# Adversarial Autoencoder and Multi-Armed Bandit for Dynamic Difficulty Adjustment in Immersive Virtual Reality for Rehabilitation: Application to Hand Movement

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## Abstract

**:**

## 1. Introduction

## 2. Related Work

## 3. Materials and Methods

#### 3.1. Experiments

#### 3.2. Database

#### 3.3. Training, Validation, and Test Dataset

#### 3.4. Autoencoder

#### 3.5. Adversarial Autoencoder

#### 3.6. Adversarial Autoencoder Training

#### 3.7. Latent Space Evaluation

#### 3.8. Adversarial Autoencoder Hyperparameters

#### 3.9. Latent Space Distance

#### 3.10. Latent Space Accuracy

## 4. Multi-Armed Bandit Problem

#### 4.1. Boltzmann

#### Upper Confidence Bound (UCB)

#### 4.2. Sibling Kalman Filter

#### 4.2.1. Innovative Upper Confidence Bound (IUCB)

#### 4.2.2. Thomson Sampling (TS)

#### 4.3. Agent Models

#### 4.4. Model Evaluation

## 5. Results

#### 5.1. Latent Space

#### 5.2. Multi-Armed Bandit Problem

## 6. Discussion

## 7. Limitations

## 8. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

VR | Virtual Reality |

HMD | Head-Mounted Display |

RL | Reinforcement Learning |

DDA | Dynamic Difficulty Adjustment |

MAB | Multi-Armed Bandits |

AAE | Adversarial Autoencoder |

UAAE | Unsupervised Adversarial Autoencoder |

SSAAE | Semi-Supervised Adversarial Autoencoder |

UCB | Upper Confidence Bound |

IUCB | Innovative Upper Confidence Bound |

TS | Thomson Sampling |

## Appendix A

Algorithm A1: MAB Algorithm—Boltzmann |

Parameter:$T=60,K=6,{Q}_{0},\alpha ,\tau ,c$Initialization: Set $\widehat{\mathbf{\mu}}\left(0\right):={Q}_{0}$, $\mathit{N}\left(0\right):=0$1: for $t=0,1,\cdots ,T$:2: if Boltzmann:4: else if Boltzmann UCB:5: $k\leftarrow $ Softmax choice with UCB vector ($\widehat{\mathbf{\mu}}\left(t\right),\tau $) with Equations (16) and (18) 6: ${R}_{k}\left(t\right)\leftarrow $ Bandits (k) 7: ${\widehat{\mu}}_{k}(t+1)\leftarrow $ Update rule - Learning rate (${\widehat{\mu}}_{k}\left(t\right),{R}_{k}\left(t\right),\alpha $) with Equation (17) |

Algorithm A2: MAB Algorithm—Sibling Kalman Filter |

Parameter:$T=60,K=6,{Q}_{0},\tau ,c,{\sigma}_{\xi}^{2},{\sigma}_{\u03f5}^{2}$Initialization: Set $\widehat{\mathbf{\mu}}\left(0\right):={Q}_{0}$, ${\widehat{\mathbf{\sigma}}}^{2}\left(0\right),\mathit{N}\left(0\right):=0$1: for $t=0,1,\cdots ,T$:2: if Sibling Kalman Filter :4: else if Sibling Kalman Filter IUCB:5: $k\leftarrow $ Softmax choice with IUCB vector ($\widehat{\mathbf{\mu}}\left(t\right),\tau ,c$) with Equations (16) and (24) 6: else if Sibling Kalman Filter TS:7: $k\leftarrow $ Softmax choice on TS ($\widehat{\mathbf{\mu}}\left(t\right),\tau $) with Equation (25) 8: ${R}_{k}\left(t\right)\leftarrow $ Bandits(j) |

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**Figure 1.**Project Overview. The adversarial autoencoder (AAE) is trained by considering a regularization component to create a set of six Gaussian distributions in a latent space. The trained AAE is then fed with a new sample (with Triangle as an example) and represented in the latent space (red cross). Finally, a distance metric is computed between a target (green centroid) and used as a reward for the MAB algorithm.

**Figure 2.**(

**a**) Example of each movement recorded with the VR wireless controller. (

**b**) Example of artificial data created for the movements “Cube”. A total of 20 movements are created based on the horizontal (Ratio X), vertical stretch (Ratio Y), and rotation factors (

**c**).

**Figure 3.**Semi-supervised adversarial autoencoder architecture with

**z**representing the latent space and

**l**as the label representing the one-hot vector. The unsupervised adversarial autoencoder will just consist of removing the input

**l**.

**Figure 4.**UAAE latent space (

**left**) and SAAE latent space (

**right**) representation with their corresponding gradient information regarding the “augmented parameters” values for horizontal stretch (

**top**), vertical stretch (

**middle**), and rotation (

**bottom**).

**Figure 5.**MAB Heatmap showing the results for UAAE (

**a**) and for SSAAE (

**b**) at iterations 5, 10, 30, and 60 for Boltzmann UCB (left) and Sibling Kalman Filter with Thomson Sampling (right) for each database (X, Y, R, XY, XR, YR, XYR).

**Figure 6.**MAB results for UAAE (

**a**) and for SSAAE (

**b**) for the Sibling Kalman Filter with Thomson Sampling for all iterations on one artificial test dataset from the database XYR. The colored area represents the uncertainty ${\widehat{\sigma}}_{k}{\left(t\right)}^{2}$ about the expected reward ${\widehat{\mu}}_{k}$.

**Table 1.**Parameters considered for the creation of the various artificial databases.

**X**represents the range of factors for the vertical stretch,

**Y**represents the range of factors for the horizontal stretch, and

**R**represents the range of factors for the rotation. Each database contained 20 artificial datasets. Each dataset contained 60 matrices for each label, thus 360 matrices in total.

Database | X | Y | R |
---|---|---|---|

${\mathit{D}}_{\mathit{x}}^{{}^{\prime}}$ | $10\ge $${p}_{h}$$\ge 50$ | $90\ge $${p}_{v}$$\ge 100$ | $-5\ge $$\theta $$\ge 5$ |

${\mathit{D}}_{\mathit{y}}^{{}^{\prime}}$ | $90\ge $${p}_{h}$$\ge 100$ | $10\ge $${p}_{v}$$\ge 50$ | $-5\ge $$\theta $$\ge 5$ |

${\mathit{D}}_{\mathit{r}}^{{}^{\prime}}$ | $90\ge $${p}_{h}$$\ge 100$ | $90\ge $${p}_{v}$$\ge 100$ | $-45\ge $$\theta $$\ge 45$ |

${\mathit{D}}_{\mathit{xy}}^{{}^{\prime}}$ | $10\ge $${p}_{h}$$\ge 50$ | $10\ge $${p}_{v}$$\ge 50$ | $-5\ge $$\theta $$\ge 5$ |

${\mathit{D}}_{\mathit{xr}}^{{}^{\prime}}$ | $10\ge $${p}_{h}$$\ge 50$ | $90\ge $${p}_{v}$$\ge 100$ | $-45\ge $$\theta $$\ge 45$ |

${\mathit{D}}_{\mathit{yr}}^{{}^{\prime}}$ | $90\ge $${p}_{h}$$\ge 100$ | $10\ge $${p}_{v}$$\ge 50$ | $-45\ge $$\theta $$\ge 45$ |

${\mathit{D}}_{\mathit{xyr}}^{{}^{\prime}}$ | $10\ge $${p}_{h}$$\ge 50$ | $10\ge $${p}_{v}$$\ge 50$ | $-45\ge $$\theta $$\ge 45$ |

Hidden Layers | Dense Size | Autoencoder Activation | Discriminator Activation | Learning Rate | Dropout Rate |
---|---|---|---|---|---|

4, 8, 16, 32, | Sigmoid, ReLU | Sigmoid | 0.01, 0.005, 0.001, | 0, 0.1, 0.2, | |

2, 3 | 64, 128, 256, 512 | Tanh | ReLU | 0.0005, 0.0001 | 0.3, 0.4 |

Model | Update Rule | Selection Rule | Parameters |
---|---|---|---|

Boltzmann | Learning rate | softmax choice: Equation (16) | optimistic Q: ${Q}_{0}$ learning rate: $\alpha $ temperature: $\tau $ |

Boltzmann UCB | Learning rate | softmax choice with UCB vector: Equations (16) and (18) | optimistic Q: ${Q}_{0}$ learning rate: $\alpha $ confidence level: c temperature: $\tau $ |

Sibling Kalman Filter | Kalman gain: Equation (22) | softmax choice: Equation (16) | optimistic Q: ${Q}_{0}$ innovative variance: ${\sigma}_{\xi}^{2}$ observation variance: ${\sigma}_{\u03f5}^{2}$ temperature: $\tau $ |

Sibling Kalman Filter IUCB | Kalman gain: Equation (22) | softmax choice with IUCB vector: Equations (16) and (24) | optimistic Q: ${Q}_{0}$ innovative variance: ${\sigma}_{\xi}^{2}$ observation variance: ${\sigma}_{\u03f5}^{2}$ temperature: $\tau $ confidence level: c |

Sibling Kalman Filter TS | Kalman gain: Equation (22) | softmax choice on TS on Normal distribution: Equation (25) | optimistic Q: ${Q}_{0}$ innovative variance: ${\sigma}_{\xi}^{2}$ observation variance: ${\sigma}_{\u03f5}^{2}$ temperature: $\tau $ |

Model | Parameters List |
---|---|

Boltzmann | ${Q}_{0}$ = [$\mathbf{5}$] |

$\alpha $ = [$0.05,0.1,0.2,0.5,\mathbf{1.0}$] | |

$\tau $ = [$\mathbf{1},2,3$] | |

Boltzmann UCB | ${Q}_{0}$ = [$\mathbf{5}$] |

$\alpha $ = [$0.05,0.1,0.2,0.5,\mathbf{1.0}$] | |

c = [$1,\mathbf{2},3$] | |

$\tau $ = [$\mathbf{1},2,3$] | |

Sibling Kalman Filter | ${Q}_{0}$ = [$\mathbf{5}$] |

${\sigma}_{\xi}^{2}$ = [$0.01,0.05,0.1,0.2,0.5,1,\mathbf{2}$] | |

${\sigma}_{\u03f5}^{2}$ = [$0.01,0.05,0.1,\mathbf{0.5},1$] | |

$\tau $ = [$\mathbf{1},2,3$] | |

Sibling Kalman Filter IUCB | ${Q}_{0}$ = [$\mathbf{5}$] |

${\sigma}_{\xi}^{2}$ = [$0.01,\mathbf{0.05},0.1,0.2,0.5,1,2$] | |

${\sigma}_{\u03f5}^{2}$ = [$0.01,\mathbf{0.05},0.1,0.5,1$] | |

$\tau $ = [$1,2,\mathbf{3},4$] | |

c = [$\mathbf{1},2,3,4$] | |

Sibling Kalman Filter TS | ${Q}_{0}$ = [$\mathbf{5}$] |

${\sigma}_{\xi}^{2}$ = [$\mathbf{0.01},0.05,0.1,0.2,0.5,1,2$] | |

${\sigma}_{\u03f5}^{2}$ = [$\mathbf{0.01},0.05,0.1,0.5,1$] | |

$\tau $ = [$\mathbf{1},2,3,4$] |

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## Share and Cite

**MDPI and ACS Style**

Kamikokuryo, K.; Haga, T.; Venture, G.; Hernandez, V.
Adversarial Autoencoder and Multi-Armed Bandit for Dynamic Difficulty Adjustment in Immersive Virtual Reality for Rehabilitation: Application to Hand Movement. *Sensors* **2022**, *22*, 4499.
https://doi.org/10.3390/s22124499

**AMA Style**

Kamikokuryo K, Haga T, Venture G, Hernandez V.
Adversarial Autoencoder and Multi-Armed Bandit for Dynamic Difficulty Adjustment in Immersive Virtual Reality for Rehabilitation: Application to Hand Movement. *Sensors*. 2022; 22(12):4499.
https://doi.org/10.3390/s22124499

**Chicago/Turabian Style**

Kamikokuryo, Kenta, Takumi Haga, Gentiane Venture, and Vincent Hernandez.
2022. "Adversarial Autoencoder and Multi-Armed Bandit for Dynamic Difficulty Adjustment in Immersive Virtual Reality for Rehabilitation: Application to Hand Movement" *Sensors* 22, no. 12: 4499.
https://doi.org/10.3390/s22124499