phi^*_{sa_\pt,\xi})]
\end{aligned}
- computational advantage of active inference to the latter is unknown
- Background to intrinsic motivations
- Formal framework for intrinsic motivations
- Perception-action loop
- Generative model
- Prediction / complete posterior
- Action selection
- Some intrinsic motivations:
- Free Energy Minimization
- Predictive information maximization
- Knowledge seeking
- Empowerment maximization
- Curiosity
- Active inference
Overview
References:
Aslanides, J., Leike, J., and Hutter, M. (2017). Universal Reinforcement Learning Algorithms: Survey and Experiments. In Proceedings of the 26th International Joint Conference on Artificial Intelligence, pages 1403–1410.
Ay, N., Bertschinger, N., Der, R., Güttler, F., and Olbrich, E. (2008). Predictive Information and Explorative Behavior of Autonomous Robots. The European Physical Journal B-Condensed Matter and Complex Systems, 63(3):329–339.
Burda, Y., Edwards, H., Pathak, D., Storkey, A., Darrell, T., and Efros, A. A. (2018). Large-Scale Study of Curiosity-Driven Learning. arXiv:1808.04355 [cs, stat]. arXiv: 1808.04355.
Friston, K. J., Parr, T., and de Vries, B. (2017). The Graphical Brain: Belief Propagation and Active Inference. Network Neuroscience, 1(4):381–414.
Klyubin, A., Polani, D., and Nehaniv, C. (2005). Empowerment: A Universal Agent-Centric Measure of Control. In The 2005 IEEE Congress on Evolutionary Computation, 2005, volume 1, pages 128–135.
Orseau, L., Lattimore, T., and Hutter, M. (2013). Universal Knowledge-Seeking Agents for Stochastic Environments. In Jain, S., Munos, R., Stephan, F., and Zeugmann, T., editors, Algorithmic Learning Theory, number 8139 in Lecture Notes in Computer Science, pages 158–172. Springer Berlin Heidelberg.
Oudeyer, P.-Y. and Kaplan, F. (2008). How can we define intrinsic motivation? In Proceedings of the 8th International Conference on Epigenetic Robotics: Modeling Cognitive Development in Robotic Systems, Lund University Cognitive Studies, Lund: LUCS, Brighton. Lund University Cognitive Studies, Lund: LUCS, Brighton.
Schmidhuber, J. (2010). Formal Theory of Creativity, Fun, and Intrinsic Motivation (1990-2010). IEEE Transactions on Autonomous Mental Development, 2(3):230–247.
Storck, J., Hochreiter, S., and Schmidhuber, J. (1995). Reinforcement Driven Information Acquisition in Non-Deterministic Environments. In Proceedings of the International Conference on Artificial Neural Networks, volume 2, pages 159–164.
Guckelsberger, C., Salge, C., & Colton, S. (2016). Intrinsically Motivated General Companion NPCs via Coupled Empowerment Maximisation. 2016 IEEE Conf. Computational Intelligence in Games (CIG’16), 150–157
2. Formal framework for intrinsic motivations
4. Recognition model / approximate prediction
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\begin{aligned}
\phi^*_{sa_\pt,\xi}:&=\text{argmin}_\phi \;\F[\phi,sa_\pt,\xi]\\
&=\text{argmin}_\phi \;\sum_{\he_\pt} \int \re(\he_\pt,\theta|\phi) \log \frac{\re(\he_\pt,\theta|\phi)} {\q(s_\pet,\he_\pt,\theta|a_\pt,\xi)} \diff \theta\\
&=\text{argmin}_\phi \;- \log \q(s_\pt|a_\pt,\xi) + \KL[\re(\hE_\pt,\Theta|\phi)||\q(\hE_\pt,\Theta|sa_\pt,\xi)]\\
&=\text{argmin}_\phi\; \KL[\re(\hE_\pt,\Theta|\phi)||\q(\hE_\pt,\Theta|sa_\pt,\xi)]\\
\end{aligned}
- Each parameter value \(\phi\) specifies a "complete posterior-like" object
- not only \(\phi^*_{sa_{\prec t},\xi}\) (the optimized one
2. Formal framework for intrinsic motivations
4. Recognition model / approximate prediction
- Complete posteriors are computationally intractable
- approximation usually necessary
- introduce approximate complete posterior
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\newcommand{\re}{\text{r}}
\re(\hs_\thT,\he_{0:\hT},\theta|\ha_\thT,\phi)
- the parameter \(\phi\) ranges over a simpler class of conditional distributions than the true posterior
- an established way of obtaining the \(\phi^*\) that parameterizes the best approximation to the true posterior is variational inference
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\newcommand{\ha}{\hat{a}}
\newcommand{\he}{\hat{e}}
\newcommand{\hE}{\hat{E}}
\newcommand{\q}{\text{q}}
\newcommand{\re}{\text{r}}
\newcommand{\F}{\mathcal{F}}
\newcommand{\diff}{\text{d}}
\newcommand{\KL}{\text{KL}}
\begin{aligned}
\phi^*_{sa_\pt,\xi}:&=\text{argmin}_\phi \F[\phi,sa_\pt,\xi]\\
&=\text{argmin}_\phi \sum_{\he_\pt} \int \re(\he_\pt,\theta|\phi) \log \frac{\re(\he_\pt,\theta|\phi)}{\q(s_\pet,\he_\pt,\theta|a_\pt,\xi)} \diff \theta\\
&=\text{argmin}_\phi - \log \q(s_\pt|a_\pt,\xi) + \KL[\re(\hE_\pt,\Theta|\phi)||\q(\hE_\pt,\Theta|sa_\pt,\xi)]
\end{aligned}
2. Formal framework for intrinsic motivations
4. Recognition model / approximate prediction
\newcommand{\tT}{{t:T}}
\newcommand{\tm}{{t-1}}
\newcommand{\hT}{{\hat{T}}}
\newcommand{\thT}{{t:\hT}}
\newcommand{\hs}{\hat{s}}
\newcommand{\pt}{{\prec t}}
\newcommand{\pet}{{\preceq t}}
\newcommand{\ha}{\hat{a}}
\newcommand{\he}{\hat{e}}
\newcommand{\q}{\text{q}}
\newcommand{\re}{\text{r}}
\re(\hs_\thT,\he_{0:\hT},\theta|\ha_\thT,\phi):= \q(\hs_\thT,\he_\thT|\ha_\thT, \he_\tm,\theta)\re(\he_\pt, \theta|\phi)
\newcommand{\tT}{{t:T}}
\newcommand{\thT}{{t:\hT}}
\newcommand{\hs}{\hat{s}}
\newcommand{\pt}{{\prec t}}
\newcommand{\pet}{{\preceq t}}
\newcommand{\ha}{\hat{a}}
\newcommand{\he}{\hat{e}}
\newcommand{\q}{\text{q}}
\newcommand{\hT}{{\hat{T}}}
\newcommand{\diff}{\text{d}}
\q(\hs_\thT,\he_{0:\hT},\theta|\ha_\thT,sa_\pt,\xi) = \q(\hs_\thT,\he_\thT|\ha_\thT,\he_{t-1},\theta) \q(\he_{\prec t},\theta|sa_\pt,\xi)
- Complete posteriors are computationally intractable
- approximation usually necessary
- one option:
- approximate the posterior factor
- approximate the posterior factor
exact complete posteriors:
approximate complete posteriors:
\newcommand{\tT}{{t:T}}
\newcommand{\tm}{{t-1}}
\newcommand{\hT}{{\hat{T}}}
\newcommand{\thT}{{t:\hT}}
\newcommand{\hs}{\hat{s}}
\newcommand{\pt}{{\prec t}}
\newcommand{\pet}{{\preceq t}}
\newcommand{\ha}{\hat{a}}
\newcommand{\he}{\hat{e}}
\newcommand{\q}{\text{q}}
\newcommand{\re}{\text{r}}
\re(\hs_\thT,\he_{0:\hT},\theta|\ha_\thT,\phi):= \q(\hs_\thT,\he_\thT|\ha_\thT, \he_\tm,\theta)\re(\he_\pt, \theta|\phi)
into posterior and predictive factor:
2. Formal framework for intrinsic motivations
4. Recognition model / approximate prediction
2. Formal framework for intrinsic motivations
4. Recognition model / approximate prediction
\newcommand{\tT}{{t:T}}
\newcommand{\tm}{{t-1}}
\newcommand{\hT}{{\hat{T}}}
\newcommand{\thT}{{t:\hT}}
\newcommand{\hs}{\hat{s}}
\newcommand{\pt}{{\prec t}}
\newcommand{\pet}{{\preceq t}}
\newcommand{\ha}{\hat{a}}
\newcommand{\he}{\hat{e}}
\newcommand{\q}{\text{q}}
\newcommand{\re}{\text{r}}
\re(\hs_\thT,\he_{0:\hT},\theta|\ha_\thT,\phi):= \q(\hs_\thT,\he_\thT|\ha_\thT, \he_\tm,\theta)\re(\he_\pt, \theta|\phi)
\newcommand{\tT}{{t:T}}
\newcommand{\thT}{{t:\hT}}
\newcommand{\hs}{\hat{s}}
\newcommand{\pt}{{\prec t}}
\newcommand{\pet}{{\preceq t}}
\newcommand{\ha}{\hat{a}}
\newcommand{\he}{\hat{e}}
\newcommand{\q}{\text{q}}
\newcommand{\hT}{{\hat{T}}}
\newcommand{\diff}{\text{d}}
\q(\hs_\thT,\he_{0:\hT},\theta|\ha_\thT,sa_\pt,\xi) = \q(\hs_\thT,\he_\thT|\ha_\thT,\he_{t-1},\theta) \q(\he_{\prec t},\theta|sa_\pt,\xi)
- Complete posteriors are computationally intractable
- approximation necessary
- one option:
- approximate the posterior factor
exact complete posteriors:
approximate complete posteriors:
- Assume independence of hidden variables i.e. \(\Theta^i\) and \(\hat{E}_{\prec t}\)
- parameterize their individual distributions by \(\phi=\{\phi^{E_0},...,\phi^{E_{t-1}},\phi^1,\phi^2,\phi^3\}\)
2. Formal framework for intrinsic motivations
4. Recognition model / approximate prediction
Call \(\text{q}(\hat{s}_{t:\hat{T}},\hat{e}_{0:\hat{T}},\theta|\hat{a}_{t:\hat{T}},sa_{\prec t},\xi)\) the complete posteriors.
2. Formal framework for intrinsic motivations
4. Recognition model / approximate prediction
3. Some intrinsic motivations
1. Free energy minimization
Intuition:
\newcommand{\hT}{\hat{T}}
\newcommand{\thT}{{t:\hT}}
\newcommand{\hs}{\hat{s}}
\newcommand{\pt}{{\prec t}}
\newcommand{\pet}{{\preceq t}}
\newcommand{\set}{{\succeq t}}
\newcommand{\ha}{\hat{a}}
\newcommand{\he}{\hat{e}}
\newcommand{\q}{\text{q}}
\newcommand{\diff}{\text{d}}
\newcommand{\ptau}{{\prec \tau}}
\newcommand{\petau}{{\preceq \tau}}
\newcommand{\stau}{{\succ \tau}}
\newcommand{\setau}{{\succeq \tau}}
\newcommand{\argmax}{\text{argmax}}
\newcommand{\mot}{\mathfrak{M}}
\newcommand{\HS}{\text{H}}
\newcommand{\hS}{\hat{S}}
\newcommand{\hE}{\hat{E}}
\q(\he_\thT|\ha_\thT,sa_\pt,\xi)=
\int \sum_{\hs_\thT,\he_\pt} \q(\hs_\thT,\he_{0:\hT},\theta|\ha_\thT,sa_\pt,\xi) \diff \theta
\newcommand{\hT}{\hat{T}}
\newcommand{\thT}{{t:\hT}}
\newcommand{\hs}{\hat{s}}
\newcommand{\pt}{{\prec t}}
\newcommand{\pet}{{\preceq t}}
\newcommand{\set}{{\succeq t}}
\newcommand{\ha}{\hat{a}}
\newcommand{\he}{\hat{e}}
\newcommand{\q}{\text{q}}
\newcommand{\diff}{\text{d}}
\newcommand{\ptau}{{\prec \tau}}
\newcommand{\petau}{{\preceq \tau}}
\newcommand{\stau}{{\succ \tau}}
\newcommand{\setau}{{\succeq \tau}}
\newcommand{\argmax}{\text{argmax}}
\newcommand{\mot}{\mathfrak{M}}
\newcommand{\HS}{\text{H}}
\newcommand{\hS}{\hat{S}}
\newcommand{\hE}{\hat{E}}
\begin{aligned}
\mot(\q(.,.,.|.,\xi),\ha_\thT) :&=-\HS_{\q}(\hS_\thT|\hE_\thT,\ha_\thT)\\
&= \sum_{\he_\thT} \q(\he_\thT|\ha_\thT) \sum_{\hs_\thT} \q(\hs_\thT|\he_\thT) \log \q(\hs_\thT|\he_\thT)\\
\end{aligned}
\newcommand{\hT}{\hat{T}}
\newcommand{\thT}{{t:\hT}}
\newcommand{\hs}{\hat{s}}
\newcommand{\pt}{{\prec t}}
\newcommand{\pet}{{\preceq t}}
\newcommand{\set}{{\succeq t}}
\newcommand{\ha}{\hat{a}}
\newcommand{\he}{\hat{e}}
\newcommand{\q}{\text{q}}
\newcommand{\diff}{\text{d}}
\newcommand{\ptau}{{\prec \tau}}
\newcommand{\petau}{{\preceq \tau}}
\newcommand{\stau}{{\succ \tau}}
\newcommand{\setau}{{\succeq \tau}}
\newcommand{\argmax}{\text{argmax}}
\newcommand{\mot}{\mathfrak{M}}
\newcommand{\HS}{\text{H}}
\newcommand{\hS}{\hat{S}}
\newcommand{\hE}{\hat{E}}
\begin{aligned}
\mot(\q(.,.,.|.,\xi),\ha_\thT) :&= \sum_{\he_\thT} \q(\he_\thT|\ha_\thT) \sum_{\hs_\thT} \q(\hs_\thT|\he_\thT) \log \q(\hs_\thT|\he_\thT)\\
&= - \sum_{\he_\thT} \q(\he_\thT|\ha_\thT)\HS_{\q}(\hS_\thT|\he_\thT)\\
&=-\HS_{\q}(\hS_\thT|\hE_\thT,\ha_\thT)\\
\end{aligned}
2. Formal framework for intrinsic motivations
2. Generative model
- In standard POMDP case all \(\Theta\) have specified values \(\Theta=\theta\)
- for \(\hat{e}_t:=\hat{s}\hat{a}_{\prec t}\) this model becomes identical to the one used in general reinforcement learning
3. Some intrinsic motivations
Remarks:
- here Occams razor / the preference of simplicity was hidden in prior
- Algorithmic information theory formalizes the idea of complexity for strings and can be used for less probabilistic intrinsic motivations (see Schmidhuber, 2010)
- Aslanides et al. (2017) have used knowledge seeking, Shannon knowledge seeking, and minimium description length improvement to augment AIXI
- great testbed and running code: http://www.hutter1.net/aixijs/
Background on intrinsic motivations
AGI:
- implement predictive model that continually improves through experience
- implement action selection / optimization that chooses according to the prediction
- drive it by open ended intrinsic motivation
2. Formal framework for intrinsic motivations
4. Action selection
- complete posterior \(\text{q}(\hat{s}_{t:\hat{T}},\hat{e}_{0:\hat{T}},\theta|\hat{a}_{t:\hat{T}},sa_{\prec t},\xi)\) captures consequences of actions for relations between
- define intrinsic motivations as functions \(\mathfrak{M}\) of this posterior and a given sequence \(\hat{a}_{t:\hat{T}}\) of future actions:
\newcommand{\tT}{{t:T}}
\newcommand{\hT}{\hat{T}}
\newcommand{\thT}{{t:\hT}}
\newcommand{\hs}{\hat{s}}
\newcommand{\pt}{{\prec t}}
\newcommand{\pet}{{\preceq t}}
\newcommand{\set}{{\succeq t}}
\newcommand{\ha}{\hat{a}}
\newcommand{\he}{\hat{e}}
\newcommand{\q}{\text{q}}
\newcommand{\diff}{\text{d}}
\newcommand{\ptau}{{\prec \tau}}
\newcommand{\petau}{{\preceq \tau}}
\newcommand{\stau}{{\succ \tau}}
\newcommand{\setau}{{\succeq \tau}}
\newcommand{\mot}{\mathfrak{M}}
Q(\ha_\thT,sa_\pt,\xi):= \mot(\q(.,.,.|.,sa_\pt,\xi),\ha_\thT)
- Requirement is a conditional probability \(\text{q}(\hat{s}_{t:\hat{T}},\hat{e}_{0:\hat{T}},\theta|\hat{a}_{t:\hat{T}})\) how this is obtained by agent does not matter.
2. Formal framework for intrinsic motivations
3. Prediction
Allows to predict consequences of future actions \(\blue{\hat{a}_{t:\hat{T}}}\):
\newcommand{\tT}{{t:T}}
\newcommand{\thT}{{t:\hT}}
\newcommand{\hs}{\hat{s}}
\newcommand{\pt}{{\prec t}}
\newcommand{\pet}{{\preceq t}}
\newcommand{\ha}{\hat{a}}
\newcommand{\he}{\hat{e}}
\newcommand{\q}{{q}}
\newcommand{\hT}{{\hat{T}}}
\newcommand{\diff}{\text{d}}
\q(\hs_\thT,\he_{0:\hT},\theta|\ha_\thT,sa_\pt,\xi)= \frac{\q(s_\pt,\hs_\thT,\he_{0:\hT},a_\pt,\ha_\thT,\theta,\xi)}{\int\sum_{\hs_\thT,\he_{0:\hT}}\q(s_\pt,\hs_\thT,\he_{0:\hT},a_\pt,\ha_\thT,\theta,\xi)\diff \theta }
2. Formal framework for intrinsic motivations
2. Generative model
Model split up into three parts:
- sensor dynamics model \(\newcommand{\q}{\text{q}}\newcommand{\hs}{\hat{s}}\q(\hat{s}|\hat{e},\theta)\)
- environment dynamics model \(\newcommand{\q}{\text{q}}\newcommand{\ha}{\hat{a}}\q(\hat{e}'|\ha,\hat{e},\theta)\)
- initial environment distribution \(\newcommand{\q}{\text{q}}\newcommand{\hs}{\hat{s}}\q(\hat{e}|\theta)\)
Extending active inference to other intrinsic motivations
By slides_martin
Extending active inference to other intrinsic motivations
Slides for presentation at Sussex University on 28 August 2018
