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parse/train/ud-WYSo9JSL/ud-WYSo9JSL.md

@@ -119,7 +119,7 @@ $$
 \ell ( v , v ^ { + } , \{ v _ { i } ^ { - } \} _ { i = 1 } ^ { m } ) = - \log \frac { e ^ { v ^ { \top } v ^ { + } / \tau } } { e ^ { v ^ { \top } v ^ { + } / \tau } + \sum _ { i = 1 } ^ { m } e ^ { v ^ { \top } v _ { i } ^ { - } / \tau } } .
 $$
 
-[Implicit feature modification] Given budget $\varepsilon \in \mathbb { R } _ { + } ^ { m }$ , and encoder $f : \mathcal { X } \to \mathbb { S } ^ { d }$ , an adversary removes features from $f$ that discriminates batch $x , \stackrel { \cdot } { x } ^ { + } , \{ x _ { i } ^ { - } \} _ { i = 1 } ^ { m }$ by maximizing the point-wise InfoNCE loss, \`"(v, v+, {vi }mi=1) = max+2B + ,{i 2B" }mi=1 \` $\begin{array} { r } { \ell _ { \varepsilon } ( v , v ^ { + } , \{ v _ { i } ^ { - } \} _ { i = 1 } ^ { m } ) = \operatorname* { m a x } _ { \delta ^ { + } \in \mathcal { B } _ { \varepsilon ^ { + } } , \{ \delta _ { i } ^ { - } \in \mathcal { B } _ { \varepsilon _ { i } } \} _ { i = 1 } ^ { m } } \ell ( v , v ^ { + } + \delta ^ { + } , \{ v _ { i } ^ { - } + \delta _ { i } ^ { - } \} _ { i = 1 } ^ { m } ) } \end{array}$ .
+[Implicit feature modification] Given budget $\varepsilon \in \mathbb { R } _ { + } ^ { m }$ , and encoder $f : \mathcal { X } \to \mathbb { S } ^ { d }$ , an adversary removes features from $f$ that discriminates batch $x , \stackrel { \cdot } { x } ^ { + } , \{ x _ { i } ^ { - } \} _ { i = 1 } ^ { m }$ by maximizing the point-wise InfoNCE loss, \`"(v, v+, {vi }mi=1) = max+2B + ,{i 2B" }mi=1 \` $\begin{array} { r } { \ell _ { \varepsilon } ( v , v ^ { + } , \{ v _ { i } ^ { - } \} _ { i = 1 } ^ { m } ) = \operatorname* { m a x } _ { \delta ^ { + } \in \mathcal { B } _ { \varepsilon ^ { + } } , \{ \delta _ { i } ^ { - } \in \mathcal { B } _ { \varepsilon _ { i } } \} _ { i = 1 } ^ { m } } \ell ( v , v ^ { + } + \delta ^ { + } , \{ v _ { i } ^ { - } + \delta _ { i } ^ { - } \} _ { i = 1 } ^ { m } ) } \end{array}$ .
 
 Here $B _ { \varepsilon }$ denotes the $\ell _ { 2 }$ -ball of radius $\varepsilon$ . Implicit feature modification (IFM) removes components of the current representations that are used to discriminate positive and negative pairs. In other words, the embeddings of positive and negative samples are modified to remove well represented features. So, if the encoder is currently using a simple shortcut solution, IFM removes the features used, thereby encouraging the encoder to also discriminate instances using other features. By applying perturbations in the embedding space IFM can modify high level semantic features (see Fig. 4), which is extremely challenging when applying perturbations in input space. In order to learn new features using the perturbed loss while still learning potentially complementary information using the original InfoNCE objective, we propose optimizing the the multi-task objective $\mathrm { m i n } _ { f } \{ \mathcal { L } ( f ) \dot { + } \alpha \mathcal { L } _ { \varepsilon } ( \dot { f } ) \} / 2$ where $\mathcal { L } _ { \varepsilon } = \mathbb { E } \ell _ { \varepsilon }$ is the adversarial perturbed loss, and $\mathcal { L }$ the standard InfoNCE loss. For simplicity, all experiments set the balancing parameter $\alpha = 1$ unless explicitly noted, and all take $\varepsilon ^ { + } , \varepsilon _ { i } ^ { - }$ to be equal, and denote this single value by $\varepsilon$ . Crucially, $\ell _ { \varepsilon }$ can be computed analytically and efficiently. For any $v , v ^ { + } , \{ v _ { i } ^ { - } \} _ { i = 1 } ^ { m } \in \bar { \mathbb { R } } ^ { d }$ we have,