The inference structures and computational complexity of existing deep neural networks, once trained, are fixed and remain the same for all test images. However, in practice, it is highly desirable to establish a progressive structure for deep neural networks which is able to adapt its inference process and complexity for images with different visual recognition complexity. In this work, we develop a multi-stage progressive structure with integrated confidence analysis and decision policy learning for deep neural networks. This new framework consists of a set of network units to be activated in a sequential manner with progressively increased complexity and visual recognition power. Our extensive experimental results on the CIFAR-10 and ImageNet datasets demonstrate that the proposed progressive deep neural network is able to obtain more than 10 fold complexity scalability while achieving the state-of-the-art performance using a single network model satisfying different complexity-accuracy requirements.
Researchers have recognized that the decision scores of existing deep neural networks are poorly calibrated @cite_28 . For example, higher scores often do not correspond to better or closer samples. @cite_23 argues that probability scores generated by should not be considered as confidence score or distance measure directly since they are based on the @math norm of inputs. To address this issue, various methods using scalar, vector, matrix and binning methods to calibrate the confidence scores produced by the function. Gaussian density modeling is proposed in @cite_23 as a post prediction calibration using prior information of the training data. Gal al @cite_35 has implemented a randomized dropout network @cite_34 to estimate the uncertainty level of network prediction. The open set deep network approach in @cite_8 attempts to measure uncertainty contributed from unknown categories. It should be noted that these methods are based on statistical modeling, being optimized on the entire validation set, and therefore not suitable for confidence analysis for each individual input image.
@cite_35 @cite_8 @cite_28 @cite_23 @cite_34
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Multi-layer neural networks have lead to remarkable performance on many kinds of benchmark tasks in text, speech and image processing. Nonlinear parameter estimation in hierarchical models is known to be subject to overfitting and misspecification. One approach to these estimation and related problems (local minima, colinearity, feature discovery etc.) is called Dropout (Hinton, et al 2012, 2016). The Dropout algorithm removes hidden units according to a Bernoulli random variable with probability @math prior to each update, creating random "shocks" to the network that are averaged over updates. In this paper we will show that Dropout is a special case of a more general model published originally in 1990 called the Stochastic Delta Rule, or SDR (Hanson, 1990). SDR redefines each weight in the network as a random variable with mean @math and standard deviation @math . Each weight random variable is sampled on each forward activation, consequently creating an exponential number of potential networks with shared weights. Both parameters are updated according to prediction error, thus resulting in weight noise injections that reflect a local history of prediction error and local model averaging. SDR therefore implements a more sensitive local gradient-dependent simulated annealing per weight converging in the limit to a Bayes optimal network. Tests on standard benchmarks (CIFAR) using a modified version of DenseNet shows the SDR outperforms standard Dropout in test error by approx. @math with DenseNet-BC 250 on CIFAR-100 and approx. @math in smaller networks. We also show that SDR reaches the same accuracy that Dropout attains in 100 epochs in as few as 35 epochs. As deep neural networks (DNNs) are applied to increasingly challenging problems, they will need to be able to represent their own uncertainty. Modeling uncertainty is one of the key features of Bayesian methods. Using Bernoulli dropout with sampling at prediction time has recently been proposed as an efficient and well performing variational inference method for DNNs. However, sampling from other multiplicative noise based variational distributions has not been investigated in depth. We evaluated Bayesian DNNs trained with Bernoulli or Gaussian multiplicative masking of either the units (dropout) or the weights (dropconnect). We tested the calibration of the probabilistic predictions of Bayesian convolutional neural networks (CNNs) on MNIST and CIFAR-10. Sampling at prediction time increased the calibration of the DNNs' probabalistic predictions. Sampling weights, whether Gaussian or Bernoulli, led to more robust representation of uncertainty compared to sampling of units. However, using either Gaussian or Bernoulli dropout led to increased test set classification accuracy. Based on these findings we used both Bernoulli dropout and Gaussian dropconnect concurrently, which we show approximates the use of a spike-and-slab variational distribution without increasing the number of learned parameters. We found that spike-and-slab sampling had higher test set performance than Gaussian dropconnect and more robustly represented its uncertainty compared to Bernoulli dropout. As deep neural networks (DNNs) are applied to increasingly challenging problems, they will need to be able to represent their own uncertainty. Modelling uncertainty is one of the key features of Bayesian methods. Bayesian DNNs that use dropout-based variational distributions and scale to complex tasks have recently been proposed. We evaluate Bayesian DNNs trained with Bernoulli or Gaussian multiplicative masking of either the units (dropout) or the weights (dropconnect). We compare these Bayesian DNNs ability to represent their uncertainty about their outputs through sampling during inference. We tested the calibration of these Bayesian fully connected and convolutional DNNs on two visual inference tasks (MNIST and CIFAR-10). By adding different levels of Gaussian noise to the test images, we assessed how these DNNs represented their uncertainty about regions of input space not covered by the training set. These Bayesian DNNs represented their own uncertainty more accurately than traditional DNNs with a softmax output. We find that sampling of weights, whether Gaussian or Bernoulli, led to more accurate representation of uncertainty compared to sampling of units. However, sampling units using either Gaussian or Bernoulli dropout led to increased convolutional neural network (CNN) classification accuracy. Based on these findings we use both Bernoulli dropout and Gaussian dropconnect concurrently, which approximates the use of a spike-and-slab variational distribution. We find that networks with spike-and-slab sampling combine the advantages of the other methods: they classify with high accuracy and robustly represent the uncertainty of their classifications for all tested architectures. State-of-the-art Deep Neural Networks can be easily fooled into providing incorrect high-confidence predictions for images with small amounts of adversarial noise. Does this expose a flaw with deep neural networks, or do we simply need a better way to estimate confidence? In this paper we consider the problem of accurately estimating predictive confidence. We formulate this problem as that of density modelling, and show how traditional methods such as softmax produce poor estimates. To address this issue, we propose a novel confidence measure based on density modelling approaches. We test these measures on images distorted by blur, JPEG compression, random noise and adversarial noise. Experiments show that our confidence measure consistently shows reduced confidence scores in the presence of such distortions - a property which softmax often lacks.