From d279c2888e7e0a00d72d364f741273f0ea023492 Mon Sep 17 00:00:00 2001 From: Pierre-Elliott Thiboud Date: Fri, 20 Jan 2023 11:15:09 +0100 Subject: [PATCH 1/4] Draft of SHAP values review --- collections/_posts/2023-01-10-SHAPValues.md | 172 ++++++++++++++++++++ 1 file changed, 172 insertions(+) create mode 100644 collections/_posts/2023-01-10-SHAPValues.md diff --git a/collections/_posts/2023-01-10-SHAPValues.md b/collections/_posts/2023-01-10-SHAPValues.md new file mode 100644 index 00000000..6d0431a7 --- /dev/null +++ b/collections/_posts/2023-01-10-SHAPValues.md @@ -0,0 +1,172 @@ +--- +layout: review +title: SHAP values +tags: explainability machine-learning black-box post-hoc +author: Pierre-Elliott Thiboud +cite: + authors: "Scott M. Lundberg, Su-In Lee" + title: "A Unified Approach to Interpreting Model Predictions" + venue: "Advances in Neural Information Processing Systems 30 (NeurIPS 2017)" +pdf: "http://papers.nips.cc/paper/by-source-2017-2493" +--- + +# Highlights + +- +- Unification of feature attribution methods under same paradigm +- Optimisation of Shapley values computations + +# Introduction + +Currently used Machine-learning and Deep-learning models are increasingly powerful and complex. However, for legal, ethical or technical reasons, there is a growing need to understand how their predictions are made. To this end, feature attribution models try to estimate how inputs and outputs are correlated. The authors propose a framework unifying existing explanation models with improved computational performances and better theoretical foundations. + +> Need more than a simple introduction +> Maybe: recap of the needs, quick overview of explanation models (just their name and the idea behind them) + +# Additive feature attribution models + +## Defining explanations + +Viewing explanations of a model's prediction as a model itself allow oneself to design this _explanation model_ to exhibit whatever properties are wanted. In this case, where we want explanations to be accurate and easily interpretable, additive feature attribution models have many advantages. + +The goal of those models is to _attribute_ an effect to all considered _features_ of the input while following the additive property: summing the effects for all features approximates the output of the original model. Thus, explanations of a model output can be reformulated as a linear function of binary variables: + +$$ g(z') = \phi_0 + \sum_{i=1}^{M} \phi_i z'_i $$ + +where $$z'$$ is a simplified input, $$z' \in \{0, 1\}^M$$, _M_ is the number of simplified input features, and $$\phi_i \in \mathbb{R}$$. + +> Need to define $$x=h_x(x')$$ +> Also need to make clear distinction between $$\,x$$, $$x'$$ and $$z'$$ (they switch from one to the other in the paper) + +## Desirable properties + +No matter the chosen class of models, some properties are desirable for our explanations: + +**Property 1 - Local Accuracy** +The explanation should accurately represent the prediction: + +$$f(x)=g(x')=\phi_0 + \sum_{i=1}^{M} \phi_i x'_i$$ + +where $$\phi_0 = f(h_x(\mathbf{0}))$$ represents the model with all simplified inputs missing. + +**Property 2 - Missingness** +If a feature is missing, no effect should be attributed to it: + +$$x'_i=0 \implies \phi_i=0$$ + +**Property 3 - Consistency** +Feature's effect should be correlated to "real" effect: + +Let $$f_x(z') = f(h_x(z'))$$ and $$z' \setminus i$$ denotes setting $$z'_i=0$$. For any two models $$f$$ and $$f'$$, if (for all inputs $$z' \in \{0, 1\}^M$$) + +$$f'_x(z') - f'_x(z'\setminus i) \geq f_x(z') - f_x(z'\setminus i)$$ + +Then, + +$$\phi_i(f', x) \geq \phi_i(f, x)$$ + +## Defined model + +With those properties and the desired additive feature attribution class of models, there exists only one possible explanation model defined as follow: + +$$\phi_i(f,x) = \sum_{z' \subseteq x'} \frac{|z'|! (M - |z'| - 1)!}{M!} \left(f_x(z') - f_x(z' \setminus i)\right)$$ + +where \|$$z'$$\| is the number of non-zero entries in $$z'$$, and $$z' \subseteq x'$$ represents all $$z'$$ vectors where the non-zero entries are a subset of the non-zero entries in $$x'$$. + +[Young (1985)](https://link.springer.com/article/10.1007/BF01769885) demonstrated that Shapley values are the only set of values that satisfy three axioms similar to properties 1 and 3 (and another property proved by the authors to be redundant in this setting). So if a method is not based on Shapley values, it violates local accuracy and/or consistency. + +> Add details about the proof? + +# Shapley values + +## Properties + +> Start from linear model example to go to cooperative game theory? + +To explain how features gave a specific prediction, we can consider each feature as a player in a cooperative game. Shapley (1952)[^1] formalized a method to fairly distribute the "payout" accross each player whom contributed to the game. In our case, it allows us to estimate how each feature (player) contributed to the current prediction (game). + +A feature value's Shapley value is its contribution to the payout, weighted and summed over all possible feature value coalition: + +$$\phi_i(val, x) = \sum_{S \subseteq \{1,...,p\} \setminus \{i\}} \frac{|S|! (p - |S| - 1)!}{p!} (val_x(S \cup \{i\}) - val_x(S))$$ + +> It means[^4]: +> $$\frac{|S|!(p-|S|-1)!}{p!} = \left( |S| \binom{p}{|S|} \right)^{-1}$$ +> But why? How? + +where $$S$$ is a subset of the features used in the model, $$x$$ is the input to be explained, $$p$$ the number of considered features and $$val_x(S)$$ is the prediction for feature values in set $$S$$ that are marginalized over features that are not included in set $$S$$. + +> Probably missing explanation or don't go into this much details + +Shapley values satisfy 4 properties: + +**Efficiency** Feature contributions sum equals the difference between prediction for $$x$$ and the average prediction: + +$$\sum_{i=1}^p \phi_i = f(x) - E_X(f(X))$$ + +**Symmetry** If two feature values contribute equally to all possible coalitions, their contributions should be the same: + +$$\forall S \subseteq \{1,...,p\} \setminus \{i,j\}, \qquad val(S \cup \{i\}) = val(S \cup \{j\}) \implies \phi_i = \phi_j$$ + +**Missingness** A feature $$i$$ which doesn't change the predicted value (no matter the coalition) should have a Shapley value of 0: + +$$\forall S \subseteq \{1,...,p\} \setminus \{i\}, \qquad val(S \cup \{i\}) = val(S) \implies \phi_i = 0$$ + +**Additivity** For a game with combined payouts ($$val_x + val_{x'}$$), the Shapley value $$i$$ is the sum of the respective Shapley values: + +$$\phi_i(val_x + val_{x'}) = \phi_i(val_x) + \phi_i(val_{x'})$$ + +## Computation + +Still Shapley values have a problem: their computation. It requires to calculate each possible coalition for each feature $$i$$, so $$2^N$$ with $$N$$ features. More than coalitions, how can we define $$z \setminus i$$, an input where feature $$i$$ is missing? How do we account for possible dependencies between features? + +To resolve this, [Strumbelj et al. (2014)](https://link.springer.com/article/10.1007/s10115-013-0679-x) proposed the _Shapley sampling_, an approximation with Monte-Carlo sampling: + +$$\hat{\phi}_i = \frac{1}{M} \sum_{m=1}^M \left( f(x_{+i}^m) - f(x_{-i}^m) \right)$$ + +where $$f(x_{+i}^m)$$ is the prediction for $$x$$ but with a random number of feature values replaced by feature values from a random data point $$z$$, except for feature $$i$$. In contrast to $$x_{+i}^m$$, in $$x_{-i}^m$$, the feature $$i$$ is also taken from the sampled $$z$$. This creates some sort of "Frankenstein monster" instance assembled from two instances which, with enough iteration $$M$$, approximates the removal of a single feature value while keeping _some_ feature interactions. + +This result in the following algorithm to compute the Shapley value for a single feature $$i$$: + +1. For all $$m=1,...,M$$: + 1. Choose a random permutation $$o$$ of the feature values + 2. Draw a random instance $$z$$ from our dataset $$X$$ + 3. Order instances with permutation $$o$$: + + $$x_o = (x_{(1)},...,x_{(i)},...,x_{(p)})$$ + + $$z_o = (z_{(1)},...,z_{(i)},...,z_{(p)})$$ + + 4. Construct two new instances: + + $$x_{+i} = (x_{(1)},...,x_{(i-1)},x_{(i)},z_{(i+1)},...,z_{(p)})$$ + + $$x_{-i} = (x_{(1)},...,x_{(i-1)},z_{(i)},z_{(i+1)},...,z_{(p)})$$ + + 5. Compute the marginal contribution: $$\hat{\phi}_i^m = f(x_{+i}) - f(x_{-i})$$ +2. Compute the Shapley value as the average of marginal contributions: + + $$\hat{\phi}_i(x) = \frac{1}{M} \sum_{m=1}^M \hat{\phi}_i^m$$ + +## Note + +> Move into limitations, with other disadvantages/risks + +Be careful when interpreting Shapley values: they do NOT represent the difference in the prediction if removed from the model, they are the average contribution of a feature value to the prediction in different features coalitions. + +# SHAP findings + +- KernelSHAP (model agnostic) +- Mix Shapley values & Additive models + - Rewrite some axioms +- Upgrade existing models to follow Shapley axioms +- Optimizations for specific models (decision trees, deep learning) + +> Talk about multiclass classification and images, how is it handled + +# References + +[^1]: Lloyd Shapley. A value for N-person games (1952). [Link](https://www.rand.org/content/dam/rand/pubs/papers/2021/P295.pdf) +[^2]: E. Štrumbelj and I. Kononenko. Explaining prediction models and individual predictions with feature contributions (2013). [Link](https://link.springer.com/article/10.1007/s10115-013-0679-x) +[^3]: christophm.github.io. [Link](https://christophm.github.io/interpretable-ml-book/shapley.html) +[^4]: Samuel Mazantti. [Link](https://towardsdatascience.com/shap-explained-the-way-i-wish-someone-explained-it-to-me-ab81cc69ef30) + From cc3804a663ce40b6cb4398f06a132f472d6628cd Mon Sep 17 00:00:00 2001 From: Pierre-Elliott Thiboud Date: Tue, 24 Jan 2023 20:05:32 +0100 Subject: [PATCH 2/4] Update SHAP review to make it clearer + Add images + Update .gitignore to remove automatic Mac images database (DS_Store) --- .gitignore | 1 + collections/_posts/2023-01-10-SHAPValues.md | 139 +++++++++--------- .../images/shap/coalitions_example.jpg | Bin 0 -> 789956 bytes .../images/shap/marginal_contribution.jpg | Bin 0 -> 449525 bytes .../images/shap/prediction_example.jpg | Bin 0 -> 238991 bytes collections/images/shap/shap_fig4.jpg | Bin 0 -> 483646 bytes collections/images/shap/shap_fig5.jpg | Bin 0 -> 481941 bytes 7 files changed, 67 insertions(+), 73 deletions(-) create mode 100644 collections/images/shap/coalitions_example.jpg create mode 100644 collections/images/shap/marginal_contribution.jpg create mode 100644 collections/images/shap/prediction_example.jpg create mode 100644 collections/images/shap/shap_fig4.jpg create mode 100644 collections/images/shap/shap_fig5.jpg diff --git a/.gitignore b/.gitignore index 23f3ce26..a9ab5cef 100644 --- a/.gitignore +++ b/.gitignore @@ -6,3 +6,4 @@ _previews .Rhistory *.swp .ruby-version +.DS_Store diff --git a/collections/_posts/2023-01-10-SHAPValues.md b/collections/_posts/2023-01-10-SHAPValues.md index 6d0431a7..2a186d26 100644 --- a/collections/_posts/2023-01-10-SHAPValues.md +++ b/collections/_posts/2023-01-10-SHAPValues.md @@ -12,120 +12,113 @@ pdf: "http://papers.nips.cc/paper/by-source-2017-2493" # Highlights -- +- Local explainability with proven desirable properties - Unification of feature attribution methods under same paradigm -- Optimisation of Shapley values computations +- Optimisation of Shapley values computations for specific models # Introduction -Currently used Machine-learning and Deep-learning models are increasingly powerful and complex. However, for legal, ethical or technical reasons, there is a growing need to understand how their predictions are made. To this end, feature attribution models try to estimate how inputs and outputs are correlated. The authors propose a framework unifying existing explanation models with improved computational performances and better theoretical foundations. +Currently used Machine-learning and Deep-learning models are increasingly powerful and complex. However, for legal, ethical or technical reasons, there is a growing need to understand how their predictions are made. To this end, there exists several methods approximating the model "prediction path", be it at the model scale or for individual predictions. _Global explanations_ focus on the model and its underlying decision process. Their goal is to get some sense of understanding of the mechanism by which the model works. On the other hand, _local explanations_ provide insights on why a particular predictions was made instead of another one. -> Need more than a simple introduction -> Maybe: recap of the needs, quick overview of explanation models (just their name and the idea behind them) +Among the different local explainability methods, which include for example _Saliency maps_, _Prototype based_ or _Counterfactuals explanations_, this article focus on _Feature importance_ through Additive feature attribution models. These models estimate how inputs and outputs are correlated, and "how much" each feature of an input impact the prediction. -# Additive feature attribution models +The authors propose a framework unifying existing _Feature importance_ based explanation model, proving their solution to be unique. With these theoritical results, the authors also proposed model-specific performance improvements through several assumptions. -## Defining explanations +# Shapley values -Viewing explanations of a model's prediction as a model itself allow oneself to design this _explanation model_ to exhibit whatever properties are wanted. In this case, where we want explanations to be accurate and easily interpretable, additive feature attribution models have many advantages. +## Games and cooperation -The goal of those models is to _attribute_ an effect to all considered _features_ of the input while following the additive property: summing the effects for all features approximates the output of the original model. Thus, explanations of a model output can be reformulated as a linear function of binary variables: +Let's assume we trained a model to predict apartment prices based on few features. This model predicted a price of €300,000 for an apartment and we want to explain the prediction. This apartment is located on the 2nd floor with an area of 50m² and a park nearby while cats are banned. -$$ g(z') = \phi_0 + \sum_{i=1}^{M} \phi_i z'_i $$ +![](/collections/images/shap/prediction_example.jpg) -where $$z'$$ is a simplified input, $$z' \in \{0, 1\}^M$$, _M_ is the number of simplified input features, and $$\phi_i \in \mathbb{R}$$. +For linear regression models, the answer is simple. The coefficients represent directly the impact of each feature and if it's a positive or negative effect. -> Need to define $$x=h_x(x')$$ -> Also need to make clear distinction between $$\,x$$, $$x'$$ and $$z'$$ (they switch from one to the other in the paper) +For more complex scenarios, Shapley values[^1] come to the rescue. They are a concept coming from game theory, describing how to fairly distribute a **payout** between **players** participating in a cooperative **game**. But how does it apply to explainability in machine learning? If we have a predictive model, then we can define: -## Desirable properties +- A **game**: reproduce the prediction of the model +- The **players**: the features of the input +- The **payout**: the individual impact of each feature to the prediction -No matter the chosen class of models, some properties are desirable for our explanations: +To determine the importance of a single feature, the idea is to compute the difference between a prediction with the feature and another prediction without. This is the "marginal contribution" of the feature to the prediction. -**Property 1 - Local Accuracy** -The explanation should accurately represent the prediction: +![](/collections/images/shap/marginal_contribution.jpg) -$$f(x)=g(x')=\phi_0 + \sum_{i=1}^{M} \phi_i x'_i$$ +But, as a cooperative game, multiple features are involved in the prediction, with possible hidden dependencies between them. So **each possible combination (or coalition) of features should be considered**, hence computing the "average marginal contribution". Thus, to explain the prediction for our apartment, we need to compute the predicted price for all coalitions **without** the `cat banned` feature AND all coalitions **with** the `cat banned` feature. -where $$\phi_0 = f(h_x(\mathbf{0}))$$ represents the model with all simplified inputs missing. +![](/collections/images/shap/coalitions_example.jpg) -**Property 2 - Missingness** -If a feature is missing, no effect should be attributed to it: +## Properties -$$x'_i=0 \implies \phi_i=0$$ +The formula to compute the Shapley value $$\phi_i$$ for feature $$i$$ is: -**Property 3 - Consistency** -Feature's effect should be correlated to "real" effect: +$$\phi_i(f, x) = \sum_{S \subseteq \{1,...,P\} \setminus \{i\}} \frac{(|S|-1)! (P - |S|)!}{P!} (f(S \cup \{i\}) - f(S))$$ -Let $$f_x(z') = f(h_x(z'))$$ and $$z' \setminus i$$ denotes setting $$z'_i=0$$. For any two models $$f$$ and $$f'$$, if (for all inputs $$z' \in \{0, 1\}^M$$) +Where $$f(x)$$ is the prediction of a model $$f$$ for the input $$x$$ with $$P$$ features, and $$S$$ being a subset of the features used by the model. We can find the "marginal contribution" of the feature $$i$$ weighted by a coefficient dependant on the number of features in the coalition, summed over possible coalitions. -$$f'_x(z') - f'_x(z'\setminus i) \geq f_x(z') - f_x(z'\setminus i)$$ +The role of these Shapley values being to fairly distribute a reward, they need to exhibit certain properties: -Then, +**Efficiency** Feature contributions sum equals the difference between prediction for $$x$$ and the average prediction: -$$\phi_i(f', x) \geq \phi_i(f, x)$$ +$$\sum_{i=1}^p \phi_i = f(x) - E_X(f(X))$$ -## Defined model +**Symmetry** If two feature values contribute equally to all possible coalitions, their contributions should be the same: -With those properties and the desired additive feature attribution class of models, there exists only one possible explanation model defined as follow: +$$\forall S \subseteq \{1,...,p\} \setminus \{i,j\}, \qquad f(S \cup \{i\}) = f(S \cup \{j\}) \implies \phi_i = \phi_j$$ -$$\phi_i(f,x) = \sum_{z' \subseteq x'} \frac{|z'|! (M - |z'| - 1)!}{M!} \left(f_x(z') - f_x(z' \setminus i)\right)$$ +**Missingness** A feature $$i$$ which doesn't change the predicted value (no matter the coalition) should have a Shapley value of 0: -where \|$$z'$$\| is the number of non-zero entries in $$z'$$, and $$z' \subseteq x'$$ represents all $$z'$$ vectors where the non-zero entries are a subset of the non-zero entries in $$x'$$. +$$\forall S \subseteq \{1,...,p\} \setminus \{i\}, \qquad f(S \cup \{i\}) = f(S) \implies \phi_i = 0$$ -[Young (1985)](https://link.springer.com/article/10.1007/BF01769885) demonstrated that Shapley values are the only set of values that satisfy three axioms similar to properties 1 and 3 (and another property proved by the authors to be redundant in this setting). So if a method is not based on Shapley values, it violates local accuracy and/or consistency. +**Additivity (or linearity)** The effect of a feature on the sum of 2 models $$f + f'$$ equals the sum of the respective Shapley values: -> Add details about the proof? +$$\phi_i(f + f') = \phi_i(f) + \phi_i(f')$$ -# Shapley values +# Additive feature attribution models -## Properties +## Defining explanations -> Start from linear model example to go to cooperative game theory? +Explanation models often use _simplified inputs_ $$x'$$ that map to the original inputs through a mapping function $$x = h_x(x')$$. And local methods' goal is to ensure $$g(z') \approx f(h_x(z'))$$ whenever $$z' \approx x'$$. Those _simplified inputs_ usually represent the presence or absence of each features in the input, as a binary vector with $$z_i' \in \{0, 1\}$$. -To explain how features gave a specific prediction, we can consider each feature as a player in a cooperative game. Shapley (1952)[^1] formalized a method to fairly distribute the "payout" accross each player whom contributed to the game. In our case, it allows us to estimate how each feature (player) contributed to the current prediction (game). +So, to summarize, our goal is to create a model $$g$$ which associates an explanation $$\phi$$ to a model's output $$f(x)$$: -A feature value's Shapley value is its contribution to the payout, weighted and summed over all possible feature value coalition: +$$g(f(x)) = \phi(f,x)$$ -$$\phi_i(val, x) = \sum_{S \subseteq \{1,...,p\} \setminus \{i\}} \frac{|S|! (p - |S| - 1)!}{p!} (val_x(S \cup \{i\}) - val_x(S))$$ +The **Efficiency** property of Shapley values allows us to define an additive feature attribution model as a linear function of binary variables: -> It means[^4]: -> $$\frac{|S|!(p-|S|-1)!}{p!} = \left( |S| \binom{p}{|S|} \right)^{-1}$$ -> But why? How? +$$ g(z') = \phi_0 + \sum_{i=1}^{M} \phi_i z'_i $$ -where $$S$$ is a subset of the features used in the model, $$x$$ is the input to be explained, $$p$$ the number of considered features and $$val_x(S)$$ is the prediction for feature values in set $$S$$ that are marginalized over features that are not included in set $$S$$. +where $$z'$$ is a simplified input (thus $$z' \in \{0, 1\}^M$$), _M_ is the number of simplified input features, $$\phi_i \in \mathbb{R}$$, and $$\phi_0$$ is the average prediction ($$E_X[f(X)]$$, by definition). -> Probably missing explanation or don't go into this much details +## Unification of existing methods -Shapley values satisfy 4 properties: +[Young (1985)](https://link.springer.com/article/10.1007/BF01769885) demonstrated that Shapley values are the only set of values that satisfy axioms related to their _local accuracy_ (of representing the prediction of the model) and their _consistency_ between models. So if a method is not based on Shapley values, it violates local accuracy and/or consistency. -**Efficiency** Feature contributions sum equals the difference between prediction for $$x$$ and the average prediction: +The authors then described other local explanation models such as LIME, DeepLIFT or Layer-wise Relevance Propagation (Saliency map), and showed that these methods follow the previously defined model. -$$\sum_{i=1}^p \phi_i = f(x) - E_X(f(X))$$ - -**Symmetry** If two feature values contribute equally to all possible coalitions, their contributions should be the same: +For a quick overview of these methods, LIME **[REF]** interprets individual model predictions by locally approximating the model around a given prediction. DeepLIFT **[REF]** recursively attributes, to Deep Learning models, effect to features comparing each feature value to a reference value (deemed to be uninformative for this specific feature), using a linear composition rule. And Layer-wise Relevance Propagation **[REF]** is showed (by **[REF]**) to be equivalent to DeepLIFT with each reference values being $$0$$. -$$\forall S \subseteq \{1,...,p\} \setminus \{i,j\}, \qquad val(S \cup \{i\}) = val(S \cup \{j\}) \implies \phi_i = \phi_j$$ +# SHAP values -**Missingness** A feature $$i$$ which doesn't change the predicted value (no matter the coalition) should have a Shapley value of 0: +## Problems with existing methods (LIME, ...) -$$\forall S \subseteq \{1,...,p\} \setminus \{i\}, \qquad val(S \cup \{i\}) = val(S) \implies \phi_i = 0$$ +> Show their graphs describing the problem (Fig 4 & 5) -**Additivity** For a game with combined payouts ($$val_x + val_{x'}$$), the Shapley value $$i$$ is the sum of the respective Shapley values: +![](/collections/images/shap/shap_fig4.jpg) -$$\phi_i(val_x + val_{x'}) = \phi_i(val_x) + \phi_i(val_{x'})$$ +![](/collections/images/shap/shap_fig5.jpg) -## Computation +> It seems their SHAP values are defined "to follow Shapley values", maybe that's the "problematic equation" -Still Shapley values have a problem: their computation. It requires to calculate each possible coalition for each feature $$i$$, so $$2^N$$ with $$N$$ features. More than coalitions, how can we define $$z \setminus i$$, an input where feature $$i$$ is missing? How do we account for possible dependencies between features? +## Shapley sampling -To resolve this, [Strumbelj et al. (2014)](https://link.springer.com/article/10.1007/s10115-013-0679-x) proposed the _Shapley sampling_, an approximation with Monte-Carlo sampling: +Shapley values have a problem: their computation. It requires to calculate each possible coalition for each feature $$i$$, so $$2^N$$ with $$N$$ features. To resolve this, [Strumbelj et al. (2014)](https://link.springer.com/article/10.1007/s10115-013-0679-x) proposed the _Shapley sampling_, an approximation with Monte-Carlo sampling: $$\hat{\phi}_i = \frac{1}{M} \sum_{m=1}^M \left( f(x_{+i}^m) - f(x_{-i}^m) \right)$$ -where $$f(x_{+i}^m)$$ is the prediction for $$x$$ but with a random number of feature values replaced by feature values from a random data point $$z$$, except for feature $$i$$. In contrast to $$x_{+i}^m$$, in $$x_{-i}^m$$, the feature $$i$$ is also taken from the sampled $$z$$. This creates some sort of "Frankenstein monster" instance assembled from two instances which, with enough iteration $$M$$, approximates the removal of a single feature value while keeping _some_ feature interactions. +where $$f(x_{+i}^m)$$ is the prediction for $$x$$ but with a random number of feature values replaced by feature values from a random data point $$z$$, except for feature $$i$$. _A contrario_, in $$x_{-i}^m$$, the feature $$i$$ is also taken from the sampled $$z$$. This creates some sort of "Frankenstein monster" instance assembled from two instances which, with enough iterations $$M$$, approximates the removal of a single feature value while keeping _some_ feature interactions. -This result in the following algorithm to compute the Shapley value for a single feature $$i$$: +This results in the following algorithm to compute the Shapley value for the feature $$i$$: 1. For all $$m=1,...,M$$: 1. Choose a random permutation $$o$$ of the feature values @@ -138,30 +131,30 @@ This result in the following algorithm to compute the Shapley value for a single 4. Construct two new instances: - $$x_{+i} = (x_{(1)},...,x_{(i-1)},x_{(i)},z_{(i+1)},...,z_{(p)})$$ + $$x_{+i} = (x_{o(1)},...,x_{o(i-1)},x_{o(i)},z_{o(i+1)},...,z_{o(p)})$$ - $$x_{-i} = (x_{(1)},...,x_{(i-1)},z_{(i)},z_{(i+1)},...,z_{(p)})$$ + $$x_{-i} = (x_{o(1)},...,x_{o(i-1)},z_{o(i)},z_{o(i+1)},...,z_{o(p)})$$ 5. Compute the marginal contribution: $$\hat{\phi}_i^m = f(x_{+i}) - f(x_{-i})$$ 2. Compute the Shapley value as the average of marginal contributions: - $$\hat{\phi}_i(x) = \frac{1}{M} \sum_{m=1}^M \hat{\phi}_i^m$$ +$$\hat{\phi}_i(x) = \frac{1}{M} \sum_{m=1}^M \hat{\phi}_i^m$$ -## Note +> Be careful when interpreting Shapley values: they do NOT represent the difference in the prediction if removed from the model, they are the average contribution of a feature value to the prediction in different features coalitions. -> Move into limitations, with other disadvantages/risks +## Other approximations -Be careful when interpreting Shapley values: they do NOT represent the difference in the prediction if removed from the model, they are the average contribution of a feature value to the prediction in different features coalitions. +> Just describe rapidly what is their solution -# SHAP findings +- KernelSHAP (LIME + Shapley values) + - Choosing adequate parameters for LIME equation allow to match Shapley values + - [Maybe just show LIME equation to point out which parameters need to be chosen correctly] +- DeppSHAP (DeepLIFT + Shapley values) + - SHAP values for smaller components of the networks -- KernelSHAP (model agnostic) -- Mix Shapley values & Additive models - - Rewrite some axioms -- Upgrade existing models to follow Shapley axioms -- Optimizations for specific models (decision trees, deep learning) +# Limitations -> Talk about multiclass classification and images, how is it handled +> Point some limitations? # References diff --git a/collections/images/shap/coalitions_example.jpg b/collections/images/shap/coalitions_example.jpg new file mode 100644 index 0000000000000000000000000000000000000000..17cd9514d69f2ec43c41081ae8babc444c21f33c GIT binary patch literal 789956 zcmeEv2V4_*oBtqc1jL9SC@8TZDqz-?qEcc7l|?KoAP7-ex{!4Ri4ZafQWdf`a8U_} z6h%O4q)6!4=`CnNf{HYmAcT=jCjarhdw2Kl-h2P=-rjoe-i6Nx5@2RB<@faO^L?b< z(mr7RUdug}fSep4w^wd207&`3#a&b{cL1=l0oDNkFb9|=w*ZjG-eH?yKQw_^e|Zl8 zd*!r#et%GI&A)Cl0|27DfSLcgjU)E@`7L(jzwG@luktB!zZ@}T#$UHqDoc_7>wCpp 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