Counterfactual explanations

6.3.10 · AI-ML › Interpretability & Explainability

Counterfactual Explanations HAIN KYA?

Yeh teen criteria kyun?

• Proximity: Agar changes bahut zyada hain, toh explanation useful nahi hai ("Approve hone ke liye, ek alag insaan ban jao").
• Validity: $f({\mathbf{x}}^{\prime })$ ko actually $y^{\prime }$ produce karna chahiye (sirf close hona kaafi nahi).
• Plausibility: ${\mathbf{x}}^{\prime }$ realistic hona chahiye (age negative nahi ho sakti, aap koi degree "un-complete" nahi kar sakte).

Counterfactuals KAISE Generate Karein

First Principles Se Derivation

Problem setup: Hamare paas $\mathbf{x}\in {\mathbb{R}}^d$ hai aur black-box $f:{\mathbb{R}}^d\to \{0,1\}$ hai jiske liye $f(\mathbf{x})=0$ (loan denied) hai. Hum chahte hain ${\mathbf{x}}^{\prime }$ jahan $f({\mathbf{x}}^{\prime })=1$ (approved) ho.

Step 1: Loss function define karo

Humein teen competing goals ko balance karna hai:

$\mathcal{L}({\mathbf{x}}^{\prime })=\underset{\text{sparsity}}{\underbrace{\Vert {\mathbf{x}}^{\prime }-\mathbf{x}{\Vert }_1}}+\lambda \underset{\text{prediction loss}}{\underbrace{\text{hinge}(f({\mathbf{x}}^{\prime }))}}+\gamma \underset{\text{realism}}{\underbrace{d_{\text{manifold}}({\mathbf{x}}^{\prime })}}$

Har term kyun?

1. $L_1$ distance ($\Vert {\mathbf{x}}^{\prime }-\mathbf{x}{\Vert }_1$): Encourage karta hai ki kam features change hon. $L_1$ sparse solutions deta hai (kaafi components zero change par rehte hain). $L_2$ kyun nahi? $L_2=\sqrt{\sum (x_i^{\prime }-x_i{)}^2}$ change ko kaafi features mein spread karta hai; $L_1=\sum |x_i^{\prime }-x_i|$ ise concentrate karta hai.

2. Prediction par Hinge loss: Binary classifier ke liye jo probability $p=f({\mathbf{x}}^{\prime })$ output karta hai: $\text{hinge}(p)=\max (0,0.5-p)$ Yeh zero hota hai jab $p>0.5$ (desired class), otherwise positive. Hinge kyun? Differentiable hai, $p$ ko decision boundary ke paar push karta hai.

3. Manifold distance: Unrealistic combinations ko penalize karta hai. Yeh ho sakta hai:

   • Nearest training point se distance: ${\min }_{{\mathbf{x}}_i\in \text{train}}\Vert {\mathbf{x}}^{\prime }-{\mathbf{x}}_i\Vert$
   • VAE reconstruction error: $\Vert \text{VAE-decode}(\text{VAE-encode}({\mathbf{x}}^{\prime }))-{\mathbf{x}}^{\prime }\Vert$
   • Feature validity constraints (age $\ge 0$, categorical levels)

Step 2: Optimization

Gradient-based (Wachter et al. 2017): ${\mathbf{x}}^{\prime (t+1)}={\mathbf{x}}^{\prime (t)}-\alpha {\nabla }_{{\mathbf{x}}^{\prime }}\mathcal{L}({\mathbf{x}}^{\prime (t)})$

${\mathbf{x}}^{\prime (0)}=\mathbf{x}$ se start karo, tab tak iterate karo jab tak $f({\mathbf{x}}^{\prime })=y^{\prime }$ na ho jaaye ya max steps tak.

Categorical features ke liye: One-hot encoding use karo, phir optimization ke baad nearest valid category par round karo.

Constraints ke liye (jaise age $\ge 18$): Har step ke baad ${\mathbf{x}}^{\prime }$ ko feasible set par project karo: ${\mathbf{x}}^{\prime }\leftarrow \text{clip}({\mathbf{x}}^{\prime },{\text{lower}}_i,{\text{upper}}_i)\text{ for each feature }i$

YEH KYU Kaam Karta Hai: Decision Boundaries ki Geometry

Ek linear classifier socho: $f(\mathbf{x})=\text{sign}({\mathbf{w}}^{\top }\mathbf{x}+b)$.

Decision boundary hai ${\mathbf{w}}^{\top }\mathbf{x}+b=0$. Agar $\mathbf{x}$ negative side par hai, toh boundary par closest point yeh hai: ${\mathbf{x}}_{\text{boundary}}=\mathbf{x}-\frac{{\mathbf{w}}^{\top }\mathbf{x}+b}{\Vert \mathbf{w}{\Vert }^2}\mathbf{w}$

Derivation: Hum $\Vert {\mathbf{x}}^{\prime }-\mathbf{x}{\Vert }^2$ minimize karna chahte hain subject to ${\mathbf{w}}^{\top }{\mathbf{x}}^{\prime }+b=0$. Lagrange multipliers use karo: $\mathcal{L}=\Vert {\mathbf{x}}^{\prime }-\mathbf{x}{\Vert }^2+\mu ({\mathbf{w}}^{\top }{\mathbf{x}}^{\prime }+b)$

${\nabla }_{{\mathbf{x}}^{\prime }}\mathcal{L}=2({\mathbf{x}}^{\prime }-\mathbf{x})+\mu \mathbf{w}=0⟹{\mathbf{x}}^{\prime }=\mathbf{x}-\frac{\mu }{2}\mathbf{w}$ lo.

Constraint mein substitute karo: ${\mathbf{w}}^{\top }(\mathbf{x}-\frac{\mu }{2}\mathbf{w})+b=0⟹\mu =\frac{2({\mathbf{w}}^{\top }\mathbf{x}+b)}{\Vert \mathbf{w}{\Vert }^2}$.

Is tarah: ${\mathbf{x}}^{\prime }=\mathbf{x}-\frac{{\mathbf{w}}^{\top }\mathbf{x}+b}{\Vert \mathbf{w}{\Vert }^2}\mathbf{w}$. ✓

Neural networks ke liye: Boundary nonlinear hai, isliye hum gradient descent use karte hain uski taraf "walk" karne ke liye.

Counterfactuals Generate Karne ke Methods

1. Gradient-Based (Wachter et al.)

Algorithm:

Initialize x' = x
for t = 1 to max_iter:
    loss = ||x' - x||₁ + λ·hinge(f(x')) + γ·plausibility(x')
    x' = x' - α·∇(loss)
    x' = project_to_valid(x')  # enforce constraints
    if f(x') == y_target: break
return x'

Pros: Fast hai, kisi bhi differentiable model ke saath kaam karta hai. Cons: Adversarial (implausible) examples dhundh sakta hai.

2. DiCE (Diverse Counterfactual Explanations)

Idea: Sirf ek ki jagah multiple diverse counterfactuals generate karo.

Kyun? Users ko options chahiye: "Ya toh income $10k badhao YA credit score 40 points improve karo."

Method: Loss mein diversity term add karo: $\mathcal{L}={\sum }_{i=1}^k\Vert {\mathbf{x}}_i^{\prime }-\mathbf{x}{\Vert }_1+\lambda {\sum }_i\text{hinge}(f({\mathbf{x}}_i^{\prime }))-\gamma {\sum }_{i<j}\Vert {\mathbf{x}}_i^{\prime }-{\mathbf{x}}_j^{\prime }{\Vert }_1$

Last term counterfactuals ke ek-doosre se similar hone ko penalize karta hai. Hum chahte hain ki ${\mathbf{x}}_1^{\prime },{\mathbf{x}}_2^{\prime },\dots$ ek-doosre se door hon.

3. FACE (Feasible and Actionable Counterfactuals)

Causal constraints add karta hai: Kuch features change nahi ho sakti (race, gender) ya sirf increase ho sakti hain (education level).

Directed acyclic graph (DAG) $G$ causal structure encode karta hai. Sirf wahi changes allow karo jo $G$ ke saath consistent hain.

Counterfactuals ke Liye Evaluation Metrics

Hum counterfactual quality kaise measure karte hain?

Trade-offs: Proximity vs. plausibility (closest counterfactual unrealistic ho sakta hai). Sparsity vs. validity (1 feature change karna kaafi nahi ho sakta).

Connections LIME: Dono local explanations provide karte hain, lekin LIME linear model se approximate karta hai, counterfactuals specific input changes dikhate hain

• SHAP: SHAP explain karta hai "yeh prediction kyun?", counterfactuals explain karte hain "kya change karein?"
• Adversarial Examples: Counterfactuals "achhe" adversarial examples hain (realistic, interpretable); adversarials "bure" hain (minimal imperceptible noise)
• Causal Inference: Counterfactuals causal structure respect karte hain (do-calculus), correlation-based methods ke unlike
• Recourse: Algorithmic recourse = actionable counterfactuals subject to constraints (cost, feasibility)
• Model Debugging: Out-of-distribution counterfactuals model flaws aur biases reveal karte hain

#flashcards/ai-ml

Ek counterfactual explanation ko kaunsi teen key properties satisfy karni chahiye? :: (1) Validity: $f({\mathbf{x}}^{\prime })=y^{\prime }$ (alag prediction), (2) Proximity: ${\mathbf{x}}^{\prime }$ close hai $\mathbf{x}$ ke (minimal change), (3) Plausibility: ${\mathbf{x}}^{\prime }$ realistic/actionable hai

Counterfactual ${\mathbf{x}}^{\prime }$ generate karne ka optimization objective $\mathbf{x}$ se likhko :: $$\min_{\mathbf{x}'} |\mathbf{x}' - \mathbf{x}|1 + \lambda \cdot \text{hinge}(f(\mathbf{x}')) + \gamma \cdot d{\text{manifold}}(\mathbf{x}') \text{ where the three terms balance proximity, prediction validity, and realism.}

DiCE (Diverse Counterfactual Explanations) single-counterfactual methods se kaise alag hai? :: DiCE multiple diverse counterfactuals generate karta hai ek diversity term -\gamma \sum_{i<j} |\mathbf{x}_i' - \mathbf{x}_j'| add karke jo counterfactuals ke beech similarity ko penalize karta hai. Isse users ko options milte hain: "Ya toh income badhao YA credit score improve karo."

Linear classifier f(\mathbf{x}) = \text{sign}(\mathbf{w}^\top \mathbf{x} + b) ke liye, \mathbf{x} se decision boundary par closest point kya hai? :::${\mathbf{x}}_{\text{boundary}}=\mathbf{x}-\frac{{\mathbf{w}}^{\top }\mathbf{x}+b}{\Vert \mathbf{w}{\Vert }^2}\mathbf{w}$ Yeh Lagrange multipliers use karke ${\mathbf{w}}^{\top }{\mathbf{x}}^{\prime }+b=0$ subject to $\Vert {\mathbf{x}}^{\prime }-\mathbf{x}{\Vert }^2$ minimize karke derive kiya jaata hai.

Concept Map