We study structured multi-armed bandits, which is the problem of online decision-making under uncertainty in the presence of structural information. In this problem, the decision-maker needs to discover the best course of action despite observing only uncertain rewards over time. The decision-maker is aware of certain structural information regarding the reward distributions and would like to minimize their regret by exploiting this information, where the regret is its performance difference against a benchmark policy that knows the best action ahead of time. In the absence of structural information, the classical upper confidence bound (UCB) and Thomson sampling algorithms are well known to suffer only minimal regret. As recently pointed out, neither algorithms are, however, capable of exploiting structural information that is commonly available in practice. We propose a novel learning algorithm that we call DUSA whose worst-case regret matches the information-theoretic regret lower bound up to a constant factor and can handle a wide range of structural information. Our algorithm DUSA solves a dual counterpart of the regret lower bound at the empirical reward distribution and follows its suggested play. Our proposed algorithm is the first computationally viable learning policy for structured bandit problems that has asymptotic minimal regret.
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Policy gradient methods can solve complex tasks but often fail when the dimensionality of the action-space or objective multiplicity grow very large. This occurs, in part, because the variance on score-based gradient estimators scales quadratically. In this paper, we address this problem through a factor baseline which exploits independence structure encoded in a novel action-target influence network. Factored policy gradients (FPGs), which follow, provide a common framework for analysing key state-of-the-art algorithms, are shown to generalise traditional policy gradients, and yield a principled way of incorporating prior knowledge of a problem domain's generative processes. We provide an analysis of the proposed estimator and identify the conditions under which variance is reduced. The algorithmic aspects of FPGs are discussed, including optimal policy factorisation, as characterised by minimum biclique coverings, and the implications for the bias-variance trade-off of incorrectly specifying the network. Finally, we demonstrate the performance advantages of our algorithm on large-scale bandit and traffic intersection problems, providing a novel contribution to the latter in the form of a spatial approximation.
We analyze the complexity of learning directed acyclic graphical models from observational data in general settings without specific distributional assumptions. Our approach is information-theoretic and uses a local Markov boundary search procedure in order to recursively construct ancestral sets in the underlying graphical model. Perhaps surprisingly, we show that for certain graph ensembles, a simple forward greedy search algorithm (i.e. without a backward pruning phase) suffices to learn the Markov boundary of each node. This substantially improves the sample complexity, which we show is at most polynomial in the number of nodes. This is then applied to learn the entire graph under a novel identifiability condition that generalizes existing conditions from the literature. As a matter of independent interest, we establish finite-sample guarantees for the problem of recovering Markov boundaries from data. Moreover, we apply our results to the special case of polytrees, for which the assumptions simplify, and provide explicit conditions under which polytrees are identifiable and learnable in polynomial time. We further illustrate the performance of the algorithm, which is easy to implement, in a simulation study. Our approach is general, works for discrete or continuous distributions without distributional assumptions, and as such sheds light on the minimal assumptions required to efficiently learn the structure of directed graphical models from data.
Recent advances in the theoretical understanding of SGD led to a formula for the optimal batch size minimizing the number of effective data passes, i.e., the number of iterations times the batch size. However, this formula is of no practical value as it depends on the knowledge of the variance of the stochastic gradients evaluated at the optimum. In this paper we design a practical SGD method capable of learning the optimal batch size adaptively throughout its iterations for strongly convex and smooth functions. Our method does this provably, and in our experiments with synthetic and real data robustly exhibits nearly optimal behaviour; that is, it works as if the optimal batch size was known a-priori. Further, we generalize our method to several new batch strategies not considered in the literature before, including a sampling suitable for distributed implementations.
Combination and aggregation techniques can significantly improve forecast accuracy. This also holds for probabilistic forecasting methods where predictive distributions are combined. There are several time-varying and adaptive weighting schemes such as Bayesian model averaging (BMA). However, the quality of different forecasts may vary not only over time but also within the distribution. For example, some distribution forecasts may be more accurate in the center of the distributions, while others are better at predicting the tails. Therefore, we introduce a new weighting method that considers the differences in performance over time and within the distribution. We discuss pointwise combination algorithms based on aggregation across quantiles that optimize with respect to the continuous ranked probability score (CRPS). After analyzing the theoretical properties of pointwise CRPS learning, we discuss B- and P-Spline-based estimation techniques for batch and online learning, based on quantile regression and prediction with expert advice. We prove that the proposed fully adaptive Bernstein online aggregation (BOA) method for pointwise CRPS online learning has optimal convergence properties. They are confirmed in simulations and a probabilistic forecasting study for European emission allowance (EUA) prices.
Value estimation is one key problem in Reinforcement Learning. Albeit many successes have been achieved by Deep Reinforcement Learning (DRL) in different fields, the underlying structure and learning dynamics of value function, especially with complex function approximation, are not fully understood. In this paper, we report that decreasing rank of $Q$-matrix widely exists during learning process across a series of continuous control tasks for different popular algorithms. We hypothesize that the low-rank phenomenon indicates the common learning dynamics of $Q$-matrix from stochastic high dimensional space to smooth low dimensional space. Moreover, we reveal a positive correlation between value matrix rank and value estimation uncertainty. Inspired by above evidence, we propose a novel Uncertainty-Aware Low-rank Q-matrix Estimation (UA-LQE) algorithm as a general framework to facilitate the learning of value function. Through quantifying the uncertainty of state-action value estimation, we selectively erase the entries of highly uncertain values in state-action value matrix and conduct low-rank matrix reconstruction for them to recover their values. Such a reconstruction exploits the underlying structure of value matrix to improve the value approximation, thus leading to a more efficient learning process of value function. In the experiments, we evaluate the efficacy of UA-LQE in several representative OpenAI MuJoCo continuous control tasks.
The difficulty in specifying rewards for many real-world problems has led to an increased focus on learning rewards from human feedback, such as demonstrations. However, there are often many different reward functions that explain the human feedback, leaving agents with uncertainty over what the true reward function is. While most policy optimization approaches handle this uncertainty by optimizing for expected performance, many applications demand risk-averse behavior. We derive a novel policy gradient-style robust optimization approach, PG-BROIL, that optimizes a soft-robust objective that balances expected performance and risk. To the best of our knowledge, PG-BROIL is the first policy optimization algorithm robust to a distribution of reward hypotheses which can scale to continuous MDPs. Results suggest that PG-BROIL can produce a family of behaviors ranging from risk-neutral to risk-averse and outperforms state-of-the-art imitation learning algorithms when learning from ambiguous demonstrations by hedging against uncertainty, rather than seeking to uniquely identify the demonstrator's reward function.
Graph neural networks (GNNs) are a popular class of machine learning models whose major advantage is their ability to incorporate a sparse and discrete dependency structure between data points. Unfortunately, GNNs can only be used when such a graph-structure is available. In practice, however, real-world graphs are often noisy and incomplete or might not be available at all. With this work, we propose to jointly learn the graph structure and the parameters of graph convolutional networks (GCNs) by approximately solving a bilevel program that learns a discrete probability distribution on the edges of the graph. This allows one to apply GCNs not only in scenarios where the given graph is incomplete or corrupted but also in those where a graph is not available. We conduct a series of experiments that analyze the behavior of the proposed method and demonstrate that it outperforms related methods by a significant margin.
Deep reinforcement learning suggests the promise of fully automated learning of robotic control policies that directly map sensory inputs to low-level actions. However, applying deep reinforcement learning methods on real-world robots is exceptionally difficult, due both to the sample complexity and, just as importantly, the sensitivity of such methods to hyperparameters. While hyperparameter tuning can be performed in parallel in simulated domains, it is usually impractical to tune hyperparameters directly on real-world robotic platforms, especially legged platforms like quadrupedal robots that can be damaged through extensive trial-and-error learning. In this paper, we develop a stable variant of the soft actor-critic deep reinforcement learning algorithm that requires minimal hyperparameter tuning, while also requiring only a modest number of trials to learn multilayer neural network policies. This algorithm is based on the framework of maximum entropy reinforcement learning, and automatically trades off exploration against exploitation by dynamically and automatically tuning a temperature parameter that determines the stochasticity of the policy. We show that this method achieves state-of-the-art performance on four standard benchmark environments. We then demonstrate that it can be used to learn quadrupedal locomotion gaits on a real-world Minitaur robot, learning to walk from scratch directly in the real world in two hours of training.
We consider the exploration-exploitation trade-off in reinforcement learning and we show that an agent imbued with a risk-seeking utility function is able to explore efficiently, as measured by regret. The parameter that controls how risk-seeking the agent is can be optimized exactly, or annealed according to a schedule. We call the resulting algorithm K-learning and show that the corresponding K-values are optimistic for the expected Q-values at each state-action pair. The K-values induce a natural Boltzmann exploration policy for which the `temperature' parameter is equal to the risk-seeking parameter. This policy achieves an expected regret bound of $\tilde O(L^{3/2} \sqrt{S A T})$, where $L$ is the time horizon, $S$ is the number of states, $A$ is the number of actions, and $T$ is the total number of elapsed time-steps. This bound is only a factor of $L$ larger than the established lower bound. K-learning can be interpreted as mirror descent in the policy space, and it is similar to other well-known methods in the literature, including Q-learning, soft-Q-learning, and maximum entropy policy gradient, and is closely related to optimism and count based exploration methods. K-learning is simple to implement, as it only requires adding a bonus to the reward at each state-action and then solving a Bellman equation. We conclude with a numerical example demonstrating that K-learning is competitive with other state-of-the-art algorithms in practice.
This work considers the problem of provably optimal reinforcement learning for episodic finite horizon MDPs, i.e. how an agent learns to maximize his/her long term reward in an uncertain environment. The main contribution is in providing a novel algorithm --- Variance-reduced Upper Confidence Q-learning (vUCQ) --- which enjoys a regret bound of $\widetilde{O}(\sqrt{HSAT} + H^5SA)$, where the $T$ is the number of time steps the agent acts in the MDP, $S$ is the number of states, $A$ is the number of actions, and $H$ is the (episodic) horizon time. This is the first regret bound that is both sub-linear in the model size and asymptotically optimal. The algorithm is sub-linear in that the time to achieve $\epsilon$-average regret for any constant $\epsilon$ is $O(SA)$, which is a number of samples that is far less than that required to learn any non-trivial estimate of the transition model (the transition model is specified by $O(S^2A)$ parameters). The importance of sub-linear algorithms is largely the motivation for algorithms such as $Q$-learning and other "model free" approaches. vUCQ algorithm also enjoys minimax optimal regret in the long run, matching the $\Omega(\sqrt{HSAT})$ lower bound. Variance-reduced Upper Confidence Q-learning (vUCQ) is a successive refinement method in which the algorithm reduces the variance in $Q$-value estimates and couples this estimation scheme with an upper confidence based algorithm. Technically, the coupling of both of these techniques is what leads to the algorithm enjoying both the sub-linear regret property and the asymptotically optimal regret.
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