# A Reduction of Imitation Learning and Structured Prediction to No-Regret Online Learning

2011 AISTATS #Content/Paper by Stéphane Ross, Geoffrey J Gordon, and J Andrew Bagnell.

In Imitation Learning (among other sequential prediction problems), future observations depend on previous actions, which violates the common i.i.d. assumption made in statistical learning. Ignoring this issue compromises learning: mistakes lead the Imitator to parts of the state space never encountered by the Target policy, leading to a compounding of errors. Here an Online Learning algorithm called DAgger is proposed to find a strong policy under these conditions. Denote $d^t_\pi$ the distribution of states at time $t$ if policy $\pi$ is followed from time $1$ to $t-1$. The average distribution of states if $\pi$ is followed for $T$ steps is therefore

$d_\pi=\frac{1}{T}\sum_{t=1}^Td_\pi^t$

Our goal is to find a policy $\hat{\pi}$ which minimises some loss function $\mathcal{L}$ relative to a Target policy $\pi^*$ on samples from $d_\pi$:

$\hat{\pi}=\arg\min_{\pi\in\Pi}\mathbb{E}_{s\sim d_\pi}[\mathcal{L(s,\pi,\pi^*)}]$

Traditional approaches to imitation learning instead train a policy to perform well under the distribution encountered by $\pi^\ast$, $d_{\pi^\ast}$. Poor performance can result. A naïve solution would be to iterative train a new policy for each timestep $t$, on the distribution of states induced by all previously-trained policies, but this is extremely computationally intensive. The generic DAgger proceeds as follows:

• Initialise an imitation policy $\hat{\pi}_1$.
• Use $\pi^*$ to gather a dataset $\mathcal{D}$.
• For each subsequent iteration $i$:
• Use $\pi^*$ to sample $N\times\beta_i$ trajectories, and $\hat{\pi}_i$ to sample $N\times(1-\beta_i$), where $\beta_i\in[0,1]$.
• For all new trajectories get the corresponding actions from $\pi^*$, and append these to $\mathcal{D}$.
• Train $\hat{\pi}_{i+1}$ on the augmented dataset.

Starting with $\beta_1=1$ is typically useful because it means we don’t have to specify an initial policy $\hat\pi_1$. The only requirement for the evolution of $\beta_i$ is that the average value across all iterations $\rightarrow0$ as $i\rightarrow\infty$. In practice, the best approach seems to be to set $\beta_1=1$ and $\beta_i=0\ \forall i>1$. A theoretical analysis in the paper demonstrates the robustness of the algorithm, and it is shown to outperform a couple of earlier alternatives for imitation learning.