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Finding Approximate POMDP solutions Through Belief Compression

N. Roy, G. Gordon, S. Thrun

arXiv:1107.0053v2cs.AI

TL;DR

Large POMDPs are difficult because exact planning spans a continuous, high-dimensional belief space. The paper compresses structured beliefs with Exponential family PCA and plans in the resulting low-dimensional space, scaling to substantially larger robot-navigation and related problems while retaining approximation and model-structure limitations.

  • Problem

    Exact POMDP value-function methods are computationally intractable for large models because they optimize over continuous, high-dimensional belief spaces.

  • Method

    The paper learns low-dimensional representations of sparse beliefs with Exponential family PCA and performs planning in the reduced belief space.

  • Results

    The method solved POMDPs an order of magnitude larger than conventional value-iteration techniques and scaled to low-dimensional surfaces in very high-dimensional spaces.

  • Takeaways & Limitations

    Structured belief spaces can support approximate POMDP policies for synthetic, robot-navigation, and factored-belief problems beyond the scale of conventional techniques.

  • Takeaways & Limitations

    E-PCA is most useful when beliefs are sparse and low-dimensional, and may fail when a POMDP lacks that structure.

Abstract

from arXiv · show

Standard value function approaches to finding policies for Partially Observable Markov Decision Processes (POMDPs) are generally considered to be intractable for large models. The intractability of these algorithms is to a large extent a consequence of computing an exact, optimal policy over the entire belief space. However, in real-world POMDP problems, computing the optimal policy for the full belief space is often unnecessary for good control even for problems with complicated policy classes. The beliefs experienced by the controller often lie near a structured, low-dimensional subspace embedded in the high-dimensional belief space. Finding a good approximation to the optimal value function for only this subspace can be much easier than computing the full value function. We introduce a new method for solving large-scale POMDPs by reducing the dimensionality of the belief space. We use Exponential family Principal Components Analysis (Collins, Dasgupta and Schapire, 2002) to represent sparse, high-dimensional belief spaces using small sets of learned features of the belief state. We then plan only in terms of the low-dimensional belief features. By planning in this low-dimensional space, we can find policies for POMDP models that are orders of magnitude larger than models that can be handled by conventional techniques. We demonstrate the use of this algorithm on a synthetic problem and on mobile robot navigation tasks.

1. Introduction

Real-world robots must make decisions under uncertainty, but conventional controllers often collapse probabilistic state estimates to a single state. The paper proposes approximate POMDP planning in a learned low-dimensional belief representation to make larger problems tractable.

  • Real-world environments are uncertain because sensors are noisy and the complete world state is not always observable.
  • Conventional controllers often extract a single best state from a full probability distribution, which can cause substantial control errors when uncertainty is high.
  • A controller that models positional uncertainty can follow walls to keep uncertainty small and arrive reliably at the goal.
  • POMDP controllers use probabilistic state estimates and can choose actions that reduce uncertainty, but exact optimal-policy computation is computationally intractable.
  • Exact POMDP techniques typically handle only a few hundred states, whereas physical robot problems may involve thousands of states, observations, and actions.
  • The paper approximates real-world POMDP solutions by learning low-dimensional belief representations and planning directly in that reduced space.

2. Partially Observable Markov Decision Processes

POMDPs represent uncertainty with beliefs and define value over a continuous, high-dimensional belief space, making exact planning infeasible for large models. The paper motivates compressing structured navigation beliefs into a low-dimensional representation while accepting approximation and loss of convexity.

  • A POMDP specifies states, actions, observations, transition and observation probabilities, rewards, a discount factor, and an initial belief.
  • Each point in a three-state belief simplex represents a probability distribution, while the value function gives a policy’s long-term expected reward from that belief.
  • The belief space has one fewer dimension than the number of states, so navigation models with thousands of states require optimization over a continuous space with many thousands of dimensions.
  • Navigation beliefs often have few modes and constrained changes in shape, suggesting concentration near a structured low-dimensional surface.
  • Representing beliefs with a small set of bases or features allows planning in terms of reduced-dimensional belief coordinates.
  • The representation trades completeness for tractability and can invalidate POMDP algorithms that rely on a convex value function after nonlinear transformation.

3. Dimensionality Reduction

The paper models experienced beliefs as lying near a structured, low-dimensional surface and evaluates dimensionality reduction on a 200-state toy maze. PCA provides a baseline, but its reconstructions poorly preserve probability-distribution structure.

  • Dimensionality Reduction: Dimensionality reduction searches for a low-dimensional surface embedded in belief space that passes near sampled belief trajectories.The representation may be lossy, requiring a trade-off between preserving important distributional structure and using few dimensions.
  • PCA: PCA finds a linear lower-dimensional representation by preserving the variance of reconstructed data after projection onto a hyperplane.Beliefs are assembled as columns of B, with U providing basis vectors and ˜B providing low-dimensional coordinates.
  • Toy Problem: The toy evaluation uses a 200-state maze with two circular corridors, four actions, and observations that separately reveal corridor identity or approximate horizontal position.The agent must use the corridor-sensing action because movement observations do not identify which corridor it occupies.
  • Toy Problem: The toy problem supplies 500 sampled beliefs, each resembling two discretized von Mises distributions with different weights for the two corridor halves.The initial belief assigns equal probability to both corridors and uses concentration parameter 1.0 for position.
  • PCA Performance: The evaluation compares average squared L2 error with average Kullback-Leibler divergence, with KL divergence treated as more appropriate for comparing probability distributions.Squared L2 is PCA’s explicit objective, whereas KL divergence measures distributional difference more directly.
  • PCA Performance: PCA performs poorly on these probability distributions, requiring between 10 and 15 basis functions despite only three degrees of freedom.Reconstructions exhibit ringing and negative values because PCA does not enforce nonnegativity or unit-sum constraints; errors are especially significant near zero probabilities.

4. Exponential Family PCA

Exponential Family PCA adapts dimensionality reduction to probability distributions through likelihood-based link functions and a generalized divergence objective. The resulting optimization alternates convex subproblems but can still encounter joint local minima.

  • E-PCA Model: E-PCA represents reconstructed data with a low-dimensional weight vector, basis matrix U, and link function f.The link function maps data into a space where linear representation is possible and is determined by the assumed likelihood model.
  • E-PCA Model: E-PCA parameters are learned by maximizing data log-likelihood, equivalently minimizing a generalized Bregman divergence using convex optimization techniques.The divergence is defined through a convex function F, its derivative f, and convex dual F∗.
  • Belief Compression: For belief compression, the exponential link function yields an unnormalized KL-divergence loss for reconstructing probability distributions.The exponential link corresponds to a Poisson error model for each reconstructed belief component.
  • Belief Compression: The exponential link constrains reconstructions to be positive and increases penalties for errors in small probabilities.These properties address PCA’s negative reconstructions and its substantial errors near zero.
  • Optimization: Unlike conventional PCA, E-PCA may have multiple local minima because optimizing U and ˜B jointly is nonconvex.With one matrix fixed, solving for the other is convex and has a unique global solution.
  • Optimization: The algorithm alternates updates of U and ˜B using Newton-style optimization, weighted least squares, regularization, and termination based on iterations or minimum error.After U is found, obtaining a low-dimensional representation for a belief is convex, and reconstructing a full-dimensional belief is straightforward.
  • Limitations: The authors expect local-minimum issues may require additional solutions when scaling E-PCA to more complicated domains.They did not encounter this problem in their experiments.

1. Collect a set of sample beliefs from the high-dimensional belief space

The first algorithm step assembles sampled beliefs into a data matrix for dimensionality reduction.

  • Assemble the sampled beliefs into the data matrix B = [b1| . . . |b|B|].

4. Fix an initial estimate for ˜B and U randomly

The algorithm initializes the low-dimensional representation and basis matrix before optimization, using random starting estimates.

  • Fix initial estimates for ˜B and U randomly before iterating the E-PCA optimization.The factorization uses a two-part representation B ≈ f(U ˜B), without requiring the factors to be orthonormal.

5. E-PCA Performance

E-PCA produced compact belief representations with substantially lower reconstruction error than conventional PCA on synthetic and robot-navigation beliefs. In navigation tasks, hundreds of state dimensions were reduced to 5 or 6 bases while retaining near-zero average KL divergence.

  • Abstract problem: 0.018 KL divergence was achieved after 4 bases for the abstract problem, with approximately 2 × 10^-9 error in a low-probability region.The reconstruction became visually indistinguishable from the original belief using 4 bases.
  • Abstract problem: Using 3 bases, E-PCA was substantially better than PCA using 10 bases, although small errors remained at the peaks of the two modes.With 4 bases, the E-PCA reconstruction was indistinguishable to the naked eye from the original belief.
  • Robot navigation tasks: In the Longwood navigation problem, average KL divergence fell very close to 0 around 6 bases, with minimal improvement thereafter.The belief space covered 53.6m × 37.9m at .5m × .5m resolution.

6. Computing POMDP policies

Because nonlinear belief compression removes the convexity needed by conventional POMDP value iteration, the paper constructs a discrete low-dimensional belief-space MDP and solves it with fitted MDP value iteration. Approximate rewards and transitions are computed by reconstructing full beliefs, applying the original POMDP dynamics, recompressing, and discretizing the result.

  • Representation scale: The compressed navigation representations reduced approximately 800-state and 2,000-state belief spaces to 5 or 6 bases, enabling more tractable value-function computation.The reported navigation reductions motivate planning in the low-dimensional space.
  • Policy computation: Nonlinear recovery of low-dimensional belief coordinates destroys value-function convexity, so the value function cannot be represented as a supremum of hyperplanes.This prevents direct use of conventional POMDP value iteration in the compressed space.
  • Policy computation: The method builds a discrete MDP over low-dimensional beliefs and applies MDP value iteration instead of conventional POMDP value iteration.The discrete space is combined with function approximation because the compressed value-function form is unknown.
  • Policy computation: Fitted value iteration uses averagers, including regular grids and 1-nearest-neighbour variable-resolution grids, to avoid exaggerating training-data errors in max-norm.The paper invokes Gordon’s bounded-error guarantee for averager-based fitted value iteration.
  • Reward function: Approximate rewards are expected immediate rewards under the reconstructed full belief, requiring recovery of the high-dimensional belief from its low-dimensional representation.The resulting reward model can favor beliefs centered on high-reward states with low uncertainty.
  • Transition function: Approximate transitions reconstruct a full belief, apply action and observation updates, recompress the resulting belief, and map it to a nearby sampled low-dimensional state.Grid and nearest-neighbour approximators determine how the recompressed belief is represented in the discrete state set.
  • Transition function: The total transition probability sums, over observations, the probability of reaching each posterior low-dimensional belief from a prior belief and action.The computation shares work across posterior beliefs reachable from the same prior belief and action.

7. Solving Large POMDPs

The E-PCA POMDP algorithm was evaluated on synthetic and simulated robot-navigation tasks, where low-dimensional belief representations supported effective policy computation. Across examples, E-PCA handled uncertainty and complex beliefs better than competing heuristics or conventional PCA in the larger or more demanding settings.

  • Toy problem: E-PCA approximately doubled policy performance compared with the Maximum-Likelihood heuristic in the regular-grid toy problem.The heuristic guessed the corridor and was correct only about half the time.
  • Algorithm: The algorithm generates a discrete low-dimensional belief space with E-PCA, computes its reward function, and applies value iteration in that compressed space.
  • Robot Navigation: With 5 basis functions, the E-PCA policy drove past the goal to verify orientation before returning, using actions that reduced uncertainty.The robot could not know that it was passing the goal because its true position was uncertain.
  • Robot Navigation: E-PCA outperformed the Maximum-Likelihood heuristic and conventional PCA in the corridor example by modeling belief uncertainty and goal status more accurately.The Maximum-Likelihood heuristic declared the goal at the wrong location about 50% of the time, while conventional PCA still selected sub-optimal actions regularly.
  • Finding People: E-PCA also produced good controllers for higher-complexity person-search beliefs, whereas conventional PCA performed very poorly even with many bases.The best E-PCA controller approached theoretical best performance, while iterative discretization refinement substantially improved performance.

8. Discussion

The experiments indicate that E-PCA scales to low-dimensional surfaces in very high-dimensional spaces, but its computational and modeling choices remain important. Performance depends on belief sampling, basis selection, discretization, and whether the POMDP has suitable structure.

  • E-PCA scales to finding low-dimensional surfaces embedded in very high-dimensional spaces.
  • Computation: For 500 sample beliefs over 20,230 states, E-PCA computation was compared with conventional PCA across discretizations and numbers of bases.The implementation used Java 1.4.0, Colt 1.0.2, a 1 GHz Athlon CPU, and 900M of RAM.
  • Computation: Computing E-PCA bases dominates runtime, while value iteration took about 50 to 100ms after the bases and discretized low-dimensional space were available.
  • Belief sampling: The method used 500 beliefs sampled by hand-coded heuristic controllers, which was sufficient for the example problems but may be improved by iterating representation construction and control.
  • Model Selection: The appropriate number of bases is difficult to determine because E-PCA does not produce informative eigenvalues; reconstruction quality is used instead.An alternative is to choose the most compact representation achieving the highest policy reward.
  • Limitations: E-PCA is unsuitable when beliefs lack the required sparse, low-dimensional structure or cannot be represented through the appropriate link function.Large KL divergence even with many bases may indicate that a POMDP lacks this structure.

9. Related Work

The paper relates E-PCA compression to reachability analysis, value-directed compression, point-based planning, and nonlinear dimensionality reduction. These approaches differ in whether they restrict beliefs, derive subspaces from model parameters, plan at selected belief points, or learn nonlinear representations.

  • Reachability analysis: Reachability methods constrain planning to reachable beliefs and can be optimal there, but enumerating reachable beliefs is uncommon in real-world problems.
  • Value-directed compression: Value-directed POMDP compression derives a low-dimensional Krylov subspace directly from reward, transition, and observation model parameters.
  • Value-directed compression: Replacing E-PCA with PCA can exploit regularities in transition matrices, but linear compression does not scale to the targeted problem sizes because beliefs rarely lie on a low-dimensional hyperplane.
  • Point-based methods: Point-based methods optimize value functions at selected beliefs, making backups polynomial or keeping value-function complexity constant in methods such as PB-DP and PBVI.PBVI still remains constrained by the curse of dimensionality.
  • Nonlinear reduction: E-PCA belongs to a broader family of nonlinear dimensionality-reduction methods that includes Self-Organizing Maps, Generative Topographic Mapping, Isomap, and Locally Linear Embedding.

10. Conclusion

The paper shows that exploiting structured, low-dimensional belief spaces enables approximate solutions for substantially larger POMDPs, while identifying assumptions and extensions that limit the method’s scope.

  • Conclusion: An order-of-magnitude increase in solvable problem size was demonstrated by exploiting structured belief spaces rather than planning over the full space.The demonstrations included synthetic, robot-navigation, and factored-belief POMDPs.
  • Conclusion: Exponential family PCA compresses sampled beliefs effectively when real-world POMDPs have sparse distributions, but it may fail on problems without that structure.Other dimensionality-reduction methods, including Isomap and Locally-Linear Embedding, are suggested for investigation.
  • Conclusion: A prediction-error objective could improve control-oriented compression by incorporating transition-probability structure, although arranging representations consistently across actions remains unresolved.The proposed loss adds a transition-prediction term and is left for future work.
  • Conclusion: The current formulation assumes that all beliefs share one low-dimensional representation, which can overestimate dimensionality when beliefs lie on multiple surfaces.A problem with two distinct low-dimensional surfaces illustrates this limitation.
  • Conclusion: Combining belief-representation methods with policy-search or hierarchical methods may support further progress on large POMDPs.E-PCA controllers depend strongly on belief complexity, whereas the alternative methods depend more strongly on policy complexity.
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