We propose novel smooth approximations to the classical rounding function, suitable for differentiable optimization and machine learning applications. Our constructions are based on two approaches: (1) localized sigmoid window functions centered at each integer, and (2) normalized weighted sums of sigmoid derivatives representing local densities. The first method approximates the step-like behavior of rounding through differences of shifted sigmoids, while the second method achieves smooth interpolation between integers via density-based weighting. Both methods converge pointwise to the classical rounding function as the sharpness parameter k tends to infinity, and allow controlled trade-offs between smoothness and approximation accuracy. We demonstrate that by restricting the summation to a small set of nearest integers, the computational cost remains low without sacrificing precision. These constructions provide fully differentiable alternatives to hard rounding, which are valuable in contexts where gradient-based methods are essential.

We study the problem of Active Simple Hypothesis Testing (ASHT) whe re an agent is faced with the problem of choosing between m different simple hypotheses after observing T samples. At the end of T samples, it has to output one of the m hypothesis. The distinguishing difference from the usual hypothes is testing scenario is the ability to choose one of K actions and observe the corresponding sample for that action. Th is ability to control the samples in this way makes the problem more interesting and difficult compared to the usual hypothesis testing with no control over the sample generation. The performance of the agent is meas ured in terms of the error probability its decision incurs. The above theoretical problem is a model for many practica l scenarios-A cosmetic drug trial often involve a testing period where the outcome of interest is to identify the best product after the trial period, choosing a channel from a set of channels before commencing communications, placeme nt of sensors in certain set of positions so as to minimize signal error. Any situation which require a period of testing b efore committing to a final decision with only certain fixed budget of samples (that is an inability to request additio nal samples) can be modeled effectively using ASHT and its more general version - Fixed Budget Best Arm Identific ation (FB-BAI). We intend to study the ASHT problem in the large deviation setting with the quantity of interest being the minimax error exponent over the hypotheses, that is, the worst case er ror exponent over the hypotheses.

Introduction to Quantum Machine Learning and Quantum Architecture Search Samuel Y en-Chi Chen 1 Zhiding Liang 2 1 Wells Fargo 2 Rensselaer Polytechnic Institute Abstract --Recent advancements in quantum computing (QC) and machine learning (ML) have fueled significant research efforts aimed at integrating these two transformative technologies. Quantum machine learning (QML), an emerging interdisciplinary field, leverages quantum principles to enhance the performance of ML algorithms. Concurrently, the exploration of systematic and automated approaches for designing high-performance quantum circuit architectures for QML tasks has gained prominence, as these methods empower researchers outside the quantum computing domain to effectively utilize quantum-enhanced tools. This tutorial will provide an in-depth overview of recent breakthroughs in both areas, highlighting their potential to expand the application landscape of QML across diverse fields. I NTRODUCTION Quantum computing (QC) offers the potential for substantial speedups in solving certain computationally challenging problems compared to classical computers. Recent advancements in quantum hardware, coupled with remarkable progress in classical AI and machine learning (ML) techniques, have sparked growing interest in merging these two technologies to further accelerate advancements in artificial intelligence.

We propose a model for making data acquisition decisions for variables in contextual stochastic optimisation problems. Data acquisition decisions are typically treated as separate and fixed. We explore problem settings in which the acquisition of contextual variables is costly and consequently constrained. The data acquisition problem is often solved heuristically for proxy objectives such as coverage. The more intuitive objective is the downstream decision quality as a result of data acquisition decisions. The whole pipeline can be characterised as an optimise-then-predict-then-optimise (OPO) problem. Analogously, much recent research has focused on how to integrate prediction and optimisation (PO) in the form of decision-focused learning. We propose leveraging differentiable optimisation to extend the integration to data acquisition. We solve the data acquisition problem with well-defined constraints by learning a surrogate linear objective function. We demonstrate an application of this model on a shortest path problem for which we first have to set a drone reconnaissance strategy to capture image segments serving as inputs to a model that predicts travel costs. We ablate the problem with a number of training modalities and demonstrate that the differentiable optimisation approach outperforms random search strategies.

Causal inference aids researchers in discovering cause-and-effect relationships, leading to scientific insights. Accurate causal estimation requires identifying confounding variables to avoid false discoveries. Pearl's causal model uses causal DAGs to identify confounding variables, but incorrect DAGs can lead to unreliable causal conclusions. However, for high dimensional data, the causal DAGs are often complex beyond human verifiability. Graph summarization is a logical next step, but current methods for general-purpose graph summarization are inadequate for causal DAG summarization. This paper addresses these challenges by proposing a causal graph summarization objective that balances graph simplification for better understanding while retaining essential causal information for reliable inference. We develop an efficient greedy algorithm and show that summary causal DAGs can be directly used for inference and are more robust to misspecification of assumptions, enhancing robustness for causal inference. Experimenting with six real-life datasets, we compared our algorithm to three existing solutions, showing its effectiveness in handling high-dimensional data and its ability to generate summary DAGs that ensure both reliable causal inference and robustness against misspecifications.

Gray-box optimization employs Walsh decomposition to obtain non-linear variable dependencies and utilize them to propose masks of variables that have a joint non-linear influence on fitness value. These masks significantly improve the effectiveness of variation operators. In some problems, all variables are non-linearly dependent, making the aforementioned masks useless. We analyze the features of the real-world instances of such problems and show that many of their dependencies may have noise-like origins. Such noise-caused dependencies are irrelevant to the optimization process and can be ignored. To identify them, we propose extending the use of Walsh decomposition by measuring variable dependency strength that allows the construction of the weighted dynamic Variable Interaction Graph (wdVIG). wdVIGs adjust the dependency strength to mixed individuals. They allow the filtering of irrelevant dependencies and re-enable using dependency-based masks by variation operators. We verify the wdVIG potential on a large benchmark suite. For problems with noise, the wdVIG masks can improve the optimizer's effectiveness. If all dependencies are relevant for the optimization, i.e., the problem is not noised, the influence of wdVIG masks is similar to that of state-of-the-art structures of this kind.

Modern society is full of computational challenges that rely on probabilistic reasoning, statistics, and combinatorics. Interestingly, many of these questions can be formulated by encoding them into propositional formulas and then asking for its number of models. With a growing interest in practical problem-solving for tasks that involve model counting, the community established the Model Counting (MC) Competition in fall of 2019 with its first iteration in 2020. The competition aims at advancing applications, identifying challenging benchmarks, fostering new solver development, and enhancing existing solvers for model counting problems and their variants. The first iteration, brought together various researchers, identified challenges, and inspired numerous new applications. In this paper, we present a comprehensive overview of the 2021-2023 iterations of the Model Counting Competition. We detail its execution and outcomes. The competition comprised four tracks, each focusing on a different variant of the model counting problem. The first track centered on the model counting problem (MC), which seeks the count of models for a given propositional formula. The second track challenged developers to submit programs capable of solving the weighted model counting problem (WMC). The third track was dedicated to projected model counting (PMC). Finally, we initiated a track that combined projected and weighted model counting (PWMC). The competition continued with a high level of participation, with seven to nine solvers submitted in various different version and based on quite diverging techniques.

Chenxuan Liu, He Li, Zongze Li, Shuai Wang, Wei Xu, Kejiang Y e, Derrick Wing Kwan Ng, Fellow, IEEE, and Chengzhong Xu, Fellow, IEEE Abstract --Realizing green communication in robotic mixed reality (RoboMR) systems presents a challenge, due to the necessity of uploading high-resolution images at high frequencies through wireless channels. This paper proposes Gaussian splatting (GS) RoboMR (GSRMR), which achieves a lower energy consumption and makes a concrete step towards green RoboMR. The crux to GSRMR is to build a GS model which enables the simulator to opportunistically render a photo-realistic view from the robot's pose, thereby reducing the need for excessive image uploads. Since the GS model may involve discrepancies compared to the actual environments, a GS cross-layer optimization (GSCLO) framework is further proposed, which jointly optimizes content switching (i.e., deciding whether to upload image or not) and power allocation across different frames. The GSCLO problem is solved by an accelerated penalty optimization (APO) algorithm. Experiments demonstrate that the proposed GSRMR reduces the communication energy by over 10 x compared with RoboMR. Furthermore, the proposed GSRMR with APO outperforms extensive baseline schemes, in terms of peak signal-to-noise ratio (PSNR) and structural similarity index measure (SSIM).

Column Generation (CG) is a popular method dedicated to enhancing computational efficiency in large scale Combinatorial Optimization (CO) problems. It reduces the number of decision variables in a problem by solving a pricing problem. For many CO problems, the pricing problem is an Elementary Shortest Path Problem with Resource Constraints (ESPPRC). Large ESPPRC instances are difficult to solve to near-optimality. Consequently, we use a Graph neural Network (GNN) to reduces the size of the ESPPRC such that it becomes computationally tractable with standard solving techniques. Our GNN is trained by Unsupervised Learning and outputs a distribution for the arcs to be retained in the reduced PP. The reduced PP is solved by a local search that finds columns with large reduced costs and speeds up convergence. We apply our method on a set of Capacitated Vehicle Routing Problems with Time Windows and show significant improvements in convergence compared to simple reduction techniques from the literature. For a fixed computational budget, we improve the objective values by over 9% for larger instances. We also analyze the performance of our CG algorithm and test the generalization of our method to different classes of instances than the training data. Introduction Real-life decision-making problems can often be modeled by Combinatorial Optimization (CO). Practical problem instances, however, usually have much larger sizes than instances often encountered in the literature (Luo et al., 2024), while requiring good solutions in limited time. This invokes the need for specialized methods capable of scaling to the large instances of these problems. Column Generation (CG) is often regarded as an efficient technique to address many CO problems such as Vehicle Routing and Crew Scheduling (Desaulniers et al., 2006), as it keeps the number To identify variables which contribute to the objective value of a problem, i.e. with non-zero reduced cost, a Pricing Problem (PP) has to be solved. As during the CG algorithm, many Pricing Problems have to be considered, the efficiency of the method lies in the ability to solve the PP efficiently(Václavík et al., 2018).

We introduce Kimina-Prover Preview, a large language model that pioneers a novel reasoning-driven exploration paradigm for formal theorem proving, as showcased in this preview release. Trained with a large-scale reinforcement learning pipeline from Qwen2.5-72B, Kimina-Prover demonstrates strong performance in Lean 4 proof generation by employing a structured reasoning pattern we term \textit{formal reasoning pattern}. This approach allows the model to emulate human problem-solving strategies in Lean, iteratively generating and refining proof steps. Kimina-Prover sets a new state-of-the-art on the miniF2F benchmark, reaching 80.7% with pass@8192. Beyond improved benchmark performance, our work yields several key insights: (1) Kimina-Prover exhibits high sample efficiency, delivering strong results even with minimal sampling (pass@1) and scaling effectively with computational budget, stemming from its unique reasoning pattern and RL training; (2) we demonstrate clear performance scaling with model size, a trend previously unobserved for neural theorem provers in formal mathematics; (3) the learned reasoning style, distinct from traditional search algorithms, shows potential to bridge the gap between formal verification and informal mathematical intuition. We open source distilled versions with 1.5B and 7B parameters of Kimina-Prover