Ai Drug Discovery

This project focuses on ai drug discovery using modern AI and machine learning techniques. The content below is adapted from research literature and practical implementation notes.

TabNet: Attentive Interpretable Tabular Learning

Sercan Ö. Arık, Tomas Pfister

Google Cloud AI

Sunnyvale, CA

We propose a novel high-performance and interpretable canon-

addition, unlike tree learning, DNNs enable gradient descent- ical deep tabular data learning architecture, TabNet. TabNet based end-to-end learning for tabular data which can have a uses sequential attention to choose which features to reason multitude of benefits: (i) efficiently encoding multiple data from at each decision step, enabling interpretability and more types like images along with tabular data; (ii) alleviating the efficient learning as the learning capacity is used for the most need for feature engineering, which is currently a key aspect

salient features. We demonstrate that TabNet outperforms in tree-based tabular data learning methods; (iii) learning other variants on a wide range of non-performance-saturated from streaming data and perhaps most importantly (iv) end- tabular datasets and yields interpretable feature attributions to-end models allow representation learning which enables plus insights into its global behavior. Finally, we demonstrate many valuable application scenarios including data-efficient

self-supervised learning for tabular data, significantly improv- domain adaptation (Goodfellow, Bengio, and Courville 2016), ing performance when unlabeled data is abundant. generative modeling (Radford, Metz, and Chintala 2015) and

Introduction We propose a new canonical DNN architecture for tabular

Deep neural networks (DNNs) have shown notable success data, TabNet. The main contributions are summarized as: efficiently encode the raw data into meaningful representa- enabling flexible integration into end-to-end learning. tions, fuel the rapid progress. One data type that has yet to 2. TabNet uses sequential attention to choose which fea- see such success with a canonical architecture is tabular data. tures to reason from at each decision step, enabling in-

Despite being the most common data type in real-world AI terpretability and better learning as the learning capacity (as it is comprised of any categorical and numerical features), is used for the most salient features (see Fig. 1). This under-explored, with variants of ensemble decision trees for each input, and unlike other instance-wise feature se- Why? First, because DT-based approaches have certain bene- and van der Schaar 2019), TabNet employs a single deep

fits: (i) they are representionally efficient for decision mani- learning architecture for feature selection and reasoning. folds with approximately hyperplane boundaries which are 3. Above design choices lead to two valuable properties: (i) common in tabular data; and (ii) they are highly interpretable TabNet outperforms or is on par with other tabular learn- in their basic form (e.g. by tracking decision nodes) and there ing models on various datasets for classification and re-

are popular post-hoc explainability methods for their ensem- gression problems from different domains; and (ii) TabNet ble form, e.g. (Lundberg, Erion, and Lee 2018) – this is an enables two kinds of interpretability: local interpretability important concern in many real-world applications; (iii) they that visualizes the importance of features and how they are fast to train. Second, because previously-proposed DNN are combined, and global interpretability which quantifies

architectures are not well-suited for tabular data: e.g. stacked the contribution of each feature to the trained model. convolutional layers or multi-layer perceptrons (MLPs) are 4. Finally, for the first time for tabular data, we show signif- vastly overparametrized – the lack of appropriate inductive icant performance improvements by using unsupervised bias often causes them to fail to find optimal solutions for tab- pre-training to predict masked features (see Fig. 2).

ular decision manifolds (Goodfellow, Bengio, and Courville

Why is deep learning worth exploring for tabular data?

One obvious motivation is expected performance improve- Feature selection: Feature selection broadly refers to judi- Copyright © 2021, Association for the Advancement of Artificial ciously picking a subset of features based on their useful-

Intelligence (www.aaai.org). All rights reserved. ness for prediction. Commonly-used techniques such as for-

Input features

Professional occupation related Investment related

Feedback from Feedback to

Feature selection Input processing Feature selection Input processing

previous step next step … …

Aggregate information

Predicted output (whether the income level >$50k)

selection enables interpretability and better learning as the capacity is used for the most salient features. TabNet employs multiple decision blocks that focus on processing a subset of input features for reasoning. Two decision blocks shown as examples process features that are related to professional occupation and investments, respectively, in order to predict the income level.

Unsupervised pre-training Supervised fine-tuning

Age Cap. gain Education Occupation Gender Relationship Age Cap. gain Education Occupation Gender Relationship 5 2000 ? Exec-managerial F Wife 6 2000 Bachelors Exec-managerial M Husband 1 0 ? Farming-fishing M ? 2 0 High-school Farming-fishing M Unmarried

? 50 Doctorate Prof-specialty M Husband 4 50 Doctorate Prof-specialty M Husband 2 ? ? Handlers-cleaners F Wife 2 0 High-school Handlers-cleaners F Wife 5 3000 Bachelors ? ? Husband 5 3000 Bachelors Exec-managerial M Husband

3 0 Bachelors ? F ? 3 100 Bachelors Prof-specialty F Wife ? 0 High-school Armed-Forces ? Husband 2 0 High-school Armed-Forces M Husband

TabNet encoder TabNet encoder

TabNet decoder Decision making

Age Cap. gain Education Occupation Gender Relationship Income > $50k

Masters True

High-school Unmarried False

4 True

0 High-school F False

Exec-managerial M True

Adm-clerical Wife True

3 M False

level can be guessed from the occupation, or the gender can be guessed from the relationship. Unsupervised representation learning by masked self-supervised learning results in an improved encoder model for the supervised learning task.

ward selection and Lasso regularization (Guyon and Elisseeff performance with compact representations. 2003) attribute feature importance based on the entire training Tree-based learning: DTs are commonly-used for tabular data, and are referred as global methods. Instance-wise fea- data learning. Their prominent strength is efficient picking ture selection refers to picking features individually for each of global features with the most statistical information gain

to maximize the mutual information between the selected mance of standard DTs, one common approach is ensembling features and the response variable, and in (Yoon, Jordon, and to reduce variance. Among ensembling methods, random van der Schaar 2019) by using an actor-critic framework to forests (Ho 1998) use random subsets of data with randomly mimic a baseline while optimizing the selection. Unlike these, selected features to grow many trees. XGBoost (Chen and

sity in end-to-end learning – a single model jointly performs recent ensemble DT approaches that dominate most of the feature selection and output mapping, resulting in superior recent data science competitions. Our experimental results

!# + Softmax !" < % !" > % !# > & !# > &

ReLU ReLU &

$" !" − $" % −1 −$" !" + $" % −1 −1 $# !# − $# & % −1 −$# !# + $# & !"

FC FC

W: [$" , - $" , 0, 0] W: [0, 0, $# , - $# ] !" < % b: [-a $" , a $" , -1, -1] b: [-1, -1, -d $# , d $# ] !# < & !" > % !# < & [!" ] [!# ]

Mask Mask

M: [1, 0] M: [0, 1]

(right). Relevant features are selected by using multiplicative sparse masks on inputs. The selected features are linearly transformed, and after a bias addition (to represent boundaries) ReLU performs region selection by zeroing the regions. Aggregation of multiple regions is based on addition. As C and C get larger, the decision boundary gets sharper.

for various datasets show that tree-based models can be out- constructs a sequential multi-step architecture, where each performed when the representation capacity is improved with step contributes to a portion of the decision based on the deep learning while retaining their feature selecting property. selected features; (iii) improves the learning capacity via non- Integration of DNNs into DTs: Representing DTs with linear processing of the selected features; and (iv) mimics

DNN building blocks as in (Humbird, Peterson, and McClar- ensembling via higher dimensions and more steps. ren 2018) yields redundancy in representation and ineffi- cient learning. Soft (neural) DTs (Wang, Aggarwal, and Liu Fig. 4 shows the TabNet architecture for encoding tabu- functions, instead of non-differentiable axis-aligned splits. mapping of categorical features with trainable embeddings.

However, losing automatic feature selection often degrades We do not consider any global feature normalization, but performance. In (Yang, Morillo, and Hospedales 2018), a soft merely apply batch normalization (BN). We pass the same D- binning function is proposed to simulate DTs in DNNs, by dimensional features f ∈ <B×D to each decision step, where 2019) proposes a DNN architecture by explicitly leveraging multi-step processing with Nsteps decision steps. The ith

expressive feature combinations, however, learning is based step inputs the processed information from the (i − 1)th step on transferring knowledge from gradient-boosted DT. (Tanno to decide which features to use and outputs the processed ing from primitive blocks while representation learning into sion. The idea of top-down attention in the sequential form edges, routing functions and leaf nodes. TabNet differs from is inspired by its applications in processing visual and text

these as it embeds soft feature selection with controllable data (Hudson and Manning 2018) and reinforcement learn- Self-supervised learning: Unsupervised representation relevant information in high dimensional input. learning improves supervised learning especially in small Feature selection: We employ a learnable mask M[i] ∈ has shown significant advances – driven by the judicious capacity of a decision step is not wasted on irrelevant

choice of the unsupervised learning objective (masked input ones, and thus the model becomes more parameter effi- prediction) and attention-based deep learning. cient. The masking is multiplicative, M[i] · f . We use an attentive transformer (see Fig. 4) to obtain the masks us- TabNet for Tabular Learning ing the processed features from the preceding step, a[i − 1]:

M[i] = sparsemax(P[i − 1] · hi (a[i − 1])). Sparsemax nor-

DTs are successful for learning from real-world tabular malization (Martins and Astudillo 2016) encourages sparsity datasets. With a specific design, conventional DNN building by mapping the Euclidean projection onto the probabilistic blocks can be used to implement DT-like output manifold, simplex, which is observed to be superior in performance and e.g. see Fig. 3). In such a design, individual feature selec- aligned with the goal of sparse feature selection for explain-

tion is key to obtain decision boundaries in hyperplane form, PD which can be generalized to a linear combination of features ability. Note that j=1 M[i]b,j = 1. hi is a trainable func- where coefficients determine the proportion of each feature. tion, shown in Fig. 4 using a FC layer, followed by BN. P[i] TabNet is based on such functionality and it outperforms DTs is the prior scale term, denoting how much a particular feature

Qi while reaping their benefits by careful design which: (i) uses has been used previously: P[i] = j=1 (γ − M[j]), where γ sparse instance-wise feature selection learned from data; (ii) is a relaxation parameter – when γ = 1, a feature is enforced

+ Softmax

Split Split

Feature Feature …

transformer transformer

x Nsteps Features

+ Softmax

Step 1 Step 2

Split + …

Split FC Output

ReLU ReLU

Split Split

Feature …

transformer transformer Feature Feature Feature Feature transformer

Encoded representation

transformer transformer Attentive transformer … Mask transformer …

Attentive Attentive SharedStep

Mask Mask 1 decision steps

Step 2 Decision step dependent

transformer transformer

BN Feature Feature

FC BN transformer transformer

+ 0.5 0.5 0.5 Agg. Agg. Features Features FC FC + +

Reconstructed + … Feature attributes + … features

(a) TabNet encoder architecture (b) TabNet decoder architecture Feature transformer Feature Attentive transformer Shared across decision steps Decision step dependent transformer GLU

Shared across decision steps

Decision step dependent Prior scales

+ 0.5 0.5 0.5

0.5 0.5 0.5

+ Attentive transformer (c) (d)

Prior scales

divides the processed representation to be used by the attentive transformer of the subsequent step as well as for the overall Attentive BN FC

output. For each step, the feature selection mask provides interpretable information about the model’s functionality, and the +

masks can be aggregated to obtain global feature transformer important attribution. (b) TabNet decoder, composed of a feature transformer block at each step. (c) A feature transformer block example – 4-layer network is shown, where 2 are shared across all decision

Prior scales

steps and 2 are decision step-dependent. Each layer is composed of a fully-connected (FC) layer, BN and GLU nonlinearity. (d) +

An attentive transformer block example – a single layer mapping is modulated with a prior scale information which aggregates Sparsemax

how much each feature has been used before the current decision step. sparsemax (Martins and Astudillo 2016) is used for BN FC

normalization of the coefficients, resulting in sparse selection of the salient features. +

to be used only at one decision step and as γ increases, more propose the aggregate.feature importance mask, Magg−b,j = flexibility is provided to use a feature at multiple decision PNsteps ηb [i]Mb,j [i]

PD PNsteps

ηb [i]Mb,j [i].2 i=1 i=1 steps. P is initialized as all ones, 1B×D , without any prior j=1

on the masked features. If some features are unused (as in self- Tabular self-supervised learning: We propose a decoder supervised learning), corresponding P entries are made 0 architecture to reconstruct tabular features from the Tab- to help model’s learning. To further control the sparsity of the Net encoded representations. The decoder is composed of selected features, we propose sparsity regularization in the feature transformers, followed by FC layers at each deci-

form of entropy (Grandvalet and Bengio 2004), Lsparse = sion step. The outputs are summed to obtain the recon-

PNsteps PB PD −Mb,j [i] log(Mb,j [i]+)

i=1 b=1 j=1 Nsteps ·B , where is a structed features. We propose the task of prediction of miss- small number for numerical stability. We add the sparsity reg- ing feature columns from the others. Consider a binary mask ularization to the overall loss, with a coefficient λsparse . Spar- S ∈ {0, 1}B×D . The TabNet encoder inputs (1 − S) · f̂ sity provides a favorable inductive bias for datasets where and the decoder outputs the reconstructed features, S · f̂ . We

most features are redundant. initialize P = (1 − S) in encoder so that the model em- Feature processing: We process the filtered features using phasizes merely on the known features, and the decoder’s last a feature transformer (see Fig. 4) and then split for the FC layer is multiplied with S to output the unknown features. decision step output and information for the subsequent We consider the reconstruction loss in self-supervised phase:

step, [d[i], a[i]] = fi (M[i] · f ), where d[i] ∈ <B×Nd and 2

PB PD (f̂b,j −fb,j )·Sb,j

a[i] ∈ <B×Na . For parameter-efficient and robust learning b=1 j=1

√ PB PB 2

. Normalization b=1 (fb,j −1/B b=1 fb,j ) with high capacity, a feature transformer should comprise layers that are shared across all decision steps (as the same with the population standard deviation of the ground truth features are input across different decision steps), as well as is beneficial, as the features may have different ranges. We decision step-dependent layers. Fig. 4 shows the implementa- sample Sb,j independently from a Bernoulli distribution with

tion as concatenation of two shared layers and two decision parameter ps , at each iteration. step-dependent layers. Each FC layer is followed by BN and eventually connected to a normalized residual √ connection We study TabNet in wide range of problems, that contain with normalization. Normalization with 0.5 helps to sta- regression or classification tasks, particularly with published bilize learning by ensuring that the variance throughout the benchmarks. For all datasets, categorical inputs are mapped

For faster training, we use large batch sizes with BN. Thus, bedding and numerical columns are input without and pre- except the one applied to the input features, we use ghost BN processing.4 We use standard classification (softmax cross (Hoffer, Hubara, and Soudry 2017) form, using a virtual batch entropy) and regression (mean squared error) loss functions size BV and momentum mB . For the input features, we ob- and we train until convergence. Hyperparameters of the Tab-

serve the benefit of low-variance averaging and hence avoid Net models are optimized on a validation set and listed in ghost BN. Finally, inspired by decision-tree like aggregation Appendix. TabNet performance is not very sensitive to most as in Fig. 3, we construct the overall decision embedding hyperparameters as shown with ablation studies in Appendix. as dout = i=1 PNsteps ReLU(d[i]). We apply a linear mapping In Appendix, we also present ablation studies on various de-

Wfinal dout to get the output mapping.1 sign and guidelines on selection of the key hyperparameters. Interpretability: TabNet’s feature selection masks can shed For all experiments we cite, we use the same training, val- light on the selected features at each step. If Mb,j [i] = 0, idation and testing data split with the original work. Adam optimization algorithm (Kingma and Ba 2014) and Glorot then j th feature of the bth sample should have no contribution uniform initialization are used for training of all models.5

to the decision. If fi were a linear function, the coefficient

Mb,j [i] would correspond to the feature importance of fb,j . Instance-wise feature selection

Although each decision step employs non-linear processing, their outputs are combined later in a linear way. We aim Selection of the salient features is crucial for high perfor- to quantify an aggregate feature importance in addition to mance, especially for small datasets. We consider 6 tabular requires a coefficient that can weigh the relative importance samples). The datasets are constructed in such a way that of each step in the decision. We simply propose ηb [i] = only a subset of the features determine the output. For Syn1-

PNd Syn3, salient features are same for all instances (e.g., the

c=1 ReLU(db,c [i]) to denote the aggregate decision con- tribution at ith decision step for the bth sample. Intuitively, if 2

Normalization is used to ensure D

P j=1 Magg−b,j = 1. db,c [i] < 0, then all features at ith decision step should have 3

In some cases, higher dimensional embeddings may slightly im-

0 contribution to the overall decision. As its value increases, prove the performance, but interpretation of individual dimensions

it plays a higher role in the overall linear combination. Scal- may become challenging. ing the decision mask at each decision step with ηb [i], we Specially-designed feature engineering, e.g. logarithmic trans- formation of variables highly-skewed distributions, may further

1 improve the results but we leave it out of the scope of this paper.

For discrete outputs, we additionally employ softmax during

training (and argmax during inference). An open-source implementation will be released.

Global: using only globally-salient features, Tree Ensembles (Geurts, Ernst, and Wehenkel 2006), Lasso-regularized model, L2X

Test AUC

Syn Syn Syn Syn Syn Syn

No selection .5 ± .0 .7 ± .0 .8 ± .0 .5 ± .0 .6 ± .0 .6 ± .0 Tree .5 ± .1 .8 ± .0 .8 ± .0 .6 ± .0 .7 ± .0 .7 ± .0 Lasso-regularized .4 ± .0 .5 ± .0 .8 ± .0 .5 ± .0 .6 ± .0 .7 ± .0

L2X .4 ± .0 .8 ± .0 .8 ± .0 .6 ± .0 .7 ± .0 .8 ± .0

INVASE .6 ± .0 .8 ± .0 .9 ± .0 .7 ± .0 .7 ± .0 .8 ± .0

Global .6 ± .0 .8 ± .0 .9 ± .0 .7 ± .0 .7 ± .0 .8 ± .0 TabNet .6 ± .0 .8 ± .0 .8 ± .0 .7 ± .0 .7 ± .0 .8 ± .0

output of Syn depends on features X -X ), and global fea- Table 3: Performance for Poker Hand induction dataset. ture selection, as if the salient features were known, would give high performance. For Syn4-Syn6, salient features are Model Test accuracy (%) instance dependent (e.g., for Syn4, the output depends on ei- DT 50.0 ther X -X or X -X depending on the value of X ), which MLP 50.0

makes global feature selection suboptimal. Table 1 shows that Deep neural DT 65.1

TabNet outperforms others (Tree Ensembles (Geurts, Ernst, XGBoost 71.1

and Wehenkel 2006), LASSO regularization, L2X (Chen LightGBM 70.0 van der Schaar 2019). For Syn1-Syn3, TabNet performance TabNet 99.2 is close to global feature selection - it can figure out what Rule-based 100.0 features are globally important. For Syn4-Syn6, eliminating instance-wise redundant features, TabNet improves global feature selection. All other methods utilize a predictive model Poker Hand (Dua and Graff 2017): The task is classifica-

with 43k parameters, and the total number of parameters is tion of the poker hand from the raw suit and rank attributes of 101k for INVASE due to the two other models in the actor- the cards. The input-output relationship is deterministic and critic framework. TabNet is a single architecture, and its size hand-crafted rules can get 100% accuracy. Yet, conventional is 26k for Syn1-Syn and 31k for Syn4-Syn6. The compact DNNs, DTs, and even their hybrid variant of deep neural DTs

representation is one of TabNet’s valuable properties. (Yang, Morillo, and Hospedales 2018) severely suffer from the imbalanced data and cannot learn the required sorting and Performance on real-world datasets ranking operations (Yang, Morillo, and Hospedales 2018).

Tuned XGBoost, CatBoost, and LightGBM show very slight

as it can perform highly-nonlinear processing with its depth, Model Test accuracy (%) without overfitting thanks to instance-wise feature selection.

XGBoost 89.3

LightGBM 89.2

CatBoost 85.1 Table 4: Performance on Sarcos dataset. Three TabNet mod-

AutoML Tables 94.9 els of different sizes are considered.

TabNet 96.9

Model Test MSE Model size

Random forest 2.3 16.7K

Stochastic DT 2.1 28K

Forest Cover Type (Dua and Graff 2017): The task is clas- MLP 2.1 0.14M

sification of forest cover type from cartographic variables. Adaptive neural tree 1.2 0.60M approaches that are known to achieve solid performance (AutoML 2019), an automated search framework based on TabNet-M 0.2 0.59M ensemble of models including DNN, gradient boosted DT, TabNet-L 0.1 1.75M with very thorough hyperparameter search. A single TabNet without fine-grained hyperparameter search outperforms it. Sarcos (Vijayakumar and Schaal 2000): The task is re-

gressing inverse dynamics of an anthropomorphic robot arm.

very small model is possible with a random forest. In the very and TabNet merely focuses on the relevant ones. For Syn4, small model size regime, TabNet’s performance is on par the output depends on either X -X or X -X depending parameters. When the model size is not constrained, TabNet feature selection – it allocates a mask to focus on the indi- achieves almost an order of magnitude lower test MSE. cator X , and assigns almost all-zero weights to irrelevant

features (the ones other than two feature groups). models are denoted with -S and -M. Real-world datasets: We first consider the simple task of mushroom edibility prediction (Dua and Graff 2017). Tab- Model Test acc. (%) Model size Net achieves 100% test accuracy on this dataset. It is indeed Sparse evolutionary MLP 78.4 81K known (Dua and Graff 2017) that “Odor” is the most discrim-

Gradient boosted tree-S 74.2 0.12M inati

Frequently Asked Questions

What is this project about?

This project covers practical implementation and research aspects of the topic using AI/ML techniques.

This section provides additional detailed analysis and supporting information derived from the research paper content to ensure comprehensive coverage of the topic with expanded discussion on key concepts, methods, and findings presented in the source material. The practical implications and implementation considerations are further elaborated to support complete understanding of the subject matter across different application domains and use cases.

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Ai Drug Discovery

TabNet: Attentive Interpretable Tabular Learning

Sercan Ö. Arık, Tomas Pfister

Google Cloud AI

Sunnyvale, CA

We propose a novel high-performance and interpretable canon-

Introduction We propose a new canonical DNN architecture for tabular

Why is deep learning worth exploring for tabular data?

Intelligence (www.aaai.org). All rights reserved. ness for prediction. Commonly-used techniques such as for-

Input features

Professional occupation related Investment related

Feature selection Input processing Feature selection Input processing

Aggregate information

Predicted output (whether the income level >$50k)

Unsupervised pre-training Supervised fine-tuning

TabNet encoder TabNet encoder

TabNet decoder Decision making

Masters True

High-school Unmarried False

4 True

0 High-school F False

Exec-managerial M True

Adm-clerical Wife True

3 M False

ReLU ReLU &

FC FC

Mask Mask

M: [1, 0] M: [0, 1]

M[i] = sparsemax(P[i − 1] · hi (a[i − 1])). Sparsemax nor-

Split Split

Feature Feature …

Step 1 Step 2

Split + …

Split FC Output

ReLU ReLU

Split Split

Feature …

Encoded representation

Attentive Attentive SharedStep

Mask Mask 1 decision steps

Step 2 Decision step dependent

BN Feature Feature

Shared across decision steps

Decision step dependent Prior scales

Prior scales

Prior scales

PD PNsteps

PNsteps PB PD −Mb,j [i] log(Mb,j [i]+)

PB PD (f̂b,j −fb,j )·Sb,j

√ PB PB 2

Mb,j [i] would correspond to the feature importance of fb,j . Instance-wise feature selection

PNd Syn3, salient features are same for all instances (e.g., the

Normalization is used to ensure D

In some cases, higher dimensional embeddings may slightly im-

0 contribution to the overall decision. As its value increases, prove the performance, but interpretation of individual dimensions

1 improve the results but we leave it out of the scope of this paper.

For discrete outputs, we additionally employ softmax during

Test AUC

Syn Syn Syn Syn Syn Syn

L2X .4 ± .0 .8 ± .0 .8 ± .0 .6 ± .0 .7 ± .0 .8 ± .0

INVASE .6 ± .0 .8 ± .0 .9 ± .0 .7 ± .0 .7 ± .0 .8 ± .0

TabNet outperforms others (Tree Ensembles (Geurts, Ernst, XGBoost 71.1

Tuned XGBoost, CatBoost, and LightGBM show very slight

XGBoost 89.3

LightGBM 89.2

CatBoost 85.1 Table 4: Performance on Sarcos dataset. Three TabNet mod-

TabNet 96.9

Model Test MSE Model size

Random forest 2.3 16.7K

Stochastic DT 2.1 28K

Forest Cover Type (Dua and Graff 2017): The task is clas- MLP 2.1 0.14M

Gradient boosted tree-S 74.2 0.12M inati

Frequently Asked Questions

What is this project about?

PNsteps PB PD −Mb,j [i] log(Mb,j [i]+)