39 research outputs found
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Όλ¬Έ (λ°μ¬) -- μμΈλνκ΅ λνμ : 곡과λν μ κΈ°Β·μ 보곡νλΆ, 2021. 2. Songhwai Oh.The problem of sequential decision making under an uncertain and complex environment is a long-standing challenging problem in robotics. In this thesis, we focus on learning a policy function of robotic systems for sequential decision making under which is called a robot learning framework. In particular, we are interested in reducing the sample complexity of the robot learning framework. Hence, we develop three sample efficient robot learning frameworks. The first one is the maximum entropy reinforcement learning. The second one is a perturbation-based exploration. The last one is learning from demonstrations with mixed qualities.
For maximum entropy reinforcement learning, we employ a generalized Tsallis entropy regularization as an efficient exploration method. Tsallis entropy generalizes Shannon-Gibbs entropy by introducing a entropic index. By changing an entropic index, we can control the sparsity and multi-modality of policy. Based on this fact, we first propose a sparse Markov decision process (sparse MDP) which induces a sparse and multi-modal optimal policy distribution. In this MDP, the sparse entropy, which is a special case of Tsallis entropy, is employed as a policy regularization. We first analyze the optimality condition of a sparse MDP. Then, we propose dynamic programming methods for the sparse MDP and prove their convergence and optimality.
We also show that the performance error of a sparse MDP has a constant bound, while the error of a soft MDP increases logarithmically with respect to the number of actions, where this performance error is caused by the introduced regularization term. Furthermore, we generalize sparse MDPs to a new class of entropy-regularized Markov decision processes (MDPs), which will be referred to as Tsallis MDPs, and analyzes different types of optimal policies with interesting properties related to the stochasticity of the optimal policy by controlling the entropic index.
Furthermore, we also develop perturbation based exploration methods to handle heavy-tailed noises. In many robot learning problems, a learning signal is often corrupted by noises such as sub-Gaussian noise or heavy-tailed noise. While most of the exploration strategies have been analyzed under sub-Gaussian noise assumption, there exist few methods for handling such heavy-tailed rewards. Hence, to overcome heavy-tailed noise, we consider stochastic multi-armed bandits with heavy-tailed rewards. First, we propose a novel robust estimator that does not require prior information about a noise distribution, while other existing robust estimators demand prior knowledge. Then, we show that an error probability of the proposed estimator decays exponentially fast. Using this estimator, we propose a perturbation-based exploration strategy and develop a generalized regret analysis scheme that provides upper and lower regret bounds by revealing the relationship between the regret and the cumulative density function of the perturbation. From the proposed analysis scheme, we obtain gap-dependent and gap-independent upper and lower regret bounds of various perturbations. We also find the optimal hyperparameters for each perturbation, which can achieve the minimax optimal regret bound with respect to total rounds.
For learning from demonstrations with mixed qualities, we develop a novel inverse reinforcement learning framework using leveraged Gaussian processes (LGP) which can handle negative demonstrations. In LGP, the correlation between two Gaussian processes is captured by a leveraged kernel function. By using properties, the proposed inverse reinforcement learning algorithm can learn from both positive and negative demonstrations. While most existing inverse reinforcement learning (IRL) methods suffer from the lack of information near low reward regions, the proposed method alleviates this issue by incorporating negative
demonstrations. To mathematically formulate negative demonstrations, we introduce a novel generative model which can generate both positive and negative demonstrations using a parameter, called proficiency.
Moreover, since we represent a reward function using a leveraged Gaussian process which can model a nonlinear function, the proposed method can effectively estimate the structure of a nonlinear reward
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νΌν©μλ²μΌλ‘ λΆν°μ νμ΅ κΈ°λ²μ κ°λ°νκΈ° μν΄μ, μ€μλ²μ λ€λ£° μ μλ μλ‘μ΄ ννμ κ°μ°μμ νλ‘μΈμ€ νκ·λΆμ λ°©μμ κ°λ°νμκ³ , μ΄ λ°©μμ νμ₯νμ¬ λ λ²λ¦¬μ§ κ°μ°μμ νλ‘μΈμ€ μκ°ννμ΅ κΈ°λ²μ κ°λ°νμλ€. κ°λ°λ κΈ°λ²μμλ μ μλ²μΌλ‘λΆν° 무μμ ν΄μΌ νλμ§μ μ€μλ²μΌλ‘λΆν° 무μμ νλ©΄ μλλμ§λ₯Ό λͺ¨λ νμ΅ν μ μλ€. κΈ°μ‘΄μ λ°©λ²μμλ μ°μΌ μ μμλ μ€μλ²μ μ¬μ© ν μ μκ² λ§λ¦μΌλ‘μ¨ μν 볡μ‘λλ₯Ό μ€μΌ μ μμκ³ μ μ λ λ°μ΄ν°λ₯Ό μμ§νμ§ μμλ λλ€λ μ μμ ν° μ₯μ μ κ°μμ μ€νμ μΌλ‘ 보μλ€.1 Introduction 1
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Organization of the Dissertation . . . . . . . . . . . . . . . . . . . 4
2 Background 5
2.1 Learning from Rewards . . . . . . . . . . . . . . . . . . . . . . . . 6
2.1.1 Multi-Armed Bandits . . . . . . . . . . . . . . . . . . . . . 7
2.1.2 Contextual Multi-Armed Bandits . . . . . . . . . . . . . . . 7
2.1.3 Markov Decision Processes . . . . . . . . . . . . . . . . . . 9
2.1.4 Soft Markov Decision Processes . . . . . . . . . . . . . . . . 10
2.2 Learning from Demonstrations . . . . . . . . . . . . . . . . . . . . 12
2.2.1 Behavior Cloning . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2.2 Inverse Reinforcement Learning . . . . . . . . . . . . . . . . 13
3 Sparse Policy Learning 19
3.1 Sparse Policy Learning for Reinforcement Learning . . . . . . . . . 19
3.1.1 Sparse Markov Decision Processes . . . . . . . . . . . . . . 23
3.1.2 Sparse Value Iteration . . . . . . . . . . . . . . . . . . . . . 29
3.1.3 Performance Error Bounds for Sparse Value Iteration . . . 30
3.1.4 Sparse Exploration and Update Rule for Sparse Deep QLearning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.1.5 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.1.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.2 Sparse Policy Learning for Imitation Learning . . . . . . . . . . . . 46
3.2.1 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.2.2 Principle of Maximum Causal Tsallis Entropy . . . . . . . . 50
3.2.3 Maximum Causal Tsallis Entropy Imitation Learning . . . 54
3.2.4 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4 Entropy-based Exploration 65
4.1 Generalized Tsallis Entropy Reinforcement Learning . . . . . . . . 65
4.1.1 Maximum Generalized Tsallis Entropy in MDPs . . . . . . 71
4.1.2 Dynamic Programming for Tsallis MDPs . . . . . . . . . . 74
4.1.3 Tsallis Actor Critic for Model-Free RL . . . . . . . . . . . . 78
4.1.4 Experiments Setup . . . . . . . . . . . . . . . . . . . . . . . 79
4.1.5 Experimental Results . . . . . . . . . . . . . . . . . . . . . 84
4.1.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
4.2 E cient Exploration for Robotic Grasping . . . . . . . . . . . . . . 92
4.2.1 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . 94
4.2.2 Shannon Entropy Regularized Neural Contextual Bandit
Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
4.2.3 Theoretical Analysis . . . . . . . . . . . . . . . . . . . . . . 99
4.2.4 Experimental Results . . . . . . . . . . . . . . . . . . . . . 104
4.2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
5 Perturbation-Based Exploration 113
5.1 Perturbed Exploration for sub-Gaussian Rewards . . . . . . . . . . 115
5.1.1 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . 115
5.1.2 Heavy-Tailed Perturbations . . . . . . . . . . . . . . . . . . 117
5.1.3 Adaptively Perturbed Exploration . . . . . . . . . . . . . . 119
5.1.4 General Regret Bound for Sub-Gaussian Rewards . . . . . . 120
5.1.5 Regret Bounds for Speci c Perturbations with sub-Gaussian Rewards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
5.1.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
5.2 Perturbed Exploration for Heavy-Tailed Rewards . . . . . . . . . . 128
5.2.1 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . 130
5.2.2 Sub-Optimality of Robust Upper Con dence Bounds . . . . 132
5.2.3 Adaptively Perturbed Exploration with A p-Robust Estimator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
5.2.4 General Regret Bound for Heavy-Tailed Rewards . . . . . . 136
5.2.5 Regret Bounds for Speci c Perturbations with Heavy-Tailed Rewards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
5.2.6 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . 144
5.2.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
6 Inverse Reinforcement Learning with Negative Demonstrations149
6.1 Leveraged Gaussian Processes Inverse Reinforcement Learning . . 151
6.1.1 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . 152
6.1.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
6.1.3 Gaussian Process Regression . . . . . . . . . . . . . . . . . 156
6.1.4 Leveraged Gaussian Processes . . . . . . . . . . . . . . . . . 159
6.1.5 Gaussian Process Inverse Reinforcement Learning . . . . . 164
6.1.6 Inverse Reinforcement Learning with Leveraged Gaussian Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
6.1.7 Simulations and Experiment . . . . . . . . . . . . . . . . . 175
6.1.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
7 Conclusion 185
Appendices 189
A Proofs of Chapter 3.1. 191
A.1 Useful Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
A.2 Sparse Bellman Optimality Equation . . . . . . . . . . . . . . . . . 192
A.3 Sparse Tsallis Entropy . . . . . . . . . . . . . . . . . . . . . . . . . 195
A.4 Upper and Lower Bounds for Sparsemax Operation . . . . . . . . . 196
A.5 Comparison to Log-Sum-Exp . . . . . . . . . . . . . . . . . . . . . 200
A.6 Convergence and Optimality of Sparse Value Iteration . . . . . . . 201
A.7 Performance Error Bounds for Sparse Value Iteration . . . . . . . . 203
B Proofs of Chapter 3.2. 209
B.1 Change of Variables . . . . . . . . . . . . . . . . . . . . . . . . . . 209
B.2 Concavity of Maximum Causal Tsallis Entropy . . . . . . . . . . . 210
B.3 Optimality Condition of Maximum Causal Tsallis Entropy . . . . . 212
B.4 Interpretation as Robust Bayes . . . . . . . . . . . . . . . . . . . . 215
B.5 Generative Adversarial Setting with Maximum Causal Tsallis Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
B.6 Tsallis Entropy of a Mixture of Gaussians . . . . . . . . . . . . . . 217
B.7 Causal Entropy Approximation . . . . . . . . . . . . . . . . . . . . 218
C Proofs of Chapter 4.1. 221
C.1 q-Maximum: Bounded Approximation of Maximum . . . . . . . . . 223
C.2 Tsallis Bellman Optimality Equation . . . . . . . . . . . . . . . . . 226
C.3 Variable Change . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228
C.4 Tsallis Bellman Optimality Equation . . . . . . . . . . . . . . . . . 230
C.5 Tsallis Policy Iteration . . . . . . . . . . . . . . . . . . . . . . . . . 234
C.6 Tsallis Bellman Expectation (TBE) Equation . . . . . . . . . . . . 234
C.7 Tsallis Bellman Expectation Operator and Tsallis Policy Evaluation235
C.8 Tsallis Policy Improvement . . . . . . . . . . . . . . . . . . . . . . 237
C.9 Tsallis Value Iteration . . . . . . . . . . . . . . . . . . . . . . . . . 239
C.10 Performance Error Bounds . . . . . . . . . . . . . . . . . . . . . . 241
C.11 q-Scheduling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
D Proofs of Chapter 4.2. 245
D.1 In nite Exploration . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
D.2 Regret Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
E Proofs of Chapter 5.1. 255
E.1 General Regret Lower Bound of APE . . . . . . . . . . . . . . . . . 255
E.2 General Regret Upper Bound of APE . . . . . . . . . . . . . . . . 257
E.3 Proofs of Corollaries . . . . . . . . . . . . . . . . . . . . . . . . . . 266
F Proofs of Chapter 5.2. 279
F.1 Regret Lower Bound for Robust Upper Con dence Bound . . . . . 279
F.2 Bounds on Tail Probability of A p-Robust Estimator . . . . . . . . 284
F.3 General Regret Upper Bound of APE2 . . . . . . . . . . . . . . . . 287
F.4 General Regret Lower Bound of APE2 . . . . . . . . . . . . . . . . 299
F.5 Proofs of Corollaries . . . . . . . . . . . . . . . . . . . . . . . . . . 302Docto
Identifiability and Generalizability in Constrained Inverse Reinforcement Learning
Two main challenges in Reinforcement Learning (RL) are designing appropriate
reward functions and ensuring the safety of the learned policy. To address
these challenges, we present a theoretical framework for Inverse Reinforcement
Learning (IRL) in constrained Markov decision processes. From a convex-analytic
perspective, we extend prior results on reward identifiability and
generalizability to both the constrained setting and a more general class of
regularizations. In particular, we show that identifiability up to potential
shaping (Cao et al., 2021) is a consequence of entropy regularization and may
generally no longer hold for other regularizations or in the presence of safety
constraints. We also show that to ensure generalizability to new transition
laws and constraints, the true reward must be identified up to a constant.
Additionally, we derive a finite sample guarantee for the suboptimality of the
learned rewards, and validate our results in a gridworld environment.Comment: Published at ICML 202
Generalized Munchausen Reinforcement Learning using Tsallis KL Divergence
Many policy optimization approaches in reinforcement learning incorporate a
Kullback-Leilbler (KL) divergence to the previous policy, to prevent the policy
from changing too quickly. This idea was initially proposed in a seminal paper
on Conservative Policy Iteration, with approximations given by algorithms like
TRPO and Munchausen Value Iteration (MVI). We continue this line of work by
investigating a generalized KL divergence -- called the Tsallis KL divergence
-- which use the -logarithm in the definition. The approach is a strict
generalization, as corresponds to the standard KL divergence;
provides a range of new options. We characterize the types of policies learned
under the Tsallis KL, and motivate when could be beneficial. To obtain a
practical algorithm that incorporates Tsallis KL regularization, we extend MVI,
which is one of the simplest approaches to incorporate KL regularization. We
show that this generalized MVI() obtains significant improvements over the
standard MVI() across 35 Atari games.Comment: Accepted by NeurIPS 202
Sparse Randomized Shortest Paths Routing with Tsallis Divergence Regularization
This work elaborates on the important problem of (1) designing optimal
randomized routing policies for reaching a target node t from a source note s
on a weighted directed graph G and (2) defining distance measures between nodes
interpolating between the least cost (based on optimal movements) and the
commute-cost (based on a random walk on G), depending on a temperature
parameter T. To this end, the randomized shortest path formalism (RSP,
[2,99,124]) is rephrased in terms of Tsallis divergence regularization, instead
of Kullback-Leibler divergence. The main consequence of this change is that the
resulting routing policy (local transition probabilities) becomes sparser when
T decreases, therefore inducing a sparse random walk on G converging to the
least-cost directed acyclic graph when T tends to 0. Experimental comparisons
on node clustering and semi-supervised classification tasks show that the
derived dissimilarity measures based on expected routing costs provide
state-of-the-art results. The sparse RSP is therefore a promising model of
movements on a graph, balancing sparse exploitation and exploration in an
optimal way