A Monte Carlo Algorithm for Universally Optimal Bayesian Sequence Prediction and Planning

01/18/2010 ∙ by Anthony Di Franco, et al. ∙ 0

The aim of this work is to address the question of whether we can in principle design rational decision-making agents or artificial intelligences embedded in computable physics such that their decisions are optimal in reasonable mathematical senses. Recent developments in rare event probability estimation, recursive bayesian inference, neural networks, and probabilistic planning are sufficient to explicitly approximate reinforcement learners of the AIXI style with non-trivial model classes (here, the class of resource-bounded Turing machines). Consideration of the effects of resource limitations in a concrete implementation leads to insights about possible architectures for learning systems using optimal decision makers as components.

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Abstract

The aim of this work is to address the question of whether we can in principle design rational decision-making agents or artificial intelligences embedded in computable physics such that their decisions are optimal in reasonable mathematical senses. Recent developments in rare event probability estimation, recursive bayesian inference, neural networks, and probabilistic planning are sufficient to explicitly approximate reinforcement learners of the AIXI style with non-trivial model classes (here, the class of resource-bounded Turing machines). Consideration of the effects of resource limitations in a concrete implementation leads to insights about possible architectures for learning systems using optimal decision makers as components.

Keywords

recurrent neural network, Turing machine, AIXI, reinforcement learning, planning, sequence prediction, Universal prior

1 Introduction

Hutter [1]

defines the universal algorithmic agent AIXI, a Pareto-optimal policy for arbitrary computable stochastic environments. Here we consider a Monte Carlo algorithm for sequence prediction and planning in a Turing-universal model class which seeks to statistically approximate resource-bounded AIXI while avoiding the enumerative search in similarly powerful algorithms replacing it with statistical approximations to the governing probability distributions.

Hutter held that “AIXI is computationally intractible, however, AIXI can serve as a gold standard for A[rtificial] G[eneral] I[ntelligence] and AGI research.” [2] The main difficulty in computing AIXI in a universal model class is obtaining the algorithmic probabilities of the hypotheses, since the algorithmic probabilities are only limit-computable. Previous approaches to learning in this class enumerate proofs or enumerate and run programs in a hypothesis space with their executions interleaved [3], [4]. We take a different approach characterized by two main points: 1) The computation runs in tandem with the environmental process, and to succeed must produce the correct result at the same time the environment demands it. Thus there is an implicit “speed prior” [5] imposed on the model parameterized by the frequency of operation of the model with respect to the environment, or, alternatively, we are approximating AIXI- with a time bound equal to the relative frequencies of the model’s cycle and the environmental inputs, and length bound determined by the space available for the model representation, but we rely directly on statistically valid approximations rather than proofs of valid approximation. 2) Otherwise, hypotheses are weighted according to their representation-length complexity as in AIXI.

This part presents the algorithm, with experimental results deferred to the second part, which is in preparation.

2 Architecture

Using the equivalence between recurrent neural networks and Turing machines [6], we choose recurrent neural networks as the model class. Additional information on the relationships between recurrent neural network architectures and computational classes is available in [7]

. The main advantages of this representation are that 1) it is parameterized by vectors of dense numbers and therefore straightforward to sample from in a meaningful and statistically efficient way using established Monte Carlo techniques, and 2) while recurrent neural networks embed the class of Turing machines, they can be counted on to produce meaningful outputs at any point in their operation, so do not suffer from the halting problem in a way that presents algorithmic difficulties.

Among the class of recurrent neural networks, we choose a modified version of Long Short-Term Memory

[8]

because it has established state-of-the-art performance on benchmark tasks and because it can easily be seen as a continuously-parameterized flip-flop fed and controlled by perceptrons, making the the interpretation of the model more straightforward, and constraining it to perform in a way known to be useful. This is a different motivation from its original one which was to prevent the premature decay of error signals in training. The modification here is to add a “bypass” gate which mixes the computation directly into the output, rather than forcing the computation to first be stored in the state before being output, so that a cell can act either as a memory or as a computational element according to circumstance.

The network outputs are made to predict the network inputs at the next time step. Network inputs are presented in a binary encoding, so that network outputs can be treated directly as a probability (that of observing a set input bit) without less general assumptions on the probability distributions of network inputs.

To plan, we present as input to the network a concatenation of the observed environmental inputs (of which some subset is interpreted as a reward signal) and the agent’s own outputs, and make the network generate samples of likely continuation sequences by predicting a distribution over inputs and outputs for the next timestep, sampling from the predicted distribution, predicting again by taking the sample as if it had been observed, and so on recursively, and generated samples from likely continuation sequences to plan by generating them out to a specified horizon and selecting the next action in the continuation that yields the most aggregate reward in the sample.

3 Algorithm

Overview

We first obtain a sample from the Universal prior over the model class. This must be done only one for a certain choice of model class (i.e. network topology). Thereafter, the algorithm operates in discrete timesteps. At each timestep, the current environmental inputs and the agent’s most recent outputs are presented as inputs to the network, and the network updates its estimate of the neural network parameters according to the probabilities it previously assigned to the observed inputs. If an action is called for at the current timestep, the current estimate of the network parameters is used to generate likely continuation sequences out to a specified time horizon, generating a likely continuation of the sequence of network inputs, which consist of (environmental input, agent output) pairs and where the environmental input contains a reward signal. The next action is chosen as the agent output that serves as a partial prefix to the set of continuation sequences with the most expected reward.

3.1 Prediction

Prediction has two components, the first being obtaining a useful sample from the Universal prior, and the second being recursively updating that sample to reflect the posterior distribution in light of the data. For the sample from the Universal prior, we face the practical difficulty that we wish to have significant numbers of samples from regions of the sample space with exponentially decreasing probability, and so turn to techniques from rare-event probability estimation. For the computation of the posterior probability, we rely on standard techniques from recursive Bayesian inference.

We seek to sample models from the model class with large weights under the following distribution so as to use

, the Universal posterior, as our predictor, that is, we take the prior probabilities of hypotheses to be their Universal probabilities and maintain their posterior likelihoods by Bayes’ rule (equations and notation from

[1]):

We measure the encoding-length (Kolmogorov) complexity of the mapping from current states and inputs to subsequent states and outputs with the quantity from Flat Minimum Search [9] which gives the number of bits required to encode the network’s parameters to maintain a certain amount of precision in the mapping it represents. In the general case of multiple inputs going to multiple outputs, the full quantity is required, but in this implementation where each weight directly influences only one output within an iteration, the following simplification obtains, (with a constant term omitted as being irrelevant,) a quantity reminiscent of the Fisher information:

We might also do this by following up to the determination of the gradient the RTRL algorithm, or an equivalent technique given a different choice of RNN architecture, in which case we might then maintain the prior and posterior terms in the weighting separately, and recompute the prior term with the recursive formula for the RTRL partial derivatives, and update the posterior term recursively according to the formula given previously for . However, the previous paragraph’s method is most appropriate to the interpretation of the recurrent neural network as a Turing machine, with the states serving as a bounded work tape, and the neural network map serving as the state-transition dynamics, and is preferred here.

Using to approximate , we initialize a sample set of size by rare-event probability estimation by strata and acceptance / rejection according to the Universal prior distribution parameterized by . That is, given a proportion between zero and one, say one half, we generate samples from according to with Metropolis-Hastings where is the encoding complexity of a state vector sampled jointly with the weights of , then within the upper

th quantile of the sample, we construct a kernel density estimator and recursively repeat the

th quantile sample with the quantile, recording the boundary value of the quantile at each step of the recursion as a sequence of levels. Given the sequence of levels, we repeat the process with the level thresholds replacing the quantiles. Notably, we obtain from this process an estimate of the normalization constant of the Universal distribution, though it is unused here. See [10].

1. Choose a sample size , number of levels desired , and quantile ; suggested are , , .

2. Generate an initial set of samples with Metropolis-Hastings by repeatedly drawing from a (say) multivariate Gaussian or Laplacian proposal centered at the current sample, accepting samples as their bounded likelihood ratio

is greater than a draw from a uniform(0,1) random variable.

3. Proceed with algorithms 2.2 and 2.1 of [10] with a transition kernel

given by a kernel density estimator over the sample, suggested is the multivariate Gaussian or Laplace distribution with the sample variances or covariances.

Algorithm 1 Sampling from the Universal Prior

Having thus obtained a sample from the Universal prior, which must only be done once per instantiation of a network of a certain architecture, we then update the posterior term recursively according to the formula given previously for .

Upon a degeneration of the sample as indicated by the variance of the weights falling below a given threshold, we may resample by replicating and replacing samples with samples from a kernel density estimate according to their weights as in the sequential Monte Carlo literature, especially [11]. We can include in each

a representation of the state vector, that is, sample states and dynamics as a block, or maintain separate estimates conditional on one another as is more common in Kalman-filtered recurrent neural networks. Here we choose the latter. We may choose to use multiple RNN simulation steps per time step of the environment to relax the time bound on the computation. Our estimate of the probability of an input bit being set is then given by the

-weighted sum of our samples from .

Denote the current observation by , the previous set of weights by , then:

1. Compute the probability of each sample in the sample set by propagating each of the current weighted state samples through the network evolution represented by , and taking the likelihood of the resulting multivariate Bernoulli with respect to the observed bits;

2. Update the weights using the resulting likelihoods according to .

3. If the effective sample size has fallen below a pre-determined threshold, say 50%, then:

3a. Sample with replacement from the set of with probabilies and assign each a unit weight to form a new sample with the same distribution;

3b. Form a kernel density estimator for the new sample and take a new sample from it, assigning each unit weight. Suggested is the shrunk kernel density estimator given, for a kernel bandwidth , by computing , translating each parameter in each sample towards the sample mean by taking the linear combination , then sampling from a kernel parameterized by the translated particles and a covariance of , being the sample variance matrix. (See [11])

4. Sample a new set of states by repeating steps 2 and 3 with the roles of the network parameters and states reversed.

Algorithm 2 Recursive Bayesian Posterior Update

3.2 Planning

This model is observable, and state-determined, so the prerequisites for the application of the approximately optimal stochastic planning algorithm in [12] are satisfied. To plan, we jointly model environmental inputs, including the reward signal, and agent actions, and sample continuations of the joint sequence of these. We obtain a distribution over the future rewards of certain actions or action sequences by grouping continuation samples according to the actions or action sequences they contain, and plan by selecting at each time step the next action with the highest expected total future reward. Arbitrary discounting is easily accomodated by applying the appropriate factor to the sum at each step in the continuation sequence. As an optimization, if the environment is sufficiently stationary and sufficiently well-approximated by the model, it is conceivable that prefixes longer than one timestep in length could be chosen and executed without re-running the planner, but this is not considered further here.

At each time step when an action is called for:

1. Generate a probability distribution over continuations of network input sequences by making a copy of the state samples, and iterating steps 2 - 4 of algorithm 2 on the state copy but without modifying the estimate of the network parameters, until a sample of continuation sequences has been produced out to the desired time horizon.

2. Group the continuation sequences according to their sharing a common next agent action, and take the expected discounted future reward within each group.

3. Perform the action with the highest expected discounted future reward.

Algorithm 3 Planning

3.3 Remarks

The algorithmic-complexity-based parameter smoothing accomplishes a form of principled dimensionality reduction within the upper bound imposed by the dimension of the representation. A reversible-jump Monte Carlo sampler could be used to select the dimensionality without an explicit upper bound [13].

To model input-output relationships, input-output pairs should be presented as network inputs, with use of the planner being unnecessary, although the more typical approach analogous to the operation of other neural networks of giving the input as the network input, calling for the output as the agent action, and giving a reward signal inversely related to the output error should also work, but waste planning effort.

3.4 Implementation notes

A prototype implementation is available at http://code.google.com/p/machine-tools in the SolomonoffNet module.

We compute the partial derivatives of network outputs with respect to weights, the key input to the computation of

, using forward-mode automatic differentiation, a choice made mostly for implementation convenience and because in the general case here, where input dimension equals (or is less than) output dimension, simply-implemented forward-mode differentiation is as efficient as (resp. more efficient than) the reverse-mode (backpropagation) more commonly encountered in neural network algorithms.

4 Outlook

This paper applies sampling to effect AIXI-style learning in a specific Turing-universal model class, but analogous techniques apply to any model class where the parameters can be sampled, with appropriate modifications to take into account the needs of sampling from another such model class, particularly the determination of the encoding complexity, and the resulting mixture estimate is as close to the true distribution as the model class permits [1].

In the recurrent network setting the map encoding complexity of the system dynamics is equivalent to the information rate of the corresponding process and the topological entropy of the corresponding dynamical system. Together with the sampling method presented here for the Universal prior, it could be used to numerically characterize the complexity of other classes of dynamical systems.

Approximately, but neglecting the encoding complexity structure of the parameter space, the size of the parameter space to be sampled grows exponentially with network dimension, as does thus the cost associated with maintaining the sample set, and the costs of other implementation details such as the Jacobian accumulation and the peephole connections across modules in LSTM grow faster than linearly. To mitigate these costs, a large network may be constructed as a set of small network modules with the ability to view one another’s outputs or states as additional environmental inputs, and a perhaps probabilistic locality structure can be imposed to limit the interconnection costs while maintaining bandwidth among computational nodes in the network, such as a topology with power-law node degrees. An analogy can be drawn to decoupled extended Kalman filter training of recurrent neural networks, though the interactions of the modules require a more powerful theoretical framework to analyze because of the bounded-Universal power of the models. Computability Logic [14] may be such a framework, wherein the network modules can take each other as oracles. Additionally, one can consider allowing these modules, capable of arbitrary computation, to interact with other special-purpose modules, such as smoothers of system input / output, pattern-associative memories, chess programs, etc. depending on the problem domain.

This motivation for decoupling the large network into modules and to view interactions of modules with their environment in terms of computabilty logic suggests a model for the mammalian brain where cortical columns are identified with resource-bounded general-purpose-computation modules and other parts of the brain are either computational oracles or filters on input / output with the environment. Suppose that the thalamus serves as the communication network by which the modules are connected with one another over long ranges and with other oracles / filters in the brain and the (perhaps filtered) environmental inputs / agent controls, and is responsible for distributing reward signal information across the computational modules; that the hippocampus serves as a pattern-associative memory in tandem with the cortical ensemble; that the cerebellum serves as motor output consolidation and smoothing. Working under that hypothesis, the functional specialization of different regions of cortex might be derived from considerations of the computational capacity of individual modules and the information bandwidth of the connections between groups of modules in a region, other regions relevant to its function, and the locations in the network of the relevant environmental inputs / outputs, rather than any sort of a priori enforced anatomical specialization. Also, the brain can thus be seen to implement a solution to the prediction and planning problems that is optimal in a resource-bounded, stochastic sense, and quite general.

Separately, but also based on the idea of an architecture for an intelligent agent built around a communication network that transmits data from the environment to a set of computational units, permits these computational units to communicate with one another, and transmits commands back to be executed upon the environment by the agent’s effectors, this agent architecture suggest a means of quantifying Boyd’s Observe-Orient-Decide-Act loop conceptual framework [15] for analyzing strategic interactions, where observation is the capacity in units of bits per second to provide sensory information to the computational units, orientation is the capacity of the computational units to inform one another of computed characterizations of the environment and is bounded by their speed of computation and the capacity of the communication network among them measured in bits per second, decision is equated with the agent’s planning capacity as manifested by the sequence of outputs it produces and the capacity of the communication network to communicate them to the effectors and is also measured in bits per second, and action is equated with actuation’s ability to affect the environment in units of joules per second. Thus Boyd’s Energy-Maneuverability theory can also be quantified in units of bit-Joules per second. It also suggests a potential approach to making mathematically rigorous his references to Heisenberg’s uncertainty principle, Goedel’s incompleteness theorem, and the second law of thermodynamics, which were merely illustrative in his presentation of his ideas, by formulating analogous statements within the context of statistics and algorithmic probability of AIXI.

Schmidhuber’s history-compressing neural networks [16] implemented, in a sense, a less general form of computational modules using one another as oracles, where the modules were arranged in a heirarchy and learned abbreviated representations of ever-longer regular input sequences.

Notes

This work presents a refined version of the algorithm described in the extended abstract “Overview of A Monte Carlo Algorithm for Universally Optimal Bayesian Sequence Prediction and Planning” [17].

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