Contextual bandits are a rich model for sequential decision making given side information, with important applications, e.g., in recommender systems. We propose novel algorithms for contextual bandits harnessing neural networks to approximate the unknown reward function. We resolve the open problem of proving sublinear regret bounds in this setting for general context sequences, considering both fully-connected and convolutional networks. To this end, we first analyze NTK-UCB, a kernelized bandit optimization algorithm employing the Neural Tangent Kernel (NTK), and bound its regret in terms of the NTK maximum information gain $\gamma_T$, a complexity parameter capturing the difficulty of learning. Our bounds on $\gamma_T$ for the NTK may be of independent interest. We then introduce our neural network based algorithm NN-UCB, and show that its regret closely tracks that of NTK-UCB. Under broad non-parametric assumptions about the reward function, our approach converges to the optimal policy at a $\tilde{\mathcal{O}}(T^{-1/2d})$ rate, where $d$ is the dimension of the context.
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In this work, we study the $k$-means cost function. Given a dataset $X \subseteq \mathbb{R}^d$ and an integer $k$, the goal of the Euclidean $k$-means problem is to find a set of $k$ centers $C \subseteq \mathbb{R}^d$ such that $\Phi(C, X) \equiv \sum_{x \in X} \min_{c \in C} ||x - c||^2$ is minimized. Let $\Delta(X,k) \equiv \min_{C \subseteq \mathbb{R}^d} \Phi(C, X)$ denote the cost of the optimal $k$-means solution. For any dataset $X$, $\Delta(X,k)$ decreases as $k$ increases. In this work, we try to understand this behaviour more precisely. For any dataset $X \subseteq \mathbb{R}^d$, integer $k \geq 1$, and a precision parameter $\varepsilon > 0$, let $L(X, k, \varepsilon)$ denote the smallest integer such that $\Delta(X, L(X, k, \varepsilon)) \leq \varepsilon \cdot \Delta(X,k)$. We show upper and lower bounds on this quantity. Our techniques generalize for the metric $k$-median problem in arbitrary metric spaces and we give bounds in terms of the doubling dimension of the metric. Finally, we observe that for any dataset $X$, we can compute a set $S$ of size $O \left(L(X, k, \varepsilon/c) \right)$ using $D^2$-sampling such that $\Phi(S,X) \leq \varepsilon \cdot \Delta(X,k)$ for some fixed constant $c$. We also discuss some applications of our bounds.
Incorporating graph side information into recommender systems has been widely used to better predict ratings, but relatively few works have focused on theoretical guarantees. Ahn et al. (2018) firstly characterized the optimal sample complexity in the presence of graph side information, but the results are limited due to strict, unrealistic assumptions made on the unknown latent preference matrix and the structure of user clusters. In this work, we propose a new model in which 1) the unknown latent preference matrix can have any discrete values, and 2) users can be clustered into multiple clusters, thereby relaxing the assumptions made in prior work. Under this new model, we fully characterize the optimal sample complexity and develop a computationally-efficient algorithm that matches the optimal sample complexity. Our algorithm is robust to model errors and outperforms the existing algorithms in terms of prediction performance on both synthetic and real data.
Network data are increasingly available in various research fields, motivating statistical analysis for populations of networks where a network as a whole is viewed as a data point. Due to the non-Euclidean nature of networks, basic statistical tools available for scalar and vector data are no longer applicable when one aims to relate networks as outcomes to Euclidean covariates, while the study of how a network changes in dependence on covariates is often of paramount interest. This motivates to extend the notion of regression to the case of responses that are network data. Here we propose to adopt conditional Fr\'{e}chet means implemented with both global least squares regression and local weighted least squares smoothing, extending the Fr\'{e}chet regression concept to networks that are quantified by their graph Laplacians. The challenge is to characterize the space of graph Laplacians so as to justify the application of Fr\'{e}chet regression. This characterization then leads to asymptotic rates of convergence for the corresponding M-estimators by applying empirical process methods. We demonstrate the usefulness and good practical performance of the proposed framework with simulations and with network data arising from NYC taxi records, as well as resting-state fMRI in neuroimaging.
User cold-start recommendation is a long-standing challenge for recommender systems due to the fact that only a few interactions of cold-start users can be exploited. Recent studies seek to address this challenge from the perspective of meta learning, and most of them follow a manner of parameter initialization, where the model parameters can be learned by a few steps of gradient updates. While these gradient-based meta-learning models achieve promising performances to some extent, a fundamental problem of them is how to adapt the global knowledge learned from previous tasks for the recommendations of cold-start users more effectively. In this paper, we develop a novel meta-learning recommender called task-adaptive neural process (TaNP). TaNP is a new member of the neural process family, where making recommendations for each user is associated with a corresponding stochastic process. TaNP directly maps the observed interactions of each user to a predictive distribution, sidestepping some training issues in gradient-based meta-learning models. More importantly, to balance the trade-off between model capacity and adaptation reliability, we introduce a novel task-adaptive mechanism. It enables our model to learn the relevance of different tasks and customize the global knowledge to the task-related decoder parameters for estimating user preferences. We validate TaNP on multiple benchmark datasets in different experimental settings. Empirical results demonstrate that TaNP yields consistent improvements over several state-of-the-art meta-learning recommenders.
Sampling methods (e.g., node-wise, layer-wise, or subgraph) has become an indispensable strategy to speed up training large-scale Graph Neural Networks (GNNs). However, existing sampling methods are mostly based on the graph structural information and ignore the dynamicity of optimization, which leads to high variance in estimating the stochastic gradients. The high variance issue can be very pronounced in extremely large graphs, where it results in slow convergence and poor generalization. In this paper, we theoretically analyze the variance of sampling methods and show that, due to the composite structure of empirical risk, the variance of any sampling method can be decomposed into \textit{embedding approximation variance} in the forward stage and \textit{stochastic gradient variance} in the backward stage that necessities mitigating both types of variance to obtain faster convergence rate. We propose a decoupled variance reduction strategy that employs (approximate) gradient information to adaptively sample nodes with minimal variance, and explicitly reduces the variance introduced by embedding approximation. We show theoretically and empirically that the proposed method, even with smaller mini-batch sizes, enjoys a faster convergence rate and entails a better generalization compared to the existing methods.
Contextual multi-armed bandit (MAB) achieves cutting-edge performance on a variety of problems. When it comes to real-world scenarios such as recommendation system and online advertising, however, it is essential to consider the resource consumption of exploration. In practice, there is typically non-zero cost associated with executing a recommendation (arm) in the environment, and hence, the policy should be learned with a fixed exploration cost constraint. It is challenging to learn a global optimal policy directly, since it is a NP-hard problem and significantly complicates the exploration and exploitation trade-off of bandit algorithms. Existing approaches focus on solving the problems by adopting the greedy policy which estimates the expected rewards and costs and uses a greedy selection based on each arm's expected reward/cost ratio using historical observation until the exploration resource is exhausted. However, existing methods are hard to extend to infinite time horizon, since the learning process will be terminated when there is no more resource. In this paper, we propose a hierarchical adaptive contextual bandit method (HATCH) to conduct the policy learning of contextual bandits with a budget constraint. HATCH adopts an adaptive method to allocate the exploration resource based on the remaining resource/time and the estimation of reward distribution among different user contexts. In addition, we utilize full of contextual feature information to find the best personalized recommendation. Finally, in order to prove the theoretical guarantee, we present a regret bound analysis and prove that HATCH achieves a regret bound as low as $O(\sqrt{T})$. The experimental results demonstrate the effectiveness and efficiency of the proposed method on both synthetic data sets and the real-world applications.
Efficient exploration remains a major challenge for reinforcement learning. One reason is that the variability of the returns often depends on the current state and action, and is therefore heteroscedastic. Classical exploration strategies such as upper confidence bound algorithms and Thompson sampling fail to appropriately account for heteroscedasticity, even in the bandit setting. Motivated by recent findings that address this issue in bandits, we propose to use Information-Directed Sampling (IDS) for exploration in reinforcement learning. As our main contribution, we build on recent advances in distributional reinforcement learning and propose a novel, tractable approximation of IDS for deep Q-learning. The resulting exploration strategy explicitly accounts for both parametric uncertainty and heteroscedastic observation noise. We evaluate our method on Atari games and demonstrate a significant improvement over alternative approaches.
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.
We consider the task of learning the parameters of a {\em single} component of a mixture model, for the case when we are given {\em side information} about that component, we call this the "search problem" in mixture models. We would like to solve this with computational and sample complexity lower than solving the overall original problem, where one learns parameters of all components. Our main contributions are the development of a simple but general model for the notion of side information, and a corresponding simple matrix-based algorithm for solving the search problem in this general setting. We then specialize this model and algorithm to four common scenarios: Gaussian mixture models, LDA topic models, subspace clustering, and mixed linear regression. For each one of these we show that if (and only if) the side information is informative, we obtain parameter estimates with greater accuracy, and also improved computation complexity than existing moment based mixture model algorithms (e.g. tensor methods). We also illustrate several natural ways one can obtain such side information, for specific problem instances. Our experiments on real data sets (NY Times, Yelp, BSDS500) further demonstrate the practicality of our algorithms showing significant improvement in runtime and accuracy.
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