1. Introduction

• concept bottleneck models
  • data point $(x,c,y)$가 주어졌을 때, $x$에 대해 concept $c$를 예측하고, 이를 이용해 $y$를 예측
  • layer를 concept 수에 맞춰 resize + intermediate loss 추가

2. Related work

3. Setup

• a target $y \in \mathbb{R}$ from input $x \in \mathbb{R}^d$
  • training point $\{(x^{(i)}, y^{(i)}, c^{(i)})\}^n_{i=1}$, $c \in \mathbb{R}^k$
  • $\hat{y} = f(g(x)) = f(\hat{c})$ ($\hat{c} = g(x)$) : concept bottleneck models
    • $g : \mathbb{R}^d \rightarrow \mathbb{R}^k$ : map an input $x$ into the concept space ("bone spurs")
    • $f : \mathbb{R}^k \rightarrow \mathbb{R}$ : map concepts into a final prediction ("arthritis severity")
  • task accuracy : how accurately $f(g(x))$ predicts $y$
  • concept accuracy : how accurately $g(x)$ predicts $c$ (averaged over each concept)
• learning concept bottleneck models
  • $L_{C_j} : \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}_+$ : a loss function that measures the disrepancy between the predicted and true $j$-th concept
  • $L_Y : \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}_+$ : the discrepancy between predicted and true targets
  • independent bottleneck (따로 학습)
    • $\hat{f} = \text{argmin}_f \sum_i L_Y(f(c^{(i)}); y^{(i)})$
    • $\hat{g} = \text{argmin}g \sum{i,j} L_{C_j} (g_j(x^{(i)}); c^{(i)}_j)$
  • sequential bottleneck (순차적으로 학습)
    • 처음에 $\hat{g}$를 학습
    • $\hat{f} = \text{argmin}_f \sum_i L_Y (f(\hat{g}(x^{(i)})); y^{(i)})$를 학습
  • joint bottleneck
    • $\hat{f}, \hat{g} = \text{argmin}{f,g} \sum_i [L_Y(f(g(x^{(i)})); y^{(i)}) + \sum_j \lambda L{C_j}(g(x^{(i)}); c^{(i)})]$ for some $\lambda > 0$
    • $\lambda$ : the tradeoff between concept vs. task loss
  • standard model
    • $\hat{f}, \hat{g} = \text{argmin}_{f,g} \sum_i L_Y(f(g(x^{(i)})); y^{(i)})$
• concept 수인 $k$에 맞춰서 레이어 하나를 수정

Classification

• independent bottleneck
  • real valued score (e.g., concept logits $\hat{l} = \hat{g}(x) \in \mathbb{R}^k$)를 probabilistic prediction (e.g., $P(\hat{c}_j = 1) = \sigma(\hat{l}_j)$ for logistic regression)로 변환
• sequential and joint bottleneck
  • $f$를 logit $\hat{l}$에 연결
  • $P(\hat{c}_j = 1) = \sigma(\hat{g}_j(x))$와 $P(\hat{y} = 1) = \sigma(\hat{f}(\hat{g}(x)))$를 모두 계산

4. Benchmarking bottleneck model accuracy

4.1. Applications

• x-ray grading과 birt identification task

X-ray grading (OAI)

• knee x-rays from the Osteoarthritis Initiative (OAI)
• Kellgren-Lawrence grade (KLG)를 예측
  • 4-level ordinal variable assessed by radiologists that measures the severity of osteoarthritis
  • $k=10$ (joint space narrowing, bone spurs, calcification, etc.)

Bird identification (CUB)

• Caltech-UCSD Birds-200-2011 (CUB) dataset