Deep neural network (DNN) classifiers are often overconfident, producing miscalibrated class probabilities. Most existing calibration methods either lack theoretical guarantees for producing calibrated outputs or reduce the classification accuracy in the process. This paper proposes a new Kernel-based calibration method called KCal. Unlike other calibration procedures, KCal does not operate directly on the logits or softmax outputs of the DNN. Instead, it uses the penultimate-layer latent embedding to train a metric space in a supervised manner. In effect, KCal amounts to a supervised dimensionality reduction of the neural network embedding, and generates a prediction using kernel density estimation on a holdout calibration set. We first analyze KCal theoretically, showing that it enjoys a provable asymptotic calibration guarantee. Then, through extensive experiments, we confirm that KCal consistently outperforms existing calibration methods in terms of both the classification accuracy and the (confidence and class-wise) calibration error.
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Networking：IFIP International Conferences on Networking。
Explanation：國際網絡會議。
Publisher：IFIP。
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Depth separation results propose a possible theoretical explanation for the benefits of deep neural networks over shallower architectures, establishing that the former possess superior approximation capabilities. However, there are no known results in which the deeper architecture leverages this advantage into a provable optimization guarantee. We prove that when the data are generated by a distribution with radial symmetry which satisfies some mild assumptions, gradient descent can efficiently learn ball indicator functions using a depth 2 neural network with two layers of sigmoidal activations, and where the hidden layer is held fixed throughout training. By building on and refining existing techniques for approximation lower bounds of neural networks with a single layer of non-linearities, we show that there are $d$-dimensional radial distributions on the data such that ball indicators cannot be learned efficiently by any algorithm to accuracy better than $\Omega(d^{-4})$, nor by a standard gradient descent implementation to accuracy better than a constant. These results establish what is to the best of our knowledge, the first optimization-based separations where the approximation benefits of the stronger architecture provably manifest in practice. Our proof technique introduces new tools and ideas that may be of independent interest in the theoretical study of both the approximation and optimization of neural networks.
Many real-world systems can be described by mathematical formulas that are human-comprehensible, easy to analyze and can be helpful in explaining the system's behaviour. Symbolic regression is a method that generates nonlinear models from data in the form of analytic expressions. Historically, symbolic regression has been predominantly realized using genetic programming, a method that iteratively evolves a population of candidate solutions that are sampled by genetic operators crossover and mutation. This gradient-free evolutionary approach suffers from several deficiencies: it does not scale well with the number of variables and samples in the training data, models tend to grow in size and complexity without an adequate accuracy gain, and it is hard to fine-tune the inner model coefficients using just genetic operators. Recently, neural networks have been applied to learn the whole analytic formula, i.e., its structure as well as the coefficients, by means of gradient-based optimization algorithms. We propose a novel neural network-based symbolic regression method that constructs physically plausible models based on limited training data and prior knowledge about the system. The method employs an adaptive weighting scheme to effectively deal with multiple loss function terms and an epoch-wise learning process to reduce the chance of getting stuck in poor local optima. Furthermore, we propose a parameter-free method for choosing the model with the best interpolation and extrapolation performance out of all models generated through the whole learning process. We experimentally evaluate the approach on the TurtleBot 2 mobile robot, the magnetic manipulation system, the equivalent resistance of two resistors in parallel, and the anti-lock braking system. The results clearly show the potential of the method to find sparse and accurate models that comply with the prior knowledge provided.
A Deep Neural Network (DNN) is a composite function of vector-valued functions, and in order to train a DNN, it is necessary to calculate the gradient of the loss function with respect to all parameters. This calculation can be a non-trivial task because the loss function of a DNN is a composition of several nonlinear functions, each with numerous parameters. The Backpropagation (BP) algorithm leverages the composite structure of the DNN to efficiently compute the gradient. As a result, the number of layers in the network does not significantly impact the complexity of the calculation. The objective of this paper is to express the gradient of the loss function in terms of a matrix multiplication using the Jacobian operator. This can be achieved by considering the total derivative of each layer with respect to its parameters and expressing it as a Jacobian matrix. The gradient can then be represented as the matrix product of these Jacobian matrices. This approach is valid because the chain rule can be applied to a composition of vector-valued functions, and the use of Jacobian matrices allows for the incorporation of multiple inputs and outputs. By providing concise mathematical justifications, the results can be made understandable and useful to a broad audience from various disciplines.
We consider the lossy quantum source coding problem where the task is to compress a given quantum source below its von Neumann entropy. Inspired by the duality connections between the rate-distortion and channel coding problems in the classical setting, we propose a new formulation for the lossy quantum source coding problem. This formulation differs from the existing quantum rate-distortion theory in two aspects. Firstly, we require that the reconstruction of the compressed quantum source fulfill a global error constraint as opposed to the sample-wise local error criterion used in the standard rate-distortion setting. Secondly, instead of a distortion observable, we employ the notion of a backward quantum channel, which we refer to as a "posterior reference map", to measure the reconstruction error. Using these, we characterize the asymptotic performance limit of the lossy quantum source coding problem in terms of single-letter coherent information of the given posterior reference map. We demonstrate a protocol to encode (at the specified rate) and decode, with the reconstruction satisfying the provided global error criterion, and therefore achieving the asymptotic performance limit. The protocol is constructed by decomposing coherent information as a difference of two Holevo information quantities, inspired from prior works in quantum communication problems. To further support the findings, we develop analogous formulations for the quantum-classical and classical variants and express the asymptotic performance limit in terms of single-letter mutual information quantities with respect to appropriately defined channels analogous to posterior reference maps. We also provide various examples for the three formulations, and shed light on their connection to the standard rate-distortion formulation wherever possible.
Sensitivity analysis measures the influence of a Bayesian network's parameters on a quantity of interest defined by the network, such as the probability of a variable taking a specific value. Various sensitivity measures have been defined to quantify such influence, most commonly some function of the quantity of interest's partial derivative with respect to the network's conditional probabilities. However, computing these measures in large networks with thousands of parameters can become computationally very expensive. We propose an algorithm combining automatic differentiation and exact inference to efficiently calculate the sensitivity measures in a single pass. It first marginalizes the whole network once, using e.g. variable elimination, and then backpropagates this operation to obtain the gradient with respect to all input parameters. Our method can be used for one-way and multi-way sensitivity analysis and the derivation of admissible regions. Simulation studies highlight the efficiency of our algorithm by scaling it to massive networks with up to 100'000 parameters and investigate the feasibility of generic multi-way analyses. Our routines are also showcased over two medium-sized Bayesian networks: the first modeling the country-risks of a humanitarian crisis, the second studying the relationship between the use of technology and the psychological effects of forced social isolation during the COVID-19 pandemic. An implementation of the methods using the popular machine learning library PyTorch is freely available.
Empirical neural tangent kernels (eNTKs) can provide a good understanding of a given network's representation: they are often far less expensive to compute and applicable more broadly than infinite width NTKs. For networks with O output units (e.g. an O-class classifier), however, the eNTK on N inputs is of size $NO \times NO$, taking $O((NO)^2)$ memory and up to $O((NO)^3)$ computation. Most existing applications have therefore used one of a handful of approximations yielding $N \times N$ kernel matrices, saving orders of magnitude of computation, but with limited to no justification. We prove that one such approximation, which we call "sum of logits", converges to the true eNTK at initialization for any network with a wide final "readout" layer. Our experiments demonstrate the quality of this approximation for various uses across a range of settings.
The adaptive processing of structured data is a long-standing research topic in machine learning that investigates how to automatically learn a mapping from a structured input to outputs of various nature. Recently, there has been an increasing interest in the adaptive processing of graphs, which led to the development of different neural network-based methodologies. In this thesis, we take a different route and develop a Bayesian Deep Learning framework for graph learning. The dissertation begins with a review of the principles over which most of the methods in the field are built, followed by a study on graph classification reproducibility issues. We then proceed to bridge the basic ideas of deep learning for graphs with the Bayesian world, by building our deep architectures in an incremental fashion. This framework allows us to consider graphs with discrete and continuous edge features, producing unsupervised embeddings rich enough to reach the state of the art on several classification tasks. Our approach is also amenable to a Bayesian nonparametric extension that automatizes the choice of almost all model's hyper-parameters. Two real-world applications demonstrate the efficacy of deep learning for graphs. The first concerns the prediction of information-theoretic quantities for molecular simulations with supervised neural models. After that, we exploit our Bayesian models to solve a malware-classification task while being robust to intra-procedural code obfuscation techniques. We conclude the dissertation with an attempt to blend the best of the neural and Bayesian worlds together. The resulting hybrid model is able to predict multimodal distributions conditioned on input graphs, with the consequent ability to model stochasticity and uncertainty better than most works. Overall, we aim to provide a Bayesian perspective into the articulated research field of deep learning for graphs.
Graph Neural Networks (GNNs), which generalize deep neural networks to graph-structured data, have drawn considerable attention and achieved state-of-the-art performance in numerous graph related tasks. However, existing GNN models mainly focus on designing graph convolution operations. The graph pooling (or downsampling) operations, that play an important role in learning hierarchical representations, are usually overlooked. In this paper, we propose a novel graph pooling operator, called Hierarchical Graph Pooling with Structure Learning (HGP-SL), which can be integrated into various graph neural network architectures. HGP-SL incorporates graph pooling and structure learning into a unified module to generate hierarchical representations of graphs. More specifically, the graph pooling operation adaptively selects a subset of nodes to form an induced subgraph for the subsequent layers. To preserve the integrity of graph's topological information, we further introduce a structure learning mechanism to learn a refined graph structure for the pooled graph at each layer. By combining HGP-SL operator with graph neural networks, we perform graph level representation learning with focus on graph classification task. Experimental results on six widely used benchmarks demonstrate the effectiveness of our proposed model.
Graph convolutional networks (GCNs) have been successfully applied in node classification tasks of network mining. However, most of these models based on neighborhood aggregation are usually shallow and lack the "graph pooling" mechanism, which prevents the model from obtaining adequate global information. In order to increase the receptive field, we propose a novel deep Hierarchical Graph Convolutional Network (H-GCN) for semi-supervised node classification. H-GCN first repeatedly aggregates structurally similar nodes to hyper-nodes and then refines the coarsened graph to the original to restore the representation for each node. Instead of merely aggregating one- or two-hop neighborhood information, the proposed coarsening procedure enlarges the receptive field for each node, hence more global information can be learned. Comprehensive experiments conducted on public datasets demonstrate the effectiveness of the proposed method over the state-of-art methods. Notably, our model gains substantial improvements when only a few labeled samples are provided.
Recently, deep learning has achieved very promising results in visual object tracking. Deep neural networks in existing tracking methods require a lot of training data to learn a large number of parameters. However, training data is not sufficient for visual object tracking as annotations of a target object are only available in the first frame of a test sequence. In this paper, we propose to learn hierarchical features for visual object tracking by using tree structure based Recursive Neural Networks (RNN), which have fewer parameters than other deep neural networks, e.g. Convolutional Neural Networks (CNN). First, we learn RNN parameters to discriminate between the target object and background in the first frame of a test sequence. Tree structure over local patches of an exemplar region is randomly generated by using a bottom-up greedy search strategy. Given the learned RNN parameters, we create two dictionaries regarding target regions and corresponding local patches based on the learned hierarchical features from both top and leaf nodes of multiple random trees. In each of the subsequent frames, we conduct sparse dictionary coding on all candidates to select the best candidate as the new target location. In addition, we online update two dictionaries to handle appearance changes of target objects. Experimental results demonstrate that our feature learning algorithm can significantly improve tracking performance on benchmark datasets.
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