3D Face Modelling

Proximal Policy Optimization, or PPO, is a policy gradient method for reinforcement learning. The motivation was to have an algorithm with the data efficiency and reliable performance of TRPO, while using only first-order optimization. Let $r\_{t}\left(\theta\right)$ denote the probability ratio $r\_{t}\left(\theta\right) = \frac{\pi\_{\theta}\left(a\_{t}\mid{s\_{t}}\right)}{\pi\_{\theta\_{old}}\left(a\_{t}\mid{s\_{t}}\right)}$, so $r\left(\theta\_{old}\right) = 1$. TRPO maximizes a “surrogate” objective: $$ L^{\text{CPI}}\left({\theta}\right) = \hat{\mathbb{E}}\_{t}\left[\frac{\pi\_{\theta}\left(a\_{t}\mid{s\_{t}}\right)}{\pi\_{\theta\_{old}}\left(a\_{t}\mid{s\_{t}}\right)})\hat{A}\_{t}\right] = \hat{\mathbb{E}}\_{t}\left[r\_{t}\left(\theta\right)\hat{A}\_{t}\right] $$ Where $CPI$ refers to a conservative policy iteration. Without a constraint, maximization of $L^{CPI}$ would lead to an excessively large policy update; hence, we PPO modifies the objective, to penalize changes to the policy that move $r\_{t}\left(\theta\right)$ away from 1: $$ J^{\text{CLIP}}\left({\theta}\right) = \hat{\mathbb{E}}\_{t}\left[\min\left(r\_{t}\left(\theta\right)\hat{A}\_{t}, \text{clip}\left(r\_{t}\left(\theta\right), 1-\epsilon, 1+\epsilon\right)\hat{A}\_{t}\right)\right] $$ where $\epsilon$ is a hyperparameter, say, $\epsilon = 0.2$. The motivation for this objective is as follows. The first term inside the min is $L^{CPI}$. The second term, $\text{clip}\left(r\_{t}\left(\theta\right), 1-\epsilon, 1+\epsilon\right)\hat{A}\_{t}$ modifies the surrogate objective by clipping the probability ratio, which removes the incentive for moving $r\_{t}$ outside of the interval $\left[1 − \epsilon, 1 + \epsilon\right]$. Finally, we take the minimum of the clipped and unclipped objective, so the final objective is a lower bound (i.e., a pessimistic bound) on the unclipped objective. With this scheme, we only ignore the change in probability ratio when it would make the objective improve, and we include it when it makes the objective worse. One detail to note is that when we apply PPO for a network where we have shared parameters for actor and critic functions, we typically add to the objective function an error term on value estimation and an entropy term to encourage exploration.

DDPG, or Deep Deterministic Policy Gradient, is an actor-critic, model-free algorithm based on the deterministic policy gradient that can operate over continuous action spaces. It combines the actor-critic approach with insights from DQNs: in particular, the insights that 1) the network is trained off-policy with samples from a replay buffer to minimize correlations between samples, and 2) the network is trained with a target Q network to give consistent targets during temporal difference backups. DDPG makes use of the same ideas along with batch normalization.

TD3 builds on the DDPG algorithm for reinforcement learning, with a couple of modifications aimed at tackling overestimation bias with the value function. In particular, it utilises clipped double Q-learning, delayed update of target and policy networks, and target policy smoothing (which is similar to a SARSA based update; a safer update, as they provide higher value to actions resistant to perturbations).

Target Policy Smoothing is a regularization strategy for the value function in reinforcement learning. Deterministic policies can overfit to narrow peaks in the value estimate, making them highly susceptible to functional approximation error, increasing the variance of the target. To reduce this variance, target policy smoothing adds a small amount of random noise to the target policy and averages over mini-batches - approximating a SARSA-like expectation/integral. The modified target update is: $$ y = r + \gamma{Q}\_{\theta'}\left(s', \pi\_{\theta'}\left(s'\right) + \epsilon \right) $$ $$ \epsilon \sim \text{clip}\left(\mathcal{N}\left(0, \sigma\right), -c, c \right) $$ where the added noise is clipped to keep the target close to the original action. The outcome is an algorithm reminiscent of Expected SARSA, where the value estimate is instead learned off-policy and the noise added to the target policy is chosen independently of the exploration policy. The value estimate learned is with respect to a noisy policy defined by the parameter $\sigma$.

Soft Actor Critic, or SAC, is an off-policy actor-critic deep RL algorithm based on the maximum entropy reinforcement learning framework. In this framework, the actor aims to maximize expected reward while also maximizing entropy. That is, to succeed at the task while acting as randomly as possible. Prior deep RL methods based on this framework have been formulated as Q-learning methods. SAC combines off-policy updates with a stable stochastic actor-critic formulation. The SAC objective has a number of advantages. First, the policy is incentivized to explore more widely, while giving up on clearly unpromising avenues. Second, the policy can capture multiple modes of near-optimal behavior. In problem settings where multiple actions seem equally attractive, the policy will commit equal probability mass to those actions. Lastly, the authors present evidence that it improves learning speed over state-of-art methods that optimize the conventional RL objective function.

ACER, or Actor Critic with Experience Replay, is an actor-critic deep reinforcement learning agent with experience replay. It can be seen as an off-policy extension of A3C, where the off-policy estimator is made feasible by: - Using Retrace Q-value estimation. - Using truncated importance sampling with bias correction. - Using a trust region policy optimization method. - Using a stochastic dueling network architecture.

A Stochastic Dueling Network, or SDN, is an architecture for learning a value function $V$. The SDN learns both $V$ and $Q$ off-policy while maintaining consistency between the two estimates. At each time step it outputs a stochastic estimate of $Q$ and a deterministic estimate of $V$.

D4PG, or Distributed Distributional DDPG, is a policy gradient algorithm that extends upon the DDPG. The improvements include a distributional updates to the DDPG algorithm, combined with the use of multiple distributed workers all writing into the same replay table. The biggest performance gain of other simpler changes was the use of $N$-step returns. The authors found that the use of prioritized experience replay was less crucial to the overall D4PG algorithm especially on harder problems.

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Mirror Descent Policy Optimization (MDPO) is a policy gradient algorithm based on the idea of iteratively solving a trust-region problem that minimizes a sum of two terms: a linearization of the standard RL objective function and a proximity term that restricts two consecutive updates to be close to each other. It is based on Mirror Descent, which is a general trust region method that attempts to keep consecutive iterates close to each other.

Primal Wasserstein Imitation Learning, or PWIL, is a method for imitation learning which ties to the primal form of the Wasserstein distance between the expert and the agent state-action distributions. The reward function is derived offline, as opposed to recent adversarial IL algorithms that learn a reward function through interactions with the environment, and requires little fine-tuning.

This method proposes first discretizing observations and calculating the action distribution distance under comparable cases (intersection states).

ACKTR, or Actor Critic with Kronecker-factored Trust Region, is an actor-critic method for reinforcement learning that applies trust region optimization using a recently proposed Kronecker-factored approximation to the curvature. The method extends the framework of natural policy gradient and optimizes both the actor and the critic using Kronecker-factored approximate curvature (K-FAC) with trust region.

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Generalized State-Dependent Exploration, or gSDE, is an exploration method for reinforcement learning that uses more general features and re-sampling the noise periodically. State-Dependent Exploration (SDE) is an intermediate solution for exploration that consists in adding noise as a function of the state $s\_{t}$, to the deterministic action $\mu\left(\mathbf{s}\_{t}\right)$. At the beginning of an episode, the parameters $\theta\_{\epsilon}$ of that exploration function are drawn from a Gaussian distribution. The resulting action $\mathbf{a}\_{t}$ is as follows: $$ \mathbf{a}\_{t}=\mu\left(\mathbf{s}\_{t} ; \theta\_{\mu}\right)+\epsilon\left(\mathbf{s}\_{t} ; \theta\_{\epsilon}\right), \quad \theta\_{\epsilon} \sim \mathcal{N}\left(0, \sigma^{2}\right) $$ This episode-based exploration is smoother and more consistent than the unstructured step-based exploration. Thus, during one episode, instead of oscillating around a mean value, the action a for a given state $s$ will be the same. In the case of a linear exploration function $\epsilon\left(\mathbf{s} ; \theta\_{\epsilon}\right)=\theta\_{\epsilon} \mathbf{s}$, by operation on Gaussian distributions, Rückstieß et al. show that the action element $\mathbf{a}\_{j}$ is normally distributed: $$ \pi]_{j}\left(\mathbf{a}\_{j} \mid \mathbf{s}\right) \sim \mathcal{N}\left(\mu\_{j}(\mathbf{s}), \hat{\sigma\_{j}}^{2}\right) $$ where $\hat{\sigma}$ is a diagonal matrix with elements $\hat{\sigma}\_{j}=\sqrt{\sum\_{i}\left(\sigma\_{i j} \mathbf{s}\_{i}\right)^{2}}$. Because we know the policy distribution, we can obtain the derivative of the log-likelihood $\log \pi(\mathbf{a} \mid \mathbf{s})$ with respect to the variance $\sigma$ : $$ \frac{\partial \log \pi(\mathbf{a} \mid \mathbf{s})}{\partial \sigma_{i j}}=\frac{\left(\mathbf{a}\_{j}-\mu\_{j}\right)^{2}-\hat{\sigma\_{j}}^{2}}{\hat{\sigma}\_{j}^{3}} \frac{\mathbf{s}\_{i}^{2} \sigma\_{i j}}{\hat{\sigma_{j}}} $$ This can be easily plugged into the likelihood ratio gradient estimator, which allows to adapt $\sigma$ during training. SDE is therefore compatible with standard policy gradient methods, while addressing most shortcomings of the unstructured exploration. For gSDE, two improvements are suggested: 1. We sample the parameters $\theta\_{\epsilon}$ of the exploration function every $n$ steps instead of every episode. 2. Instead of the state s, we can in fact use any features. We chose policy features $\mathbf{z}\_{\mu}\left(\mathbf{s} ; \theta\_{\mathbf{z}\_{\mu}}\right)$ (last layer before the deterministic output $\left.\mu(\mathbf{s})=\theta\_{\mu} \mathbf{z}\_{\mu}\left(\mathbf{s} ; \theta_{\mathbf{z}\_{\mu}}\right)\right)$ as input to the noise function $\epsilon\left(\mathbf{s} ; \theta\_{\epsilon}\right)=\theta\_{\epsilon} \mathbf{z}\_{\mu}(\mathbf{s})$