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Uncertainty-Aware Learning from Demonstration using Mixture Density Networks with Sampling-Free Variance Modeling

Sungjoon Choi, Kyungjae Lee, Sungbin Lim, Songhwai Oh

arXiv:1709.02249v2cs.CVcs.AIcs.LGcs.RO

TL;DR

Deep-network uncertainty estimates can be costly to acquire with Monte Carlo sampling, while safe physical control requires attention to predictive uncertainty. This paper uses an MDN to separate uncertainty components and guide a learning-from-demonstration driving controller; on a real-world driving dataset, the method outperforms compared methods in safety.

  • Problem

    Monte Carlo sampling for deep-network uncertainty estimation is computationally heavy, limiting its suitability for real-time applications where physical-system safety matters.

    Source: §I · p. 1

  • Method

    The method uses an MDN to estimate explained and unexplained variances, then uses explained variance to switch from an aggressive learned driving controller to a rule-based conservative controller.

    Source: §I · p. 1

  • Results

    On a complex real-world driving dataset, the proposed uncertainty-aware method outperforms compared methods in safety, and the paper also reports improved safety and efficiency.

    Source: §I · p. 1, §VI · p. 7

  • Takeaways & Limitations

    The authors conclude that a single-forward-path MDN uncertainty measure can distinguish model uncertainty from measurement noise and support uncertainty-aware driving control.

    Source: §I · p. 1, §VII · p. 9

Abstract

from arXiv · show

In this paper, we propose an uncertainty-aware learning from demonstration method by presenting a novel uncertainty estimation method utilizing a mixture density network appropriate for modeling complex and noisy human behaviors. The proposed uncertainty acquisition can be done with a single forward path without Monte Carlo sampling and is suitable for real-time robotics applications. The properties of the proposed uncertainty measure are analyzed through three different synthetic examples, absence of data, heavy measurement noise, and composition of functions scenarios. We show that each case can be distinguished using the proposed uncertainty measure and presented an uncertainty-aware learn- ing from demonstration method of an autonomous driving using this property. The proposed uncertainty-aware learning from demonstration method outperforms other compared methods in terms of safety using a complex real-world driving dataset.

I. INTRODUCTION

The paper introduces an MDN-based uncertainty measure for learning from demonstration and applies it to autonomous driving. It aims to make uncertainty estimation practical for real-time robotics while improving driving safety.

I. INTRODUCTION

  • Uncertainty estimation matters in physical systems because deep-learning mispredictions can create safety risks, as illustrated by a fatal ADAS accident.The cited accident involved an image-processing component that failed to distinguish a trailer's white side from a bright sky.Source: §I · p. 1
  • A single MDN forward path acquires uncertainty without Monte Carlo sampling, making the method suitable for real-time robotic applications.The paper contrasts this with methods requiring multiple models or sampling.Source: §I · p. 1
  • On a real-world US Highway 101 driving dataset, the uncertainty-aware method outperforms compared methods in driving safety by using explained variance to detect out-of-distribution inputs.The controller switches from an aggressive learned policy to a rule-based conservative controller.Source: §I · p. 1
  • The proposed MDN uncertainty measure separates explained variance, indicating model ignorance, from unexplained variance, indicating measurement noise.The paper analyzes the measure in absence-of-data, heavy-noise, and function-composition scenarios.Source: §I · p. 1

II. RELATED WORK

Prior work estimates predictive uncertainty using stochastic forward paths, density networks, or ensembles, often with added computational demands. The paper positions its MDN variance method as a sampling-free way to model both reducible and irreducible uncertainty.

II. RELATED WORK

  • MC dropout estimates predictive variance from stochastic forward paths and has been applied to regression, classification, and reinforcement learning.The method connects approximate Bayesian networks with sparse Gaussian processes.Source: §II · pp. 1–2
  • Density networks predict means and variances, while ensembles can estimate uncertainty from mixture-distribution sample variance; MDNs are encouraged for more complex distributions, [9].Source: §II · p. 2
  • A robotics approach combines bootstrapped networks and dropout, switching to a risk-averse cost when predictive variance exceeds a threshold, but requires multiple networks.The added networks increase training and test computational complexity.Source: §II · p. 2
  • A visual-navigation method switches to rule-based collision estimation for novel inputs, whereas this paper models uncertainty over policy input-output pairs.Source: §II · p. 2
  • A single MDN without Monte Carlo sampling can model both reducible and irreducible uncertainty, extending earlier uncertainty-estimation approaches.The paper reports better modeling of both uncertainty types in synthetic examples.Source: §II · p. 2

A. Uncertainty Acquisition in Deep Learning

Predictive uncertainty includes noise inherent in the data and uncertainty about the model. Distinguishing these components can inform whether a prediction reflects multiple plausible outputs or insufficient training coverage.

A. Uncertainty Acquisition in Deep Learning

  • Aleatoric uncertainty reflects irreducible data-generating noise, whereas epistemic uncertainty reflects model ignorance that can be reduced with sufficient training data [6].The paper represents these components as σ²_a and σ²_e, respectively.Source: A. Uncertainty Acquisition in Deep… · p. 2
  • Under the stated model, measurement error η is zero-mean Gaussian with variance σ²_η, which corresponds to aleatoric uncertainty.Source: A. Uncertainty Acquisition in Deep… · p. 2
  • For autonomous steering, high aleatoric and low epistemic uncertainty can indicate multiple reasonable steering choices, while low aleatoric and high epistemic uncertainty can indicate insufficient training data.In the latter case, switching to a risk-averse controller or alerting the driver is described as reasonable.Source: A. Uncertainty Acquisition in Deep… · p. 3

B. Mixture Density Network

An MDN maps inputs to the parameters of a Gaussian mixture distribution over outputs. The paper describes its output constraints, negative-log-likelihood training, and a highest-weight mixture choice for prediction.

B. Mixture Density Network

  • An MDN maps an input x to Gaussian-mixture parameters, including mixture probabilities, means, and variances.Source: B. Mixture Density Network · p. 3
  • The method assumes independent output dimensions and uses diagonal mixture variances, with constraints to keep mixture weights on a simplex and variances positive definite.The variance output uses a sigmoid multiplied by a manually selected σ_max of five; subtracting the maximum mixture value addresses numerical instability.Source: B. Mixture Density Network · p. 3
  • The MDN is trained by minimizing negative log likelihood, with ε = 10^-6 included for logarithm numerical stability.The architecture can be built on a multilayer perceptron, convolutional neural network, or recurrent neural network.Source: B. Mixture Density Network · p. 3
  • After training, prediction uses the mean and variance of the mixture component with the highest mixture weight, treating weights as a gating network for local experts.Source: B. Mixture Density Network · p. 3

IV. PROPOSED UNCERTAINTY ESTIMATION METHOD

The method acquires regression uncertainty from an MDN using the law of total variance. Modeling the output as a Gaussian mixture makes the approach suitable for complex, noisy distributions.

  • The proposed regression uncertainty acquisition uses an MDN and the law of total variance.Source: §IV · p. 3
  • An MDN represents the output y given input x as a Gaussian mixture model.Source: §IV · p. 3
  • The paper notes that an MDN is more suitable than a density network for fitting complex and noisy distributions.Source: §IV · p. 3

A. Uncertainty Acquisition for a Mixture Density Network

This subsection develops total variance for a Gaussian mixture model and connects its two terms to epistemic and aleatoric uncertainty.

A. Uncertainty Acquisition for a Mixture Density Network

B. Connection to Aleatoric and Epistemic Uncertainties

The GMM predictive variance separates into explained and unexplained components associated with epistemic and aleatoric uncertainty. The MDN computes this total variance without Monte Carlo sampling.

B. Connection to Aleatoric and Epistemic Uncertainties

V. ANALYSIS OF THE PROPOSED UNCERTAINTY MODELING WITH SYNTHETIC EXAMPLES

The synthetic examples test whether the proposed variance components distinguish missing data, heavy measurement noise, and competing function outputs. They show distinct uncertainty patterns across these cases, and the sampling-free method is reported to be 21.15 times faster than the comparison method.

Heavy noise

  • In heavy-noise regions, E[V(y|x, k)] and the compared method’s epistemic uncertainty capture the noise, which the authors relate to density networks modeling measurement noise.The proposed variance components are illustrated in Fig. 2(a), alongside the comparison method in Fig. 2(c).Source: §V · p. 5, Figure 2 · p. 5

Absence of data

  • For absent-data regions, V(E[y|x, k]) assigns high uncertainty, while the compared method’s aleatoric and epistemic estimates hardly capture them.The authors interpret V(E[y|x, k]) as related to epistemic uncertainty, or model ignorance.Source: §V · pp. 5–6, Figure 2 · p. 5

Summary

  • Table I summarizes the variance patterns as E[V(y|x, k)] / V(E[y|x, k)]: heavy noise, High / High or Low; absence of data, High / High; composition of functions, Low / High.The table summarizes the two proposed components across the three synthetic scenarios.Source: §V · p. 6, Table I · p. 6

Weight decay

  • As weight decay increases, regions without training data are captured more accurately, and E[V(y|x, k)] is more affected by weight decay.The authors connect this sensitivity to E[V(y|x, k)] corresponding to aleatoric uncertainty.Source: §V · p. 6, Figure 3 · p. 6

Composition of functions

  • With two competing function outputs, the 10-mixture MDN models them accurately; E[V(y|x, k)] stays low while V(E[y|x, k]) varies with their difference.The training data contain no measurement noise, and the authors attribute the low E[V(y|x, k)] values in the input domain to that condition.Source: §V · p. 6, Figure 3 · p. 6

Analysis overview

  • The proposed method was about 21.15 times faster than [6] for uncertainty estimation because it does not require Monte Carlo sampling.Table II compares the proposed variance estimates with the aleatoric and epistemic estimates in [6], using 50 MC samples for the latter.Source: Table II · p. 6, §V · p. 7

VI. UNCERTAINTY-AWARE LEARNING FROM DEMONSTRATION TO DRIVE

UALfD uses explained variance to switch between learned and rule-based safe control. In the NGSIM driving evaluation, the authors report improved safety and stability, with the shortest average elapsed time among the compared methods.

VI. UNCERTAINTY-AWARE LEARNING FROM DEMONSTRATION TO DRIVE

  • The evaluation uses NGSIM US Highway 101 vehicle trajectories recorded at 10 Hz and counts leaving the track or colliding as a collision.Testing uses the second road segment, with the goal of reaching the third segment.Source: §VI · p. 7
  • Driving demonstrations are generated by varying ego-car starts and other-car configurations; collision trajectories are excluded from the training set.The authors also note that demonstrations could be collected manually using human-in-the-loop simulations.Source: §VI · p. 7
  • The learning policy maps seven track features—three front gaps, three rear gaps, and lane-deviation distance—to the desired heading encoded as (cos θdev, sin θdev).Source: §VI · p. 7
  • UALfD switches to the safe policy when log explained variance is higher than −2; because variance is unscaled, the authors manually tune the threshold, while UALfD2 utilizes E[V(y|x,k)] instead of V(E[y|x,k]).Source: §VI · p. 7
  • UALfD improves driving safety and stability, has the shortest average elapsed time, and avoids collisions; MDN (K=1) and RegNet collide and leave the track.UALfD2, MDN (K=10), and Safe Mode also navigate without collision.Source: §VI · pp. 7–8, Table III · p. 8, Table IV · p. 8
  • Explainable variance better captures possible model uncertainty in data-sparse regions, while heading errors—including collision-prone directions—occur in higher-variance regions.The figure compares total, explainable, and unexplained variance estimated with an MDN using ten mixtures.Source: Figure 5 · p. 7, §VI · p. 8
  • UALfD and UALfD2 both avoid collisions, but UALfD2's measurement-noise estimate makes control more conservative, like Safe Mode, affecting elapsed time and lane changes.Source: §VI · p. 9

VII. CONCLUSION

The conclusion presents an MDN-based uncertainty measure that runs without Monte Carlo sampling and separates explained from unexplained variance. It reports distinguishing three uncertainty scenarios and applying the method to real-world driving.

VII. CONCLUSION

  • The proposed MDN uncertainty measure runs with a single feedforward model without computationally heavy sampling and decomposes uncertainty into explained and unexplained variances.The paper analyzes absence of data, heavy measurement noise, and composition-of-functions scenarios, then applies uncertainty-aware LfD to a real-world driving dataset.Source: §VII · p. 9

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