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We study the optimal batch-regret tradeoff for batch linear contextual bandits. For any batch number $M$, number of actions $K$, time horizon $T$, and dimension $d$, we provide an algorithm and prove its regret guarantee, which, due to technical reasons, features a two-phase expression as the time horizon $T$ grows. We also prove a lower bound theorem that surprisingly shows the optimality of our two-phase regret upper bound (up to logarithmic factors) in the \emph{full range} of the problem parameters, therefore establishing the exact batch-regret tradeoff. Compared to the recent work \citep{ruan2020linear} which showed that $M = O(\log \log T)$ batches suffice to achieve the asymptotically minimax-optimal regret without the batch constraints, our algorithm is simpler and easier for practical implementation. Furthermore, our algorithm achieves the optimal regret for all $T \geq d$, while \citep{ruan2020linear} requires that $T$ greater than an unrealistically large polynomial of $d$. Along our analysis, we also prove a new matrix concentration inequality with dependence on their dynamic upper bounds, which, to the best of our knowledge, is the first of its kind in literature and maybe of independent interest.

This paper introduces and studies the $k\texttt{-experts}$ problem -- a generalization of the classic Prediction with Expert's Advice (i.e., the $\texttt{Experts}$) problem. Unlike the $\texttt{Experts}$ problem, where the learner chooses exactly one expert, in this problem, the learner selects a subset of $k$ experts from a pool of $N$ experts at each round. The reward obtained by the learner at any round depends on the rewards of the selected experts. The $k\texttt{-experts}$ problem arises in many practical settings, including online ad placements, personalized news recommendations, and paging. Our primary goal is to design an online learning policy having a small regret. In this pursuit, we propose $\texttt{SAGE}$ ($\textbf{Sa}$mpled Hed$\textbf{ge}$) - a framework for designing efficient online learning policies by leveraging statistical sampling techniques. We show that, for many related problems, $\texttt{SAGE}$ improves upon the state-of-the-art bounds for regret and computational complexity. Furthermore, going beyond the notion of regret, we characterize the mistake bounds achievable by online learning policies for a class of stable loss functions. We conclude the paper by establishing a tight regret lower bound for a variant of the $k\texttt{-experts}$ problem and carrying out experiments with standard datasets.

Nonlinear metrics, such as the F1-score, Matthews correlation coefficient, and Fowlkes-Mallows index, are often used to evaluate the performance of machine learning models, in particular, when facing imbalanced datasets that contain more samples of one class than the other. Recent optimal decision tree algorithms have shown remarkable progress in producing trees that are optimal with respect to linear criteria, such as accuracy, but unfortunately nonlinear metrics remain a challenge. To address this gap, we propose a novel algorithm based on bi-objective optimisation, which treats misclassifications of each binary class as a separate objective. We show that, for a large class of metrics, the optimal tree lies on the Pareto frontier. Consequently, we obtain the optimal tree by using our method to generate the set of all nondominated trees. To the best of our knowledge, this is the first method to compute provably optimal decision trees for nonlinear metrics. Our approach leads to a trade-off when compared to optimising linear metrics: the resulting trees may be more desirable according to the given nonlinear metric at the expense of higher runtimes. Nevertheless, the experiments illustrate that runtimes are reasonable for majority of the tested datasets.

Decision tree learning is a widely used approach in machine learning, favoured in applications that require concise and interpretable models. Heuristic methods are traditionally used to quickly produce models with reasonably high accuracy. A commonly criticised point, however, is that the resulting trees may not necessarily be the best representation of the data in terms of accuracy and size. In recent years, this motivated the development of optimal classification tree algorithms that globally optimise the decision tree in contrast to heuristic methods that perform a sequence of locally optimal decisions. We follow this line of work and provide a novel algorithm for learning optimal classification trees based on dynamic programming and search. Our algorithm supports constraints on the depth of the tree and number of nodes. The success of our approach is attributed to a series of specialised techniques that exploit properties unique to classification trees. Whereas algorithms for optimal classification trees have traditionally been plagued by high runtimes and limited scalability, we show in a detailed experimental study that our approach uses only a fraction of the time required by the state-of-the-art and can handle datasets with tens of thousands of instances, providing several orders of magnitude improvements and notably contributing towards the practical realisation of optimal decision trees.

Ensuring group fairness among groups of individuals in our society is desirable and crucial for many application domains. A social planner's typical medium of achieving group fair outcomes is through solving an optimization problem under a given objective for a particular domain. When the input is provided by strategic agents, the planner is facing a difficult situation of achieving fair outcomes while ensuring agent truthfulness without using incentive payment. To address this challenge, we consider the approximate mechanism design without money paradigm with group-fair objectives. We first consider the group-fair facility location problems where agents are divided into groups. The agents are located on a real line, modeling agents' private ideal preferences/points for the facility's location. Our aim is to locate a facility to approximately minimize the costs of groups of agents to the facility fairly while eliciting the agents' private locations truthfully. We consider various group-fair objectives and show that many objectives have an unbounded approximation ratio. We then consider the objectives of minimizing the maximum total group cost and the average group cost. For the first objective, we show that the approximation ratio of the median mechanism depends on the number of groups and provide a new group-based mechanism with an approximation ratio of 3. For the second objective, the median mechanism obtains a ratio of 3, and we propose a randomized mechanism that obtains a better approximation ratio. We also provide lower bounds for both objectives. We then study the notion of intergroup and intragroup fairness that measures fairness between groups and within each group. We consider various objectives and provide mechanisms with tight approximation ratios.

We consider the phase retrieval problem, in which the observer wishes to recover a $n$-dimensional real or complex signal $\mathbf{X}^\star$ from the (possibly noisy) observation of $|\mathbf{\Phi} \mathbf{X}^\star|$, in which $\mathbf{\Phi}$ is a matrix of size $m \times n$. We consider a \emph{high-dimensional} setting where $n,m \to \infty$ with $m/n = \mathcal{O}(1)$, and a large class of (possibly correlated) random matrices $\mathbf{\Phi}$ and observation channels. Spectral methods are a powerful tool to obtain approximate observations of the signal $\mathbf{X}^\star$ which can be then used as initialization for a subsequent algorithm, at a low computational cost. In this paper, we extend and unify previous results and approaches on spectral methods for the phase retrieval problem. More precisely, we combine the linearization of message-passing algorithms and the analysis of the \emph{Bethe Hessian}, a classical tool of statistical physics. Using this toolbox, we show how to derive optimal spectral methods for arbitrary channel noise and right-unitarily invariant matrix $\mathbf{\Phi}$, in an automated manner (i.e. with no optimization over any hyperparameter or preprocessing function).

In many contexts, lying -- the use of verbal falsehoods to deceive -- is harmful. While lying has traditionally been a human affair, AI systems that make sophisticated verbal statements are becoming increasingly prevalent. This raises the question of how we should limit the harm caused by AI "lies" (i.e. falsehoods that are actively selected for). Human truthfulness is governed by social norms and by laws (against defamation, perjury, and fraud). Differences between AI and humans present an opportunity to have more precise standards of truthfulness for AI, and to have these standards rise over time. This could provide significant benefits to public epistemics and the economy, and mitigate risks of worst-case AI futures. Establishing norms or laws of AI truthfulness will require significant work to: (1) identify clear truthfulness standards; (2) create institutions that can judge adherence to those standards; and (3) develop AI systems that are robustly truthful. Our initial proposals for these areas include: (1) a standard of avoiding "negligent falsehoods" (a generalisation of lies that is easier to assess); (2) institutions to evaluate AI systems before and after real-world deployment; and (3) explicitly training AI systems to be truthful via curated datasets and human interaction. A concerning possibility is that evaluation mechanisms for eventual truthfulness standards could be captured by political interests, leading to harmful censorship and propaganda. Avoiding this might take careful attention. And since the scale of AI speech acts might grow dramatically over the coming decades, early truthfulness standards might be particularly important because of the precedents they set.

In this paper, we propose a one-stage online clustering method called Contrastive Clustering (CC) which explicitly performs the instance- and cluster-level contrastive learning. To be specific, for a given dataset, the positive and negative instance pairs are constructed through data augmentations and then projected into a feature space. Therein, the instance- and cluster-level contrastive learning are respectively conducted in the row and column space by maximizing the similarities of positive pairs while minimizing those of negative ones. Our key observation is that the rows of the feature matrix could be regarded as soft labels of instances, and accordingly the columns could be further regarded as cluster representations. By simultaneously optimizing the instance- and cluster-level contrastive loss, the model jointly learns representations and cluster assignments in an end-to-end manner. Extensive experimental results show that CC remarkably outperforms 17 competitive clustering methods on six challenging image benchmarks. In particular, CC achieves an NMI of 0.705 (0.431) on the CIFAR-10 (CIFAR-100) dataset, which is an up to 19\% (39\%) performance improvement compared with the best baseline.

Node classification is an important problem in graph data management. It is commonly solved by various label propagation methods that work iteratively starting from a few labeled seed nodes. For graphs with arbitrary compatibilities between classes, these methods crucially depend on knowing the compatibility matrix that must be provided by either domain experts or heuristics. Can we instead directly estimate the correct compatibilities from a sparsely labeled graph in a principled and scalable way? We answer this question affirmatively and suggest a method called distant compatibility estimation that works even on extremely sparsely labeled graphs (e.g., 1 in 10,000 nodes is labeled) in a fraction of the time it later takes to label the remaining nodes. Our approach first creates multiple factorized graph representations (with size independent of the graph) and then performs estimation on these smaller graph sketches. We define algebraic amplification as the more general idea of leveraging algebraic properties of an algorithm's update equations to amplify sparse signals. We show that our estimator is by orders of magnitude faster than an alternative approach and that the end-to-end classification accuracy is comparable to using gold standard compatibilities. This makes it a cheap preprocessing step for any existing label propagation method and removes the current dependence on heuristics.

Degradation of image quality due to the presence of haze is a very common phenomenon. Existing DehazeNet [3], MSCNN [11] tackled the drawbacks of hand crafted haze relevant features. However, these methods have the problem of color distortion in gloomy (poor illumination) environment. In this paper, a cardinal (red, green and blue) color fusion network for single image haze removal is proposed. In first stage, network fusses color information present in hazy images and generates multi-channel depth maps. The second stage estimates the scene transmission map from generated dark channels using multi channel multi scale convolutional neural network (McMs-CNN) to recover the original scene. To train the proposed network, we have used two standard datasets namely: ImageNet [5] and D-HAZY [1]. Performance evaluation of the proposed approach has been carried out using structural similarity index (SSIM), mean square error (MSE) and peak signal to noise ratio (PSNR). Performance analysis shows that the proposed approach outperforms the existing state-of-the-art methods for single image dehazing.