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Score-based diffusion models, which generate new data by learning to reverse a diffusion process that perturbs data from the target distribution into noise, have achieved remarkable success across various generative tasks. Despite their superior empirical performance, existing theoretical guarantees are often constrained by stringent assumptions or suboptimal convergence rates. In this paper, we establish a fast convergence theory for a popular SDE-based sampler under minimal assumptions. Our analysis shows that, provided $\ell_{2}$-accurate estimates of the score functions, the total variation distance between the target and generated distributions is upper bounded by $O(d/T)$ (ignoring logarithmic factors), where $d$ is the data dimensionality and $T$ is the number of steps. This result holds for any target distribution with finite first-order moment. To our knowledge, this improves upon existing convergence theory for both the SDE-based sampler and another ODE-based sampler, while imposing minimal assumptions on the target data distribution and score estimates. This is achieved through a novel set of analytical tools that provides a fine-grained characterization of how the error propagates at each step of the reverse process.

We present a 1.8334-approximation algorithm for Vertex Cover on string graphs given with a representation, which takes polynomial time in the size of the representation; the exact approximation factor is $11/6$. Recently, the barrier of 2 was broken by Lokshtanov et al. [SoGC '24] with a 1.9999-approximation algorithm. Thus we increase by three orders of magnitude the distance of the approximation ratio to the trivial bound of 2. Our algorithm is very simple. The intricacies reside in its analysis, where we mainly establish that string graphs without odd cycles of length at most 11 are 8-colorable. Previously, Chudnovsky, Scott, and Seymour [JCTB '21] showed that string graphs without odd cycles of length at most 7 are 80-colorable, and string graphs without odd cycles of length at most 5 have bounded chromatic number.

Fog computing is of particular interest to Internet of Things (IoT), where inexpensive simple devices can offload their computation tasks to nearby Fog Nodes. Online scheduling in such fog networks is challenging due to stochastic network states such as task arrivals, wireless channels and location of nodes. In this paper, we focus on the problem of optimizing computation offloading management, arrival data admission control and resource scheduling, in order to improve the overall system performance, in terms of throughput fairness, power efficiency, and average mean of queue backlogs. We investigate this problem for a fog network with homogeneous mobile Fog Nodes, serving multiple wireless devices, controlled by a Fog Control Node. By formulating the problem as a stochastic optimization problem, maximizing utility-power efficiency, defined as achievable utility per-unit power consumption, subject to queue backlog stability, we modify Lyapunov optimization techniques to deal with the fractional form of utility-power efficiency function. Then we propose an online utility-power efficient task scheduling algorithm, which is asymptotically optimal. Our online task scheduling algorithm can achieve the theoretical [O(1/V), O(V)] trade-off between utility-power efficiency and average mean of queue backlogs,

Sparse matrix-vector multiplication (SpMV) is a fundamental operation in machine learning, scientific computing, and graph algorithms. In this paper, we investigate the space, time, and energy efficiency of SpMV using various compressed formats for large sparse matrices, focusing specifically on Boolean matrices and real-valued vectors. Through extensive analysis and experiments conducted on server and edge devices, we found that different matrix compression formats offer distinct trade-offs among space usage, execution time, and energy consumption. Notably, by employing the appropriate compressed format, we can reduce energy consumption by an order of magnitude on both server and single-board computers. Furthermore, our experiments indicate that while data parallelism can enhance execution speed and energy efficiency, achieving simultaneous time and energy efficiency presents partially distinct challenges. Specifically, we show that for certain compression schemes, the optimal degree of parallelism for time does not align with that for energy, thereby challenging prevailing assumptions about a straightforward linear correlation between execution time and energy consumption. Our results have significant implications for software engineers in all domains where SpMV operations are prevalent. They also suggest that similar studies exploring the trade-offs between time, space, and energy for other compressed data structures can substantially contribute to designing more energy-efficient software components.

We study the question of whether submodular functions of random variables satisfying various notions of negative dependence satisfy Chernoff-like concentration inequalities. We prove such a concentration inequality for the lower tail when the random variables satisfy negative association or negative regression, partially resolving an open problem raised in (Qiu and Singla [QS22]). Previous work showed such concentration results for random variables that come from specific dependent-rounding algorithms (Chekuri, Vondrak, and Zenklusen [CVZ10] and Harvey and Olver [HO14]). We discuss some applications of our results to combinatorial optimization and beyond. We also show applications to the concentration of read-k families [Gav+15] under certain forms of negative dependence; we further show a simplified proof of the entropy-method approach of [Gav+15].

The circuits comprising superconducting optoelectronic synapses, dendrites, and neurons are described by numerically cumbersome and formally opaque coupled differential equations. Reference 1 showed that a phenomenological model of superconducting loop neurons eliminates the need to solve the Josephson circuit equations that describe synapses and dendrites. The initial goal of the model was to decrease the time required for simulations, yet an additional benefit of the model was increased transparency of the underlying neural circuit operations and conceptual clarity regarding the connection of loop neurons to other physical systems. Whereas the original model simplified the treatment of the Josephson-junction dynamics, essentially by only considering low-pass versions of the dendritic outputs, the model resorted to an awkward treatment of spikes generated by semiconductor transmitter circuits that required explicitly checking for threshold crossings and distinct treatment of time steps wherein somatic threshold is reached. Here we extend that model to simplify the treatment of spikes coming from somas, again making use of the fact that in neural systems the downstream recipients of spike events almost always perform low-pass filtering. We provide comparisons between the first and second phenomenological models, quantifying the accuracy of the additional approximations. We identify regions of circuit parameter space in which the extended model works well and regions where it works poorly. For some circuit parameters it is possible to represent the downstream dendritic response to a single spike as well as coincidences or sequences of spikes, indicating the model is not simply a reduction to rate coding. The governing equations are shown to be nearly identical to those ubiquitous in the neuroscience literature for modeling leaky-integrator dendrites and neurons.

Classical convergence analyses for optimization algorithms rely on the widely-adopted uniform smoothness assumption. However, recent experimental studies have demonstrated that many machine learning problems exhibit non-uniform smoothness, meaning the smoothness factor is a function of the model parameter instead of a universal constant. In particular, it has been observed that the smoothness grows with respect to the gradient norm along the training trajectory. Motivated by this phenomenon, the recently introduced $(L_0, L_1)$-smoothness is a more general notion, compared to traditional $L$-smoothness, that captures such positive relationship between smoothness and gradient norm. Under this type of non-uniform smoothness, existing literature has designed stochastic first-order algorithms by utilizing gradient clipping techniques to obtain the optimal $\mathcal{O}(\epsilon^{-3})$ sample complexity for finding an $\epsilon$-approximate first-order stationary solution. Nevertheless, the studies of quasi-Newton methods are still lacking. Considering higher accuracy and more robustness for quasi-Newton methods, in this paper we propose a fast stochastic quasi-Newton method when there exists non-uniformity in smoothness. Leveraging gradient clipping and variance reduction, our algorithm can achieve the best-known $\mathcal{O}(\epsilon^{-3})$ sample complexity and enjoys convergence speedup with simple hyperparameter tuning. Our numerical experiments show that our proposed algorithm outperforms the state-of-the-art approaches.

Developing efficient traffic models is essential for optimizing transportation systems, yet current approaches remain time-intensive and susceptible to human errors due to their reliance on manual processes. Traditional workflows involve exhaustive literature reviews, formula optimization, and iterative testing, leading to inefficiencies in research. In response, we introduce the Traffic Research Agent (TR-Agent), an AI-driven system designed to autonomously develop and refine traffic models through an iterative, closed-loop process. Specifically, we divide the research pipeline into four key stages: idea generation, theory formulation, theory evaluation, and iterative optimization; and construct TR-Agent with four corresponding modules: Idea Generator, Code Generator, Evaluator, and Analyzer. Working in synergy, these modules retrieve knowledge from external resources, generate novel ideas, implement and debug models, and finally assess them on the evaluation datasets. Furthermore, the system continuously refines these models based on iterative feedback, enhancing research efficiency and model performance. Experimental results demonstrate that TR-Agent achieves significant performance improvements across multiple traffic models, including the Intelligent Driver Model (IDM) for car following, the MOBIL lane-changing model, and the Lighthill-Whitham-Richards (LWR) traffic flow model. Additionally, TR-Agent provides detailed explanations for its optimizations, allowing researchers to verify and build upon its improvements easily. This flexibility makes the framework a powerful tool for researchers in transportation and beyond. To further support research and collaboration, we have open-sourced both the code and data used in our experiments, facilitating broader access and enabling continued advancements in the field.

Datasets often contain values that naturally reside in a metric space: numbers, strings, geographical locations, machine-learned embeddings in a Euclidean space, and so on. We study the computational complexity of repairing inconsistent databases that violate integrity constraints, where the database values belong to an underlying metric space. The goal is to update the database values to retain consistency while minimizing the total distance between the original values and the repaired ones. We consider what we refer to as \emph{coincidence constraints}, which include key constraints, inclusion, foreign keys, and generally any restriction on the relationship between the numbers of cells of different labels (attributes) coinciding in a single value, for a fixed attribute set. We begin by showing that the problem is APX-hard for general metric spaces. We then present an algorithm solving the problem optimally for tree metrics, which generalize both the line metric (i.e., where repaired values are numbers) and the discrete metric (i.e., where we simply count the number of changed values). Combining our algorithm for tree metrics and a classic result on probabilistic tree embeddings, we design a (high probability) logarithmic-ratio approximation for general metrics. We also study the variant of the problem where each individual value's allowed change is limited. In this variant, it is already NP-complete to decide the existence of any legal repair for a general metric, and we present a polynomial-time repairing algorithm for the case of a line metric.

It is important to detect anomalous inputs when deploying machine learning systems. The use of larger and more complex inputs in deep learning magnifies the difficulty of distinguishing between anomalous and in-distribution examples. At the same time, diverse image and text data are available in enormous quantities. We propose leveraging these data to improve deep anomaly detection by training anomaly detectors against an auxiliary dataset of outliers, an approach we call Outlier Exposure (OE). This enables anomaly detectors to generalize and detect unseen anomalies. In extensive experiments on natural language processing and small- and large-scale vision tasks, we find that Outlier Exposure significantly improves detection performance. We also observe that cutting-edge generative models trained on CIFAR-10 may assign higher likelihoods to SVHN images than to CIFAR-10 images; we use OE to mitigate this issue. We also analyze the flexibility and robustness of Outlier Exposure, and identify characteristics of the auxiliary dataset that improve performance.