Deeper neural networks are more difficult to train. We present a residual learning framework to ease the training of networks that are substantially deeper than those used previously. We explicitly reformulate the layers as learning residual functions with reference to the layer inputs, instead of learning unreferenced functions. We provide comprehensive empirical evidence showing that these residual networks are easier to optimize, and can gain accuracy from considerably increased depth. On the ImageNet dataset we evaluate residual nets with a depth of up to 152 layers—8× deeper than VGG nets [40] but still having lower complexity. An ensemble of these residual nets achieves 3.57% error on the ImageNet test set. This result won the 1st place on the ILSVRC 2015 classification task. We also present analysis on CIFAR-10 with 100 and 1000 layers. The depth of representations is of central importance for many visual recognition tasks. Solely due to our extremely deep representations, we obtain a 28% relative improvement on the COCO object detection dataset. Deep residual nets are foundations of our submissions to ILSVRC & COCO 2015 competitions1, where we also won the 1st places on the tasks of ImageNet detection, ImageNet localization, COCO detection, and COCO segmentation.

/pdf/deep-residual-learning-for-image-recognition-b75jg61qrq.pdf

Deep Residual Learning for Image Recognition

In this work we investigate the effect of the convolutional network depth on its accuracy in the large-scale image recognition setting. Our main contribution is a thorough evaluation of networks of increasing depth using an architecture with very small (3x3) convolution filters, which shows that a significant improvement on the prior-art configurations can be achieved by pushing the depth to 16-19 weight layers. These findings were the basis of our ImageNet Challenge 2014 submission, where our team secured the first and the second places in the localisation and classification tracks respectively. We also show that our representations generalise well to other datasets, where they achieve state-of-the-art results. We have made our two best-performing ConvNet models publicly available to facilitate further research on the use of deep visual representations in computer vision.

Very Deep Convolutional Networks for Large-Scale Image Recognition

Deeper neural networks are more difficult to train. We present a residual learning framework to ease the training of networks that are substantially deeper than those used previously. We explicitly reformulate the layers as learning residual functions with reference to the layer inputs, instead of learning unreferenced functions. We provide comprehensive empirical evidence showing that these residual networks are easier to optimize, and can gain accuracy from considerably increased depth. On the ImageNet dataset we evaluate residual nets with a depth of up to 152 layers---8x deeper than VGG nets but still having lower complexity. An ensemble of these residual nets achieves 3.57% error on the ImageNet test set. This result won the 1st place on the ILSVRC 2015 classification task. We also present analysis on CIFAR-10 with 100 and 1000 layers. 
The depth of representations is of central importance for many visual recognition tasks. Solely due to our extremely deep representations, we obtain a 28% relative improvement on the COCO object detection dataset. Deep residual nets are foundations of our submissions to ILSVRC & COCO 2015 competitions, where we also won the 1st places on the tasks of ImageNet detection, ImageNet localization, COCO detection, and COCO segmentation.

We propose a deep convolutional neural network architecture codenamed Inception that achieves the new state of the art for classification and detection in the ImageNet Large-Scale Visual Recognition Challenge 2014 (ILSVRC14). The main hallmark of this architecture is the improved utilization of the computing resources inside the network. By a carefully crafted design, we increased the depth and width of the network while keeping the computational budget constant. To optimize quality, the architectural decisions were based on the Hebbian principle and the intuition of multi-scale processing. One particular incarnation used in our submission for ILSVRC14 is called GoogLeNet, a 22 layers deep network, the quality of which is assessed in the context of classification and detection.

/pdf/going-deeper-with-convolutions-1yobw2o2ds.pdf

Going deeper with convolutions

Matroids admitting an odd ear-decomposition can be viewed as natural generalizations of factor-critical graphs. We prove that a matroid representable over a field of characteristic 2 admits an odd ear-decomposition if and only if it can be represented by some space on which the induced scalar product is a non-degenerate symplectic form. We also show that, for a matroid representable over a field of characteristic 2, the independent sets whose contraction admits an odd ear-decomposition form the family of feasible sets of a representable Δ-matroid.

Symplectic Spaces And Ear-Decomposition Of Matroids

The method of cyclic relaxation for the minimization of a function depending on several variables cyclically updates the value of each of the variables to its optimum subject to the condition that the remaining variables are fixed. 

We present a simple and transparent proof for the fact that cyclic relaxation converges linearly to an optimum solution when applied to the minimization of functions of the form $f(x_{1},\ldots,x_{n})=\sum_{i=1}^{n}\sum_{j=1}^{n}a_{i,j}\frac{x_{i}}{x_{j}}+\sum_{i=1}^{n}(b_{i}x_{i}+\frac{c_{i}}{x_{i}})$ for a i,j ,b i ,c i ???0 with max?{min?{b 1,b 2,?,b n },min?{c 1,c 2,?,c n }}>0 over the n-dimensional interval [l 1,u 1]×[l 2,u 2]×???×[l n ,u n ] with 0<l i <u i for 1?i?n. Our result generalizes several convergence results that have been observed for algorithms applied to gate- and wire-sizing problems that arise in chip design.

A class of problems for which cyclic relaxation converges linearly

Traditionally, research in global placement has focused on relatively few simple metrics, such as pure wirelength or routability estimates. However, in the real world today, designs are driven by not-so-simple issues such as timing and crosstalk. The future holds even more difficulties as physical models for devices and interconnects become increasingly complex and unpredictable. Adoption of an iterative methodology, where one incrementally fixes design errors, is a basic approach to tackling these problems. However, developers of placement algorithms have long neglected the need for an tool which can be easily adopted into an incremental design flow.We propose a novel placement approach called grid morphing, which is specifically tailored for an incremental approach to placement. In particular, our technique focuses on the stability of the placement, which is critical for minimization of perturbation of the final placement under changes applied to the input netlist. We comparethe stability of our approach to existing placement tools, and show through experiments that our approach still delivers good results under traditional placement metrics.

A morphing approach to address placement stability

Aspects of the present disclosure relate generally to model fitting. A target model having a large number of inputs is fit using a performance model having relatively few inputs. The performance model is learned during the fitting process. Optimal optimization parameters including a sample size, a damping factor, and an iteration count are selected for an optimization round. A random subset of data is sampled based on the selected sample size. The optimization round is conducted using the iteration count and the sampled data to produce optimized parameters. The performance model is updated based on the performance of the optimization round. The parameters of the target model are then updated based on the damping factor and the parameters computed by the optimization round. The aforementioned steps are performed in a loop in order to obtain optimized parameters and fit of the data to the target model.

Christian Szegedy

Papers

Symplectic Spaces And Ear-Decomposition Of Matroids

A class of problems for which cyclic relaxation converges linearly

A morphing approach to address placement stability

Robust and fast model fitting by adaptive sampling

The delay of circuits whose inputs have specified arrival times