The minimum sum-of-squares clustering (MSSC), or k-means type clustering, has been recently extended to exploit prior knowledge on the cardinality of each cluster. Such knowledge is used to increase performance as well as solution quality. In this paper, we propose a global optimization approach based on the branch-and-cut technique to solve the cardinality-constrained MSSC. For the lower bound routine, we use the semidefinite programming (SDP) relaxation recently proposed by Rujeerapaiboon et al. [SIAM J. Optim. 29(2), 1211-1239, (2019)]. However, this relaxation can be used in a branch-and-cut method only for small-size instances. Therefore, we derive a new SDP relaxation that scales better with the instance size and the number of clusters. In both cases, we strengthen the bound by adding polyhedral cuts. Benefiting from a tailored branching strategy which enforces pairwise constraints, we reduce the complexity of the problems arising in the children nodes. For the upper bound, instead, we present a local search procedure that exploits the solution of the SDP relaxation solved at each node. Computational results show that the proposed algorithm globally solves, for the first time, real-world instances of size 10 times larger than those solved by state-of-the-art exact methods.

We introduce two new extensions to the beam search algorithm based on conformal predictions (CP) to produce sets of sequences with theoretical coverage guarantees. The first method is very simple and proposes dynamically-sized subsets of beam search results but, unlike typical CP procedures, has an upper bound on the achievable guarantee depending on a post-hoc calibration measure. Our second algorithm introduces the conformal set prediction procedure as part of the decoding process, producing a variable beam width which adapts to the current uncertainty. While more complex, this procedure can achieve coverage guarantees selected a priori. We provide marginal coverage bounds for each method, and evaluate them empirically on a selection of tasks drawing from natural language processing and chemistry.

The goal of cryptocurrencies is decentralization. In principle, all currencies have equal status. Unlike traditional stock markets, there is no default currency of denomination (fiat), thus the trading pairs can be set freely. However, it is impractical to set up a trading market between every two currencies. In order to control management costs and ensure sufficient liquidity, we must give priority to covering those large-volume trading pairs and ensure that all coins are reachable. We note that this is an optimization problem. Its particularity lies in: 1) the trading volume between most (>99.5%) possible trading pairs cannot be directly observed. 2) It satisfies the connectivity constraint, that is, all currencies are guaranteed to be tradable. To solve this problem, we use a two-stage process: 1) Fill in missing values based on a regularized, truncated eigenvalue decomposition, where the regularization term is used to control what extent missing values should be limited to zero. 2) Search for the optimal trading pairs, based on a branch and bound process, with heuristic search and pruning strategies. The experimental results show that: 1) If the number of denominated coins is not limited, we will get a more decentralized trading pair settings, which advocates the establishment of trading pairs directly between large currency pairs. 2) There is a certain room for optimization in all exchanges. The setting of inappropriate trading pairs is mainly caused by subjectively setting small coins to quote, or failing to track emerging big coins in time. 3) Too few trading pairs will lead to low coverage; too many trading pairs will need to be adjusted with markets frequently. Exchanges should consider striking an appropriate balance between them.

Reinforcement learning policy evaluation problems are often modeled as finite or discounted/averaged infinite-horizon MDPs. In this paper, we study undiscounted off-policy policy evaluation for absorbing MDPs. Given the dataset consisting of the i.i.d episodes with a given truncation level, we propose a so-called MWLA algorithm to directly estimate the expected return via the importance ratio of the state-action occupancy measure. The Mean Square Error (MSE) bound for the MWLA method is investigated and the dependence of statistical errors on the data size and the truncation level are analyzed. With an episodic taxi environment, computational experiments illustrate the performance of the MWLA algorithm.

In online ranking, a learning algorithm sequentially ranks a set of items and receives feedback on its ranking in the form of relevance scores. Since obtaining relevance scores typically involves human annotation, it is of great interest to consider a partial feedback setting where feedback is restricted to the top-$k$ items in the rankings. Chaudhuri and Tewari [2017] developed a framework to analyze online ranking algorithms with top $k$ feedback. A key element in their work was the use of techniques from partial monitoring. In this paper, we further investigate online ranking with top $k$ feedback and solve some open problems posed by Chaudhuri and Tewari [2017]. We provide a full characterization of minimax regret rates with the top $k$ feedback model for all $k$ and for the following ranking performance measures: Pairwise Loss, Discounted Cumulative Gain, and Precision@n. In addition, we give an efficient algorithm that achieves the minimax regret rate for Precision@n.

Random Search is one of the most widely-used method for Hyperparameter Optimization, and is critical to the success of deep learning models. Despite its astonishing performance, little non-heuristic theory has been developed to describe the underlying working mechanism. This paper gives a theoretical accounting of Random Search. We introduce the concept of scattering dimension that describes the landscape of the underlying function, and quantifies the performance of random search. We show that, when the environment is noise-free, the output of random search converges to the optimal value in probability at rate $ \widetilde{\mathcal{O}} \left( \left( \frac{1}{T} \right)^{ \frac{1}{d_s} } \right) $, where $ d_s \ge 0 $ is the scattering dimension of the underlying function. When the observed function values are corrupted by bounded $iid$ noise, the output of random search converges to the optimal value in probability at rate $ \widetilde{\mathcal{O}} \left( \left( \frac{1}{T} \right)^{ \frac{1}{d_s + 1} } \right) $. In addition, based on the principles of random search, we introduce an algorithm, called BLiN-MOS, for Lipschitz bandits in doubling metric spaces that are also endowed with a probability measure, and show that under certain conditions, BLiN-MOS achieves a regret rate of order $ \widetilde{\mathcal{O}} \left( T^{ \frac{d_z}{d_z + 1} } \right) $, where $d_z$ is the zooming dimension of the problem instance.

With the success of Vision Transformers (ViTs) in computer vision tasks, recent arts try to optimize the performance and complexity of ViTs to enable efficient deployment on mobile devices. Multiple approaches are proposed to accelerate attention mechanism, improve inefficient designs, or incorporate mobile-friendly lightweight convolutions to form hybrid architectures. However, ViT and its variants still have higher latency or considerably more parameters than lightweight CNNs, even true for the years-old MobileNet. In practice, latency and size are both crucial for efficient deployment on resource-constraint hardware. In this work, we investigate a central question, can transformer models run as fast as MobileNet and maintain a similar size? We revisit the design choices of ViTs and propose a novel supernet with low latency and high parameter efficiency. We further introduce a novel fine-grained joint search strategy for transformer models that can find efficient architectures by optimizing latency and number of parameters simultaneously. The proposed models, EfficientFormerV2, achieve 3.5% higher top-1 accuracy than MobileNetV2 on ImageNet-1K with similar latency and parameters. This work demonstrate that properly designed and optimized vision transformers can achieve high performance even with MobileNet-level size and speed.

In the past few years, AlphaZero's exceptional capability in mastering intricate board games has garnered considerable interest. Initially designed for the game of Go, this revolutionary algorithm merges deep learning techniques with the Monte Carlo tree search (MCTS) to surpass earlier top-tier methods. In our study, we broaden the use of AlphaZero to Gomoku, an age-old tactical board game also referred to as "Five in a Row." Intriguingly, Gomoku has innate challenges due to a bias towards the initial player, who has a theoretical advantage. To add value, we strive for a balanced game-play. Our tests demonstrate AlphaZero's versatility in adapting to games other than Go. MCTS has become a predominant algorithm for decision processes in intricate scenarios, especially board games. MCTS creates a search tree by examining potential future actions and uses random sampling to predict possible results. By leveraging the best of both worlds, the AlphaZero technique fuses deep learning from Reinforcement Learning with the balancing act of MCTS, establishing a fresh standard in game-playing AI. Its triumph is notably evident in board games such as Go, chess, and shogi.

This work is inspired by the problem of planning sequences of operations, as welding, in car manufacturing stations where multiple industrial robots cooperate. The goal is to minimize the station cycle time, \emph{i.e.} the time it takes for the last robot to finish its cycle. This is done by dispatching the tasks among the robots, and by routing and scheduling the robots in a collision-free way, such that they perform all predefined tasks. We propose an iterative and decoupled approach in order to cope with the high complexity of the problem. First, collisions among robots are neglected, leading to a min-max Multiple Generalized Traveling Salesman Problem (MGTSP). Then, when the sets of robot loads have been obtained and fixed, we sequence and schedule their tasks, with the aim to avoid conflicts. The first problem (min-max MGTSP) is solved by an exact branch and bound method, where different lower bounds are presented by combining the solutions of a min-max set partitioning problem and of a Generalized Traveling Salesman Problem (GTSP). The second problem is approached by assuming that robots move synchronously: a novel transformation of this synchronous problem into a GTSP is presented. Eventually, in order to provide complete robot solutions, we include path planning functionalities, allowing the robots to avoid collisions with the static environment and among themselves. These steps are iterated until a satisfying solution is obtained. Experimental results are shown for both problems and for their combination. We even show the results of the iterative method, applied to an industrial test case adapted from a stud welding station in a car manufacturing line.

This research concerns Learned Data Structures, a recent area that has emerged at the crossroad of Machine Learning and Classic Data Structures. It is methodologically important and with a high practical impact. We focus on Learned Indexes, i.e., Learned Sorted Set Dictionaries. The proposals available so far are specific in the sense that they can boost, indeed impressively, the time performance of Table Search Procedures with a sorted layout only, e.g., Binary Search. We propose a novel paradigm that, complementing known specialized ones, can produce Learned versions of any Sorted Set Dictionary, for instance, Balanced Binary Search Trees or Binary Search on layouts other that sorted, i.e., Eytzinger. Theoretically, based on it, we obtain several results of interest, such as (a) the first Learned Optimum Binary Search Forest, with mean access time bounded by the Entropy of the probability distribution of the accesses to the Dictionary; (b) the first Learned Sorted Set Dictionary that, in the Dynamic Case and in an amortized analysis setting, matches the same time bounds known for Classic Dictionaries. This latter under widely accepted assumptions regarding the size of the Universe. The experimental part, somewhat complex in terms of software development, clearly indicates the nonobvious finding that the generalization we propose can yield effective and competitive Learned Data Structural Booster, even with respect to specific benchmark models.