Active expansion sampling for learning feasible domains in an unbounded input space

Abstract

Many engineering problems require identifying feasible domains under implicit constraints. One example is finding acceptable car body styling designs based on constraints like aesthetics and functionality. Current active-learning based methods learn feasible domains for bounded input spaces. However, we usually lack prior knowledge about how to set those input variable bounds. Bounds that are too small will fail to cover all feasible domains; while bounds that are too large will waste query budget. To avoid this problem, we introduce Active Expansion Sampling (AES), a method that identifies (possibly disconnected) feasible domains over an unbounded input space. AES progressively expands our knowledge of the input space, and uses successive exploitation and exploration stages to switch between learning the decision boundary and searching for new feasible domains. We show that AES has a misclassification loss guarantee within the explored region, independent of the number of iterations or labeled samples. Thus it can be used for real-time prediction of samples’ feasibility within the explored region. We evaluate AES on three test examples and compare AES with two adaptive sampling methods — the Neighborhood-Voronoi algorithm and the straddle heuristic — that operate over fixed input variable bounds.

Appendices

Appendix A: Theorem proofs

1.1 A1 Proof of theorem 3

According to (2), given an optimal query x^∗, we have

$$\begin{array}{lllllllllll} |\bar{f}(\boldsymbol{x}^{*})| &= |\boldsymbol{k}(\boldsymbol{x}^{*})^{T} \nabla \log p(\boldsymbol{y}|\hat{\boldsymbol{f}})| \\ &= \left|\boldsymbol{k}(\boldsymbol{x}^{*})^{T} \nabla \log \left[\begin{array}{llllllllll} {\Phi}(y_{1}f_{1})\\ \vdots\\ {\Phi}(y_{t-1}f_{t-1}) \end{array}\right]\right| \\ &= \left|\boldsymbol{k}(\boldsymbol{x}^{*})^{T} \left[\begin{array}{llllllllll} y_{1}\mathcal{N}(f_{1})/{\Phi}(y_{1}f_{1})\\ \vdots\\ y_{t-1}\mathcal{N}(f_{t-1})/{\Phi}(y_{t-1}f_{t-1}) \end{array}\right]\right| \\ &= \left| {\sum}_{i = 1}^{t-1} k(\boldsymbol{x}^{*},\boldsymbol{x}^{(i)})y_{i}\frac{\mathcal{N}(f_{i})}{\Phi(y_{i}f_{i})} \right| \\ &\leq {\sum}_{i = 1}^{t-1} \left| k(\boldsymbol{x}^{*},\boldsymbol{x}^{(i)})y_{i}\frac{\mathcal{N}(f_{i})}{\Phi(y_{i}f_{i})} \right| \\ &< {\sum}_{i = 1}^{t-1} k_{m} \text{sign}(y_{i})y_{i}\frac{\mathcal{N}(f_{i})}{\Phi(y_{i}f_{i})} \\ &= k_{m} \text{sign}(\boldsymbol{y})^{T} \left[\begin{array}{llllllllll} y_{1}\mathcal{N}(f_{1})/{\Phi}(y_{1}f_{1})\\ \vdots\\ y_{t-1}\mathcal{N}(f_{t-1})/{\Phi}(y_{t-1}f_{t-1}) \end{array}\right] \\ &= k_{m} \mu \end{array} $$

where

$$ \begin{array}{llllllllll} k_{m} &= \max_{\boldsymbol{x}^{(i)}\in X_{L}} k(\boldsymbol{x}^{*},\boldsymbol{x}^{(i)}) \\ &= \exp\left( -\frac{\min_{\boldsymbol{x}^{(i)}\in X_{L}} \|\boldsymbol{x}^{*}-\boldsymbol{x}^{(i)}\|^{2}}{2l^{2}}\right) \\ &= e^{-\delta^{2}/(2l^{2})} \end{array} $$

(A1)

and

$$ \mu = \text{sign}(\boldsymbol{y})^{T} \nabla \log p(\boldsymbol{y}|\hat{\boldsymbol{f}}) $$

(A2)

Similarly,

$$ \begin{array}{llllllll} V(\boldsymbol{x}^{*}) &= 1-\boldsymbol{k}(\boldsymbol{x}^{*})^{T}(K+W^{-1})^{-1}\boldsymbol{k}(\boldsymbol{x}^{*}) \\ &> 1-(k_{m}\boldsymbol{1})^{T} (K+W^{-1})^{-1} (k_{m}\boldsymbol{1}) \\ &= 1-{k_{m}^{2}}\boldsymbol{1}^{T}(K+W^{-1})^{-1}\boldsymbol{1} \\ &= 1-{k_{m}^{2}} \nu \end{array} $$

(A3)

where

$$ \nu = \boldsymbol{1}^{T}(K+W^{-1})^{-1}\boldsymbol{1} $$

(A4)

Therefore for the optimal query x^∗ we have

$$p_{\epsilon}(\boldsymbol{x}^{*}) = {\Phi}\left( -\frac{|\bar{f}(\boldsymbol{x}^{*})|+\epsilon}{\sqrt{V(\boldsymbol{x}^{*})}}\right) > {\Phi}\left( -\frac{k_{m} \mu+\epsilon}{\sqrt{1-{k_{m}^{2}} \nu}}\right) $$

Both Theorem 1 and 2 state that p _𝜖 (x^∗) = τ, thus

$${\Phi}\left( -\frac{k_{m} \mu+\epsilon}{\sqrt{1-{k_{m}^{2}} \nu}}\right) < \tau $$

When τ = Φ(−η𝜖), we have

$$ \frac{k_{m} \mu+\epsilon}{\sqrt{1-{k_{m}^{2}} \nu}} > \eta\epsilon $$

(A5)

Plugging (A1) into (A5) and solving for the distance δ, we get

$$\delta < \beta l $$

where

$$ \beta=\sqrt{2\log\frac{\mu^{2}+\eta^{2}\epsilon^{2}\nu}{\eta\epsilon\sqrt{\mu^{2}+(\eta^{2}-1)\epsilon^{2}\nu}-\epsilon\mu}} $$

(A6)

1.2 A2 Proof of theorem 5

Theorem 1 states that the optimal query in the exploitation stage lies at the intersection of $\bar {f}(\boldsymbol {x})= 0$ and p _𝜖 (x) = τ. By substituting Φ(−η𝜖) for τ, we have

$$ V(\boldsymbol{x}^{*}) = \frac{1}{\eta^{2}} $$

(A7)

According to (A3), we have $V(\boldsymbol {x}^{*})>1-k_m^2\nu$. Combining (A1), (A4), and (A7), we get

$$\delta < \delta_{exploit} = \gamma l $$

where

$$ \gamma = \sqrt{\log\frac{\eta^{2}\nu}{\eta^{2}-1}} $$

(A8)

1.3 A3 Proof of theorem 6

According to (A7), the predictive variance of an optimal query x _{e x p l o i t} in the exploitation stage is

$$V(\boldsymbol{x}_{exploit}) = \frac{1}{\eta^{2}} $$

While in the exploration stage, we have p _𝜖 (x _{e x p l o r e} ) = τ at the optimal query x _{e x p l o r e} (Theorem 2). And by applying (4) and setting τ = Φ(−η𝜖), we have

$$V(\boldsymbol{x}_{explore}) = \frac{1}{\eta^{2}}\left( 1+\frac{|\bar{f}(\boldsymbol{x}_{explore})|}{\epsilon}\right)^{2} $$

Appendix B: Additional experimental results

1.1 B1 Hosaki example

We use the Hosaki example as an additional 2-dimensional example to demonstrate the performance of our proposed method. Different from the Branin example, the Hosaki example has feasible domains of different scales. Its feasible domains resemble two isolated feasible regions — a large “island” and a small one (Fig. 15a). The Hosaki function is

$$g(\boldsymbol{x})=\left( 1-8x_{1}+ 7{x_{1}^{2}}-\frac{7}{3}{x_{1}^{3}}+\frac{1}{4}{x_{1}^{4}}\right){x_{2}^{2}}e^{-x_{2}} $$

We define the label y =  1 if x ∈ {x|g(x) ≤ −  1, 0 < x₁, x₂ < 5}; and y = − 1 otherwise.

Fig. 15

AES with different 𝜖 and η on the Hosaki example

For AES, we set the initial point x⁽⁰⁾ = (3, 3). We use a Gaussian kernel with a length scale l =  0.4. The test set to compute F1 scores is generated along a 100 × 100 grid in the region where x₁ ∈ [− 3, 9] and x₂ ∈ [− 3.5, 8.5]. For NV and straddle, the input space bounds are shown in Table 3.

Table 3 Input space bounds for the NV algorithm and the straddle heuristic (Hosaki example)

Table 4 shows the final F1 scores and running time of AES, NV, and the straddle heuristic. Fig. 15 shows the F1 scores and queries under different 𝜖 and η. Fig. 16 compares the performance of AES and NV with different boundary sizes. Fig. 17 shows the performance of AES and NV under Bernoulli and Gaussian noise.

Table 4 Final F1 scores and running time (Hosaki example)

Fig. 16

AES and NV (with different input variable bounds) on the Hosaki example

Fig. 17

AES and NV on the Hosaki example using noisy labels

1.2 B2 Results of straddle heuristic

In this section we list experimental results related to the straddle heuristic. Specifically, Fig. 18 shows straddle’s F1 scores and queries using different sizes of input variable bounds, and the comparison with AES. Fig. 19 shows the comparison of AES and straddle under noisy labels.

Fig. 18

AES and straddle (with different input variable bounds)

Fig. 19

AES and straddle under noisy labels