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md/train/-bdp_8Itjwp/-bdp_8Itjwp.md

@@ -128,7 +128,7 @@ $$
 \bar { P } _ { e } \overset { < } { \le } 1 - \exp \bigg ( - \Big ( H ( T ) + I ( X ; S | T ) + I ( Z ; X | S , T ) - \hat { I } _ { \theta ^ { * } } ^ { ( n ) } ( Z _ { X } ; S ) + O \big ( \sqrt { \frac { d + \log ( 1 / \delta ) } { n } } \big ) \Big ) \bigg ) .
 $$
 
-Given arbitrary learned representations $( Z _ { X } )$ , Theorem 3 suggests the corresponding Bayes error rate $( P _ { e } )$ is small when: 1) the estimated mutual information $\big ( \widehat { I } _ { \theta ^ { * } } ^ { ( n ) } ( Z _ { X } ; S ) \big )$ is large; 2) a larger number of samples $n$ are used for estimating the mutual information; and 3) the task-irrelevant information the compression gap $I ( X ; S | T )$ and the superfluous information $I ( Z ; X | S , T )$ , defined in Theorem 2
 is small. The first and the second results supports the claim that maximizing $I ( Z _ { X } ; S )$ may learn the representations that are beneficial to downstream tasks. The third result implies the learned representations may perform better on the downstream task when the compression gap is small. Additionally, $Z ^ { \mathrm { s s l } _ { \mathrm { m i n } } }$ is preferable than $Z ^ { \mathrm { s s l } }$ since $I ( Z ^ { \mathrm { s s l } _ { \mathrm { m i n } } } ; X | S , T ) = 0$ and $I ( Z ^ { \mathrm { s s l } } ; \bar { X } | \bar { S } , T ) \ge 0$ .
+Given arbitrary learned representations $( Z _ { X } )$ , Theorem 3 suggests the corresponding Bayes error rate $( P _ { e } )$ is small when: 1) the estimated mutual information $\big ( \widehat { I } _ { \theta ^ { * } } ^ { ( n ) } ( Z _ { X } ; S ) \big )$ is large; 2) a larger number of samples $n$ are used for estimating the mutual information; and 3) the task-irrelevant information the compression gap $I ( X ; S | T )$ and the superfluous information $I ( Z ; X | S , T )$ , defined in Theorem 2
 is small. The first and the second results supports the claim that maximizing $I ( Z _ { X } ; S )$ may learn the representations that are beneficial to downstream tasks. The third result implies the learned representations may perform better on the downstream task when the compression gap is small. Additionally, $Z ^ { \mathrm { s s l } _ { \mathrm { m i n } } }$ is preferable than $Z ^ { \mathrm { s s l } }$ since $I ( Z ^ { \mathrm { s s l } _ { \mathrm { m i n } } } ; X | S , T ) = 0$ and $I ( Z ^ { \mathrm { s s l } } ; \bar { X } | \bar { S } , T ) \ge 0$ .
 
 Theorem 4 (Bayes Error Rates for Self-supervised Learned Representations). Let $P _ { e } ^ { \mathrm { s u p } } / P _ { e } ^ { \mathrm { s s l } } / P _ { e } ^ { \mathrm { s s l } _ { \mathrm { m i n } } }$ be the Bayes error rate of the supervised or the self-supervised learned representations $Z _ { X } ^ { \mathrm { s u p } } / Z _ { X } ^ { \mathrm { s s l } } / Z _ { X } ^ { \mathrm { s s l } _ { \mathrm { m i n } } }$ . Then, $P _ { e } ^ { \mathrm { s s l } } = \mathrm { T h } ( \bar { P } _ { e } ^ { \mathrm { s s l } } )$ and $P _ { e } ^ { \mathrm { s s l } _ { \mathrm { m i n } } } = \mathrm { T h } ( \bar { P } _ { e } ^ { \mathrm { s s l } _ { \mathrm { m i n } } } ) ~ w$ ith
 

md/train/AYAgKFl78z/AYAgKFl78z.md

@@ -139,7 +139,7 @@ Under the negative comonotonicity with $\begin{array} { r } { - \frac { 1 } { 8
 
 # 4.2 Comparison to EAG under the monotonicity $( \rho = 0$ )
 
-For an $L$ -Lipschitz continuous and monotone operator $\pmb { F }$ , [43] proposed two EAG methods, named EAG-C and EAG-V, with same βk = 1k+2 but with different choices of $\alpha _ { k }$ . EAG-C sets $\alpha _ { k }$ to be a constant $\frac { 1 } { 8 L }$ for all $k \geq 0$ in (EAG), and has a large constant 260 in its convergence rate, $\begin{array} { r } { \| \pmb { F } \pmb { z } _ { k } \| ^ { 2 } \le \frac { 2 6 0 L ^ { 2 } \| \pmb { z } _ { 0 } - \pmb { z } _ { * } \| ^ { 2 } } { ( k + 1 ) ^ { 2 } } } \end{array}$ 260L2kz0−z∗k22 for all k ≥ 0. On the other hand, while EAG-V requires a complicated recursive update for {αk}, αk+1 = αk1−α2L2  $\begin{array} { r } { \alpha _ { k + 1 } = \frac { \alpha _ { k } } { 1 - \alpha _ { k } ^ { 2 } L ^ { 2 } } \big ( 1 - \frac { ( k + 2 ) ^ { 2 } } { ( k + 1 ) ( k + 3 ) } \alpha _ { k } ^ { 2 } L ^ { 2 } \big ) } \end{array}$ for all $k \geq 0$ , with $\begin{array} { r } { \alpha _ { 0 } = \frac { 0 . 6 1 8 } { L } } \end{array}$ , its rate has a smaller constant 27.
+For an $L$ -Lipschitz continuous and monotone operator $\pmb { F }$ , [43] proposed two EAG methods, named EAG-C and EAG-V, with same βk = 1k+2 but with different choices of $\alpha _ { k }$ . EAG-C sets $\alpha _ { k }$ to be a constant $\frac { 1 } { 8 L }$ for all $k \geq 0$ in (EAG), and has a large constant 260 in its convergence rate, $\begin{array} { r } { \| \pmb { F } \pmb { z } _ { k } \| ^ { 2 } \le \frac { 2 6 0 L ^ { 2 } \| \pmb { z } _ { 0 } - \pmb { z } _ { * } \| ^ { 2 } } { ( k + 1 ) ^ { 2 } } } \end{array}$ 260L2kz0−z∗k22 for all k ≥ 0. On the other hand, while EAG-V requires a complicated recursive update for {αk}, αk+1 = αk1−α2L2  $\begin{array} { r } { \alpha _ { k + 1 } = \frac { \alpha _ { k } } { 1 - \alpha _ { k } ^ { 2 } L ^ { 2 } } \big ( 1 - \frac { ( k + 2 ) ^ { 2 } } { ( k + 1 ) ( k + 3 ) } \alpha _ { k } ^ { 2 } L ^ { 2 } \big ) } \end{array}$ for all $k \geq 0$ , with $\begin{array} { r } { \alpha _ { 0 } = \frac { 0 . 6 1 8 } { L } } \end{array}$ , its rate has a smaller constant 27.
 
 ![](images/380353bbf2580099fbe770143af658fb4debcb8120afd75852aaa71dc5266deb.jpg)  
 Figure 2: Numerical result with $\begin{array} { r } { f ( x , y ) = - \frac { 1 } { 6 } x ^ { 2 } + \frac { 2 \sqrt { 2 } } { 3 } x y + \frac { 1 } { 6 } y ^ { 2 } } \end{array}$ . The dashed line represents the theoretical bound (3) of FEG.

md/train/H1lj0nNFwB/H1lj0nNFwB.md

@@ -584,7 +584,7 @@ $$
 
 While we were able to generalize our result to gradient descent for $N = 2$ , our proof technique relies on the ability to get a non-implicit solution for $\sigma ( t )$ which we discretized and bounded. This is harder to generalize to larger values of $N$ , where the solution is implicit. Still, we can informally illustrate the effect of depth on the dynamics of gradient descent by approximating the update rule of the values.
 
-We start by reminding ourselves of the gradient descent update rule for $\sigma$ , for a learning rate $\eta =$ c  1σ ∗ 
 2 − 2N :
+We start by reminding ourselves of the gradient descent update rule for $\sigma$ , for a learning rate $\eta =$ c  1σ ∗ 
 2 − 2N :
 
 $$
 \sigma _ { i } ( t + 1 ) = \sigma _ { i } ( t ) \Big ( 1 + \frac { c } { N } \big ( \frac { 1 } { \sigma _ { 1 } ^ { * } } \big ) ^ { 2 - \frac { 2 } { N } } \sigma _ { i } ( t ) ^ { 1 - \frac { 2 } { N } } \big ( \sigma _ { i } ^ { * } - \sigma _ { i } ( t ) \big ) \Big ) ^ { N }

md/train/ryeK6nNFDr/ryeK6nNFDr.md

@@ -36,7 +36,7 @@ $$
 \mathcal { P } _ { \mathcal { X } } ( x ) = A \cdot e ^ { - \lvert \beta x \rvert ^ { c } } ,
 $$
 
-where β = 1σ  Γ(3/c)Γ(1/c) 
 and $\begin{array} { r } { A = \frac { \beta c } { 2 \Gamma ( 1 / c ) } } \end{array}$ , with $\Gamma ( \cdot )$ being the Gamma function. Note that the mean parameter $\mu$ is omitted above, as $\mu$ has no relation with the shape of distribution and we set it as 0 without loss of generality. A nice characteristic of GGD is that it covers many popular distributions
+where β = 1σ  Γ(3/c)Γ(1/c) 
 and $\begin{array} { r } { A = \frac { \beta c } { 2 \Gamma ( 1 / c ) } } \end{array}$ , with $\Gamma ( \cdot )$ being the Gamma function. Note that the mean parameter $\mu$ is omitted above, as $\mu$ has no relation with the shape of distribution and we set it as 0 without loss of generality. A nice characteristic of GGD is that it covers many popular distributions
 
 with varied shape factors. For example, when $c = 1$ , then it becomes the Laplacian distribution; when $c = 2$ , then it is the Gaussian distribution with a variance of √ √ $\sigma ^ { 2 }$ ; when $c \to + \infty$ , then it is specified as a uniform distribution on $( - \sqrt { 2 } \sigma , \sqrt { 2 } \sigma )$ .
 

md/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,
 

md/train/v_4XcXsAZUn/v_4XcXsAZUn.md

@@ -92,7 +92,7 @@ Figure 3: Influence of block size $b$ on PR-BCD (dashed $L _ { 0 }$ PGD $\mathbb
 12: pt[maskres.] 0   
 13: Resample it[maskres.]   
 14: P ⇠ Bernoulli(pE) s.t. $\sum P \leq \Delta$   
-15: Return A  P
+15: Return A  P
 
 PR-BCD. For $L _ { 0 }$ -norm PGD we relax the discrete edge perturbations $_ { P }$ from $\{ 0 , 1 \} ^ { ( n \times n ) }$ to $[ 0 , 1 ] ^ { ( n \times n ) }$ as proposed by $\mathrm { X u }$ et al. [43]. Each entry of $_ { r }$ denotes the probability for flipping it. In each epoch we only look at a randomly sampled, non-contiguous block of $_ { r }$ of size $b$ (line 3, line 10-13) and additionally ignore the diagonal elements (i.e. self-loops). If using an undirected graph, the potential edges are restricted to the upper/lower triangular $n \times n$ matrix. In each epoch $t \in \{ \bar { 1 } , \bar { 2 } , \dots \}$ , $\pmb { p }$ is added to $/$ subtracted from the discrete edge weight (line 6). Note, we overload $\oplus$ s.t. $\pmb { A } _ { i j } \oplus p _ { i j } = A _ { i j } + p _ { i j }$ if $A _ { i j } = 0$ and $A _ { i j } - p _ { i j }$ otherwise. We use $\pmb { p }$ and $_ { P }$ interchangeably while $\pmb { p }$ only corresponds to the current subset/block of $\bar { P } _ { i _ { t } }$ . After each gradient update (line 7), the projection $\Pi _ { \mathbb { E } [ \mathrm { B e r n o u l l i } ( p ) ] \leq \Delta } ( p )$ adjusts the probability mass such that $\begin{array} { r } { \mathbb { E } [ \bar { \mathrm { B e r n o u l l i } } ( \bar { p } ) ] = \sum _ { i \in b } p _ { i } \le \bar { \Delta } } \end{array}$ and that $\pmb { p } \in [ 0 , 1 ]$ (line 8). In the end we draw $b$ sample s.t. $P \in \{ 0 , 1 \} ^ { ( n \times n ) }$ via $P \sim \mathrm { B e r n o u l l i } ( p )$ (line 14).