This paper introduces a new online learning framework for multiclass classification called learning with diluted bandit feedback. At every time step, the algorithm predicts a candidate label set instead of a single label for the observed example. It then receives feedback from the environment whether the actual label lies in this candidate label set or not. This feedback is called "diluted bandit feedback". Learning in this setting is even more challenging than the bandit feedback setting, as there is more uncertainty in the supervision. We propose an algorithm for multiclass classification using dilute bandit feedback (MC-DBF), which uses the exploration-exploitation strategy to predict the candidate set in each trial. We show that the proposed algorithm achieves O(T^{1-\frac{1}{m+2}}) mistake bound if candidate label set size (in each step) is m. We demonstrate the effectiveness of the proposed approach with extensive simulations.
We consider two scenarios of multiclass online learning of a hypothesis class $H\subseteq Y^X$. In the {\em full information} scenario, the learner is exposed to instances together with their labels. In the {\em bandit} scenario, the true label is not exposed, but rather an indication whether the learner's prediction is correct or not. We show that the ratio between the error rates in the two scenarios is at most $8\cdot|Y|\cdot \log(|Y|)$ in the realizable case, and $\tilde{O}(\sqrt{|Y|})$ in the agnostic case. The results are tight up to a logarithmic factor and essentially answer an open qu…
University of Michigan（密歇根大学）；Department of Computer Science（计算机科学系）；Department of Statistics（统计系）
We present online boosting algorithms for multiclass classification with bandit feedback, where the learner only receives feedback about the correctness of its prediction. We propose an unbiased estimate of the loss using a randomized prediction, allowing the model to update its weak learners with limited information. Using the unbiased estimate, we extend two full information boosting algorithms (Jung et al., 2017) to the bandit setting. We prove that the asymptotic error bounds of the bandit algorithms exactly match their full information counterparts. The cost of restricted feedback is refl…
Yahoo Research（雅虎研究院）；Stony Brook University（石溪大学）；University of California, San Diego（加利福尼亚大学圣迭戈分校）
We present an efficient second-order algorithm with $\tilde{O}(\frac{1}η\sqrt{T})$ regret for the bandit online multiclass problem. The regret bound holds simultaneously with respect to a family of loss functions parameterized by $η$, for a range of $η$ restricted by the norm of the competitor. The family of loss functions ranges from hinge loss ($η=0$) to squared hinge loss ($η=1$). This provides a solution to the open problem of (J. Abernethy and A. Rakhlin. An efficient bandit algorithm for $\sqrt{T}$-regret in online multiclass prediction? In COLT, 2009). We test our algorithm experimental…
This paper addresses the problem of multiclass classification with corrupted or noisy bandit feedback. In this setting, the learner may not receive true feedback. Instead, it receives feedback that has been flipped with some non-zero probability. We propose a novel approach to deal with noisy bandit feedback based on the unbiased estimator technique. We further offer a method that can efficiently estimate the noise rates, thus providing an end-to-end framework. The proposed algorithm enjoys a mistake bound of the order of $O(\sqrt{T})$ in the high noise case and of the order of $O(T^{\nicefrac…
Beyond Bandit Feedback in Online Multiclass Classification
Università degli Studi di Milano（米兰大学）；Sapienza Università di Roma（罗马大学）；Dept. of Computer Science（计算机科学系）；Dept. of Computer, Control and Management Engineering（计算机、控制与管理工程系）；DSRC
We consider the online multiclass linear classification under the bandit feedback setting. Beygelzimer, Pál, Szörényi, Thiruvenkatachari, Wei, and Zhang [ICML'19] considered two notions of linear separability, weak and strong linear separability. When examples are strongly linearly separable with margin $γ$, they presented an algorithm based on Multiclass Perceptron with mistake bound $O(K/γ^2)$, where $K$ is the number of classes. They employed rational kernel to deal with examples under the weakly linearly separable condition, and obtained the mistake bound of $\min(K\cdot 2^{\tilde{O}(K\log…
Yahoo Research（雅虎研究院）；New York University（纽约大学）；University of Southern California（南加州大学）；Microsoft Research（微软研究院）
We study the problem of efficient online multiclass linear classification with bandit feedback, where all examples belong to one of $K$ classes and lie in the $d$-dimensional Euclidean space. Previous works have left open the challenge of designing efficient algorithms with finite mistake bounds when the data is linearly separable by a margin $γ$. In this work, we take a first step towards this problem. We consider two notions of linear separability: strong and weak. 1. Under the strong linear separability condition, we design an efficient algorithm that achieves a near-optimal mistake bound o…
Bandit-Feedback Online Multiclass Classification: Variants and Tradeoffs
The Henry and Marilyn Taub Faculty of Computer Science, Technion（以色列理工学院亨利与玛丽莲·陶布计算机科学学院）；Faculty of Mathematics, Technion（以色列理工学院数学学院）；Department of Computer Science, Purdue University（普渡大学计算机科学系）；Faculty of Data and Decision Sciences, Technion（以色列理工学院数据与决策科学学院）；Google Research（谷歌研究院）
The Sample Complexity of Multiclass and Sparse Contextual Bandits
Tel Aviv University（特拉维夫大学）；Massachusetts Institute of Technology（麻省理工学院）；Google Research Tel Aviv（谷歌研究特拉维夫）；Technion—Israel Institute of Technology（技术学院—以色列理工学院）
In this paper, we present online algorithm called {\it Delaytron} for learning multi class classifiers using delayed bandit feedbacks. The sequence of feedback delays $\{d_t\}_{t=1}^T$ is unknown to the algorithm. At the $t$-th round, the algorithm observes an example $\mathbf{x}_t$ and predicts a label $\tilde{y}_t$ and receives the bandit feedback $\mathbb{I}[\tilde{y}_t=y_t]$ only $d_t$ rounds later. When $t+d_t>T$, we consider that the feedback for the $t$-th round is missing. We show that the proposed algorithm achieves regret of $\mathcal{O}\left(\sqrt{\frac{2 K}γ\left[\frac{T}{2}+\left(…
The standard model and the bandit model are two generalizations of the mistake-bound model to online multiclass classification. In both models the learner guesses a classification in each round, but in the standard model the learner recieves the correct classification after each guess, while in the bandit model the learner is only told whether or not their guess is correct in each round. For any set $F$ of multiclass classifiers, define $opt_{std}(F)$ and $opt_{bandit}(F)$ to be the optimal worst-case number of prediction mistakes in the standard and bandit models respectively. Long (Theoretic…
Deep reinforcement learning has enabled human-level or even super-human performance in various types of games. However, the amount of exploration required for learning is often quite large. Deep reinforcement learning also has super-human performance in that no human being would be able to achieve such amounts of exploration. To address this problem, we focus on the \textit{satisficing} policy, which is a qualitatively different approach from that of existing optimization algorithms. Thus, we propose Linear RS (LinRS), which is a type of satisficing algorithm and a linear extension of risk-sen…
We present Gaptron, a randomized first-order algorithm for online multiclass classification. In the full information setting we show expected mistake bounds with respect to the logistic loss, hinge loss, and the smooth hinge loss with constant regret, where the expectation is with respect to the learner's randomness. In the bandit classification setting we show that Gaptron is the first linear time algorithm with $O(K\sqrt{T})$ expected regret, where $K$ is the number of classes. Additionally, the expected mistake bound of Gaptron does not depend on the dimension of the feature vector, contrar…
Multiclass logistic regression is a fundamental task in machine learning with applications in classification and boosting. Previous work (Foster et al., 2018) has highlighted the importance of improper predictors for achieving "fast rates" in the online multiclass logistic regression problem without suffering exponentially from secondary problem parameters, such as the norm of the predictors in the comparison class. While Foster et al. (2018) introduced a statistically optimal algorithm, it is in practice computationally intractable due to its run-time complexity being a large polynomial in th…
