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

Systems, methods, and apparatus, including computer programs encoded on a computer storage medium. In one aspect, a system includes initial neural network layers configured to: receive an input image, and process the input image to generate a plurality of first feature maps that characterize the input image; a location generating convolutional neural network layer configured to perform a convolution on the representation of the first plurality of feature maps to generate data defining a respective location of each of a predetermined number of bounding boxes in the input image, wherein each bounding box identifies a respective first region of the input image; and a confidence score generating convolutional neural network layer configured to perform a convolution on the representation of the first plurality of feature maps to generate a confidence score for each of the predetermined number of bounding boxes in the input image.

Object detection using neural network systems

Over the recent years deep learning has found successful applications in mathematical reasoning. Today, we can predict fine-grained proof steps, relevant premises, and even useful conjectures using neural networks. This extended abstract summarizes recent developments of machine learning in mathematical reasoning and the vision of the N2Formal group at Google Research to create an automatic mathematician. The second part discusses the key challenges on the road ahead.

/pdf/towards-the-automatic-mathematician-303fljufcc.pdf

Towards the Automatic Mathematician

While designing inductive bias in neural architectures has been widely studied, we hypothesize that transformer networks are flexible enough to learn inductive bias from suitable generic tasks. Here, we replace architecture engineering by encoding inductive bias in the form of datasets. Inspired by Peirce's view that deduction, induction, and abduction form an irreducible set of reasoning primitives, we design three synthetic tasks that are intended to require the model to have these three abilities. We specifically design these synthetic tasks in a way that they are devoid of mathematical knowledge to ensure that only the fundamental reasoning biases can be learned from these tasks. This defines a new pre-training methodology called "LIME" (Learning Inductive bias for Mathematical rEasoning). Models trained with LIME significantly outperform vanilla transformers on three very different large mathematical reasoning benchmarks. Unlike dominating the computation cost as traditional pre-training approaches, LIME requires only a small fraction of the computation cost of the typical downstream task.

LIME: Learning Inductive Bias for Primitives of Mathematical Reasoning.

For some k>=2 let d=(d"1,d"2,...,d"k)@?R">"0^k. We denote the concatenation of k vectors a"1,a"2,...,a"[email protected][email protected]?"n">="0R^n by a"1a"2...a"k and use @e to denote the empty vector. We consider a recursively defined function D"d:@?"n">="0R^n->[email protected]?{-~} with D"d(@e)=-~, D"d((a))=a for [email protected]?R and D"d(a)=min{max{D"d(a"i)+d"i|[email protected][email protected]?k}|a=a"1a"2...a"k with a"[email protected][email protected]?m=0n-1R^m for [email protected][email protected]?k} for [email protected]?R^n with n>=2. The function D"d equals the cost of an optimal alphabetic code tree with unequal letter costs and the above recursion naturally generalizes a recursion studied by Kapoor and Reingold [S. Kapoor, E.M. Reingold, Optimum lopsided binary trees, J. Assoc. Comput. Mach. 36 (1989) 573-590]. If z(n) denotes the vector consisting of n>=0 zeros, then let f(@a)=max{[email protected]?N"0|D"d(z(i))@[email protected]} for @[email protected]?R. Let d=min{d"1,d"2,...,d"k} and D=max{d"1,d"2,...,d"k}. Our main result is that D"d(z(@?i=1nf(a"i)))@?D"d(a)@?D"d(z(@?i=1nf(a"i)))+6D-2d for a=(a"1,a"2,...,a"n)@?R">="0^n. This result is useful for the analysis of the asymptotic growth of D"d.

On the cost of optimal alphabetic code trees with unequal letter costs

We examine whether language modeling applied to mathematical formulas enables logical reasoning. We suggest several logical reasoning tasks that can be used to evaluate language models trained on formal mathematical statements, such as type inference, suggesting missing assumptions and completing equalities. To train language models for formal mathematics, we propose a novel skip-tree task, which outperforms standard language modeling tasks on our reasoning benchmarks. We also analyze the models' ability to formulate new conjectures by measuring how often the predictions that do not fit the ground truth or any training data turn out to be true and useful statements.

Christian Szegedy

Papers

Object detection using neural network systems

Towards the Automatic Mathematician

LIME: Learning Inductive Bias for Primitives of Mathematical Reasoning.

On the cost of optimal alphabetic code trees with unequal letter costs

Language Modeling for Formal Mathematics