Testing Semantic Importance via Betting
Jeremias Sulam
Statistics of Trustworthy ML
@
2024 IMS International Conference on Statistics and Data Science (ICSDS)
"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. [...]
We want AI agents that can discover like we can, not which contain what we have discovered."The Bitter Lesson, Rich Sutton 2019
Interpretability in Image Classification
\((X,Y) \in \mathcal X \times \mathcal Y\)
\((X,Y) \sim P_{X,Y}\)
\(\hat{Y} = f(X) : \mathcal X \to \mathcal Y\)
Setting:
-
What features are important for this prediction?
-
What does importance mean, exactly?
-
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
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Adebayo et al, Sanity checks for saliency maps, 2018
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Ghorbani et al, Interpretation of neural networks is fragile, 2019
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Shah et al, Do input gradients highlight discriminative features? 2021
Is the piano important for \(\hat Y = \text{cat}\)?
Semantic Interpretability of classifiers
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?
Is the piano important for \(\hat Y = \text{cat}\)?
Semantic Interpretability of classifiers
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?
Post-hoc Interpretability Methods
Interpretable by
construction
Is the piano important for \(\hat Y = \text{cat}\)?
Semantic Interpretability of classifiers
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?
Post-hoc Interpretability Methods
Interpretable by
construction
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: \(Z = C^\top H \in R^m\)
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... )
Concept Bank: \(C = [c_1, c_2, \dots, c_m] \in \mathbb R^{d\times m}\)
Semantic Interpretability of classifiers
Vision-language models
(training)
[Radford et al, 2021]
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
\[H^{j,S}_0:~ g({\tilde H_{S \cup \{j\}}}) \overset{d}{=} g(\tilde H_S), \qquad \tilde H_S \sim P_{H|Z_S = C_S^\top f(x)} \]
Tightly related to Shapley values
[Teneggi et al, The Shapley Value Meets Conditional Independence Testing, 2023]
Precise notions of semantic importance
\(\hat{Y}_\text{gas pump}\)
\(Z_S\cup Z_{j}\)
\(Z_{S}\)
\(Z_j=\)
Local Conditional Importance
\[H^{j,S}_0:~ g({\tilde H_{S \cup \{j\}}}) \overset{d}{=} g(\tilde H_S), \qquad \tilde H_S \sim P_{H|Z_S = C_S^\top f(x)} \]
\(\tilde{Z}_S = [z_\text{text}, z_\text{old}, Z_\text{dispenser}, Z_\text{trumpet}, Z_\text{fire}, \dots ] \)
\(S\)
\(\tilde{Z}_{S\cup j} = [z_\text{text}, z_\text{old}, z_\text{dispenser}, Z_\text{trumpet}, Z_\text{Fire}, \dots ] \)
\(S\)
\(j\)
Precise notions of semantic importance
\(\hat{Y}_\text{gas pump}\)
\(\hat{Y}_\text{gas pump}\)
\(Z_S\cup Z_{j}\)
\(Z_{S}\)
\(Z_S\cup Z_{j}\)
\(Z_{S}\)
Local Conditional Importance
\(Z_j=\)
\(Z_j=\)
\[H^{j,S}_0:~ g({\tilde H_{S \cup \{j\}}}) \overset{d}{=} g(\tilde H_S), \qquad \tilde H_S \sim P_{H|Z_S = C_S^\top f(x)} \]
\(\tilde{Z}_S = [z_\text{text}, z_\text{old}, Z_\text{dispenser}, Z_\text{trumpet}, Z_\text{fire}, \dots ] \)
\(\tilde{Z}_{S\cup j} = [z_\text{text}, z_\text{old}, Z_\text{dispenser}, z_\text{trumpet}, Z_\text{Fire}, \dots ] \)
\(S\)
\(S\)
\(j\)
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
[Grünwald 2019, Shaer et al. 2023, Shekhar and Ramdas 2023]
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 \(e\geq 1/\alpha\)
- Consider a wealth process
\(K_0 = 1;\)
\(\text{for}~ t = 1, \dots \\ \quad K_t = K_{t-1}(1+\kappa_t v_t)\)
\(\mathbb P_{H_0}[\exists t\in \mathbb N : K_t \geq 1/\alpha ] \leq \alpha \)
Online testing by e-values
[Shafer 2021, Shaer et al. 2023, Shekhar and Ramdas 2023 ]
Fair game (test martingale): \(~~\mathbb E_{H_0}[\kappa_t | \text{Everything seen}_{t-1}] = 0\)
\(v_t \in (0,1):\) betting fraction
\(\kappa_t \in [-1,1]\) payoff
Testing by betting via SKIT (Podkopaev et al., 2023)
Testing by betting via SKIT (Podkopaev et al., 2023)
Online testing by e-values
\(v_t \in (0,1):\) betting fraction
\(H_0: ~ P = Q\)
\(\kappa_t = \text{tanh}({\color{teal}\rho(X_t)} - {\color{teal}\rho(Y_t)})\)
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
\(X_t \sim P, Y_t \sim Q\)
rejection time
rejection rate
Important Semantic Concepts
(Reject \(H_0\))
Unimportant Semantic Concepts
(fail to reject \(H_0\))
Results
- Type 1 error control
- False discovery rate control
Results: Local Testing
Important Semantic Concepts
(Reject \(H_0\))
Unimportant Semantic Concepts
(Fail to reject)
- Type 1 error control
- False discovery rate control
rejection time
rejection rate
0.0
1.0
Results: Local Testing
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
Results: Imagenette
Semantic comparison of vision-language models
...that's it!
Jacopo Teneggi
JHU
Appendix
Concept Bank: \(C = [c_1, c_2, \dots, c_m] \in \mathbb R^{d\times m}\)
Semantic Interpretability of classifiers
Vision-language models
(training)
[Radford et al, 2021]
(inference)
