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Title: Log-DenseNet: How to Sparsify a DenseNet

Paper content:
Skip connections are increasingly utilized by deep neural networks to improve accuracy and cost-efficiency.
In particular, the recent DenseNet is efficient in computation and parameters, and achieves state-of-the-art predictions by directly connecting each feature layer to all previous ones.
However, DenseNet's extreme connectivity pattern may hinder its scalability to high depths, and in applications like fully convolutional networks, full DenseNet connections are prohibitively expensive.
This work first experimentally shows that one key advantage of skip connections is to have short distances among feature layers during backpropagation.
Specifically, using a fixed number of skip connections, the connection patterns with shorter backpropagation distance among layers have more accurate predictions.
Following this insight, we propose a connection template, Log-DenseNet, which, in comparison to DenseNet,  only slightly increases the backpropagation distances among layers from 1 to  ($1 + \log_2 L$), but uses only $L\log_2 L$ total connections instead of $O(L^2)$.
Hence, \logdenses are easier to scale than DenseNets, and no longer require careful GPU memory management.
We demonstrate the effectiveness of our design principle by showing better performance than DenseNets on tabula rasa semantic segmentation, and competitive results on visual recognition.
Deep neural networks have been improving performance for many machine learning tasks, scaling from networks like AlexNet BID17 to increasingly more complex and expensive networks, like VGG BID30 , ResNet BID8 and Inception BID5 .
Continued hardware and software advances will enable us to build deeper neural networks, which have higher representation power than shallower ones.
However, the payoff from increasing the depth of the networks only holds in practice if the networks can be trained effectively.
It has been shown that naïvely scaling up the depth of networks actually decreases the performance BID8 , partially because of vanishing/exploding gradients in very deep networks.
Furthermore, in certain tasks such as semantic segmentation, it is common to take a pre-trained network and fine-tune, because training from scratch is difficult in terms of both computational cost and reaching good solutions.
Overcoming the vanishing gradient problem and being able to train from scratch are two active areas of research.Recent works attempt to overcome these training difficulties in deeper networks by introducing skip, or shortcut, connections BID25 BID7 BID31 BID8 BID19 so the gradient reaches earlier layers and compositions of features at varying depth can be combined for better performance.
In particular, DenseNet is the extreme example of this, concatenating all previous layers to form the input of each layer, i.e., connecting each layer to all previous ones.
However, this incurs an O(L 2 ) run-time complexity for a depth L network, and may hinder the scaling of networks.
Specifically, in fully convolutional networks (FCNs), where the final feature maps have high resolution so that full DenseNet connections are prohibitively expensive, BID14 propose to cut most of connections from the mid-depth.
To combat the scaling issue, propose to halve the total channel size a number of times.
Futhermore, cut 40% of the channels in DenseNets while maintaining the accuracy, suggesting that much of the O(L 2 ) computation is redundant.
Therefore, it is both necessary and natural to consider a more efficient design principle for placing shortcut connections in deep neural networks.1In this work, we address the scaling issue of skip connections by answering the question: if we can only afford the computation of a limited number of skip connections and we believe the network needs to have at least a certain depth, where should the skip connections be placed?
We design experiments to show that with the same number of skip connections at each layer, the networks can have drastically different performance based on where the skip connections are.
In particular, we summarize this result as the following design principle, which we formalize in Sec. 3.2: given a fixed number of shortcut connections to each feature layer, we should choose these shortcut connections to minimize the distance among layers during backpropagation.Following this principle, we design a network template, Log-DenseNet.
In comparison to DenseNets at depth L, Log-DenseNets cost only L log L, instead of O(L 2 ) run-time complexity.
Furthermore, Log-DenseNets only slightly increase the short distances among layers during backpropagation from 1 to 1 + log L. Hence, Log-DenseNets can scale to deeper and wider networks, even without custom GPU memory managements that DenseNets require.
In particular, we show that Log-DenseNets outperform DenseNets on tabula rasa semantic segmentation on CamVid BID2 , while using only half of the parameters, and similar computation.
Log-DenseNets also achieve comparable performance to DenseNet with the same computations on visual recognition data-sets, including ILSVRC2012 BID29 .
In short, our contributions are as follows:• We experimentally support the design principle that with a fixed number of skip connections per layer, we should place them to minimize the distance among layers during backpropagation.•
The proposed Log-DenseNets achieve small 1 + log 2 L between-layer distances using few connections (L log 2 L), and hence, are scalable for deep networks and applications like FCNs.•
The proposed network outperforms DenseNet on CamVid for tabula rasa semantic segmentation, and achieves comparable performance on ILSVRC2012 for recognition.
We show that short backpropagation distances are important for networks that have shortcut connections: if each layer has a fixed number of shortcut inputs, they should be placed to minimize MBD.
Based on this principle, we design Log-DenseNet, which uses O(L log L) total shortcut connections on a depth-L network to achieve 1 + log L MBD.
We show that Log-DenseNets improve the performance and scalability of tabula rasa fully convolutional DenseNets on CamVid.
Log-DenseNets also achieve competitive results in visual recognition data-sets, offering a trade-off between accuracy and network depth.
Our work provides insights for future network designs, especially those that cannot afford full dense shortcut connections and need high depths, like FCNs.
8 smallest interval in the recursion tree such that i, j ∈ [s, t].
Then we can continue the path to x j by following the recursion calls whose input segments include j until j is in a key location set.
The longest path is then the depth of the recursion tree plus one initial jump, i.e., 2 + log log L. Figure 5a shows the average number of input layers for each feature layer in LogLog-DenseNet.
Without augmentations, lglg_conn on average has 3 to 4 connections per layer.
With augmentations using Log-DenseNet, we desire each layer to have four inputs if possible.
On average, this increases the number of inputs by 1 to 1.5 for L ∈ (10, 2000).

assistant: text: We show shortcut connections should be placed in patterns that minimize between-layer distances during backpropagation, and design networks that achieve log L distances using L log(L) connections.