markdown_text stringlengths 1 2.5k | pdf_metadata dict | header_metadata dict | chunk_metadata dict |
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### **8 Conclusions**
Our proposed Hogwild! algorithm takes advantage of sparsity in machine learning problems
to enable near linear speedups on a variety of applications. Empirically, our implementations
outperform our theoretical analysis. For instance, *ρ* is quite large in the RCV1 SVM problem, yet
we still obtai... | {
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"Header 2": "Hogwild!: A Lock-Free Approach to Parallelizing Stochastic Gradient Descent",
"Header 3": "**8 Conclusions**",
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### **References**
[1] Max-flow problem instances in vision. From `http://vision.csd.uwo.ca/data/maxflow/` .
[2] K. Asanovic and *et al* . The landscape of parallel computing research: A view from berkeley. Technical
Report UCB/EECS-2006-183, Electrical Engineering and Computer Sciences, University of California
at... | {
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"Header 1": null,
"Header 2": "Hogwild!: A Lock-Free Approach to Parallelizing Stochastic Gradient Descent",
"Header 3": "**References**",
"Header 4": null
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"chunk_type": "references"
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*Learning Research*, 10:777–801, 2009.
[17] J. Langford, A. J. Smola, and M. Zinkevich. Slow learners are fast. In *Advances in Neural Information*
*Processing Systems*, 2009.
[18] J. Lee,, B. Recht, N. Srebro, R. R. Salakhutdinov, and J. A. Tropp. Practical large-scale optimization
for max-norm regularization. In ... | {
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"Header 3": "**References**",
"Header 4": null
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*Information Processing Systems*, 2004.
[29] P. Tseng. An incremental gradient(-projection) method with momentum term and adaptive stepsize
rule. *SIAM Joural on Optimization*, 8(2):506–531, 1998.
[30] J. Tsitsiklis, D. P. Bertsekas, and M. Athans. Distributed asynchronous deterministic and stochastic
gradient opti... | {
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"Header 4": null
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### **A Analysis of Hogwild!**
It follows by rearrangement of (4.3) that
( *x −* *x* *[′]* ) *[T]* *∇f* ( *x* ) *≥* *f* ( *x* ) *−* *f* ( *x* *[′]* ) + [1] (A.1)
2 *[c][∥][x][ −]* *[x]* *[′]* *[∥]* [2] *[,]* [ for all] *[ x][ ∈]* *[X]* [.]
In particular, by setting *x* *[′]* = *x* *⋆* (the minimizer) we have
... | {
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"Header 3": "**A Analysis of Hogwild!**",
"Header 4": null
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Let *a* *j* = [1] 2 [E][[] *[∥][x]* *[j]* *[ −]* *[x]* *[⋆]* *[∥]* 2 [2] []. By taking expectations of both sides and using the bound (4.4), we obtain]
*a* *j* +1 *≤* *a* *j* *−* *γ* E[( *x* *j* *−* *x* *k* ( *j* ) ) *[T]* *G* *e* *j* ( *x* *j* )] *−* *γ* E[( *x* *j* *−* *x* *k* ( *j* ) ) *[T]* ( *G* *e* *j* ( *x* *k... | {
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= E �( *x* *j* *−* *x* *k* ( *j* ) ) *[T]* E[ *G* *e* *j* ( *x* *j* ) *| e* [ *j−* 1] ]�
= E[( *x* *j* *−* *x* *k* ( *j* ) ) *[T]* *∇f* ( *x* *j* )]
*≥* E[ *f* ( *x* *j* ) *−* *f* ( *x* *k* ( *j* ) )] + *[c]* (A.5)
2 [E][[] *[∥][x]* *[j]* *[ −]* *[x]* *[k]* [(] *[j]* [)] *[∥]* [2] []]
17
-----
where the fin... | {
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*j−* 1
*≥−* E � 2Ω *M* [2] *γ*
*i* = *k* ( *j* )
*e* *i* *∩e* *j* = *̸* *∅*
*≥−* 2Ω *M* [2] *γρτ* (A.8)
18
-----
where *ρ* is defined by (2.6). Here, the third line follows from our definition of the gradient update.
The fourth line is tautological: only the edges where *e* *i* and *e* *j* intersect... | {
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*≤τγ* Ω *M* ∆ [1] *[/]* [2] E[ *∥x* *k* ( *j* ) *−* *x* *⋆* *∥* 2 ]
*≤τγ* Ω *M* ∆ [1] *[/]* [2] (E[ *∥x* *j* *−* *x* *⋆* *∥* 2 ] + *τγ* Ω *M* )
*≤τγ* Ω *M* ∆ [1] *[/]* [2] ( *√* 2 *a* [1] *j* *[/]* [2] + *τγ* Ω *M* ) *,*
where ∆is defined in (2.6). The first inequality is Cauchy-Schwartz. The next inequality is J... | {
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"Header 3": "**A Analysis of Hogwild!**",
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*a* *j* +1 *≤* (1 *−* *cγ* (1 *−* *δ* ( *τ, ρ,* ∆ *,* Ω)))( *a* *j* *−* *a* *∞* ) + *a* *∞* (A.13)
with *a* *∞* *≤* *C* ( *τ, ρ,* ∆ *,* Ω) *[M]* 2 [2] *c* *[γ]* [.] In the case that *τ* = 0 (the serial case), *C* ( *τ, ρ,* ∆ *,* Ω) = Ωand
*δ* ( *τ, ρ,* ∆ *,* Ω) = 0. Note that if *τ* is non-zero, but *ρ* and ∆are *o... | {
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1 *−* *δ* ( *τ, ρ,* ∆ *,* Ω) [= Ω] [2] *[τ]* [ 2] [∆] *[·]* 1 *−* 1
1+ 1+ *Q*
~~�~~ Ω [2] *τ* [2] ∆
2
*Q*
1 + 1 +
= Ω [2] *τ* [2] ∆ *·* � ~~�~~ Ω [2] *τ* [2] ∆ �
*Q*
1 +
~~�~~ Ω [2] *τ* [2] ∆
*≤* 8Ω [2] *τ* [2] ∆+ 2 *Q*
= 8Ω [2] *τ* [2] ∆+ 2Ω+ 4 *τρ* + 8Ω *ρτ* + 4 *τ* [2] Ω [2] ∆ [1] *[/]* [2]
*≤* 2... | {
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## C Y CADA: C YCLE -C ONSISTENT A DVERSARIAL D OMAIN A DAPTATION
**Judy Hoffman, Eric Tzeng, Taesung Park, Jun-Yan Zhu**
BAIR, UC Berkeley
{jhoffman,etzeng,taesung_park,junyanz}@eecs.berkeley
**Alexei A. Efros, Trevor Darrell**
BAIR, UC Berkeley
{efros,trevor}@eecs.berkeley
**Phillip Isola**
OpenAI *[∗]*
isola... | {
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#### A BSTRACT
Domain adaptation is critical for success in new, unseen environments. Adversarial
adaptation models applied in feature spaces discover domain invariant representations, but are difficult to visualize and sometimes fail to capture pixel-level and
low-level domain shifts. Recent work has shown that gene... | {
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#### 1 I NTRODUCTION
Deep neural networks excel at learning from large amounts of data, but can be poor at generalizing
learned knowledge to new datasets or environments. Even a slight departure from a network’s training
domain can cause it to make spurious predictions and significantly hurt its performance (Tzeng et... | {
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|Pixel accuracy on target Source-only: 54.0% Adapted (ours): 83.6% Source image (GTA5) Adapted source image (Ours) Target image (CityScapes)|Col2|Col3|Col4|Col5|Col6|Col7|Col8|Col9|Pixel accuracy on target Source-only: 54.0% Adapted (ours): 83.6%|
|---|---|---|---|---|---|---|---|---|---|
||||||||||Accuracy on target S... | {
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techniques (Zhu et al., 2017), as illustrated in Table 1. It is applicable across a range of deep
architectures and/or representation levels, and has several advantages over existing unsupervised
domain adaptation methods. We use a reconstruction (cycle-consistency) loss to encourage the
cross-domain transformation to ... | {
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#### 2 R ELATED W ORK
The problem of visual domain adaptation was introduced along with a pairwise metric transform
solution by Saenko et al. (2010) and was further popularized by the broad study of visual dataset
bias (Torralba & Efros, 2011). Early deep adaptive works focused on feature space alignment through
mini... | {
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domain to encourage alignment in the unsupervised adaptation setting.
In contrast, another approach is to directly convert the target image into a source style image (or
visa versa), largely based on Generative Adversarial Networks (GANs) (Goodfellow et al., 2014).
Researchers have successfully applied GANs to variou... | {
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Adversarial Networks (CycleGAN) (Zhu et al., 2017) produced compelling image translation results
such as generating photorealistic images from impressionism paintings or transforming horses into
zebras at high resolution using the cycle-consistency loss. This loss was simultaneously proposed
by Yi et al. (2017) and Kim... | {
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#### 3 C YCLE -C ONSISTENT A DVERSARIAL D OMAIN A DAPTION
We consider the problem of unsupervised adaptation, where we are provided source data *X* *S*, source
labels *Y* *S*, and target data *X* *T*, but no target labels. The goal is to learn a model *f* that can correctly
predict the label for the target data *X* *... | {
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ensures that *G* *S→T* ( *x* *s* ) for some *x* *s* will resemble data drawn from *X* *T*, there is no way to guarantee
that *G* *S→T* ( *x* *s* ) preserves the structure or content of the original sample *x* *s* .
In order to encourage the source content to be preserved during the conversion process, we impose
a cyc... | {
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Taken together, these loss functions form our complete objective:
*L* CyCADA ( *f* *T* *, X* *S* *, X* *T* *, Y* *S* *, G* *S→T* *, G* *T →S* *, D* *S* *, D* *T* ) (5)
= *L* task ( *f* *T* *, G* *S→T* ( *X* *S* ) *, Y* *S* )
+ *L* GAN ( *G* *S→T* *, D* *T* *, X* *T* *, X* *S* ) + *L* GAN ( *G* *T →S* *, D* *S* *, X* ... | {
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#### 4 E XPERIMENTS
We evaluate CyCADA on several unsupervised adaptation scenarios. We first focus on adaptation for
digit classification using the MNIST (LeCun et al., 1998), USPS and Street View House Numbers
(SVHN) (Netzer et al., 2011) datasets. After which we present results for the task of semantic
image segme... | {
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GAN and cycle constraints are satisfied (translated image matches MNIST style and reconstructed
image matches original), the image translated to the target domain lacks the proper semantics.
level adaptation outperforms the pixel level adaptation, and both may be combined to produce an
overall model which outperforms... | {
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Cycle-consistent adversarial adaptation is general and can be applied at any layer of a network. Since
optimizing the full CyCADA objective in Equation 5 end-to-end is memory-intensive in practice,
we train our model in stages. First, we perform image-space adaptation and map our source data
into the target domain. Nex... | {
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performance on average across all categories. *[∗]* FCNs in the wild is by Hoffman et al. (2016).
for this at the time of submission, but leave it to future work to deploy model parallelism or experiment
with larger GPU memory.
For our first evaluation, we consider the SYNTHIA dataset (Ros et al., 2016a), which con... | {
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This process is diagrammed in Figure 2.
For cycle-consistent image translation, we followed the network architecture and hyperparameters of
CycleGAN(Zhu et al., 2017). All images were resized to have width of 1024 pixels while keeping
the aspect ratio, and the training was performed with randomly cropped patches of s... | {
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|---|---|---|---|---|
|||34.8|73.1|82.8|
|||35.4|73.8|83.6|
|Oracle (Train on target)|96.4 74.5 87.1 35.3 37.8 36.4 46.9 60.1 89.0 54.3 89.8 65.6 35.9 89.4 38.6 64.1 38.6 40.5 65.1|60.3|87.6|93.1|
|---|---|---|---|---|
Table 4: Adaptation between GTA5 and Cityscapes, showing IoU for each class and mean IoU,
freq-we... | {
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under the fwIoU and pixel accuracy metrics, CyCADA approaches oracle performance, falling short
by only a few points, despite being entirely unsupervised. This indicates that CyCADA is extremely
effective at correcting the most common classes in the dataset. This conclusion is supported by
inspection of the individual ... | {
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to compensate. We also observe texture changes, which are perhaps most apparent in the road:
in-game, the roads appear rough with many blemishes, but Cityscapes roads tend to be fairly uniform
in appearance, so in converting from GTA5 to Cityscapes, our model removes most of the texture.
Somewhat amusingly, our model h... | {
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#### 5 C ONCLUSION
We presented a cycle-consistent adversarial domain adaptation method that unifies cycle-consistent
adversarial models with adversarial adaptation methods. CyCADA is able to adapt even in the absence
of target labels and is broadly applicable at both the pixel-level and in feature space. An image-sp... | {
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#### R EFERENCES
Martin Arjovsky, Soumith Chintala, and Léon Bottou. Wasserstein gan. *CoRR*, abs/1701.07875,
[2017. URL http://arxiv.org/abs/1701.07875.](http://arxiv.org/abs/1701.07875)
Konstantinos Bousmalis, George Trigeorgis, Nathan Silberman, Dilip Krishnan, and Dumitru Erhan.
Domain separation networks. In *... | {
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"title": "CyCADA: Cycle-Consistent Adversarial Domain Adaptation",
"categories": [
"cs.CV"
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} | {
"Header 1": null,
"Header 2": "C Y CADA: C YCLE -C ONSISTENT A DVERSARIAL D OMAIN A DAPTATION",
"Header 3": null,
"Header 4": "R EFERENCES"
} | {
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Leon A Gatys, Alexander S Ecker, and Matthias Bethge. Image style transfer using convolutional
neural networks. In *Computer Vision and Pattern Recognition (CVPR)*, pp. 2414–2423, 2016.
Muhammad Ghifary, W Bastiaan Kleijn, Mengjie Zhang, David Balduzzi, and Wen Li. Deep
reconstruction-classification networks for unsu... | {
"id": "1711.03213",
"title": "CyCADA: Cycle-Consistent Adversarial Domain Adaptation",
"categories": [
"cs.CV"
]
} | {
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"Header 2": "C Y CADA: C YCLE -C ONSISTENT A DVERSARIAL D OMAIN A DAPTATION",
"Header 3": null,
"Header 4": "R EFERENCES"
} | {
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} |
URL [http://scalable.mpi-inf.mpg.de/files/2013/10/levinkov13iccv.](http://scalable.mpi-inf.mpg.de/files/2013/10/levinkov13iccv.pdf http://www.d2.mpi-inf.mpg.de/sequential-bayesian-update)
[pdfhttp://www.d2.mpi-inf.mpg.de/sequential-bayesian-update.](http://scalable.mpi-inf.mpg.de/files/2013/10/levinkov13iccv.pdf http:/... | {
"id": "1711.03213",
"title": "CyCADA: Cycle-Consistent Adversarial Domain Adaptation",
"categories": [
"cs.CV"
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} | {
"Header 1": null,
"Header 2": "C Y CADA: C YCLE -C ONSISTENT A DVERSARIAL D OMAIN A DAPTATION",
"Header 3": null,
"Header 4": "R EFERENCES"
} | {
"chunk_type": "references"
} |
*of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR)*, 2016a.
Germán Ros, Simon Stent, Pablo F. Alcantarilla, and Tomoki Watanabe. Training constrained
deconvolutional networks for road scene semantic segmentation. *CoRR*, abs/1604.01545, 2016b.
[URL http://arxiv.org/abs/1604.01545.](http://arxiv... | {
"id": "1711.03213",
"title": "CyCADA: Cycle-Consistent Adversarial Domain Adaptation",
"categories": [
"cs.CV"
]
} | {
"Header 1": null,
"Header 2": "C Y CADA: C YCLE -C ONSISTENT A DVERSARIAL D OMAIN A DAPTATION",
"Header 3": null,
"Header 4": "R EFERENCES"
} | {
"chunk_type": "references"
} |
[abs/1412.3474.](http://arxiv.org/abs/1412.3474)
Eric Tzeng, Judy Hoffman, Trevor Darrell, and Kate Saenko. Simultaneous deep transfer across
domains and tasks. In *International Conference in Computer Vision (ICCV)*, 2015.
Eric Tzeng, Judy Hoffman, Kate Saenko, and Trevor Darrell. Adversarial discriminative domain... | {
"id": "1711.03213",
"title": "CyCADA: Cycle-Consistent Adversarial Domain Adaptation",
"categories": [
"cs.CV"
]
} | {
"Header 1": null,
"Header 2": "C Y CADA: C YCLE -C ONSISTENT A DVERSARIAL D OMAIN A DAPTATION",
"Header 3": null,
"Header 4": "R EFERENCES"
} | {
"chunk_type": "references"
} |
## Binarized Neural Networks
#### Itay Hubara Daniel Soudry Dept. of Computer Science Dept. of Statistics Technion – Israel Institute of Technology Columbia University Ran El-Yaniv Dept. of Computer Science Technion – Israel Institute of Technology
**Abstract**
In this work we introduce a binarized deep neural ne... | {
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"Header 4": "Itay Hubara Daniel Soudry Dept. of Computer Science Dept. of Statistics Technion – Israel Institute of Technology Columbia University Ran El-Yaniv Dept. of Computer Science Technion – Israel Institute of Technology"
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### **1 Introduction**
The success of deep neural networks (DNNs) and, in particular, convolutional neural networks (CNNs) for large scale object recognition (Krizhevsky et al., 2012) has
motivated on going exploration of alternative architectures, optimization and regularization techniques, that enable better accura... | {
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"Header 3": "**1 Introduction**",
"Header 4": null
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*population count* ( *i.e.,* counting the number of ones in the binary number) operations.
The proposed method is particularly beneficial for implementing large convolutional
networks whose neuron to weight ratio is very large.
We argue that the proposed BBP algorithm can be implemented in hardware and argue that it is... | {
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"Header 1": null,
"Header 2": "Binarized Neural Networks",
"Header 3": "**1 Introduction**",
"Header 4": null
} | {
"chunk_type": "body"
} |
### **2 Related Work**
Until recently, the use of extremely low-precision networks (binary in the extreme
case)was believed to be highly destructive to the network performance (Courbariaux
et al., 2015b). Soudry et al. (2014) proved the contrary by using a variational Bayesian
approach, that one can infer networks wi... | {
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"Header 2": "Binarized Neural Networks",
"Header 3": "**2 Related Work**",
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back propagation process, and not in forward propagation. Other research (Judd et al.,
2015; Gong et al., 2014) aimed to compress a fully trained high precision network by
using a quantization or matrix factorization method. These methods required training
the network with full precision weights and neurons thus requir... | {
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"Header 2": "Binarized Neural Networks",
"Header 3": "**2 Related Work**",
"Header 4": null
} | {
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#### **2.1 Binary Connect**
Our work expands the BinaryConnect approach of Courbariaux et al. (2015a). We now
summarize their ideas, and introduce our extension in the next section. BinaryConnect
(Courbariaux et al., 2015a), as well as DropConnect (Wan et al., 2013) which share the
same idea. During the training phas... | {
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"Header 2": "Binarized Neural Networks",
"Header 3": "**2 Related Work**",
"Header 4": "**2.1 Binary Connect**"
} | {
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} |
### **3 Binarized Back Propagation**
In this section the BBP algorithm ispresenteds It shows the procedures that were used
including: how to binarize the neurons (deterministic vs. stochastic implementation);
how to reduce the impact of the weights and hidden units binarization without batch
normalization; and finall... | {
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"Header 2": "Binarized Neural Networks",
"Header 3": "**3 Binarized Back Propagation**",
"Header 4": null
} | {
"chunk_type": "body"
} |
#### **3.1 Stochastic and Deterministic Binarization**
The binarization operation used in the present work transforms real-valued weights into
two possible values. At training time a stochastic binarization is applied to facilitate a
finer, more informative binarization noise in comparison to the standard sign functi... | {
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"Header 3": "**3 Binarized Back Propagation**",
"Header 4": "**3.1 Stochastic and Deterministic Binarization**"
} | {
"chunk_type": "body"
} |
#### **3.2 Forward and Backward Propagation**
In the process of forward propagation we clipped the input via HT( *x* ) defined in Eq.
(4) and then binarieze it using Eq. (3) (or Eq. (5) for inference). However, in order
to implement the backward propagation phase, we needed to first differentiate through
these binary... | {
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"Header 2": "Binarized Neural Networks",
"Header 3": "**3 Binarized Back Propagation**",
"Header 4": "**3.2 Forward and Backward Propagation**"
} | {
"chunk_type": "body"
} |
#### **3.3 Batch Normalization and Clipping**
As shown by Ioffe & Szegedy (2015), due to the constant change of the distribution
of each layer’s input, training neural networks can be a very noisy procedure which
strongly depends on the weights’ initialization and the learning rate, and requires long
convergence time... | {
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"Header 2": "Binarized Neural Networks",
"Header 3": "**3 Binarized Back Propagation**",
"Header 4": "**3.3 Batch Normalization and Clipping**"
} | {
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of weights. In our experiments we also noticed that BN improved accuracy and accelerated the convergence speed. To proxy BN we used shift based batch normalization
technique which approximates BN almost without multiplications.
Standard BN perform the following normalization:
*C* ( *x* ) = *x −⟨x⟩*
1
*σ* *[−]* [1] ... | {
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"Header 2": "Binarized Neural Networks",
"Header 3": "**3 Binarized Back Propagation**",
"Header 4": "**3.3 Batch Normalization and Clipping**"
} | {
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} |
#### **3.4 Additional Implementation Details**
Throughout our work we restricted ourselves to use only adders, bitwise and shift operations. The comparison operation is also cheap, since adding and comparing two
variables requires the same amount of energy. One of most common ways to compare
two values is by subtract... | {
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"Header 2": "Binarized Neural Networks",
"Header 3": "**3 Binarized Back Propagation**",
"Header 4": "**3.4 Additional Implementation Details**"
} | {
"chunk_type": "body"
} |
### **4 Expected Efficiency Gains**
Recently new technologies enable us to increase computing performance greatly. Improving computing performance has always been a challenge, a number of factors have
made this process difficult in this last decade, and made power the main constraint
on performance(Horowitz, 2014). T... | {
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"Header 2": "Binarized Neural Networks",
"Header 3": "**4 Expected Efficiency Gains**",
"Header 4": null
} | {
"chunk_type": "body"
} |
#### **4.1 Energy Efficiency Estimates**
. Horowitz (2014) provides rough numbers for the energy consumption [1] as summarized in table 1 and 2. As can be seen in Table 1, while floating-point multipicators
demand 1.1pJ-3.7pJ, floating point adders require only 0.4pJ-0.9pJ. Courbariaux et al.
(2015b) replaced approxi... | {
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} | {
"Header 1": null,
"Header 2": "Binarized Neural Networks",
"Header 3": "**4 Expected Efficiency Gains**",
"Header 4": "**4.1 Energy Efficiency Estimates**"
} | {
"chunk_type": "body"
} |
#### **4.2 Exploiting Kernel Repetitions**
When using a CNN architecture with binary weights, the maximum amount of unique
kernels is bounded by the kernel size. For example, in our implementation we use
kernels of size 3 *×* 3, so the maximum number of unique 2D kernels is 2 [9] = 512.
1 The given number are for 4... | {
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"Header 3": "**4 Expected Efficiency Gains**",
"Header 4": "**4.2 Exploiting Kernel Repetitions**"
} | {
"chunk_type": "body"
} |
### **5 Benchmark Results**
This section, report on empirical results showing that BBP obtains near state-of-theart results with fully binary networks on the permutation-invariant MNIST, CIFAR-10
and SVHN datasets. In all of our experiments we used an identical architecture as
the BinarryConnect does. We use the L2-S... | {
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"Header 2": "Binarized Neural Networks",
"Header 3": "**5 Benchmark Results**",
"Header 4": null
} | {
"chunk_type": "body"
} |
#### **5.1 Datasets**
**5.1.1** **CIFAR-10**
The well known CIFAR-10 is an image classification benchmark dataset. Containing
50,000 training images and 10,000 test images of 32 × 32 color images in 10 classes
(airplanes, automobiles, birds, cats, deers, dogs, frogs, horses, ships and trucks). For
this dataset, we ... | {
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"Header 2": "Binarized Neural Networks",
"Header 3": "**5 Benchmark Results**",
"Header 4": "**5.1 Datasets**"
} | {
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considered significantly more difficult. It consists of a training set of 604K instances
and a test set of 26K instances where each instance is a 32 × 32 color images. We
applied the same procedure we used for CIFAR-10, with an architecture similar to that
of BinnaryConnect. We report results after 500 iterations. | {
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"Header 2": "Binarized Neural Networks",
"Header 3": "**5 Benchmark Results**",
"Header 4": "**5.1 Datasets**"
} | {
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#### **5.2 Results**
As can be seen in Table 3, BBP algorithm using the architecture stated above received
10.15% error rate on CIFAR10, 2.53% on SVHN and 1.4% on permutation invariant
MNIST. It is somehow surprising that despite the binarization noise and the other rough
power-of-2 estimation (shift base BN and S-Ad... | {
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"Header 3": "**5 Benchmark Results**",
"Header 4": "**5.2 Results**"
} | {
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} |
![](/content/images/1... | {
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"Header 3": "**5 Benchmark Results**",
"Header 4": "**5.2 Results**"
} | {
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Hwang & Sung (2014)[1bit] 1.38%
Kim & Paris (2015) 1.33%
Standard DNN results ( without binarization )
No reg 1.3 *±* 0.2% 2.44% 10.94%
Maxout NetsGoodfellow et al. (2013) 0.94% 2.47% 11.68%
Network in NetworkLin et al. (2013) 2.35% 10.41%
DropConnectWan et al. (2013) - 1.94%
Deeply-Supervised-Networks ... | {
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"Header 2": "Binarized Neural Networks",
"Header 3": "**5 Benchmark Results**",
"Header 4": "**5.2 Results**"
} | {
"chunk_type": "body"
} |
### **6 Discussion and Future Work**
In this work, we have introduced binary back propagation (BBP), a novel binarization
scheme for weights and neurons during forward and backward propagation. We have
shown that it is possible to train BDNNs on the permutation invariant MNIST, CIFAR10 and SVHN data sets and achieve ... | {
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"Header 3": "**6 Discussion and Future Work**",
"Header 4": null
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et al. (2015)). As shown in figure 5.2 many weights (75-90%) reach the saturation level.
Those weights can theoretically be saved with one bit which simply means that we do
not care about the accuracy of the other bits. Furthermore, as mentioned in section
4.2, approximately 63% of the *and* and *popcount* operations c... | {
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"Header 2": "Binarized Neural Networks",
"Header 3": "**6 Discussion and Future Work**",
"Header 4": null
} | {
"chunk_type": "body"
} |
### **References**
Cheng, Zhiyong, Soudry, Daniel, Mao, Zexi, and Lan, Zhenzhong. Training Binary
Multilayer Neural Networks for Image Classification using Expectation Backpropgation. *arXiv:1503.03562*, (2012):8, 2015. 3
Courbariaux, Matthieu, Bengio, Yoshua, and David, Jean-Pierre. BinaryConnect:
Training Deep Ne... | {
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"title": "Binarized Neural Networks",
"categories": [
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} | {
"Header 1": null,
"Header 2": "Binarized Neural Networks",
"Header 3": "**References**",
"Header 4": null
} | {
"chunk_type": "references"
} |
*Systems (SiPS)*, pp. 1–6, 2014. doi: 10.1109/SiPS.2014.6986082. 2, 3, 6
Ioffe, Sergey and Szegedy, Christian. Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift. *arXiv*, 2015. 3.3
Judd, Patrick, Albericio, Jorge, Hetherington, Tayler, Aamodt, Tor, Jerger, Natalie Enright,... | {
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"title": "Binarized Neural Networks",
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} | {
"Header 1": null,
"Header 2": "Binarized Neural Networks",
"Header 3": "**References**",
"Header 4": null
} | {
"chunk_type": "references"
} |
## **Visualizing and Understanding Recurrent Networks**
**Andrej Karpathy** *[∗]* **Justin Johnson** *[∗]* **Li Fei-Fei**
Department of Computer Science, Stanford University
*{* karpathy,jcjohns,feifeili *}* @cs.stanford.edu | {
"id": "1506.02078",
"title": "Visualizing and Understanding Recurrent Networks",
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"Header 1": null,
"Header 2": "**Visualizing and Understanding Recurrent Networks**",
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### **Abstract**
Recurrent Neural Networks (RNNs), and specifically a variant with Long ShortTerm Memory (LSTM), are enjoying renewed interest as a result of successful
applications in a wide range of machine learning problems that involve sequential
data. However, while LSTMs provide exceptional results in practice,... | {
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"Header 1": null,
"Header 2": "**Visualizing and Understanding Recurrent Networks**",
"Header 3": "**Abstract**",
"Header 4": null
} | {
"chunk_type": "body"
} |
### **1 Introduction**
Recurrent Neural Networks, and specifically a variant with Long Short-Term Memory (LSTM) (14),
have recently emerged as an effective model in a wide variety of applications that involve sequential
data. These include language modeling (22), handwriting recognition and generation (9), machine
tr... | {
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"title": "Visualizing and Understanding Recurrent Networks",
"categories": [
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"cs.CL",
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} | {
"Header 1": null,
"Header 2": "**Visualizing and Understanding Recurrent Networks**",
"Header 3": "**1 Introduction**",
"Header 4": null
} | {
"chunk_type": "body"
} |
### **2 Related Work**
**Recurrent Networks** . Recurrent Neural Networks (RNNs) have a long history of applications in
various sequence learning tasks (31; 26; 25). Despite their early successes, the difficulty of training
*∗* Both authors contributed equally to this work.
1
-----
simple recurrent networks (... | {
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"title": "Visualizing and Understanding Recurrent Networks",
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} | {
"Header 1": null,
"Header 2": "**Visualizing and Understanding Recurrent Networks**",
"Header 3": "**2 Related Work**",
"Header 4": null
} | {
"chunk_type": "body"
} |
### **3 Experimental Setup**
We first describe three variants of a deep recurrent network (RNN, LSTM and the GRU models),
then explain how they are used in sequence learning and finally describe the optimization.
**3.1** **Recurrent Neural Network Models**
The simplest instantiation of a deep recurrent network arra... | {
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} | {
"Header 1": null,
"Header 2": "**Visualizing and Understanding Recurrent Networks**",
"Header 3": "**3 Experimental Setup**",
"Header 4": null
} | {
"chunk_type": "body"
} |
be alleviated with a heuristic of clipping the gradients at some maximum value (24). On the other
hand, LSTMs were designed to mitigate the vanishing gradient problem. In addition to a hidden
state vector *h* *[l]* *t* [, LSTMs also maintain a memory vector] *[ c]* *[l]* *t* [. At each time step the LSTM can choose]
to... | {
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} | {
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"Header 2": "**Visualizing and Understanding Recurrent Networks**",
"Header 3": "**3 Experimental Setup**",
"Header 4": null
} | {
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} |
a *candidate* hidden vector *h* [˜] *[l]* *t* [and then smoothly interpolating towards it, as gated by] *[ z]* [.]
**3.2** **Character-level Language Modeling**
We use character-level language modeling as an interpretable testbed for sequence learning. In this
setting, the input to the network is a sequence of charac... | {
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"title": "Visualizing and Understanding Recurrent Networks",
"categories": [
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} | {
"Header 1": null,
"Header 2": "**Visualizing and Understanding Recurrent Networks**",
"Header 3": "**3 Experimental Setup**",
"Header 4": null
} | {
"chunk_type": "body"
} |
### **4 Experiments**
**Datasets** . Two datasets previously used in the context of character-level language models are the
Penn Treebank dataset (20) and the Hutter Prize 100MB of Wikipedia dataset (16) . However, both
datasets contain a mix of common language and special markup. Our goal is not to compete with
prev... | {
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} | {
"Header 1": null,
"Header 2": "**Visualizing and Understanding Recurrent Networks**",
"Header 3": "**4 Experiments**",
"Header 4": null
} | {
"chunk_type": "body"
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|Size War and Peace Dataset||||
|64 1.449 1.442 1.540 128 1.277 1.227 1.279 256 1.189 1.137 1.141 512 1.161 1.092 1.082|1.446 1.401 1.396 1.417 1.286 1.277 1.342 1.256 1.239 - - -|1.398 1.373 1.47 1.230 1.226 1.25 1.198 1.164 1.13 1.170 1.201 1.07||
|Linux Kernel Dataset||||
|64 1.355 1.331 1.366 1.407 1.371 1.383 1.33... | {
"id": "1506.02078",
"title": "Visualizing and Understanding Recurrent Networks",
"categories": [
"cs.LG",
"cs.CL",
"cs.NE"
]
} | {
"Header 1": null,
"Header 2": "**Visualizing and Understanding Recurrent Networks**",
"Header 3": "**4 Experiments**",
"Header 4": null
} | {
"chunk_type": "body"
} |
dynamics, or approximate gradients due to truncated backpropagation through time) might prevent
it from discovering these solutions. In this experiment we verify that multiple interpretable cells do
in fact exist in these networks. We show several examples in Figure 2. Note that truncated backpropagation prevents the g... | {
"id": "1506.02078",
"title": "Visualizing and Understanding Recurrent Networks",
"categories": [
"cs.LG",
"cs.CL",
"cs.NE"
]
} | {
"Header 1": null,
"Header 2": "**Visualizing and Understanding Recurrent Networks**",
"Header 3": "**4 Experiments**",
"Header 4": null
} | {
"chunk_type": "body"
} |
utilize information from a fixed number of previous steps. In particular, we consider two baselines:
4
-----
Figure 2: Several examples of cells with interpretable activations discovered in our best Linux Kernel and War and Peace LSTMs. Text color corresponds to *ta... | {
"id": "1506.02078",
"title": "Visualizing and Understanding Recurrent Networks",
"categories": [
"cs.LG",
"cs.CL",
"cs.NE"
]
} | {
"Header 1": null,
"Header 2": "**Visualizing and Understanding Recurrent Networks**",
"Header 3": "**4 Experiments**",
"Header 4": null
} | {
"chunk_type": "body"
} |
**Error Analysis.** It is instructive to delve deeper into the errors made by both recurrent networks and
*n* -gram models. In particular, we define a character to be an error if the probability assigned to it
by a model is below 0.5. Figure 4 (left) shows the overlap between the test-set errors for the 3-layer
LSTM, a... | {
"id": "1506.02078",
"title": "Visualizing and Understanding Recurrent Networks",
"categories": [
"cs.LG",
"cs.CL",
"cs.NE"
]
} | {
"Header 1": null,
"Header 2": "**Visualizing and Understanding Recurrent Networks**",
"Header 3": "**4 Experiments**",
"Header 4": null
} | {
"chunk_type": "body"
} |
features an interesting long-term dependency with the carriage return, which occurs approximately
every 70 characters. Figure 4 (right) shows that the LSTM has a distinct advantage on this character.
To accurately predict the presence of the carriage return the model likely needs to keep track of
its distance since the... | {
"id": "1506.02078",
"title": "Visualizing and Understanding Recurrent Networks",
"categories": [
"cs.LG",
"cs.CL",
"cs.NE"
]
} | {
"Header 1": null,
"Header 2": "**Visualizing and Understanding Recurrent Networks**",
"Header 3": "**4 Experiments**",
"Header 4": null
} | {
"chunk_type": "body"
} |
loss in Figure 5 (right). Notably, we see that in the first few iterations the LSTM behaves like the 1NN model but then diverges from it soon after. The LSTM then behaves most like the 2-NN, 3-NN,
and 4-NN models in turn. This experiment suggests that the LSTM “grows” its competence over
increasingly longer dependencie... | {
"id": "1506.02078",
"title": "Visualizing and Understanding Recurrent Networks",
"categories": [
"cs.LG",
"cs.CL",
"cs.NE"
]
} | {
"Header 1": null,
"Header 2": "**Visualizing and Understanding Recurrent Networks**",
"Header 3": "**4 Experiments**",
"Header 4": null
} | {
"chunk_type": "body"
} |
fixed with better modeling of the previous *n* characters. In particular, we evaluate our *n* -gram model
( *n* = 1 *, . . .,* 9) and remove a character error if it is correctly classified by any of these models.
**Dynamic** *n* **-long memory oracle.** Consider the string *“Jon yelled at Mary but Mary couldn’t hear*
*... | {
"id": "1506.02078",
"title": "Visualizing and Understanding Recurrent Networks",
"categories": [
"cs.LG",
"cs.CL",
"cs.NE"
]
} | {
"Header 1": null,
"Header 2": "**Visualizing and Understanding Recurrent Networks**",
"Header 3": "**4 Experiments**",
"Header 4": null
} | {
"chunk_type": "body"
} |
an oracle that boosts the probability of the correct letter by a fixed amount. These oracles allow us
to understand the distribution of the difficulty of the remaining errors.
We now subject two LSTM models to the error analysis: First, our best LSTM model and second,
the best LSTM model in the smallest model categor... | {
"id": "1506.02078",
"title": "Visualizing and Understanding Recurrent Networks",
"categories": [
"cs.LG",
"cs.CL",
"cs.NE"
]
} | {
"Header 1": null,
"Header 2": "**Visualizing and Understanding Recurrent Networks**",
"Header 3": "**4 Experiments**",
"Header 4": null
} | {
"chunk_type": "body"
} |
(32), where the model is allowed to attend to a recent history of the sequence while making its next
prediction. Finally, the rare words oracle accounts for 9% of the errors. This error type could be
mitigated with transfer learning, or by increasing the size of the training set. The majority of the
remaining errors (3... | {
"id": "1506.02078",
"title": "Visualizing and Understanding Recurrent Networks",
"categories": [
"cs.LG",
"cs.CL",
"cs.NE"
]
} | {
"Header 1": null,
"Header 2": "**Visualizing and Understanding Recurrent Networks**",
"Header 3": "**4 Experiments**",
"Header 4": null
} | {
"chunk_type": "body"
} |
### **5 Conclusion**
We have presented a comprehensive analysis of Recurrent Neural Networks and their representations, predictions and error types. In particular, qualitative visualization experiments, cell activation
statistics and in-depth comparisons to finite horizon *n* -gram models demonstrate that these netwo... | {
"id": "1506.02078",
"title": "Visualizing and Understanding Recurrent Networks",
"categories": [
"cs.LG",
"cs.CL",
"cs.NE"
]
} | {
"Header 1": null,
"Header 2": "**Visualizing and Understanding Recurrent Networks**",
"Header 3": "**5 Conclusion**",
"Header 4": null
} | {
"chunk_type": "body"
} |
### **References**
[1] D. Bahdanau, K. Cho, and Y. Bengio. Neural machine translation by jointly learning to align and translate.
*arXiv preprint arXiv:1409.0473*, 2014.
[2] Y. Bengio, P. Simard, and P. Frasconi. Learning long-term dependencies with gradient descent is difficult.
*Neural Networks, IEEE Transactions... | {
"id": "1506.02078",
"title": "Visualizing and Understanding Recurrent Networks",
"categories": [
"cs.LG",
"cs.CL",
"cs.NE"
]
} | {
"Header 1": null,
"Header 2": "**Visualizing and Understanding Recurrent Networks**",
"Header 3": "**References**",
"Header 4": null
} | {
"chunk_type": "references"
} |
[14] S. Hochreiter and J. Schmidhuber. Long short-term memory. *Neural computation*, 9(8):1735–1780, 1997.
[15] X. Huang, A. Acero, H.-W. Hon, and R. Foreword By-Reddy. *Spoken language processing: A guide to*
*theory, algorithm, and system development* . Prentice Hall PTR, 2001.
[16] M. Hutter. The human knowledge... | {
"id": "1506.02078",
"title": "Visualizing and Understanding Recurrent Networks",
"categories": [
"cs.LG",
"cs.CL",
"cs.NE"
]
} | {
"Header 1": null,
"Header 2": "**Visualizing and Understanding Recurrent Networks**",
"Header 3": "**References**",
"Header 4": null
} | {
"chunk_type": "references"
} |
*in Neural Information Processing Systems*, pages 3104–3112, 2014.
[29] L. Van der Maaten and G. Hinton. Visualizing data using t-sne. *Journal of Machine Learning Research*,
9(2579-2605):85, 2008.
[30] O. Vinyals, A. Toshev, S. Bengio, and D. Erhan. Show and tell: A neural image caption generator. *CVPR*,
2015.
... | {
"id": "1506.02078",
"title": "Visualizing and Understanding Recurrent Networks",
"categories": [
"cs.LG",
"cs.CL",
"cs.NE"
]
} | {
"Header 1": null,
"Header 2": "**Visualizing and Understanding Recurrent Networks**",
"Header 3": "**References**",
"Header 4": null
} | {
"chunk_type": "references"
} |
### **Supplementary Material**
**Exact Model Sizes**
**LSTM** **RNN** **GRU**
|Layers 1 2 3|1 2 3|1 2 3|
|---|---|---|
Size War and Peace Dataset
|64 64 45 37 128 128 89 68 256 256 159 126 512 512 308 241|141 96 78 273 174 139 531 325 257 1045 623 487|77 53 44 151 98 79 300 185 146 596 357 280|
|---|---|---| ... | {
"id": "1506.02078",
"title": "Visualizing and Understanding Recurrent Networks",
"categories": [
"cs.LG",
"cs.CL",
"cs.NE"
]
} | {
"Header 1": null,
"Header 2": "**Visualizing and Understanding Recurrent Networks**",
"Header 3": "**Supplementary Material**",
"Header 4": null
} | {
"chunk_type": "body"
} |
We experimented with multiple settings of this hyperparameter on multiple architectures and found its sensitivity to be relatively low. In particular, It is safer to err on the side of making this range smaller, since some
models start to diverge when it is greater than 1, but the performance degrades gracefully, even ... | {
"id": "1506.02078",
"title": "Visualizing and Understanding Recurrent Networks",
"categories": [
"cs.LG",
"cs.CL",
"cs.NE"
]
} | {
"Header 1": null,
"Header 2": "**Visualizing and Understanding Recurrent Networks**",
"Header 3": "**Supplementary Material**",
"Header 4": null
} | {
"chunk_type": "body"
} |
reduction in performance for *α <* 0 *.* 25. Using *α* = 2 allows the LSTM to saturate the cell in one step. In this
case we observed a consistently faster convergence (e.g. 700 vs. 900 iterations to reach loss of 2.0) across all
architectures. However, the final performance at epoch 50 remains unchanged.
10
----- ... | {
"id": "1506.02078",
"title": "Visualizing and Understanding Recurrent Networks",
"categories": [
"cs.LG",
"cs.CL",
"cs.NE"
]
} | {
"Header 1": null,
"Header 2": "**Visualizing and Understanding Recurrent Networks**",
"Header 3": "**Supplementary Material**",
"Header 4": null
} | {
"chunk_type": "body"
} |
1-gram 1581 0.011 1697 0.009
2-gram 4001 0.029 4637 0.025
3-gram 2959 0.021 8200 0.045
4-gram 3425 0.025 14053 0.076
5-gram 3183 0.023 11602 0.063
6-gram 2974 0.021 8257 0.045
7-gram 2825 0.020 5859 0.032
8-gram 2406 0.017 4268 0.023
9- g ram 1635 0.012 2634 0.014
100 memory 989 0.007 1074 0.006
500 memory 2714 0.019 2... | {
"id": "1506.02078",
"title": "Visualizing and Understanding Recurrent Networks",
"categories": [
"cs.LG",
"cs.CL",
"cs.NE"
]
} | {
"Header 1": null,
"Header 2": "**Visualizing and Understanding Recurrent Networks**",
"Header 3": "**Supplementary Material**",
"Header 4": null
} | {
"chunk_type": "body"
} |
## **What Uncertainties Do We Need in Bayesian Deep Learning** **for Computer Vision?**
**Alex Kendall** [1] **Yarin Gal** [1] | {
"id": "1703.04977",
"title": "What Uncertainties Do We Need in Bayesian Deep Learning for Computer\n Vision?",
"categories": [
"cs.CV"
]
} | {
"Header 1": null,
"Header 2": "**What Uncertainties Do We Need in Bayesian Deep Learning** **for Computer Vision?**",
"Header 3": null,
"Header 4": null
} | {
"chunk_type": "title"
} |
### **Abstract**
There are two major types of uncertainty one can
model. *Aleatoric* uncertainty captures noise inherent in the observations. On the other hand,
*epistemic* uncertainty accounts for uncertainty in
the model – uncertainty which can be explained
away given enough data. Traditionally it has
been difficul... | {
"id": "1703.04977",
"title": "What Uncertainties Do We Need in Bayesian Deep Learning for Computer\n Vision?",
"categories": [
"cs.CV"
]
} | {
"Header 1": null,
"Header 2": "**What Uncertainties Do We Need in Bayesian Deep Learning** **for Computer Vision?**",
"Header 3": "**Abstract**",
"Header 4": null
} | {
"chunk_type": "body"
} |
### **1. Introduction**
Understanding what a model does not know is a critical
part of many machine learning systems. Today, deep learning algorithms are able to learn powerful representations
which can map high dimensional data to an array of outputs. However these mappings are often taken blindly and
assumed to be ... | {
"id": "1703.04977",
"title": "What Uncertainties Do We Need in Bayesian Deep Learning for Computer\n Vision?",
"categories": [
"cs.CV"
]
} | {
"Header 1": null,
"Header 2": "**What Uncertainties Do We Need in Bayesian Deep Learning** **for Computer Vision?**",
"Header 3": "**1. Introduction**",
"Header 4": null
} | {
"chunk_type": "body"
} |
other hand, *epistemic* uncertainty accounts for uncertainty
in the model parameters – uncertainty which captures our
ignorance about which model generated our collected data.
This uncertainty can be explained away given enough data,
and is often referred to as *model uncertainty* . Aleatoric
uncertainty can further be... | {
"id": "1703.04977",
"title": "What Uncertainties Do We Need in Bayesian Deep Learning for Computer\n Vision?",
"categories": [
"cs.CV"
]
} | {
"Header 1": null,
"Header 2": "**What Uncertainties Do We Need in Bayesian Deep Learning** **for Computer Vision?**",
"Header 3": "**1. Introduction**",
"Header 4": null
} | {
"chunk_type": "body"
} |
segment the footpath, and the corresponding increased epistemic uncertainty.
have very high uncertainty.
In this paper we make the observation that in many big data
regimes (such as the ones common to deep learning with
image data), it is most effective to model aleatoric uncertainty, uncertainty which cannot be ex... | {
"id": "1703.04977",
"title": "What Uncertainties Do We Need in Bayesian Deep Learning for Computer\n Vision?",
"categories": [
"cs.CV"
]
} | {
"Header 1": null,
"Header 2": "**What Uncertainties Do We Need in Bayesian Deep Learning** **for Computer Vision?**",
"Header 3": "**1. Introduction**",
"Header 4": null
} | {
"chunk_type": "body"
} |
### **2. Related Work**
Existing approaches to Bayesian deep learning capture either epistemic uncertainty alone, or aleatoric uncertainty
alone (Gal, 2016). These uncertainties are formalised as
probability distributions over either the model parameters,
or model outputs, respectively. Epistemic uncertainty is
model... | {
"id": "1703.04977",
"title": "What Uncertainties Do We Need in Bayesian Deep Learning for Computer\n Vision?",
"categories": [
"cs.CV"
]
} | {
"Header 1": null,
"Header 2": "**What Uncertainties Do We Need in Bayesian Deep Learning** **for Computer Vision?**",
"Header 3": "**2. Related Work**",
"Header 4": null
} | {
"chunk_type": "body"
} |
*p* ( **y** *|* **f** **[W]** ( **x** )) = *N* ( **f** **[W]** ( **x** ) *, σ* [2] )
with an observation noise scalar *σ* . For classification, on
the other hand, we often squash the model output through a
softmax function, and sample from the resulting probability
vector:
*p* ( **y** *|* **f** **[W]** ( **x** ))... | {
"id": "1703.04977",
"title": "What Uncertainties Do We Need in Bayesian Deep Learning for Computer\n Vision?",
"categories": [
"cs.CV"
]
} | {
"Header 1": null,
"Header 2": "**What Uncertainties Do We Need in Bayesian Deep Learning** **for Computer Vision?**",
"Header 3": "**2. Related Work**",
"Header 4": null
} | {
"chunk_type": "body"
} |
*i* =1
with *N* data points, dropout probability *p*, samples **W** [�] *i* *∼*
*q* *θ* *[∗]* [(] **[W]** [)][, and] *[ θ]* [ the set of the simple distribution’s parame-]
ters to be optimised (weight matrices in dropout’s case). In
regression, for example, the negative log likelihood can be
further simplified as
*... | {
"id": "1703.04977",
"title": "What Uncertainties Do We Need in Bayesian Deep Learning for Computer\n Vision?",
"categories": [
"cs.CV"
]
} | {
"Header 1": null,
"Header 2": "**What Uncertainties Do We Need in Bayesian Deep Learning** **for Computer Vision?**",
"Header 3": "**2. Related Work**",
"Header 4": null
} | {
"chunk_type": "body"
} |
**What Uncertainties Do We Need in Ba** **y** **esian Dee** **p** **Learnin** **g** **for Com** **p** **uter Vision?**
The first term in the predictive variance, *σ* [2], corresponds
to the amount of noise inherent in the data (which will be
explained in more detail soon). The second part of the predictive variance m... | {
"id": "1703.04977",
"title": "What Uncertainties Do We Need in Bayesian Deep Learning for Computer\n Vision?",
"categories": [
"cs.CV"
]
} | {
"Header 1": null,
"Header 2": "**What Uncertainties Do We Need in Bayesian Deep Learning** **for Computer Vision?**",
"Header 3": "**2. Related Work**",
"Header 4": null
} | {
"chunk_type": "body"
} |
### **3. Combining Aleatoric Uncertainty and** **Epistemic Uncertainty in One Model**
In the previous section we described existing Bayesian
deep learning techniques. In this section we present novel
contributions which extend this existing literature. We develop models that will allow us to study the effects of
mode... | {
"id": "1703.04977",
"title": "What Uncertainties Do We Need in Bayesian Deep Learning for Computer\n Vision?",
"categories": [
"cs.CV"
]
} | {
"Header 1": null,
"Header 2": "**What Uncertainties Do We Need in Bayesian Deep Learning** **for Computer Vision?**",
"Header 3": "**3. Combining Aleatoric Uncertainty and** **Epistemic Uncertainty in One Model**",
"Header 4": null
} | {
"chunk_type": "body"
} |
*D* = 1 for image-level regression tasks, or *D* equal to the
number of pixels for dense prediction tasks (predicting a
unary corresponding to each input image pixel). ˆ *σ* *i* [2] [is the]
BNN output for the predicted variance for pixel *i* .
This loss consists of two components; the residual regression obtained wi... | {
"id": "1703.04977",
"title": "What Uncertainties Do We Need in Bayesian Deep Learning for Computer\n Vision?",
"categories": [
"cs.CV"
]
} | {
"Header 1": null,
"Header 2": "**What Uncertainties Do We Need in Bayesian Deep Learning** **for Computer Vision?**",
"Header 3": "**3. Combining Aleatoric Uncertainty and** **Epistemic Uncertainty in One Model**",
"Header 4": null
} | {
"chunk_type": "body"
} |
logit value), and the corrupted vector is then squashed with
the softmax function to obtain **p** *i*, the probability vector for
pixel *i* .
Our expected log likelihood for this model is given by:
*E* *N* (ˆ **x** *i* ; **f** *i* **W** *,* ( *σ* *i* **[W]** ) [2] ) [[log ˆ] **[p]** *[i,c]* []]
with *c* the obser... | {
"id": "1703.04977",
"title": "What Uncertainties Do We Need in Bayesian Deep Learning for Computer\n Vision?",
"categories": [
"cs.CV"
]
} | {
"Header 1": null,
"Header 2": "**What Uncertainties Do We Need in Bayesian Deep Learning** **for Computer Vision?**",
"Header 3": "**3. Combining Aleatoric Uncertainty and** **Epistemic Uncertainty in One Model**",
"Header 4": null
} | {
"chunk_type": "body"
} |
for pixel *x* in this combined model can be approximated
using:
*T*
� *x* ˆ *t*
*t* =1
*T*
� *σ* ˆ *t* [2]
*t* =1
Var( *x* ) *≈* [1]
*T*
*T*
� *t* =1 *x* ˆ [2] *t* *[−]* � *T* 1
2
+ [1]
� *T*
with *{x* ˆ *t* *,* ˆ *σ* *t* [2] *[}]* *[T]* *t* =1 [a set of] *[ T]* [ sampled outputs:][ ˆ] *[x]* *[t]* *[,... | {
"id": "1703.04977",
"title": "What Uncertainties Do We Need in Bayesian Deep Learning for Computer\n Vision?",
"categories": [
"cs.CV"
]
} | {
"Header 1": null,
"Header 2": "**What Uncertainties Do We Need in Bayesian Deep Learning** **for Computer Vision?**",
"Header 3": "**3. Combining Aleatoric Uncertainty and** **Epistemic Uncertainty in One Model**",
"Header 4": null
} | {
"chunk_type": "body"
} |
heteroscedastic NNs to classification heteroscedastic NNs,
discussed next.
-----
**What Uncertainties Do We Need in Ba** **y** **esian Dee** **p** **Learnin** **g** **for Com** **p** **uter Vision?**
|CamVid|IoU|
|---|---|
|SegNet (Badrinarayanan et al., 2017) FCN-8 (Shelhamer et al., 2016) DeepLab-LFOV (Chen et ... | {
"id": "1703.04977",
"title": "What Uncertainties Do We Need in Bayesian Deep Learning for Computer\n Vision?",
"categories": [
"cs.CV"
]
} | {
"Header 1": null,
"Header 2": "**What Uncertainties Do We Need in Bayesian Deep Learning** **for Computer Vision?**",
"Header 3": "**3. Combining Aleatoric Uncertainty and** **Epistemic Uncertainty in One Model**",
"Header 4": null
} | {
"chunk_type": "body"
} |
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