We study the problem of $K$-armed dueling bandit for both stochastic and adversarial environments, where the goal of the learner is to aggregate information through relative preferences of pair of decisions points queried in an online sequential manner. We first propose a novel reduction from any (general) dueling bandits to multi-armed bandits and despite the simplicity, it allows us to improve many existing results in dueling bandits. In particular, \emph{we give the first best-of-both world result for the dueling bandits regret minimization problem} -- a unified framework that is guaranteed to perform optimally for both stochastic and adversarial preferences simultaneously. Moreover, our algorithm is also the first to achieve an optimal $O(\sum_{i = 1}^K \frac{\log T}{\Delta_i})$ regret bound against the Condorcet-winner benchmark, which scales optimally both in terms of the arm-size $K$ and the instance-specific suboptimality gaps $\{\Delta_i\}_{i = 1}^K$. This resolves the long-standing problem of designing an instancewise gap-dependent order optimal regret algorithm for dueling bandits (with matching lower bounds up to small constant factors). We further justify the robustness of our proposed algorithm by proving its optimal regret rate under adversarially corrupted preferences -- this outperforms the existing state-of-the-art corrupted dueling results by a large margin. In summary, we believe our reduction idea will find a broader scope in solving a diverse class of dueling bandits setting, which are otherwise studied separately from multi-armed bandits with often more complex solutions and worse guarantees. The efficacy of our proposed algorithms is empirically corroborated against the existing dueling bandit methods.
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The design of effective online caching policies is an increasingly important problem for content distribution networks, online social networks and edge computing services, among other areas. This paper proposes a new algorithmic toolbox for tackling this problem through the lens of optimistic online learning. We build upon the Follow-the-Regularized-Leader (FTRL) framework which is developed further here to include predictions for the file requests, and we design online caching algorithms for bipartite networks with fixed-size caches or elastic leased caches subject to time-average budget constraints. The predictions are provided by a content recommendation system that influences the users viewing activity, and hence can naturally reduce the caching network's uncertainty about future requests. We prove that the proposed optimistic learning caching policies can achieve sub-zero performance loss (regret) for perfect predictions, and maintain the best achievable regret bound $O(\sqrt T)$ even for arbitrary-bad predictions. The performance of the proposed algorithms is evaluated with detailed trace-driven numerical tests.
Minimum cut/maximum flow (min-cut/max-flow) algorithms solve a variety of problems in computer vision and thus significant effort has been put into developing fast min-cut/max-flow algorithms. As a result, it is difficult to choose an ideal algorithm for a given problem. Furthermore, parallel algorithms have not been thoroughly compared. In this paper, we evaluate the state-of-the-art serial and parallel min-cut/max-flow algorithms on the largest set of computer vision problems yet. We focus on generic algorithms, i.e., for unstructured graphs, but also compare with the specialized GridCut implementation. When applicable, GridCut performs best. Otherwise, the two pseudoflow algorithms, Hochbaum pseudoflow and excesses incremental breadth first search, achieves the overall best performance. The most memory efficient implementation tested is the Boykov-Kolmogorov algorithm. Amongst generic parallel algorithms, we find the bottom-up merging approach by Liu and Sun to be best, but no method is dominant. Of the generic parallel methods, only the parallel preflow push-relabel algorithm is able to efficiently scale with many processors across problem sizes, and no generic parallel method consistently outperforms serial algorithms. Finally, we provide and evaluate strategies for algorithm selection to obtain good expected performance. We make our dataset and implementations publicly available for further research.
Learning Markov decision processes (MDPs) in the presence of the adversary is a challenging problem in reinforcement learning (RL). In this paper, we study RL in episodic MDPs with adversarial reward and full information feedback, where the unknown transition probability function is a linear function of a given feature mapping, and the reward function can change arbitrarily episode by episode. We propose an optimistic policy optimization algorithm POWERS and show that it can achieve $\tilde{O}(dH\sqrt{T})$ regret, where $H$ is the length of the episode, $T$ is the number of interactions with the MDP, and $d$ is the dimension of the feature mapping. Furthermore, we also prove a matching lower bound of $\tilde{\Omega}(dH\sqrt{T})$ up to logarithmic factors. Our key technical contributions are two-fold: (1) a new value function estimator based on importance weighting; and (2) a tighter confidence set for the transition kernel. They together lead to the nearly minimax optimal regret.
We study reinforcement learning for two-player zero-sum Markov games with simultaneous moves in the finite-horizon setting, where the transition kernel of the underlying Markov games can be parameterized by a linear function over the current state, both players' actions and the next state. In particular, we assume that we can control both players and aim to find the Nash Equilibrium by minimizing the duality gap. We propose an algorithm Nash-UCRL based on the principle "Optimism-in-Face-of-Uncertainty". Our algorithm only needs to find a Coarse Correlated Equilibrium (CCE), which is computationally efficient. Specifically, we show that Nash-UCRL can provably achieve an $\tilde{O}(dH\sqrt{T})$ regret, where $d$ is the linear function dimension, $H$ is the length of the game and $T$ is the total number of steps in the game. To assess the optimality of our algorithm, we also prove an $\tilde{\Omega}( dH\sqrt{T})$ lower bound on the regret. Our upper bound matches the lower bound up to logarithmic factors, which suggests the optimality of our algorithm.
Persistent homology is an important methodology from topological data analysis which adapts theory from algebraic topology to data settings and has been successfully implemented in many applications. It produces a statistical summary in the form of a persistence diagram, which captures the shape and size of the data. Despite its widespread use, persistent homology is simply impossible to implement when a dataset is very large. In this paper we address the problem of finding a representative persistence diagram for prohibitively large datasets. We adapt the classical statistical method of bootstrapping, namely, drawing and studying smaller multiple subsamples from the large dataset. We show that the mean of the persistence diagrams of subsamples -- taken as a mean persistence measure computed from the subsamples -- is a valid approximation of the true persistent homology of the larger dataset. We give the rate of convergence of the mean persistence diagram to the true persistence diagram in terms of the number of subsamples and size of each subsample. Given the complex algebraic and geometric nature of persistent homology, we adapt the convexity and stability properties in the space of persistence diagrams together with random set theory to achieve our theoretical results for the general setting of point cloud data. We demonstrate our approach on simulated and real data, including an application of shape clustering on complex large-scale point cloud data.
We study streaming algorithms in the white-box adversarial model, where the stream is chosen adaptively by an adversary who observes the entire internal state of the algorithm at each time step. We show that nontrivial algorithms are still possible. We first give a randomized algorithm for the $L_1$-heavy hitters problem that outperforms the optimal deterministic Misra-Gries algorithm on long streams. If the white-box adversary is computationally bounded, we use cryptographic techniques to reduce the memory of our $L_1$-heavy hitters algorithm even further and to design a number of additional algorithms for graph, string, and linear algebra problems. The existence of such algorithms is surprising, as the streaming algorithm does not even have a secret key in this model, i.e., its state is entirely known to the adversary. One algorithm we design is for estimating the number of distinct elements in a stream with insertions and deletions achieving a multiplicative approximation and sublinear space; such an algorithm is impossible for deterministic algorithms. We also give a general technique that translates any two-player deterministic communication lower bound to a lower bound for {\it randomized} algorithms robust to a white-box adversary. In particular, our results show that for all $p\ge 0$, there exists a constant $C_p>1$ such that any $C_p$-approximation algorithm for $F_p$ moment estimation in insertion-only streams with a white-box adversary requires $\Omega(n)$ space for a universe of size $n$. Similarly, there is a constant $C>1$ such that any $C$-approximation algorithm in an insertion-only stream for matrix rank requires $\Omega(n)$ space with a white-box adversary. Our algorithmic results based on cryptography thus show a separation between computationally bounded and unbounded adversaries. (Abstract shortened to meet arXiv limits.)
The stochastic gradient Langevin Dynamics is one of the most fundamental algorithms to solve sampling problems and non-convex optimization appearing in several machine learning applications. Especially, its variance reduced versions have nowadays gained particular attention. In this paper, we study two variants of this kind, namely, the Stochastic Variance Reduced Gradient Langevin Dynamics and the Stochastic Recursive Gradient Langevin Dynamics. We prove their convergence to the objective distribution in terms of KL-divergence under the sole assumptions of smoothness and Log-Sobolev inequality which are weaker conditions than those used in prior works for these algorithms. With the batch size and the inner loop length set to $\sqrt{n}$, the gradient complexity to achieve an $\epsilon$-precision is $\tilde{O}((n+dn^{1/2}\epsilon^{-1})\gamma^2 L^2\alpha^{-2})$, which is an improvement from any previous analyses. We also show some essential applications of our result to non-convex optimization.
There are many important high dimensional function classes that have fast agnostic learning algorithms when strong assumptions on the distribution of examples can be made, such as Gaussianity or uniformity over the domain. But how can one be sufficiently confident that the data indeed satisfies the distributional assumption, so that one can trust in the output quality of the agnostic learning algorithm? We propose a model by which to systematically study the design of tester-learner pairs $(\mathcal{A},\mathcal{T})$, such that if the distribution on examples in the data passes the tester $\mathcal{T}$ then one can safely trust the output of the agnostic learner $\mathcal{A}$ on the data. To demonstrate the power of the model, we apply it to the classical problem of agnostically learning halfspaces under the standard Gaussian distribution and present a tester-learner pair with a combined run-time of $n^{\tilde{O}(1/\epsilon^4)}$. This qualitatively matches that of the best known ordinary agnostic learning algorithms for this task. In contrast, finite sample Gaussian distribution testers do not exist for the $L_1$ and EMD distance measures. A key step in the analysis is a novel characterization of concentration and anti-concentration properties of a distribution whose low-degree moments approximately match those of a Gaussian. We also use tools from polynomial approximation theory. In contrast, we show strong lower bounds on the combined run-times of tester-learner pairs for the problems of agnostically learning convex sets under the Gaussian distribution and for monotone Boolean functions under the uniform distribution over $\{0,1\}^n$. Through these lower bounds we exhibit natural problems where there is a dramatic gap between standard agnostic learning run-time and the run-time of the best tester-learner pair.
Proactive dialogue system is able to lead the conversation to a goal topic and has advantaged potential in bargain, persuasion and negotiation. Current corpus-based learning manner limits its practical application in real-world scenarios. To this end, we contribute to advance the study of the proactive dialogue policy to a more natural and challenging setting, i.e., interacting dynamically with users. Further, we call attention to the non-cooperative user behavior -- the user talks about off-path topics when he/she is not satisfied with the previous topics introduced by the agent. We argue that the targets of reaching the goal topic quickly and maintaining a high user satisfaction are not always converge, because the topics close to the goal and the topics user preferred may not be the same. Towards this issue, we propose a new solution named I-Pro that can learn Proactive policy in the Interactive setting. Specifically, we learn the trade-off via a learned goal weight, which consists of four factors (dialogue turn, goal completion difficulty, user satisfaction estimation, and cooperative degree). The experimental results demonstrate I-Pro significantly outperforms baselines in terms of effectiveness and interpretability.
Convolutional neural networks (CNNs) have shown dramatic improvements in single image super-resolution (SISR) by using large-scale external samples. Despite their remarkable performance based on the external dataset, they cannot exploit internal information within a specific image. Another problem is that they are applicable only to the specific condition of data that they are supervised. For instance, the low-resolution (LR) image should be a "bicubic" downsampled noise-free image from a high-resolution (HR) one. To address both issues, zero-shot super-resolution (ZSSR) has been proposed for flexible internal learning. However, they require thousands of gradient updates, i.e., long inference time. In this paper, we present Meta-Transfer Learning for Zero-Shot Super-Resolution (MZSR), which leverages ZSSR. Precisely, it is based on finding a generic initial parameter that is suitable for internal learning. Thus, we can exploit both external and internal information, where one single gradient update can yield quite considerable results. (See Figure 1). With our method, the network can quickly adapt to a given image condition. In this respect, our method can be applied to a large spectrum of image conditions within a fast adaptation process.
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