We study query and computationally efficient planning algorithms with linear function approximation and a simulator. We assume that the agent only has local access to the simulator, meaning that the agent can only query the simulator at states that have been visited before. This setting is more practical than many prior works on reinforcement learning with a generative model. We propose an algorithm named confident Monte Carlo least square policy iteration (Confident MC-LSPI) for this setting. Under the assumption that the Q-functions of all deterministic policies are linear in known features of the state-action pairs, we show that our algorithm has polynomial query and computational complexities in the dimension of the features, the effective planning horizon and the targeted sub-optimality, while these complexities are independent of the size of the state space. One technical contribution of our work is the introduction of a novel proof technique that makes use of a virtual policy iteration algorithm. We use this method to leverage existing results on $\ell_\infty$-bounded approximate policy iteration to show that our algorithm can learn the optimal policy for the given initial state even only with local access to the simulator. We believe that this technique can be extended to broader settings beyond this work.
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We study the optimization landscape and the stability properties of training problems with squared loss for neural networks and general nonlinear conic approximation schemes. It is demonstrated that, if a nonlinear conic approximation scheme is considered that is (in an appropriately defined sense) more expressive than a classical linear approximation approach and if there exist unrealizable label vectors, then a training problem with squared loss is necessarily unstable in the sense that its solution set depends discontinuously on the label vector in the training data. We further prove that the same effects that are responsible for these instability properties are also the reason for the emergence of saddle points and spurious local minima, which may be arbitrarily far away from global solutions, and that neither the instability of the training problem nor the existence of spurious local minima can, in general, be overcome by adding a regularization term to the objective function that penalizes the size of the parameters in the approximation scheme. The latter results are shown to be true regardless of whether the assumption of realizability is satisfied or not. We demonstrate that our analysis in particular applies to training problems for free-knot interpolation schemes and deep and shallow neural networks with variable widths that involve an arbitrary mixture of various activation functions (e.g., binary, sigmoid, tanh, arctan, soft-sign, ISRU, soft-clip, SQNL, ReLU, leaky ReLU, soft-plus, bent identity, SILU, ISRLU, and ELU). In summary, the findings of this paper illustrate that the improved approximation properties of neural networks and general nonlinear conic approximation instruments are linked in a direct and quantifiable way to undesirable properties of the optimization problems that have to be solved in order to train them.
We consider the problem of spatial path planning. In contrast to the classical solutions which optimize a new plan from scratch and assume access to the full map with ground truth obstacle locations, we learn a planner from the data in a differentiable manner that allows us to leverage statistical regularities from past data. We propose Spatial Planning Transformers (SPT), which given an obstacle map learns to generate actions by planning over long-range spatial dependencies, unlike prior data-driven planners that propagate information locally via convolutional structure in an iterative manner. In the setting where the ground truth map is not known to the agent, we leverage pre-trained SPTs in an end-to-end framework that has the structure of mapper and planner built into it which allows seamless generalization to out-of-distribution maps and goals. SPTs outperform prior state-of-the-art differentiable planners across all the setups for both manipulation and navigation tasks, leading to an absolute improvement of 7-19%.
We consider the problem of learning an episodic safe control policy that minimizes an objective function, while satisfying necessary safety constraints -- both during learning and deployment. We formulate this safety constrained reinforcement learning (RL) problem using the framework of a finite-horizon Constrained Markov Decision Process (CMDP) with an unknown transition probability function. Here, we model the safety requirements as constraints on the expected cumulative costs that must be satisfied during all episodes of learning. We propose a model-based safe RL algorithm that we call the Optimistic-Pessimistic Safe Reinforcement Learning (OPSRL) algorithm, and show that it achieves an $\tilde{\mathcal{O}}(S^{2}\sqrt{A H^{7}K}/ (\bar{C} - \bar{C}_{b}))$ cumulative regret without violating the safety constraints during learning, where $S$ is the number of states, $A$ is the number of actions, $H$ is the horizon length, $K$ is the number of learning episodes, and $(\bar{C} - \bar{C}_{b})$ is the safety gap, i.e., the difference between the constraint value and the cost of a known safe baseline policy. The scaling as $\tilde{\mathcal{O}}(\sqrt{K})$ is the same as the traditional approach where constraints may be violated during learning, which means that our algorithm suffers no additional regret in spite of providing a safety guarantee. Our key idea is to use an optimistic exploration approach with pessimistic constraint enforcement for learning the policy. This approach simultaneously incentivizes the exploration of unknown states while imposing a penalty for visiting states that are likely to cause violation of safety constraints. We validate our algorithm by evaluating its performance on benchmark problems against conventional approaches.
We study variants of the mean problem under the $p$-Dynamic Time Warping ($p$-DTW) distance, a popular and robust distance measure for sequential data. In our setting we are given a set of finite point sequences over an arbitrary metric space and we want to compute a mean point sequence of given length that minimizes the sum of $p$-DTW distances, each raised to the $q$th power, between the input sequences and the mean sequence. In general, the problem is $\mathrm{NP}$-hard and known not to be fixed-parameter tractable in the number of sequences. We show that it is even hard to approximate within any constant factor unless $\mathrm{P} = \mathrm{NP}$ and moreover if there exists a $\delta>0$ such that there is a $(\log n)^{\delta}$-approximation algorithm for DTW mean then $\mathrm{NP} \subseteq \mathrm{QP}$. On the positive side, we show that restricting the length of the mean sequence significantly reduces the hardness of the problem. We give an exact algorithm running in polynomial time for constant-length means. We explore various approximation algorithms that provide a trade-off between the approximation factor and the running time. Our approximation algorithms have a running time with only linear dependency on the number of input sequences. In addition, we use our mean algorithms to obtain clustering algorithms with theoretical guarantees.
We investigate the optimal design of experimental studies that have pre-treatment outcome data available. The average treatment effect is estimated as the difference between the weighted average outcomes of the treated and control units. A number of commonly used approaches fit this formulation, including the difference-in-means estimator and a variety of synthetic-control techniques. We propose several methods for choosing the set of treated units in conjunction with the weights. Observing the NP-hardness of the problem, we introduce a mixed-integer programming formulation which selects both the treatment and control sets and unit weightings. We prove that these proposed approaches lead to qualitatively different experimental units being selected for treatment. We use simulations based on publicly available data from the US Bureau of Labor Statistics that show improvements in terms of mean squared error and statistical power when compared to simple and commonly used alternatives such as randomized trials.
This paper presents an efficient reversible algorithm for linear regression, both with and without ridge regression. Our reversible algorithm matches the asymptotic time and space complexity of standard irreversible algorithms for this problem. Needed for this result is the expansion of the analysis of efficient reversible matrix multiplication to rectangular matrices and matrix inversion.
Retrosynthetic planning is a fundamental problem in chemistry for finding a pathway of reactions to synthesize a target molecule. Recently, search algorithms have shown promising results for solving this problem by using deep neural networks (DNNs) to expand their candidate solutions, i.e., adding new reactions to reaction pathways. However, the existing works on this line are suboptimal; the retrosynthetic planning problem requires the reaction pathways to be (a) represented by real-world reactions and (b) executable using "building block" molecules, yet the DNNs expand reaction pathways without fully incorporating such requirements. Motivated by this, we propose an end-to-end framework for directly training the DNNs towards generating reaction pathways with the desirable properties. Our main idea is based on a self-improving procedure that trains the model to imitate successful trajectories found by itself. We also propose a novel reaction augmentation scheme based on a forward reaction model. Our experiments demonstrate that our scheme significantly improves the success rate of solving the retrosynthetic problem from 86.84% to 96.32% while maintaining the performance of DNN for predicting valid reactions.
Many important real-world problems have action spaces that are high-dimensional, continuous or both, making full enumeration of all possible actions infeasible. Instead, only small subsets of actions can be sampled for the purpose of policy evaluation and improvement. In this paper, we propose a general framework to reason in a principled way about policy evaluation and improvement over such sampled action subsets. This sample-based policy iteration framework can in principle be applied to any reinforcement learning algorithm based upon policy iteration. Concretely, we propose Sampled MuZero, an extension of the MuZero algorithm that is able to learn in domains with arbitrarily complex action spaces by planning over sampled actions. We demonstrate this approach on the classical board game of Go and on two continuous control benchmark domains: DeepMind Control Suite and Real-World RL Suite.
In order to avoid the curse of dimensionality, frequently encountered in Big Data analysis, there was a vast development in the field of linear and nonlinear dimension reduction techniques in recent years. These techniques (sometimes referred to as manifold learning) assume that the scattered input data is lying on a lower dimensional manifold, thus the high dimensionality problem can be overcome by learning the lower dimensionality behavior. However, in real life applications, data is often very noisy. In this work, we propose a method to approximate $\mathcal{M}$ a $d$-dimensional $C^{m+1}$ smooth submanifold of $\mathbb{R}^n$ ($d \ll n$) based upon noisy scattered data points (i.e., a data cloud). We assume that the data points are located "near" the lower dimensional manifold and suggest a non-linear moving least-squares projection on an approximating $d$-dimensional manifold. Under some mild assumptions, the resulting approximant is shown to be infinitely smooth and of high approximation order (i.e., $O(h^{m+1})$, where $h$ is the fill distance and $m$ is the degree of the local polynomial approximation). The method presented here assumes no analytic knowledge of the approximated manifold and the approximation algorithm is linear in the large dimension $n$. Furthermore, the approximating manifold can serve as a framework to perform operations directly on the high dimensional data in a computationally efficient manner. This way, the preparatory step of dimension reduction, which induces distortions to the data, can be avoided altogether.
UMAP (Uniform Manifold Approximation and Projection) is a novel manifold learning technique for dimension reduction. UMAP is constructed from a theoretical framework based in Riemannian geometry and algebraic topology. The result is a practical scalable algorithm that applies to real world data. The UMAP algorithm is competitive with t-SNE for visualization quality, and arguably preserves more of the global structure with superior run time performance. Furthermore, UMAP has no computational restrictions on embedding dimension, making it viable as a general purpose dimension reduction technique for machine learning.
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