Adversarial Reweighting with $\alpha$-Power Maximization for Domain Adaptation

Abstract

The practical Domain Adaptation (DA) tasks, e.g., Partial DA (PDA), open-set DA, universal DA, and test-time adaptation, have gained increasing attention in the machine learning community. In this paper, we propose a novel approach, dubbed Adversarial Reweighting with $\alpha$-Power Maximization (ARPM), for PDA where the source domain contains private classes absent in target domain. In ARPM, we propose a novel adversarial reweighting model that adversarially learns to reweight source domain data to identify source-private class samples by assigning smaller weights to them, for mitigating potential negative transfer. Based on the adversarial reweighting, we train the transferable recognition model on the reweighted source distribution to be able to classify common class data. To reduce the prediction uncertainty of the recognition model on the target domain for PDA, we present an $\alpha$-power maximization mechanism in ARPM, which enriches the family of losses for reducing the prediction uncertainty for PDA. Extensive experimental results on five PDA benchmarks, e.g., Office-31, Office-Home, VisDA-2017, ImageNet-Caltech, and DomainNet, show that our method is superior to recent PDA methods. Ablation studies also confirm the effectiveness of components in our approach. To theoretically analyze our method, we deduce an upper bound of target domain expected error for PDA, which is approximately minimized in our approach. We further extend ARPM to open-set DA, universal DA, and test time adaptation, and verify the usefulness through experiments.

Appendix

We first give some lemmas and then provide the proof of Theorem 1.

Lemma 1

Divide $\mathcal {Y}_\textrm{com}$ into $S_1$ and $S_2$ such that $S_1 = \{i\in \mathcal {Y}_\textrm{com}:\mathbb {E}_{(\textbf{x},y)\sim \frac{P^\textrm{c}_i+Q_i}{2}}{\mathbb {I}}(\exists \textbf{x}'\in {\mathcal {N}}(\textbf{x}), {\tilde{f}}(\textbf{x})\ne {\tilde{f}}(\textbf{x}'))<\min \{\epsilon ,q\}\}$ and $S_2 = \{i\in \mathcal {Y}_\textrm{com}:\mathbb {E}_{(\textbf{x},y)\sim \frac{P^\textrm{c}_i+Q_i}{2}}{\mathbb {I}}(\exists \textbf{x}'\in {\mathcal {N}}(\textbf{x}),{\tilde{f}}(\textbf{x}) \ne {\tilde{f}}(\textbf{x}'))\ge \min \{\epsilon ,q\}\}$. Under the condition of Theorem 1, we have

$$\begin{aligned} \sum _{i\in S_2} \frac{P^\textrm{c}+Q}{2}(y=i) \le \frac{R_{\frac{P^\textrm{c}+Q}{2}}(f)}{\min \{\epsilon ,q\}}. \end{aligned}$$

(16)

Proof

Suppose $\sum _{i\in S_2} \frac{P^\textrm{c}+Q}{2}(y=i) > \frac{R(f)}{\min \{\epsilon ,q\}}$, which implies

$$\begin{aligned} \begin{aligned}&R_{\frac{P^\textrm{c}+Q}{2}}(f) \\&\quad = \frac{P^\textrm{c}+Q}{2}(\{\textbf{x},\exists \textbf{x}'\in {\mathcal {N}}(\textbf{x}),{\tilde{f}}(\textbf{x})\ne {\tilde{f}}(\textbf{x}')\})\\&\quad =\mathbb {E}_{(\textbf{x},y)\sim \frac{P^\textrm{c}+Q}{2}}{\mathbb {I}}(\exists \textbf{x}'\in {\mathcal {N}}(\textbf{x}),{\tilde{f}}(\textbf{x})\ne {\tilde{f}}(\textbf{x}'))\\&\quad = \sum _{i\in \mathcal {Y}_\textrm{com}}\Big \{\mathbb {E}_{\textbf{x}\sim \frac{P^\textrm{c}_i+Q_i}{2}}{\mathbb {I}}(\exists \textbf{x}'\in {\mathcal {N}}(\textbf{x}), {\tilde{f}}(\textbf{x})\ne \\&\qquad {\tilde{f}}(\textbf{x}'))\frac{P^\textrm{c}+Q}{2}(y=i)\Big \}\\&\quad \ge \sum _{i\in S_2}\Big \{\mathbb {E}_{\textbf{x}\sim \frac{P^\textrm{c}_i+Q_i}{2}}{\mathbb {I}}(\exists \textbf{x}'\in {\mathcal {N}}(\textbf{x}),{\tilde{f}}(\textbf{x})\ne \\&\qquad {\tilde{f}}(\textbf{x}'))\frac{P^\textrm{c}+Q}{2}(y=i)\Big \}\\&\quad \ge \min \{\epsilon ,q\} \sum _{i\in S_2} \frac{P^\textrm{c}+Q}{2}(y=i) > R_{\frac{P^\textrm{c}+Q}{2}}(f). \end{aligned} \end{aligned}$$

(17)

$R_{\frac{P^\textrm{c}+Q}{2}}(f)>R_{\frac{P^\textrm{c}+Q}{2}}(f)$ forms a contradiction. $\square$

Lemma 2

(Lemma 2 in Liu et al. (2021)) Under the condition of Theorem 1, if sub-populations $P^\textrm{c}_i$ and $Q_i$ satisfy $\mathbb {E}_{(\textbf{x},y)\sim \frac{P^\textrm{c}_i+Q_i}{2}}{\mathbb {I}}(\exists \textbf{x}'\in {\mathcal {N}}(\textbf{x}),{\tilde{f}}(\textbf{x}) \ne {\tilde{f}}(\textbf{x}'))<\min \{\epsilon ,q\}$, we have

$$\begin{aligned} |\varepsilon _{P^\textrm{c}_i}(f)-\varepsilon _{Q_i}(f)|\le 2q. \end{aligned}$$

(18)

Lemma 3

(Lemma 3 in Liu et al. (2021)) For any distribution P, if f is L-Lipschiz w.r.t. $d(\cdot ,\cdot )$, we have

$$\begin{aligned} R_P(f) \le \frac{1}{(1-2L\xi )}(1-{M}_P(f)). \end{aligned}$$

(19)

Proof of Theorem 1

From the definition of $\varepsilon _Q(f)$ in PDA, we have

$$\begin{aligned} \varepsilon _Q(f)= & {} \sum _{i\in \mathcal {Y}_\textrm{com}}\varepsilon _{Q_i}(f)Q(y=i) \nonumber \\\le & {} \sum _{i\in S_1}\varepsilon _{Q_i}(f)Q(y=i) + \sum _{i\in S_2} Q(y=i)\nonumber \\\le & {} \sum _{i\in S_1}(\varepsilon _{P_i}(f)+2q)rP^\textrm{c}(y=i) \nonumber \\{} & {} + \sum _{i\in S_2} Q(y=i)\nonumber \\\le & {} \sum _{i\in \mathcal {Y}_\textrm{com}}(\varepsilon _{P_i}(f)+2q)rP^\textrm{c}(y=i) \nonumber \\{} & {} + \sum _{i\in S_2} Q(y=i)\nonumber \\= & {} r\varepsilon _{P^\textrm{c}}(f) + 2qr + \sum _{i\in S_2} Q(y=i). \end{aligned}$$

(20)

The second inequality uses Lemma 2 and $\frac{Q(y=i)}{P^\textrm{c}(y=i)}\le r$ for $i\in \mathcal {Y}_\textrm{com}$. Since

$$\begin{aligned} \begin{aligned} \frac{P^\textrm{c}+Q}{2}(y=i)&= \frac{1}{2}(P^\textrm{c}(y=i)+Q(y=i))\\&\ge \frac{1}{2}(\frac{1}{r}Q(y=i)+Q(y=i)) \\&= \frac{1+r}{2r}Q(y=i), \end{aligned} \end{aligned}$$

(21)

we have

$$\begin{aligned} \sum _{i\in S_2} Q(y=i) \le \frac{2r}{1+r}\sum _{i\in S_2} \frac{P^\textrm{c}+Q}{2}(y=i). \end{aligned}$$

(22)

Using Lemma 1, we have

$$\begin{aligned} \begin{aligned} \sum _{i\in S_2} Q(y=i) \le \frac{2r}{\min \{\epsilon ,q\}(1+r)}R_{\frac{P^\textrm{c}+Q}{2}}(f). \end{aligned} \end{aligned}$$

(23)

Applying Lemma 3, for any $\eta \in [0,1]$, we have

$$\begin{aligned} \begin{aligned}&\sum _{i\in S_2} Q(y=i) \\&\quad \le \frac{2r\eta }{\min \{\epsilon ,q\}(1+r)}R_{\frac{P^\textrm{c}+Q}{2}}(f)\\&\qquad + \frac{2r(1-\eta )}{\min \{\epsilon ,q\}(1+r)(1-2L\xi )}(1-{\mathcal {M}}_{\frac{P^\textrm{c}+Q}{2}}(f)). \end{aligned} \end{aligned}$$

(24)

Combining Eqs. (20) and (24), we have

$$\begin{aligned} \begin{aligned} \varepsilon _Q(f)&\le r\varepsilon _{P^\textrm{c}}(f) + c_1 R_{\frac{P^\textrm{c}+Q}{2}}(f)\\&\quad +c_2(1-{M}_{\frac{P^\textrm{c}+Q}{2}}(f)) + 2rq, \end{aligned} \end{aligned}$$

(25)

where the coeffcients $c_1 = \frac{2\eta r}{\min \{\epsilon ,q\}(1+r)}$ and $c_2 = \frac{2r(1-\eta ) }{\min \{\epsilon ,q\}(1-2\,L\xi )(1+r)}$ are constants to f. $\square$