[reading] Reinforcement and Imitation Learning via Interactive No-Regret Learning

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The paper presents new methods for imitation and reinforcement learning that leverage cost-to-go information within an interactive no-regret learning framework. Key contributions include the AGGREVATE algorithm for imitation learning and a related algorithm for reinforcement learning, both offering strong theoretical guarantees.

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  • Authors: Stéphane Ross, J. Andrew Bagnell
  • Institution: The Robotics Institute, Carnegie Mellon University
  • Date: June 23, 2014
  • Link: https://arxiv.org/pdf/1406.5979

Key Contributions

  1. AGGREVATE Algorithm:
    • Purpose: Extends DAGGER by minimizing long-term cost-to-go instead of just immediate classification loss.
    • Method:
      1. Collect data by observing the expert and performing random explorations.
      2. Observe the resulting cost-to-go \(Q\).
      3. Train policies \(\hat{\pi}_n\) to minimize these costs.
    • Algorithm Steps:
      Initialize $$D \leftarrow \emptyset$$, $$\hat{\pi}_1 \leftarrow any policy in \Pi$$
      for $$i = 1$$ to $$N$$ do
          Collect $$m$$ data points as follows:
              for $$j = 1$$ to $$m$$ do
                  Sample uniformly $$t \in \{1, 2, \ldots, T\}$$
                  Start new trajectory in initial state $$s_1$$
                  Execute current policy $$\hat{\pi}_i$$ up to time $$t-1$$
                  Explore action $$a_t$$ in state $$s_t$$
                  Execute expert from time $$t+1$$ to $$T$$, observe cost-to-go $$Q$$
                  Add $$(s, t, a, Q)$$ to dataset $$D$$
          Train cost-sensitive classifier $$\hat{\pi}_{i+1}$$ on $$D$$
      Return best $$\hat{\pi}_i$$ on validation
      
  2. No-Regret Policy Iteration (NRPI):
    • Purpose: Adapts AGGREVATE for reinforcement learning using cost-to-go estimates.
    • Method:
      1. Collect cost-to-go examples by sampling from state distributions \(u_t\).
      2. Explore actions and apply the current policy to estimate future costs.
      3. Train policies using a no-regret online learning algorithm.
    • Algorithm Steps:
      Initialize $$D \leftarrow \emptyset$$, $$\hat{\pi}_1 \leftarrow any policy in \Pi$$
      for $$i = 1$$ to $$N$$ do
          Collect $$m$$ data points as follows:
              for $$j = 1$$ to $$m$$ do
                  Sample uniformly $$t \in \{1, 2, \ldots, T\}$$
                  Sample state $$s_t$$ from exploration distribution $$
      u_t$$
                  Execute exploration action $$a_t$$ in state $$s_t$$
                  Execute current policy $$\hat{\pi}_i$$ from time $$t+1$$ to $$T$$, observe cost-to-go $$Q$$
                  Add $$(s, a, t, Q)$$ to dataset $$D$$
          Train cost-sensitive classifier $$\hat{\pi}_{i+1}$$ on $$D$$
      Return best $$\hat{\pi}_i$$ on validation
      

Key Equations and Theoretical Guarantees

  1. Performance Metric:
    • Total cost of executing policy \(\pi\) over \(T\) steps: \(J(\pi) = \sum_{t=1}^{T} \mathbb{E}_{s \sim d_t^\pi} [C(s, \pi(s))]\)
  2. Cost-to-Go Function:
    • Expected future cost-to-go of executing action \(a\) in state \(s\) followed by policy \(\pi\) for \(t-1\) steps: \(Q^\pi_t(s, a)\)
  3. AGGREVATE Theoretical Guarantee:
    • If a no-regret algorithm is used, as \(N \to \infty\): \(\lim_{N o \infty} J(\pi) \leq J(\pi^*) + T \epsilon_{ ext{class}}\)
  4. NRPI Theoretical Guarantee:
    • For any policy \(\pi'\): \(J(\hat{\pi}) \leq J(\pi) \leq J(\pi') + T \epsilon_{ ext{regret}} + T Q_{\max} D( u, \pi')\)
    • If a no-regret algorithm is used, as \(N \to \infty\): \(\lim_{N o \infty} J(\pi) \leq J(\pi') + T Q_{\max} D( u, \pi')\)

Conclusion

By integrating cost-to-go information into no-regret learning frameworks, the paper presents significant advancements in both imitation learning and reinforcement learning. The proposed algorithms, AGGREVATE and NRPI, offer a unified approach with strong theoretical guarantees, enhancing the stability and effectiveness of learned policies in practical applications.