What has my model learned?

Jeremias Sulam

On Shapley  Values and Testing for Semantic Importance via Betting

Theory of Interpretable AI Seminar

"The biggest lesson that can be read from 70 years of AI research is that general methods that leverage computation are ultimately the most effective, and by a large margin. [...] Seeking an improvement that makes a difference in the shorter term, researchers seek to leverage their human knowledge of the domain, but the only thing that matters in the long run is the leveraging of computation. [...]
We want AI agents that can discover like we can, not which contain what we have discovered."

The Bitter Lesson, Rich Sutton 2019

{f}\huge(
{\huge)} = \text{\texttt{sick}}

  • What parts of the image are important for this prediction?

  • What are the subsets of the input                so that 

{f}(x_C) \approx {f}(x) ?
C \subseteq [d]

Interpretability in Image Classification

  • Sensitivity or Gradient-based perturbations

  • Shapley coefficients

  • Variational formulations

  • Counterfactual & causal explanations

LIME [Ribeiro et al, '16], CAM [Zhou et al, '16], Grad-CAM [Selvaraju et al, '17]

Shap [Lundberg & Lee, '17], ...

RDE [Macdonald et al, '19], ...

[Sani et al, 2020] [Singla et al '19],..

Post-hoc Interpretability in Image Classification

  • Sensitivity or Gradient-based perturbations

  • Shapley coefficients

  • Variational formulations

  • Counterfactual & causal explanations

LIME [Ribeiro et al, '16], CAM [Zhou et al, '16], Grad-CAM [Selvaraju et al, '17]

Shap [Lundberg & Lee, '17], ...

RDE [Macdonald et al, '19], ...

[Sani et al, 2020] [Singla et al '19],..

Post-hoc Interpretability in Image Classification

Post-hoc Interpretability Methods

Interpretable by

construction

  • Sensitivity or Gradient-based perturbations

  • Shapley coefficients

  • Variational formulations

  • Counterfactual & causal explanations

LIME [Ribeiro et al, '16], CAM [Zhou et al, '16], Grad-CAM [Selvaraju et al, '17]

Shap [Lundberg & Lee, '17], ...

RDE [Macdonald et al, '19], ...

[Sani et al, 2020] [Singla et al '19],..

Post-hoc Interpretability in Image Classification

Post-hoc Interpretability Methods

Interpretable by

construction

  • efficiency

  • nullity

  • symmetry

  • exponential complexity

Lloyd S Shapley. A value for n-person games. Contributions to the Theory of Games, 2(28):307–317, 1953.

Let                       be an    -person cooperative game with characteristic function 

G = ([n],f)
n
f : \mathcal P([n]) \mapsto \mathbb R

How important is each player for the outcome of the game?

\displaystyle \phi_i = \sum_{S_j\subseteq [n]\setminus \{i\} } w_{S_j} \left[ f(S_j\cup \{i\}) - f(S_j) \right]

marginal contribution of player i with coalition S

Shapley values

f \approx \mathbb E[Y|X=x]
X \in \mathcal X \subset \mathbb R^n
Y\in \mathcal Y = \{0,1\}

inputs

responses

f:\mathcal X \to \mathcal Y
\text{For any}~ S \subset [n],~ \text{and a sample } { x} \newline \text{ define }\tilde{X}_S = [{x_S},X_{S^c}], \text{ where } X_{S^c}\sim \mathcal D_{X_S={x_S}}

predictor

Shap-Explanations

x_S
x:
x_{S^c}
f \approx \mathbb E[Y|X=x]
X \in \mathcal X \subset \mathbb R^n
Y\in \mathcal Y = \{0,1\}

inputs

responses

f:\mathcal X \to \mathcal Y
\text{For any}~ S \subset [n],~ \text{and a sample } { x} \newline \text{ define }\tilde{X}_S = [{x_S},X_{S^c}], \text{ where } X_{S^c}\sim \mathcal D_{X_S={x_S}}

predictor

Shap-Explanations

x_S
x:
x_{S^c}
\tilde{X}_S:

Shap-Explanations

x_S
x
\tilde{X}_S
f \approx \mathbb E[Y|X=x]
X \in \mathcal X \subset \mathbb R^n
Y\in \mathcal Y = \{0,1\}

inputs

responses

f:\mathcal X \to \mathcal Y
\text{For any}~ S \subset [n],~ \text{and a sample } { x} \newline \text{ define }\tilde{X}_S = [{x_S},X_{S^c}], \text{ where } X_{S^c}\sim \mathcal D_{X_S={x_S}}

predictor

\displaystyle \phi_i = \sum_{S_j\subseteq [n]\setminus \{i\}} w_j ~ \mathbb E \left[ f(\tilde X_{S_j\cup \{i\}}) - f(\tilde X_{S_j}) \right]
Scott Lundberg and Su-In Lee. A Unified Approach to Interpreting Model Predictions, NeurIPS , 2017

Shap-Explanations

f \approx \mathbb E[Y|X=x]
X \in \mathcal X \subset \mathbb R^n
Y\in \mathcal Y = \{0,1\}

inputs

responses

f:\mathcal X \to \mathcal Y
\text{For any}~ S \subset [n],~ \text{and a sample } { x} \newline \text{ define }\tilde{X}_S = [{x_S},X_{S^c}], \text{ where } X_{S^c}\sim \mathcal D_{X_S={x_S}}

predictor

Guiding Questions

Question 1)

Can we resolve the computational bottleneck (and when)?

Question 2)

What do these coefficients mean statistically?

Question 3)

How to go beyond input-features explanations?

We focus on data with certain structure:

f\huge(
{\huge)} = 0
{f}\huge(
{\huge)} = 1
{f}\huge(
{\huge)} = 0

Example:

f(x) = 1

if     contains a sick cell

x

Hierarchical Shap (h-Shap)

\text{\textbf{Assumption 1:}}~ f(x) = 1 \Leftrightarrow \exist~ i: f(\tilde X_i) = 1

Question 1) Can we resolve the computational bottleneck (and when) ?

\gamma = 2

We focus on data with certain structure:

\text{\textbf{Assumption 1:}}~ f(x) = 1 \Leftrightarrow \exist~ i: f(\tilde X_i) = 1

Theorem (informal)

  • h-Shap runs in linear time

  • Under A1, h-Shap \(\to\) Shapley

\mathcal O(2^\gamma k \log n)

Hierarchical Shap (h-Shap)

Hierarchical Shap (h-Shap)

Fast hierarchical games for image explanations, Teneggi, Luster & S., IEEE Transactions on Pattern Analysis and Machine Intelligence, 2022

Hierarchical Shap (h-Shap)

Question 2) What do these coefficients mean, statistically?

Precise notions of importance

Formal Feature Importance 

H_{0,S}:~ X_S \perp\!\!\!\perp Y | X_{[n]\setminus S}

[Candes et al, 2018]

Question 2) What do these coefficients mean, statistically?

Precise notions of importance

XRT: eXplanation Randomization Test

returns a \(\hat{p}_{i,S}\) for the test above

\text{reject} ~\Rightarrow~ i=2 \text{: important}
H^0_{i=2,S=\{1,3,4\}}
i=1
i=2
i=3
i=4

How do we test?

\text{For any } S \subseteq [n]\setminus \{{\color{Red}i}\}, \text{ and a sample } x\sim \mathcal p_X
H^0_{{\color{red}i},S}:~ (f(\tilde X_{S\cup \{{\color{red}i}\}}) ) \overset{d}{=} (f(\tilde X_{S}) )

Local Feature Importance 

Precise notions of importance

Local Feature Importance 

\text{For any } S \subseteq [n]\setminus \{{\color{Red}i}\}, \text{ and a sample } x\sim \mathcal D_X
H^0_{{\color{red}i},S}:~ (f(\tilde X_{S\cup \{{\color{red}i}\}}) ) \overset{d}{=} (f(\tilde X_{S}) )
\gamma_{i,S}
\displaystyle \phi_{\color{red}i}(x) = \sum_{S\subseteq [n]\setminus \{{\color{red}i}\}} w_{S} ~ \mathbb E \left[ f(\tilde X_{S\cup \{{\color{red}i}\}}) - f(\tilde X_S) \right]

Given the Shapley coefficient of any feature 

Then

p_{i,S}

and the (expected) p-value obtained for         , i.e.         ,

H^0_{i,S}

Theorem:

p_{i,S} \leq 1 - \gamma_{i,S}.
i\in[n],
Teneggi, Bharti, Romano, and S. "SHAP-XRT: The Shapley Value Meets Conditional Independence Testing." TMLR (2023).

Question 3)

How to go beyond input-features explanations?

Is the piano important for \(\hat Y = \text{cat}\)?

How can we explain black-box predictors with semantic features?

Is the piano important for \(\hat Y = \text{cat}\), given that there is a cute mammal in the image?

Semantic Interpretability of classifiers

Is the presence of \(\color{Blue}\texttt{edema}\) important for \(\hat Y = \text{lung opacity}\)?

How can we explain black-box predictors with semantic features?

Is the presence of \(\color{magenta}\texttt{devices}\) important for \(\hat Y = \texttt{lung opacity}\), given that there is \(\color{blue}\texttt{edema}\) in the image?

model-agnostic interpretability

lung opacity
cardiomegaly
fracture
no findding

Semantic Interpretability of classifiers

Concept Bank:     \(C = [c_1, c_2, \dots, c_m] \in \mathbb R^{d\times m}\)

Embeddings:     \(H = f(X) \in \mathbb R^d\)

Semantics:     \(Z = C^\top H \in \mathbb R^m\)

Concept Bank:     \(C = [c_1, c_2, \dots, c_m] \in \mathbb R^{d\times m}\)

Concept Activation Vectors

(Kim et al, 2018)

\(c_\text{cute}\)

Semantic Interpretability of classifiers

Vision-language models

(CLIP, BLIP, etc... )

Semantic Interpretability of classifiers

[Bhalla et al, "Splice", 2024]

Concept Bottleneck Models (CMBs)

[Koh et al '20, Yang et al '23, Yuan et al '22 ]

  • Need to engineer a (large) concept bank
  • Performance hit w.r.t. original predictor

\(\tilde{Y} = \hat w^\top Z\)

\(\hat w_j\) is the importance of the \(j^{th}\) concept

Desiderata

  • Fixed original predictor (post-hoc)
  • Global and local importance notions
  • Testing for any concepts (no need for large concept banks)
  • Precise testing with guarantees (Type 1 error/FDR control)

Precise notions of semantic importance

\(C = \{\text{``cute''}, \text{``whiskers''}, \dots \}\)

Global Importance

\(H^G_{0,j} : \hat{Y} \perp\!\!\!\perp Z_j \)

Global Conditional Importance

\(H^{GC}_{0,j} : \hat{Y} \perp\!\!\!\perp Z_j | Z_{-j}\)

Precise notions of semantic importance

Global Importance

\(C = \{\text{``cute''}, \text{``whiskers''}, \dots \}\)

\(H^G_{0,j} : g(f(X)) \perp\!\!\!\perp c_j^\top f(X) \)

Global Conditional Importance

\(H^{GC}_{0,j} : g(f(X)) \perp\!\!\!\perp c_j^\top f(X) | C_{-j}^\top f(X)\)

\(H^G_{0,j} : \hat{Y} \perp\!\!\!\perp Z_j \)

\(H^{GC}_{0,j} : \hat{Y} \perp\!\!\!\perp Z_j | Z_{-j}\)

Precise notions of semantic importance

"The classifier (its distribution) does not change if we condition

on concepts \(S\) vs on concepts \(S\cup\{j\} \)"

\(C = \{\text{``cute''}, \text{``whiskers''}, \dots \}\)

Local Conditional Importance

\[H^{j,S}_0:~ g({\tilde H_{S \cup \{j\}}}) \overset{d}{=} g(\tilde H_S), \qquad \tilde H_S \sim P_{H|Z_S = C_S^\top f(x)} \]

Precise notions of semantic importance

"The classifier (its distribution) does not change if we condition

on concepts \(S\) vs on concepts \(S\cup\{j\} \)"

\(\hat{Y}_\text{gas pump}\)

\(Z_S\cup Z_{j}\)

\(Z_{S}\)

\(Z_j=\)

Local Conditional Importance

\[H^{j,S}_0:~ g({\tilde H_{S \cup \{j\}}}) \overset{d}{=} g(\tilde H_S), \qquad \tilde H_S \sim P_{H|Z_S = C_S^\top f(x)} \]

Precise notions of semantic importance

"The classifier (its distribution) does not change if we condition

on concepts \(S\) vs on concepts \(S\cup\{j\} \)"

\(\hat{Y}_\text{gas pump}\)

\(\hat{Y}_\text{gas pump}\)

\(Z_S\cup Z_{j}\)

\(Z_{S}\)

\(Z_S\cup Z_{j}\)

\(Z_{S}\)

Local Conditional Importance

\(Z_j=\)

\(Z_j=\)

\[H^{j,S}_0:~ g({\tilde H_{S \cup \{j\}}}) \overset{d}{=} g(\tilde H_S), \qquad \tilde H_S \sim P_{H|Z_S = C_S^\top f(x)} \]

Testing by betting 

\(H^G_{0,j} : \hat{Y} \perp\!\!\!\perp Z_j     \iff      P_{\hat{Y},Z_j} = P_{\hat{Y}} \times P_{Z_j}\)

Testing importance via two-sample tests

\(H^{GC}_{0,j} : \hat{Y} \perp\!\!\!\perp Z_j | Z_{-j}     \iff      P_{\hat{Y}Z_jZ_{-j}} = P_{\hat{Y}\tilde{Z}_j{Z_{-j}}}\)

\(\tilde{Z_j} \sim P_{Z_j|Z_{-j}}\)

[Shaer et al, 2023]

[Teneggi et al, 2023]

\[H^{j,S}_0:~ g({\tilde H_{S \cup \{j\}}}) \overset{d}{=} g(\tilde H_S), \qquad \tilde H_S \sim P_{H|Z_S = C_S^\top f(x)} \]

Testing by betting

Goal: Test a null hypothesis \(H_0\) at significance level \(\alpha\)

Standard testing by p-values

Collect data, then test, and reject if \(p \leq \alpha\)

Online testing by e-values

Any-time valid inference, monitor online and reject when \(e\geq 1/\alpha\)

[Shaer et al. 2023, Shekhar and Ramdas 2023, Podkopaev et al 2023]

  • Consider a wealth process
    \(K_0 = 1;\)
    \(\text{for}~ t = 1, \dots \\ \)

Online testing by e-values

Fair game: \(~~\mathbb E_{H_0}[\kappa_t | \text{Everything seen}_{t-1}] = 0\)

\(v_t \in (0,1):\) betting fraction

\(\kappa_t \in [-1,1]\) payoff

\( K_t = K_{t-1}(1+\kappa_t v_t)\)

Testing by betting via SKIT (Podkopaev et al., 2023)

[Shaer et al. 2023, Shekhar and Ramdas 2023, Podkopaev et al 2023]

Lemma: For a fair game, \(\mathbb P_{H_0}[\exists t \in \mathbb N : K_t \geq 1/\alpha ]\leq\alpha\)

Online testing by e-values

\(v_t \in (0,1):\) betting fraction

\(H_0: ~ P = Q\)

\(\kappa_t = \text{tahn}({\color{teal}\rho(X_t)} - {\color{teal}\rho(Y_t)})\)

Payoff function

\({\color{black}\text{MMD}(P,Q)} : \text{ Maximum Mean Discrepancy}\)

\({\color{teal}\rho} = \underset{\rho\in \mathcal R:\|\rho\|_\mathcal R\leq 1}{\arg\sup} ~\mathbb E_P [\rho(X)] - \mathbb E_Q[\rho(Y)]\)

\( K_t = K_{t-1}(1+\kappa_t v_t)\)

     Data efficient 

     Rank induced by rejection time

Testing by betting via SKIT (Podkopaev et al., 2023)

[Shaer et al. 2023, Shekhar and Ramdas 2023, Podkopaev et al 2023]

rejection time

rejection rate

Important Semantic Concepts
(Reject \(H_0\))

Unimportant Semantic Concepts

(fail to reject \(H_0\))

Results:     Imagenette

Type 1 error control

   False discovery rate control

Results:     Imagenette

Results:     CUB dataset

CheXpert: validating BiomedVLP 

What concepts does BiomedVLP find important to predict                                 ? 

lung opacity

Results: RSNA Brain CT Hemorrhage Challenge

Hemorrhage
No Hemorrhage
Hemorrhage
Hemorrhage
intraparenchymal
subdural
subarachnoid
intraventricular
epidural
intraparenchymal
subarachnoid
intraventricular
epidural
subdural
intraparenchymal
subarachnoid
subdural
epidural
intraventricular
intraparenchymal
subarachnoid
intraventricular
epidural
subdural
(+)
(-)
(-)
(-)
(-)
(+)
(-)
(+)
(-)
(-)
(+)
(+)
(-)
(-)
(-)
(-)
(-)
(-)
(-)
(-)

Results:     Imagenette

Global Importance

Results:     Imagenette

Global Conditional Importance

Results:     Imagenette

Results:     Imagenette

Semantic comparison of vision-language models

Question 1)

Can we resolve the computational bottleneck (and when)?

Question 2)

What do these coefficients mean statistically?

Question 3)

How to go beyond input-features explanations?

Concluding

Distributional assumptions + hierarchical extensions

Allow us to conclude on differences in distributions

Use online testing by betting for semantic concepts

Jacopo Teneggi
JHU

Beepul Bharti
JHU

Teneggi et al, SHAP-XRT: The Shapley Value Meets Conditional Independence Testing, TMLR (2023).
Teneggi et al, Fast hierarchical games for image explanations, Teneggi, Luster & S., IEEE TPAMI (2022)
Teneggi & S., Testing Semantic Importance via Betting, Neurips (2024).
Yaniv Romano Technion

Appendix