Incorporating Second-Order Functional Knowledge for Better Option Pricing

By Charles Dugas, Yoshua Bengio, François Bélisle, Claude Nadeau, René Garcia. 2000

541 citations recorded by the source corpus.

Subfields: Probabilistic & Bayesian Methods, Economics & Operations Research

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Probabilistic & Bayesian Methods

In Economics & Operations Research

Citations in the corpus

Large Scale Adversarial Representation Learning (2019, neutral): “μ and σ̂ are given by a linear layer at the output of the model, and the final standard deviation σ is computed from σ̂ using a non-negative “softplus” non-linearity σ = log(1 + exp(σ̂)) [7].”
A Sensitivity Analysis of (and Practitioners’ Guide to) Convolutional Neural Networks for Sentence Classification (2015, background): “…different activation functions in the convolution layer, including: ReLU (as per the baseline conﬁguration), hyperbolic tangent (tanh), Sigmoid function (Maas et al., 2013), SoftPlus function (Dugas et al., 2001), Cube function (Chen and Manning, 2014), and tanh cube function (Pei et al., 2015).”
Lipschitz-Margin Training: Scalable Certification of Perturbation Invariance for Deep Neural Networks (2018, methodology): “ReLU 1 Leaky ReLU [22] max(1 , | α | ) sigmoid 1 / 4 tanh 1 soft plus [5] 1 ELU [9] max(1 , | α | ) average-pooling: Before bounding the Lipschitz constant, we note that the following inequality holds for a vector X : This can be proved using Now, we bound the Lipschitz constant of the average…”
Interpolation Consistency Training for Semi-Supervised Learning (2019, methodology): “Moreover, ICT worked the best after replacing the ReLU activation function with the softplus activation function (Dugas et al., 2001): φ(z) = ln(1 + exp(κz))/κ with κ = 100.”
Activation Functions: Comparison of trends in Practice and Research for Deep Learning (2018, background): “It was proposed by [21] and it is a primitive of the sigmoid, given by”
SMIL: Multimodal Learning with Severely Missing Modality (2021, methodology): “Hence, we adopt Softplus (Dugas et al. 2000) activation to weaken the regularization.”
Faster CryptoNets: Leveraging Sparsity for Real-World Encrypted Inference (2018, methodology): “In our experiments, we consider ReLU [30], Softplus [21], and Swish [55].”
Deep learning classifiers for hyperspectral imaging: A review (2019, background): “Other interesting functions are the softplus (Dugas et al., 2001) and softmax activation functions, with equations = + x e ( ) ln(1 ) x and”
A survey on modern trainable activation functions (2020, background): “, 2015), and similar activation functions (for example, (Dugas et al., 2000; Clevert et al., 2016)), has shown to improve the network performances significantly, thanks to properties as the no saturation feature that helps to avoid typical learning problems as vanishing gradient (Bishop, 2006).”
Differentiable Abstract Interpretation for Provably Robust Neural Networks (2018, methodology): “We found that applying a smooth version of ReLU – softplus (Dugas et al., 2001) – to this loss helped to avoid this issue.”
Theoretical Analysis of Auto Rate-Tuning by Batch Normalization (2018, methodology): “We can instead analyse smooth activations, such as sigmoid, tanh, softplus (Dugas et al., 2001), etc.”
Dynamic ReLU (2020, unknown): “A number of smooth activation functions have been developed to address this, such as softplus [6], ELU [4], SELU [20], Mish [27].”
Single image portrait relighting (2019, methodology): “…activation is used in the layer before the target portrait image prediction (as images are assumed to be in [ 0 , 1 ] ), a softplus activation [Dugas et al. 2000] is used immediately before the weighted average that predicts the lighting of the source portrait (as confidence must be…”
Safety Alignment Should Be Made More Than Just a Few Tokens Deep (2024, unknown): “To see why β t can be used to control the deviation of the generative distribution at each token position, we can rewrite the fine-tuning objective as: where S ( z ) := log(1 + e z ) is the softplus function (Dugas et al., 2000).”
Free-Lunch Saliency via Attention in Atari Agents (2019, methodology): “We have verified experimentally that normalizing the output of SoftPlus makes the model perform worse (see Section 4.2).”
Deep Sparse Rectifier Neural Networks (2011, background): “To evaluate the potential impact of this effect we also investigate the softplus activation: softplus(x) = log(1+e) (Dugas et al., 2001), a smooth version of the rectifying non-linearity.”
Large-Scale Image Retrieval with Attentive Deep Local Features (2016, methodology): “The score function is designed using a 2-layer CNN with a softplus [9] activation at the top.”