Visual Out-of-Distribution Detection in Open-Set Noisy Environments

Abstract

The presence of noisy examples in the training set inevitably hampers the performance of out-of-distribution (OOD) detection. In this paper, we investigate a previously overlooked problem called OOD detection under asymmetric open-set noise, which is frequently encountered and significantly reduces the identifiability of OOD examples. We analyze the generating process of asymmetric open-set noise and observe the influential role of the confounding variable, entangling many open-set noisy examples with partial in-distribution (ID) examples referred to as hard-ID examples due to spurious-related characteristics. To address the issue of the confounding variable, we propose a novel method called Adversarial Confounder REmoving (ACRE) that utilizes progressive optimization with adversarial learning to curate three collections of potential examples (easy-ID, hard-ID, and open-set noisy) while simultaneously developing invariant representations and reducing spurious-related representations. Specifically, by obtaining easy-ID examples with minimal confounding effect, we learn invariant representations from ID examples that aid in identifying hard-ID and open-set noisy examples based on their similarity to the easy-ID set. By triplet adversarial learning, we achieve the joint minimization and maximization of distribution discrepancies across the three collections, enabling the dual elimination of the confounding variable. We also leverage potential open-set noisy examples to optimize a K+1-class classifier, further removing the confounding variable and inducing a tailored K+1-Guided scoring function. Theoretical analysis establishes the feasibility of ACRE, and extensive experiments demonstrate its effectiveness and generalization. Code is available at https://github.com/Anonymous-re-ssl/ACRE0.

Appendix A: The Proof of Theorem 1

Proof

First, we fix the feature extractor G, and minimize the distribution discrimination loss $\mathcal {L}_D$.

$$\begin{aligned} \min _{D} \mathcal {L}_D (x) =&\mathbb {E}_{P_E(x)}[ -\log D_0(G(x))]\nonumber \\&\quad + \mathbb {E}_{P_H(x)} [ -\log D_1(G(x))]\nonumber \\&\quad \quad + \mathbb {E}_{P_O(x)} [ -\log D_2(G(x))] \nonumber \\ =&-\int _{x \sim P_E(x)} \log D_0(G(x)) d x \nonumber \\&\quad {-} \int _{x \sim P_H(x)} \log D_{1}(G(x)) d x\nonumber \\&\quad - \int _{x \sim P_O(x)} \log D_{2}(G(x)) d x \nonumber \\ =&{-}\int _{z \sim P_E(z)} \log D_0(z) d z \nonumber \\ {}&\quad - \int _{z \sim P_H(z)} \log D_{1}(z) d z\nonumber \\&\quad - \int _{z \sim P_O(z)} \log D_{2}(z) d z \nonumber \\ =&\int _z\left( -P_E(z) \log D_0(z)- P_H(z) \log D_{1}(z)\right. \nonumber \\&\quad \left. - P_O(z) \log D_{2}(z)\right) d z \end{aligned}$$

(A1)

$D_0(z) + D_1(z) + D_2(z) = 1$ for all z. Therefore, we transform the above optimization problem into an optimization problem with constraints as follows:

$$\begin{aligned}&\min _{D} -P_E(z) \log D_0(z)- P_H(z) \log D_{1}(z)\nonumber \\&\quad - P_O(z) \log D_{2}(z) \nonumber \\&s.t. \quad D_0(z) + D_1(z) + D_2(z) = 1 \end{aligned}$$

(A2)

To solve the optimization problem with constraints, we use the Lagrange multiplier method.

$$\begin{aligned} \min _{D} \tilde{\mathcal {L}}_D&:= -P_E(z) \log D_0(z) -P_H(z) \log D_{1}(z)\nonumber \\&\quad - P_O(z) \log D_{2}(z) \nonumber \\&\quad + v(D_0(z) + D_1(z) + D_2(z) - 1)\, \end{aligned}$$

(A3)

where v denotes the Lagrange variable.

We compute the derivative of $\tilde{\mathcal {L}}_D$ with respect to D and v as follows:

$$\begin{aligned}&\frac{\partial \tilde{\mathcal {L}}_D}{\partial D_0(z)} =\frac{-P_E(z)}{D_0(z)}+v=0 \quad \Leftrightarrow \quad D_0(z) =\frac{P_E(z)}{v} \nonumber \\&\frac{\partial \tilde{\mathcal {L}}_D}{\partial D_{1}(z)} =\frac{- P_H(z)}{D_{1}(z)}+v=0 \quad \Leftrightarrow \quad D_{1}(z)=\frac{P_H(z)}{v} \nonumber \\&\frac{\partial \tilde{\mathcal {L}}_D}{\partial D_{2}(z)} =\frac{- P_O(z)}{D_{2}(z)}+v=0 \quad \Leftrightarrow \quad D_{2}(z)=\frac{P_O(z)}{v} \nonumber \\&\frac{\partial \tilde{\mathcal {L}}_D}{\partial v}=D_0(z) +D_{1}(z)+D_{2}(z)-1=0 \nonumber \\&\quad \Leftrightarrow \quad D_0(z)+D_{1}(z)+D_{2}(z)=1 \end{aligned}$$

(A4)

According to the above equations, we can know

$$\begin{aligned} D_0(z)+D_{1}(z)+D_{2}(z) = \frac{P_E(z)}{v} + \frac{P_H(z)}{v} + \frac{P_O(z)}{v} = 1\,, \end{aligned}$$

(A5)

where

$$\begin{aligned} v = P_E(z) + P_H(z) + P_O(z) = 3P_{avg}\,. \end{aligned}$$

(A6)

Thus, we obtain optimal $D^{*}$ as

$$\begin{aligned} D^{*}(z)&= [D^{*}_0(z), D^{*}_1(z), D^{*}_2(z)] \nonumber \\&= \left[ \frac{P_E(z)}{3P_{avg}(z)}, \frac{P_H(z)}{3P_{avg}(z)}, \frac{P_O(z)}{3P_{avg}(z)}\right] \,. \end{aligned}$$

(A7)

Then, during optimizing G through minimizing $\mathcal {L}_{OTA}$, we fix D with $D^*$.

$$\begin{aligned}&\min _{G} \mathcal {L}_{OTA} (x) = \mathbb {E}_{P_E (x)}[ -\log D_1^{*}(G(x))] \nonumber \\&\qquad + \mathbb {E}_{P_H(x)}[ -\log D_0^{*}(G(x))] + \mathbb {E}_{P_O(x)} [ -\log D_2^{*}(G(x))] \nonumber \\&\quad =\int _z\left( -P_E(z) \log D_1^*(z) - P_H(z) \log D_{0}^*(z)\right. \nonumber \\&\qquad \left. -P_O(z) \log D_{2}^*(z)\right) d z \nonumber \\&\quad =\int _z\left( -P_E(z) \log \frac{P_H(z)}{3P_{avg}(z)} - P_H(z) \log \frac{P_E(z)}{3P_{avg}(z)}\right. \nonumber \\&\qquad \left. -P_O(z) \log \frac{P_O(z)}{3P_{avg}(z)}\right) d z \nonumber \\&\quad =\int _z\left( -P_E(z) \log \frac{P_H(z)}{3P_{avg}(z)} - P_H(z) \log \frac{P_E(z)}{3P_{avg}(z)}\right. \nonumber \\&\qquad \left. -P_O(z) \log \frac{P_O(z)}{3P_{avg}(z)}\right) d z \nonumber \\&\quad =\int _z\left( (P_H(z)+P_O(z)-3P_{avg}) \log \frac{P_H(z)}{3P_{avg}(z)}\right. \nonumber \\&\qquad \left. +(P_E(z)+P_O(z)-3P_{avg}) \log \frac{P_E(z)}{3P_{avg}(z)}\right. \nonumber \\&\qquad \left. -P_O(z) \log \frac{P_O(z)}{3P_{avg}(z)}\right) d z \nonumber \\&\quad =\int _z\left( P_H(z)\log \frac{P_H(z)}{3P_{avg}(z)} +P_O(z)\log \frac{P_H(z)}{3P_{avg}(z)}\right. \nonumber \\&\qquad \left. -3P_{avg}\log \frac{P_H(z)}{3P_{avg}(z)} + P_E(z)\log \frac{P_E(z)}{3P_{avg}(z)}\right. \nonumber \\&\qquad \left. +P_O(z)\log \frac{P_E(z)}{3P_{avg}(z)} -3P_{avg}\log \frac{P_E(z)}{3P_{avg}(z)} \right. \nonumber \\&\qquad \left. -P_O(z) \log \frac{P_O(z)}{3P_{avg}(z)}\right) d z \nonumber \\&\quad {=}KL\left( P_H \Vert 3P_{avg}\right) {+} KL\left( 3P_{avg} \Vert P_{H}\right) {+} KL\left( P_E \Vert 3P_{avg}\right) \nonumber \\&\qquad + KL\left( 3P_{avg} \Vert P_{E}\right) - KL\left( P_O \Vert 3P_{avg}\right) \nonumber \\&\qquad + \int _z\left( P_O(z)\log \frac{P_H(z)}{3P_{avg}(z)} + P_O(z)\log \frac{P_E(z)}{3P_{avg}(z)}\right) d z\nonumber \\&\quad =KL\left( P_H \Vert P_{avg}\right) + 3KL\left( P_{avg} \Vert P_{H}\right) + KL\left( P_E \Vert P_{avg}\right) \nonumber \\&\qquad + 3KL\left( P_{avg} \Vert P_{E}\right) - KL\left( P_O \Vert P_{avg}\right) + 5 \log 3\nonumber \\&\qquad + \int _z\left( P_O(z)\log \frac{P_H(z)}{3P_{avg}(z)} + P_O(z)\log \frac{P_E(z)}{3P_{avg}(z)}\right) d z \nonumber \\&\quad = KL\left( P_H \Vert P_{avg}\right) + 3KL\left( P_{avg} \Vert P_{H}\right) + KL\left( P_E \Vert P_{avg}\right) \nonumber \\&\qquad + 3KL\left( P_{avg} \Vert P_{E}\right) - KL\left( P_O \Vert P_{avg}\right) \nonumber \\&\qquad + O_{EH} + 5 \log 3\,, \end{aligned}$$

(A8)

where $O_{EH}$ denotes $\int _z\left( P_O(z)\log \frac{P_H(z)}{3P_{avg}(z)}+ P_O(z) \log \right. \left. \frac{P_E(z)}{3P_{avg}(z)}\right) d z$ for convenience. Then, we analyze $KL\left( P_H \Vert P_{avg}\right) + 3KL \left( P_{avg} \Vert P_{H}\right) + KL\left( P_E \Vert P_{avg}\right) + 3KL\left( P_{avg} \Vert P_{E}\right) - KL \left( P_O \Vert P_{avg}\right)$ by the analysis of forces in the field of physics. Since the KL dispersion is asymmetric, it can be viewed as a force approximately. As shown in Fig. 5, we use $F_{ea}, F_{ha}, F_{ah}, F_{ae}$ to denote $KL\left( P_E \Vert P_{avg}\right)$, $KL\left( P_H \Vert P_{avg}\right)$, $KL\left( P_{avg} \Vert P_{H}\right)$, $\left( P_{avg} \Vert P_{E}\right)$, respectively. $\mathcal {E}, \mathcal {H}, \mathcal {O}, \mathcal {A}$ denote $P_E, P_H, P_O, P_A$, located at the three vertices and the center of the triangle, respectively. $F_{aeh}$ denotes the resultant force, and its direction represents the direction $\mathcal {A}$ moves. $F_{ha}$ and $F_{ea}$ will keep $\mathcal {E}$ and $\mathcal {H}$ moving closer to $\mathcal {A}$. By optimizing $\mathcal {L}_{OTA}$, $KL\left( P_E \Vert P_{avg}\right)$, $KL\left( P_H \Vert P_{avg}\right)$, $KL\left( P_{avg} \Vert P_{H}\right)$, $\left( P_{avg} \Vert P_{E}\right)$ will keep decreasing until $P_E \approx P_H \approx P_{avg}$. We use $F_a$ denote $- KL\left( P_O \Vert P_{avg}\right)$. Minimizing $\mathcal {L}_{OTA}$ increases $KL\left( P_O \Vert P_{avg}\right)$, resulting in $\mathcal {O}$ constantly moving away from $\mathcal {A}$. $D_{AO}$ denotes the distance of $\mathcal {A}$ and $\mathcal {O}$ in the optimal G. Moreover, minimizing $\mathcal {L}_{OTA}$ will decrease $O_{EH}$ and the output of the open-set data on $D_0$ and $D_1$, contributing to enhancing the separability of ID and OOD distribution as well. $\square$