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A Reduction of Imitation Learning and Structured Prediction to No-Regret Online Learning
Stephane Ross, Geoffrey J. Gordon, J. Andrew Bagnell
TL;DR
Sequential prediction creates a non-i.i.d., policy-dependent learning problem with difficult performance guarantees. The paper introduces DAGGER, a no-regret reduction that trains stationary deterministic policies, and reports better practical performance than prior approaches on challenging imitation-learning tasks.
Problem
Policy-dependent state distributions make sequential prediction non-i.i.d. and optimization difficult, while standard supervised imitation can have quadratic-in-T performance growth.
Method
DAGGER aggregates expert-labeled states visited by iteratively mixed policies, reuses supervised learners, and returns a stationary deterministic policy.
Results
DAGGER outperforms prior approaches in two challenging imitation-learning tasks, including steering in SuperTuxKart and playing Super Mario Bros.
Takeaways & Limitations
The approach provides a practical route to policies evaluated under their own induced state distributions while reusing existing supervised-learning algorithms.
Takeaways & Limitations
Forward training is impractical for large or undefined horizons because it requires training T different policies sequentially and completing all T iterations.
Abstract
from arXiv · showhide
Sequential prediction problems such as imitation learning, where future observations depend on previous predictions (actions), violate the common i.i.d. assumptions made in statistical learning. This leads to poor performance in theory and often in practice. Some recent approaches provide stronger guarantees in this setting, but remain somewhat unsatisfactory as they train either non-stationary or stochastic policies and require a large number of iterations. In this paper, we propose a new iterative algorithm, which trains a stationary deterministic policy, that can be seen as a no regret algorithm in an online learning setting. We show that any such no regret algorithm, combined with additional reduction assumptions, must find a policy with good performance under the distribution of observations it induces in such sequential settings. We demonstrate that this new approach outperforms previous approaches on two challenging imitation learning problems and a benchmark sequence labeling problem.
1 INTRODUCTION
The introduction frames imitation learning as sequential prediction under changing, learner-induced observations, where expert-distribution accuracy can fail badly. It motivates DAGGER as a reduction-based method for learning a stationary deterministic policy with linear-in-horizon performance guarantees.
- Motivation: Imitation learning trains controllers to predict expert actions from observations in sequential robotic systems.Such sequence prediction problems arise when systems must act over observations revealed over time.
- Problem: A classifier with expert-state error probability ϵ can incur as many as T^2ϵ expected mistakes over T steps under its induced state distribution.A single mistake can lead the learner to encounter substantially different observations.
- Limitations of prior work: Prior approaches obtain linear or nearly linear dependence on T and ϵ by iterating and letting learner controls determine where expert demonstrations are collected.One approach learns a non-stationary policy mixture, which may contain poor policies and produce an unstable controller.
- Contribution: DAGGER is proposed as a reduction-based meta-algorithm that learns a stationary deterministic policy performing well under its induced state distribution.Its guarantees have mistakes or costs growing linearly in T and classification cost ϵ.
- Analysis and evaluation: The paper analyzes DAGGER through no-regret, reduction, and online-to-batch techniques, and evaluates it on two challenging imitation learning problems.The introduction also states that the method is scalable and outperforms previous approaches in practice.
2 PRELIMINARIES
The paper formulates imitation learning as minimizing surrogate loss under the learner’s induced state distribution, which is difficult because that distribution depends on the policy. Prior methods address this shift with guarantees ranging from quadratic to near-linear dependence on the task horizon, but use non-stationary or stochastic policies.
- Problem setup: Policies operate for horizon T, inducing time-indexed state distributions d_t^π and average distribution d^π, with bounded immediate costs and total cost J(π).The expected immediate policy cost is C^π(s), while C(s,a) lies in [0, 1].
- Problem setup: Imitation learning minimizes an observed surrogate loss ℓ against expert policy π*, seeking a policy with low loss under its own induced state distribution.The true task cost C may be unavailable, so demonstrations provide the basis for optimizing ℓ, such as 0-1, squared, or hinge loss.
- Problem setup: Because system dynamics are unknown and policy-dependent, d^π can only be sampled by execution, making the problem non-i.i.d. and generally non-convex.The objective can remain non-convex even when ℓ(s, ·) is convex for every state.
- Prior approaches: Traditional supervised learning trains on expert-induced states d^π* and guarantees J(π) ≤ J(π*) + T^2ϵ when the expert-distribution loss is ϵ.This quadratic dependence is tight in some sequence-prediction problems and can yield poor performance as T grows.
- Prior approaches: Forward training uses a distinct policy per time step, while SMILe trains a stochastic stationary policy; both target learner-induced states, and SMILe achieves near-linear regret with α in O(1/T^2) and N in O(T^2 log T).Forward training can improve over supervised learning when the recovery-cost parameter u is O(1) or sub-linear in T, but not when u is O(T).
3 DATASET AGGREGATION
DAGGER iteratively aggregates trajectories from the current policy and trains deterministic policies to mimic the expert on the combined dataset. Under stated assumptions, it provides performance guarantees under the learned policy’s induced state distribution, including finite-sample guarantees.
- Algorithm: DAGGER trains a deterministic policy by collecting expert-labeled trajectories under the current policy and fitting the next policy on all aggregated datasets.At iteration i, trajectories are sampled under π_i, expert actions label visited states, datasets are aggregated, and a classifier ˆπ_i+1 is trained.
- Algorithm: The algorithm builds inputs likely to arise during execution and can be interpreted as Follow-The-Leader over the aggregate dataset.An optional mixture π_i = β_iπ∗ + (1 − β_i)ˆπ_i queries the expert during data collection, which is useful when early learned policies make many mistakes.
- Infinite-sample guarantees: If N is ˜O(T), DAGGER returns a policy with induced loss at most ϵ_N + O(1/T).The guarantee holds in the infinite-sample case under the section’s stated assumptions, including strongly convex, bounded loss.
- Infinite-sample guarantees: If N is ˜O(uT), DAGGER returns a policy whose task cost is at most J(π∗) + uTϵ_N + O(1).This applies when the surrogate loss conditions allow the theorem to relate induced loss to arbitrary task cost.
- Finite-sample guarantees: With probability at least 1 − δ, finite-sample DAGGER achieves induced loss ˆϵ_N + O(1/T) when N is O(T^2 log(1/δ)) and m is O(1).Here m trajectories are sampled at each iteration.
- Finite-sample guarantees: With probability at least 1 − δ, finite-sample DAGGER achieves J(π∗) + uTˆϵ_N + O(1) when N is O(u^2T^2 log(1/δ)) and m is O(1).A refined analysis may reduce the required iteration order to ˜O(T log(1/δ)).
4 THEORETICAL ANALYSIS
DAGGER reduces imitation learning to no-regret online learning by choosing losses under the current policy’s induced state distribution. Under error-reduction and sampling conditions, it guarantees low self-distribution loss, with iteration and trajectory requirements tied to horizon, regret, and confidence.
- No-regret reduction: DAGGER’s analysis relies only on the no-regret property of Follow-The-Leader on strongly convex losses, so it extends to any applicable no-regret online learner.The reduction treats mini-batches of trajectories as online-learning loss observations.
- No-regret reduction: The online-learning losses are defined as ℓ_i(π) = E_s∼d_πi[ℓ(s, π)], aligning optimization with the state distribution induced by the current policy.This choice enables guarantees for performance under the policy’s own state distribution.
- Infinite-sample guarantee: If an error-reduction assumption provides a policy with surrogate loss ϵ for every input distribution, DAGGER finds one with ϵ self-distribution loss as β_N → 0.With β_i = (1 − α)^(i−1), the distribution-mismatch penalty becomes negligible after ˜O(T) iterations.
- Infinite-sample guarantee: ˜O(T) iterations are needed for the penalty and average regret to become negligible, matching the iteration scale of no-regret algorithms.The guarantee is weaker than general classification-only reductions because it requires a no-regret method or strongly convex surrogate loss.
- Finite-sample guarantee: With probability at least 1−δ, DAGGER contains a policy whose self-distribution loss is at most ˆϵ_N + 2 log(1/δ)/(mN), where ˆϵ_N is the finite-sample best-policy loss.The bound applies when each iteration samples m trajectories from the current policy.
- Finite-sample guarantee: Nm in O(T^2 log(1/δ)) trajectories suffices for generalization error O(1/T), while strong convexity may reduce this to O(T log(T/δ)).These requirements follow from applying Azuma-Hoeffding’s inequality to finite trajectory samples.
5 EXPERIMENTS
The experiments evaluate DAGGER on two challenging imitation-learning tasks—steering in Super Tux Kart and playing Super Mario Bros.—and on handwritten-word sequence labeling. Across these settings, the passages report limitations of supervised learning and competing reductions, while DAGGER improves performance in the described comparisons.
- Experimental scope: DAGGER is evaluated on two challenging imitation-learning problems and a handwriting-recognition sequence-labeling task.The tasks are Super Tux Kart steering, Super Mario Bros. gameplay, and handwritten-word recognition.
- Super Tux Kart: In Super Tux Kart, supervised training on expert trajectories does not improve with more data, while SMILe still falls about twice per lap after 20 iterations.The passage attributes these limitations to similar expert trajectories and SMILe’s stochastic policy making bad choices.
- Super Mario Bros.: Super Mario Bros. is evaluated by average distance travelled per stage on randomly generated difficulty-1 stages with a 60-second time limit.Stage distances average around 4200-4300, so performance ranges roughly from 0 to 4300.
- Super Mario Bros.: In Mario, supervised performance stagnates because expert data does not address controller-specific errors, whereas occasional expert intervention exposes useful locations while unsticking the character.The passage contrasts indicator-based data collection, which can leave Mario stuck, with using the expert a small fraction of the time.
- Handwriting sequence labeling: For handwritten-word recognition, DAGGER is compared with SMILe, SEARN, and non-structured baselines while predicting characters left to right using previous predictions.SEARN and SMILe with α = 0.1 perform similarly but significantly worse than DAGGER.
6 FUTURE WORK
The paper identifies future directions beyond simple greedy forward decoding for structured prediction and toward base classifiers using Inverse Optimal Control techniques. It also emphasizes that batching interaction iterations enables no-regret methods, including DAGGER, to provide strong performance guarantees.
- Future Directions: Future work will explore more sophisticated structured-prediction strategies than simple greedy forward decoding.The passage specifically identifies this as a future direction.
- Future Directions: Future work will consider base classifiers that rely on Inverse Optimal Control techniques.The passage cites Abbeel and Ng (2004) and Ratliff et al. (2006) as examples of these techniques.
- Established Result: Batching over interaction iterations allows no-regret methods, including DAGGER, to provide strong performance guarantees in imitation learning and structured prediction.The passage presents this as the paper’s demonstrated learning reduction.