Maximum subarray is a classical problem in computer science that given an array of numbers aims to find a contiguous subarray with the largest sum. We focus on its use for a noisy statistical problem of localizing an interval with a mean different from background. While a naive application of maximum subarray fails at this task, both a penalized and a constrained version can succeed. We show that the penalized version can be derived for common exponential family distributions, in a manner similar to the change-point detection literature, and we interpret the resulting optimal penalty value. The failure of the naive formulation is then explained by an analysis of the estimated interval boundaries. Experiments further quantify the effect of deviating from the optimal penalty. We also relate the penalized and constrained formulations and show that the solutions to the former lie on the convex hull of the solutions to the latter.
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Quantum counting is a key quantum algorithm that aims to determine the number of marked elements in a database. This algorithm is based on the quantum phase estimation algorithm and uses the evolution operator of Grover's algorithm because its non-trivial eigenvalues are dependent on the number of marked elements. Since Grover's algorithm can be viewed as a quantum walk on a complete graph, a natural way to extend quantum counting is to use the evolution operator of quantum-walk-based search on non-complete graphs instead of Grover's operator. In this paper, we explore this extension by analyzing the coined quantum walk on the complete bipartite graph with an arbitrary number of marked vertices. We show that some eigenvalues of the evolution operator depend on the number of marked vertices and using this fact we show that the quantum phase estimation can be used to obtain the number of marked vertices. The time complexity for estimating the number of marked vertices in the bipartite graph with our algorithm aligns closely with that of the original quantum counting algorithm.
The multiobjective evolutionary optimization algorithm (MOEA) is a powerful approach for tackling multiobjective optimization problems (MOPs), which can find a finite set of approximate Pareto solutions in a single run. However, under mild regularity conditions, the Pareto optimal set of a continuous MOP could be a low dimensional continuous manifold that contains infinite solutions. In addition, structure constraints on the whole optimal solution set, which characterize the patterns shared among all solutions, could be required in many real-life applications. It is very challenging for existing finite population based MOEAs to handle these structure constraints properly. In this work, we propose the first model-based algorithmic framework to learn the whole solution set with structure constraints for multiobjective optimization. In our approach, the Pareto optimality can be traded off with a preferred structure among the whole solution set, which could be crucial for many real-world problems. We also develop an efficient evolutionary learning method to train the set model with structure constraints. Experimental studies on benchmark test suites and real-world application problems demonstrate the promising performance of our proposed framework.
We construct a quasi-polynomial time deterministic approximation algorithm for computing the volume of an independent set polytope with restrictions. Randomized polynomial time approximation algorithms for computing the volume of a convex body have been known now for several decades, but the corresponding deterministic counterparts are not available, and our algorithm is the first of this kind. The class of polytopes for which our algorithm applies arises as linear programming relaxation of the independent set problem with the additional restriction that each variable takes value in the interval $[0,1-\alpha]$ for some $\alpha<1/2$. (We note that the $\alpha\ge 1/2$ case is trivial). We use the correlation decay method for this problem applied to its appropriate and natural discretization. The method works provided $\alpha> 1/2-O(1/\Delta^2)$, where $\Delta$ is the maximum degree of the graph. When $\Delta=3$ (the sparsest non-trivial case), our method works provided $0.488<\alpha<0.5$. Interestingly, the interpolation method, which is based on analyzing complex roots of the associated partition functions, fails even in the trivial case when the underlying graph is a singleton.
A common method to study deep learning systems is to use simplified model representations -- for example, using singular value decomposition to visualize the model's hidden states in a lower dimensional space. This approach assumes that the results of these simplified are faithful to the original model. Here, we illustrate an important caveat to this assumption: even if the simplified representations can accurately approximate the full model on the training set, they may fail to accurately capture the model's behavior out of distribution -- the understanding developed from simplified representations may be an illusion. We illustrate this by training Transformer models on controlled datasets with systematic generalization splits. First, we train models on the Dyck balanced-parenthesis languages. We simplify these models using tools like dimensionality reduction and clustering, and then explicitly test how these simplified proxies match the behavior of the original model on various out-of-distribution test sets. We find that the simplified proxies are generally less faithful out of distribution. In cases where the original model generalizes to novel structures or deeper depths, the simplified versions may fail, or generalize better. This finding holds even if the simplified representations do not directly depend on the training distribution. Next, we study a more naturalistic task: predicting the next character in a dataset of computer code. We find similar generalization gaps between the original model and simplified proxies, and conduct further analysis to investigate which aspects of the code completion task are associated with the largest gaps. Together, our results raise questions about the extent to which mechanistic interpretations derived using tools like SVD can reliably predict what a model will do in novel situations.
Optimization pipelines targeting polyhedral programs try to maximize the compute throughput. Traditional approaches favor reuse and temporal locality; while the communicated volume can be low, failure to optimize spatial locality may cause a low I/O performance. Memory allocation schemes using data partitioning such as data tiling can improve the spatial locality, but they are domain-specific and rarely applied by compilers when an existing allocation is supplied. In this paper, we propose to derive a partitioned memory allocation for tiled polyhedral programs using their data flow information. We extend the existing MARS partitioning to handle affine dependences, and determine which dependences can lead to a regular, simple control flow for communications. While this paper consists in a theoretical study, previous work on data partitioning in inter-node scenarios has shown performance improvements due to better bandwidth utilization.
A fundamental question is whether one can maintain a maximum independent set in polylogarithmic update time for a dynamic collection of geometric objects in Euclidean space. Already, for a set of intervals, it is known that no dynamic algorithm can maintain an exact maximum independent set in sublinear update time. Therefore, the typical objective is to explore the trade-off between update time and solution size. Substantial efforts have been made in recent years to understand this question for various families of geometric objects, such as intervals, hypercubes, hyperrectangles, and fat objects. We present the first fully dynamic approximation algorithm for disks of arbitrary radii in the plane that maintains a constant-factor approximate maximum independent set in polylogarithmic expected amortized update time. Moreover, for a fully dynamic set of $n$ disks of unit radius in the plane, we show that a $12$-approximate maximum independent set can be maintained with worst-case update time $O(\log n)$, and optimal output-sensitive reporting. This result generalizes to fat objects of comparable sizes in any fixed dimension $d$, where the approximation ratio depends on the dimension and the fatness parameter. Further, we note that, even for a dynamic set of disks of unit radius in the plane, it is impossible to maintain $O(1+\varepsilon)$-approximate maximum independent set in truly sublinear update time, under standard complexity assumptions.
The Knapsack Problem is a classic problem in combinatorial optimisation. Solving these problems may be computationally expensive. Recent years have seen a growing interest in the use of deep learning methods to approximate the solutions to such problems. A core problem is how to enforce or encourage constraint satisfaction in predicted solutions. A promising approach for predicting solutions to constrained optimisation problems is the Lagrangian Dual Framework which builds on the method of Lagrangian Relaxation. In this paper we develop neural network models to approximate Knapsack Problem solutions using the Lagrangian Dual Framework while improving constraint satisfaction. We explore the problems of output interpretation and model selection within this context. Experimental results show strong constraint satisfaction with a minor reduction of optimality as compared to a baseline neural network which does not explicitly model the constraints.
Kernel methods are a popular class of nonlinear predictive models in machine learning. Scalable algorithms for learning kernel models need to be iterative in nature, but convergence can be slow due to poor conditioning. Spectral preconditioning is an important tool to speed-up the convergence of such iterative algorithms for training kernel models. However computing and storing a spectral preconditioner can be expensive which can lead to large computational and storage overheads, precluding the application of kernel methods to problems with large datasets. A Nystrom approximation of the spectral preconditioner is often cheaper to compute and store, and has demonstrated success in practical applications. In this paper we analyze the trade-offs of using such an approximated preconditioner. Specifically, we show that a sample of logarithmic size (as a function of the size of the dataset) enables the Nystrom-based approximated preconditioner to accelerate gradient descent nearly as well as the exact preconditioner, while also reducing the computational and storage overheads.
As soon as abstract mathematical computations were adapted to computation on digital computers, the problem of efficient representation, manipulation, and communication of the numerical values in those computations arose. Strongly related to the problem of numerical representation is the problem of quantization: in what manner should a set of continuous real-valued numbers be distributed over a fixed discrete set of numbers to minimize the number of bits required and also to maximize the accuracy of the attendant computations? This perennial problem of quantization is particularly relevant whenever memory and/or computational resources are severely restricted, and it has come to the forefront in recent years due to the remarkable performance of Neural Network models in computer vision, natural language processing, and related areas. Moving from floating-point representations to low-precision fixed integer values represented in four bits or less holds the potential to reduce the memory footprint and latency by a factor of 16x; and, in fact, reductions of 4x to 8x are often realized in practice in these applications. Thus, it is not surprising that quantization has emerged recently as an important and very active sub-area of research in the efficient implementation of computations associated with Neural Networks. In this article, we survey approaches to the problem of quantizing the numerical values in deep Neural Network computations, covering the advantages/disadvantages of current methods. With this survey and its organization, we hope to have presented a useful snapshot of the current research in quantization for Neural Networks and to have given an intelligent organization to ease the evaluation of future research in this area.
We propose a novel approach to multimodal sentiment analysis using deep neural networks combining visual analysis and natural language processing. Our goal is different than the standard sentiment analysis goal of predicting whether a sentence expresses positive or negative sentiment; instead, we aim to infer the latent emotional state of the user. Thus, we focus on predicting the emotion word tags attached by users to their Tumblr posts, treating these as "self-reported emotions." We demonstrate that our multimodal model combining both text and image features outperforms separate models based solely on either images or text. Our model's results are interpretable, automatically yielding sensible word lists associated with emotions. We explore the structure of emotions implied by our model and compare it to what has been posited in the psychology literature, and validate our model on a set of images that have been used in psychology studies. Finally, our work also provides a useful tool for the growing academic study of images - both photographs and memes - on social networks.
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