Sequential learning with feedback graphs is a natural extension of the multi-armed bandit problem where the problem is equipped with an underlying graph structure that provides additional information - playing an action reveals the losses of all the neighbors of the action. This problem was introduced by \citet{mannor2011} and received considerable attention in recent years. It is generally stated in the literature that the minimax regret rate for this problem is of order $\sqrt{\alpha T}$, where $\alpha$ is the independence number of the graph, and $T$ is the time horizon. However, this is proven only when the number of rounds $T$ is larger than $\alpha^3$, which poses a significant restriction for the usability of this result in large graphs. In this paper, we define a new quantity $R^*$, called the \emph{problem complexity}, and prove that the minimax regret is proportional to $R^*$ for any graph and time horizon $T$. Introducing an intricate exploration strategy, we define the \mainAlgorithm algorithm that achieves the minimax optimal regret bound and becomes the first provably optimal algorithm for this setting, even if $T$ is smaller than $\alpha^3$.
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Attention models are typically learned by optimizing one of three standard loss functions that are variously called -- soft attention, hard attention, and latent variable marginal likelihood (LVML) attention. All three paradigms are motivated by the same goal of finding two models -- a `focus' model that `selects' the right \textit{segment} of the input and a `classification' model that processes the selected segment into the target label. However, they differ significantly in the way the selected segments are aggregated, resulting in distinct dynamics and final results. We observe a unique signature of models learned using these paradigms and explain this as a consequence of the evolution of the classification model under gradient descent when the focus model is fixed. We also analyze these paradigms in a simple setting and derive closed-form expressions for the parameter trajectory under gradient flow. With the soft attention loss, the focus model improves quickly at initialization and splutters later on. On the other hand, hard attention loss behaves in the opposite fashion. Based on our observations, we propose a simple hybrid approach that combines the advantages of the different loss functions and demonstrates it on a collection of semi-synthetic and real-world datasets
A central issue lying at the heart of online reinforcement learning (RL) is data efficiency. While a number of recent works achieved asymptotically minimal regret in online RL, the optimality of these results is only guaranteed in a ``large-sample'' regime, imposing enormous burn-in cost in order for their algorithms to operate optimally. How to achieve minimax-optimal regret without incurring any burn-in cost has been an open problem in RL theory. We settle this problem for the context of finite-horizon inhomogeneous Markov decision processes. Specifically, we prove that a modified version of Monotonic Value Propagation (MVP), a model-based algorithm proposed by \cite{zhang2020reinforcement}, achieves a regret on the order of (modulo log factors) \begin{equation*} \min\big\{ \sqrt{SAH^3K}, \,HK \big\}, \end{equation*} where $S$ is the number of states, $A$ is the number of actions, $H$ is the planning horizon, and $K$ is the total number of episodes. This regret matches the minimax lower bound for the entire range of sample size $K\geq 1$, essentially eliminating any burn-in requirement. It also translates to a PAC sample complexity (i.e., the number of episodes needed to yield $\varepsilon$-accuracy) of $\frac{SAH^3}{\varepsilon^2}$ up to log factor, which is minimax-optimal for the full $\varepsilon$-range. Further, we extend our theory to unveil the influences of problem-dependent quantities like the optimal value/cost and certain variances. The key technical innovation lies in the development of a new regret decomposition strategy and a novel analysis paradigm to decouple complicated statistical dependency -- a long-standing challenge facing the analysis of online RL in the sample-hungry regime.
In reinforcement learning (RL), rewards of states are typically considered additive, and following the Markov assumption, they are $\textit{independent}$ of states visited previously. In many important applications, such as coverage control, experiment design and informative path planning, rewards naturally have diminishing returns, i.e., their value decreases in light of similar states visited previously. To tackle this, we propose $\textit{submodular RL}$ (SubRL), a paradigm which seeks to optimize more general, non-additive (and history-dependent) rewards modelled via submodular set functions which capture diminishing returns. Unfortunately, in general, even in tabular settings, we show that the resulting optimization problem is hard to approximate. On the other hand, motivated by the success of greedy algorithms in classical submodular optimization, we propose SubPO, a simple policy gradient-based algorithm for SubRL that handles non-additive rewards by greedily maximizing marginal gains. Indeed, under some assumptions on the underlying Markov Decision Process (MDP), SubPO recovers optimal constant factor approximations of submodular bandits. Moreover, we derive a natural policy gradient approach for locally optimizing SubRL instances even in large state- and action- spaces. We showcase the versatility of our approach by applying SubPO to several applications, such as biodiversity monitoring, Bayesian experiment design, informative path planning, and coverage maximization. Our results demonstrate sample efficiency, as well as scalability to high-dimensional state-action spaces.
Conditional Monte Carlo or pre-integration is a powerful tool for reducing variance and improving the regularity of integrands when using Monte Carlo and quasi-Monte Carlo (QMC) methods. To select the variable to pre-integrate, one must consider both the variable's importance and the tractability of the conditional expectation. For integrals over a Gaussian distribution, any linear combination of variables can potentially be pre-integrated. Liu and Owen (2022) propose to select the linear combination based on an active subspace decomposition of the integrand. However, pre-integrating the selected direction might be intractable. In this work, we address this issue by finding the active subspace subject to constraints such that pre-integration can be easily carried out. The proposed algorithm also provides a computationally-efficient alternative to dimension reduction for pre-integrated functions. The method is applied to examples from computational finance, density estimation, and computational chemistry, and is shown to achieve smaller errors than previous methods.
We consider the problem of contextual bandits and imitation learning, where the learner lacks direct knowledge of the executed action's reward. Instead, the learner can actively query an expert at each round to compare two actions and receive noisy preference feedback. The learner's objective is two-fold: to minimize the regret associated with the executed actions, while simultaneously, minimizing the number of comparison queries made to the expert. In this paper, we assume that the learner has access to a function class that can represent the expert's preference model under appropriate link functions, and provide an algorithm that leverages an online regression oracle with respect to this function class for choosing its actions and deciding when to query. For the contextual bandit setting, our algorithm achieves a regret bound that combines the best of both worlds, scaling as $O(\min\{\sqrt{T}, d/\Delta\})$, where $T$ represents the number of interactions, $d$ represents the eluder dimension of the function class, and $\Delta$ represents the minimum preference of the optimal action over any suboptimal action under all contexts. Our algorithm does not require the knowledge of $\Delta$, and the obtained regret bound is comparable to what can be achieved in the standard contextual bandits setting where the learner observes reward signals at each round. Additionally, our algorithm makes only $O(\min\{T, d^2/\Delta^2\})$ queries to the expert. We then extend our algorithm to the imitation learning setting, where the learning agent engages with an unknown environment in episodes of length $H$ each, and provide similar guarantees for regret and query complexity. Interestingly, our algorithm for imitation learning can even learn to outperform the underlying expert, when it is suboptimal, highlighting a practical benefit of preference-based feedback in imitation learning.
The Independent Cutset problem asks whether there is a set of vertices in a given graph that is both independent and a cutset. Such a problem is $\textsf{NP}$-complete even when the input graph is planar and has maximum degree five. In this paper, we first present a $\mathcal{O}^*(1.4423^{n})$-time algorithm for the problem. We also show how to compute a minimum independent cutset (if any) in the same running time. Since the property of having an independent cutset is MSO$_1$-expressible, our main results are concerned with structural parameterizations for the problem considering parameters that are not bounded by a function of the clique-width of the input. We present $\textsf{FPT}$-time algorithms for the problem considering the following parameters: the dual of the maximum degree, the dual of the solution size, the size of a dominating set (where a dominating set is given as an additional input), the size of an odd cycle transversal, the distance to chordal graphs, and the distance to $P_5$-free graphs. We close by introducing the notion of $\alpha$-domination, which allows us to identify more fixed-parameter tractable and polynomial-time solvable cases.
We study the problem of collaboratively learning least squares estimates for $m$ agents. Each agent observes a different subset of the features$\unicode{x2013}$e.g., containing data collected from sensors of varying resolution. Our goal is to determine how to coordinate the agents in order to produce the best estimator for each agent. We propose a distributed, semi-supervised algorithm Collab, consisting of three steps: local training, aggregation, and distribution. Our procedure does not require communicating the labeled data, making it communication efficient and useful in settings where the labeled data is inaccessible. Despite this handicap, our procedure is nearly asymptotically local minimax optimal$\unicode{x2013}$even among estimators allowed to communicate the labeled data such as imputation methods. We test our method on real and synthetic data.
In this monograph, I introduce the basic concepts of Online Learning through a modern view of Online Convex Optimization. Here, online learning refers to the framework of regret minimization under worst-case assumptions. I present first-order and second-order algorithms for online learning with convex losses, in Euclidean and non-Euclidean settings. All the algorithms are clearly presented as instantiation of Online Mirror Descent or Follow-The-Regularized-Leader and their variants. Particular attention is given to the issue of tuning the parameters of the algorithms and learning in unbounded domains, through adaptive and parameter-free online learning algorithms. Non-convex losses are dealt through convex surrogate losses and through randomization. The bandit setting is also briefly discussed, touching on the problem of adversarial and stochastic multi-armed bandits. These notes do not require prior knowledge of convex analysis and all the required mathematical tools are rigorously explained. Moreover, all the proofs have been carefully chosen to be as simple and as short as possible.
Graph Neural Networks (GNNs), which generalize deep neural networks to graph-structured data, have drawn considerable attention and achieved state-of-the-art performance in numerous graph related tasks. However, existing GNN models mainly focus on designing graph convolution operations. The graph pooling (or downsampling) operations, that play an important role in learning hierarchical representations, are usually overlooked. In this paper, we propose a novel graph pooling operator, called Hierarchical Graph Pooling with Structure Learning (HGP-SL), which can be integrated into various graph neural network architectures. HGP-SL incorporates graph pooling and structure learning into a unified module to generate hierarchical representations of graphs. More specifically, the graph pooling operation adaptively selects a subset of nodes to form an induced subgraph for the subsequent layers. To preserve the integrity of graph's topological information, we further introduce a structure learning mechanism to learn a refined graph structure for the pooled graph at each layer. By combining HGP-SL operator with graph neural networks, we perform graph level representation learning with focus on graph classification task. Experimental results on six widely used benchmarks demonstrate the effectiveness of our proposed model.
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.
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