There is currently a large gap in performance between the statistically rigorous methods like linear regression or additive splines and the powerful deep methods using neural networks. Previous works attempting to close this gap have failed to fully investigate the exponentially growing number of feature combinations which deep networks consider automatically during training. In this work, we develop a tractable selection algorithm to efficiently identify the necessary feature combinations by leveraging techniques in feature interaction detection. Our proposed Sparse Interaction Additive Networks (SIAN) construct a bridge from these simple and interpretable models to fully connected neural networks. SIAN achieves competitive performance against state-of-the-art methods across multiple large-scale tabular datasets and consistently finds an optimal tradeoff between the modeling capacity of neural networks and the generalizability of simpler methods.
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We develop a general theory to optimize the frequentist regret for sequential learning problems, where efficient bandit and reinforcement learning algorithms can be derived from unified Bayesian principles. We propose a novel optimization approach to generate "algorithmic beliefs" at each round, and use Bayesian posteriors to make decisions. The optimization objective to create "algorithmic beliefs," which we term "Algorithmic Information Ratio," represents an intrinsic complexity measure that effectively characterizes the frequentist regret of any algorithm. To the best of our knowledge, this is the first systematical approach to make Bayesian-type algorithms prior-free and applicable to adversarial settings, in a generic and optimal manner. Moreover, the algorithms are simple and often efficient to implement. As a major application, we present a novel algorithm for multi-armed bandits that achieves the "best-of-all-worlds" empirical performance in the stochastic, adversarial, and non-stationary environments. And we illustrate how these principles can be used in linear bandits, bandit convex optimization, and reinforcement learning.
Most of existing correspondence pruning methods only concentrate on gathering the context information as much as possible while neglecting effective ways to utilize such information. In order to tackle this dilemma, in this paper we propose Graph Context Transformation Network (GCT-Net) enhancing context information to conduct consensus guidance for progressive correspondence pruning. Specifically, we design the Graph Context Enhance Transformer which first generates the graph network and then transforms it into multi-branch graph contexts. Moreover, it employs self-attention and cross-attention to magnify characteristics of each graph context for emphasizing the unique as well as shared essential information. To further apply the recalibrated graph contexts to the global domain, we propose the Graph Context Guidance Transformer. This module adopts a confident-based sampling strategy to temporarily screen high-confidence vertices for guiding accurate classification by searching global consensus between screened vertices and remaining ones. The extensive experimental results on outlier removal and relative pose estimation clearly demonstrate the superior performance of GCT-Net compared to state-of-the-art methods across outdoor and indoor datasets. The source code will be available at: //github.com/guobaoxiao/GCT-Net/.
In stochastic zeroth-order optimization, a problem of practical relevance is understanding how to fully exploit the local geometry of the underlying objective function. We consider a fundamental setting in which the objective function is quadratic, and provide the first tight characterization of the optimal Hessian-dependent sample complexity. Our contribution is twofold. First, from an information-theoretic point of view, we prove tight lower bounds on Hessian-dependent complexities by introducing a concept called energy allocation, which captures the interaction between the searching algorithm and the geometry of objective functions. A matching upper bound is obtained by solving the optimal energy spectrum. Then, algorithmically, we show the existence of a Hessian-independent algorithm that universally achieves the asymptotic optimal sample complexities for all Hessian instances. The optimal sample complexities achieved by our algorithm remain valid for heavy-tailed noise distributions, which are enabled by a truncation method.
In the near term, quantum approximate optimization algorithms (QAOAs) hold great potential to solve combinatorial optimization problems. These are hybrid algorithms, i.e., a combination of quantum and classical algorithms. Several proof-of-concept applications of QAOAs for solving combinatorial problems, such as portfolio optimization, energy optimization in power systems, and job scheduling, have been demonstrated. However, whether QAOAs can efficiently solve optimization problems from classical software engineering, such as test optimization, remains unstudied. To this end, we present the first effort to formulate a software test case optimization problem as a QAOA problem and solve it on quantum computer simulators. To solve bigger test optimization problems that require many qubits, which are unavailable these days, we integrate a problem decomposition strategy with the QAOA. We performed an empirical evaluation with five test case optimization problems and four industrial datasets from ABB, Google, and Orona to compare various configurations of our approach, assess its decomposition strategy of handling large datasets, and compare its performance with classical algorithms (i.e., Genetic Algorithm (GA) and Random Search). Based on the evaluation results, we recommend the best configuration of our approach for test case optimization problems. Also, we demonstrate that our strategy can reach the same effectiveness as GA and outperform GA in two out of five test case optimization problems we conducted.
We consider the problem of detecting jumps in an otherwise smoothly evolving trend whilst the covariance and higher-order structures of the system can experience both smooth and abrupt changes over time. The number of jump points is allowed to diverge to infinity with the jump sizes possibly shrinking to zero. The method is based on a multiscale application of an optimal jump-pass filter to the time series, where the scales are dense between admissible lower and upper bounds. For a wide class of non-stationary time series models and trend functions, the proposed method is shown to be able to detect all jump points within a nearly optimal range with a prescribed probability asymptotically under mild conditions. For a time series of length $n$, the computational complexity of the proposed method is $O(n)$ for each scale and $O(n\log^{1+\epsilon} n)$ overall, where $\epsilon$ is an arbitrarily small positive constant. Numerical studies show that the proposed jump testing and estimation method performs robustly and accurately under complex temporal dynamics.
We consider optimal experimental design (OED) for nonlinear Bayesian inverse problems governed by large-scale partial differential equations (PDEs). For the optimality criteria of Bayesian OED, we consider both expected information gain and summary statistics including the trace and determinant of the information matrix that involves the evaluation of the parameter-to-observable (PtO) map and its derivatives. However, it is prohibitive to compute and optimize these criteria when the PDEs are very expensive to solve, the parameters to estimate are high-dimensional, and the optimization problem is combinatorial, high-dimensional, and non-convex. To address these challenges, we develop an accurate, scalable, and efficient computational framework to accelerate the solution of Bayesian OED. In particular, the framework is developed based on derivative-informed neural operator (DINO) surrogates with proper dimension reduction techniques and a modified swapping greedy algorithm. We demonstrate the high accuracy of the DINO surrogates in the computation of the PtO map and the optimality criteria compared to high-fidelity finite element approximations. We also show that the proposed method is scalable with increasing parameter dimensions. Moreover, we demonstrate that it achieves high efficiency with over 1000X speedup compared to a high-fidelity Bayesian OED solution for a three-dimensional PDE example with tens of thousands of parameters, including both online evaluation and offline construction costs of the surrogates.
We explore some connections between association schemes and the analyses of the semidefinite programming (SDP) based convex relaxations of combinatorial optimization problems in the Lov\'{a}sz--Schrijver lift-and-project hierarchy. Our analysis of the relaxations of the stable set polytope leads to bounds on the clique and stability numbers of some regular graphs reminiscent of classical bounds by Delsarte and Hoffman, as well as the notion of deeply vertex-transitive graphs -- highly symmetric graphs that we show arise naturally from some association schemes. We also study relaxations of the hypergraph matching problem, and determine exactly or provide bounds on the lift-and-project ranks of these relaxations. Our proofs for these results also inspire the study of the general hypermatching pseudo-scheme, which is an association scheme except it is generally non-commutative. We then illustrate the usefulness of obtaining commutative subschemes from non-commutative pseudo-schemes via contraction in this context.
Recent contrastive representation learning methods rely on estimating mutual information (MI) between multiple views of an underlying context. E.g., we can derive multiple views of a given image by applying data augmentation, or we can split a sequence into views comprising the past and future of some step in the sequence. Contrastive lower bounds on MI are easy to optimize, but have a strong underestimation bias when estimating large amounts of MI. We propose decomposing the full MI estimation problem into a sum of smaller estimation problems by splitting one of the views into progressively more informed subviews and by applying the chain rule on MI between the decomposed views. This expression contains a sum of unconditional and conditional MI terms, each measuring modest chunks of the total MI, which facilitates approximation via contrastive bounds. To maximize the sum, we formulate a contrastive lower bound on the conditional MI which can be approximated efficiently. We refer to our general approach as Decomposed Estimation of Mutual Information (DEMI). We show that DEMI can capture a larger amount of MI than standard non-decomposed contrastive bounds in a synthetic setting, and learns better representations in a vision domain and for dialogue generation.
Embedding entities and relations into a continuous multi-dimensional vector space have become the dominant method for knowledge graph embedding in representation learning. However, most existing models ignore to represent hierarchical knowledge, such as the similarities and dissimilarities of entities in one domain. We proposed to learn a Domain Representations over existing knowledge graph embedding models, such that entities that have similar attributes are organized into the same domain. Such hierarchical knowledge of domains can give further evidence in link prediction. Experimental results show that domain embeddings give a significant improvement over the most recent state-of-art baseline knowledge graph embedding models.
Multi-relation Question Answering is a challenging task, due to the requirement of elaborated analysis on questions and reasoning over multiple fact triples in knowledge base. In this paper, we present a novel model called Interpretable Reasoning Network that employs an interpretable, hop-by-hop reasoning process for question answering. The model dynamically decides which part of an input question should be analyzed at each hop; predicts a relation that corresponds to the current parsed results; utilizes the predicted relation to update the question representation and the state of the reasoning process; and then drives the next-hop reasoning. Experiments show that our model yields state-of-the-art results on two datasets. More interestingly, the model can offer traceable and observable intermediate predictions for reasoning analysis and failure diagnosis, thereby allowing manual manipulation in predicting the final answer.
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