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@@ -189,7 +189,7 @@ Main insights. Theorem 2 justifies CST under the expansion assumption. The gener
 
 # 4.2 Hard Case for Feature Adaptation and Standard Self-Training
 
-To gain more insight, we study UDA in a quadratic neural network $f _ { \theta , \phi } ( x ) = \theta ^ { \top } ( \phi ^ { \top } x ) ^ { \odot 2 }$ , where $\odot$ is element-wise power. In UDA, the source can have multiple solutions but we aim to learn the one working on the target [34]. We design the underlying distributions $p$ and $q$ in Table 6 to reflect this. Consider the following $P$ and $Q$ . $x _ { [ 1 ] }$ and $x _ { [ 2 ] }$ are sampled i.i.d. from distribution $p$ on $P$ , and from $q$ on $Q$ . For $i \in [ 3 , d ]$ , $x _ { [ i ] } = \sigma _ { i } x _ { [ 2 ] }$ on $P$ and $x _ { [ i ] } = \sigma _ { i } x _ { [ 1 ] }$ on $Q$ . $\sigma _ { i } \in \{ \pm 1 \}$ are i.i.d. and uniform. We also assume realizability: for all $y = x _ { [ 1 ] } ^ { 2 } - x _ { [ 2 ] } ^ { 2 }$ $i \in [ 2 , d ]$ for both source and target. Note that are solutions to $P$ but only $y = x _ { [ 1 ] } ^ { 2 } - x _ { [ 2 ] } ^ { 2 }$ $y = x _ { [ 1 ] } ^ { 2 } - x _ { [ i ] } ^ { 2 }$ [1]  [i]works on $Q$ . We visualize this specialized setting in Figure 4.
+To gain more insight, we study UDA in a quadratic neural network $f _ { \theta , \phi } ( x ) = \theta ^ { \top } ( \phi ^ { \top } x ) ^ { \odot 2 }$ , where $\odot$ is element-wise power. In UDA, the source can have multiple solutions but we aim to learn the one working on the target [34]. We design the underlying distributions $p$ and $q$ in Table 6 to reflect this. Consider the following $P$ and $Q$ . $x _ { [ 1 ] }$ and $x _ { [ 2 ] }$ are sampled i.i.d. from distribution $p$ on $P$ , and from $q$ on $Q$ . For $i \in [ 3 , d ]$ , $x _ { [ i ] } = \sigma _ { i } x _ { [ 2 ] }$ on $P$ and $x _ { [ i ] } = \sigma _ { i } x _ { [ 1 ] }$ on $Q$ . $\sigma _ { i } \in \{ \pm 1 \}$ are i.i.d. and uniform. We also assume realizability: for all $y = x _ { [ 1 ] } ^ { 2 } - x _ { [ 2 ] } ^ { 2 }$ $i \in [ 2 , d ]$ for both source and target. Note that are solutions to $P$ but only $y = x _ { [ 1 ] } ^ { 2 } - x _ { [ 2 ] } ^ { 2 }$ $y = x _ { [ 1 ] } ^ { 2 } - x _ { [ i ] } ^ { 2 }$ [1]  [i]works on $Q$ . We visualize this specialized setting in Figure 4.
 
 Table 1: The design of $p$ and $q$ .   
 
@@ -198,7 +198,7 @@ Table 1: The design of $p$ and $q$ .
 ![](images/c5b098da78c5868e7858cc364d018d8dfcb55a8acb4b0000cd134653d0e02a42.jpg)  
 Figure 4: The hard case where $d = 3$ . Green dots for $y = 1$ , red dots for $y = 0$ , and blue dots for $y = - 1$ . The grey curve is the classification boundary of different features. The good feature $x _ { [ 1 ] } ^ { 2 } - x _ { [ 2 ] } ^ { 2 }$ works on the target domain (shown in (a) and (c)), whereas the spurious feature $x _ { [ 1 ] } ^ { 2 } - x _ { [ 3 ] } ^ { 2 }$ only works on the source domain (shown in (b) andwhile CST learns tion 4.2, we show that feature adaptation and standard self-training learn . $x _ { [ 1 ] } ^ { 2 } - x _ { [ 3 ] } ^ { 2 }$ , $x _ { [ 1 ] } ^ { 2 } - x _ { [ 2 ] } ^ { 2 }$
 
-To make the features more tractable, we study the norm-constrained version of the algorithms (details are deferred to Section A.3.2). We compare the features learned by feature adaptation, standard selftraining, and CST. Intuitively, feature adaptation fails because the ideal target solution $y = x _ { [ 1 ] } ^ { 2 } - x _ { [ 2 ] } ^ { 2 }$ has larger distance in the feature space than other spurious solutions y = x2[1] $y = x _ { [ 1 ] } ^ { 2 } - x _ { [ i ] } ^ { 2 }$  x2[i] . Standard selftraining also fails since it will choose randomly among all solutions. In comparison, CST can recover the ground truth, because it can distinguish the spurious solution resulting in bad pseudo-labels. A classifier trained with those pseudo-labels cannot work on the source domain in turn. This intuition is rigorously justified in the following two theorems.
+To make the features more tractable, we study the norm-constrained version of the algorithms (details are deferred to Section A.3.2). We compare the features learned by feature adaptation, standard selftraining, and CST. Intuitively, feature adaptation fails because the ideal target solution $y = x _ { [ 1 ] } ^ { 2 } - x _ { [ 2 ] } ^ { 2 }$ has larger distance in the feature space than other spurious solutions y = x2[1] $y = x _ { [ 1 ] } ^ { 2 } - x _ { [ i ] } ^ { 2 }$  x2[i] . Standard selftraining also fails since it will choose randomly among all solutions. In comparison, CST can recover the ground truth, because it can distinguish the spurious solution resulting in bad pseudo-labels. A classifier trained with those pseudo-labels cannot work on the source domain in turn. This intuition is rigorously justified in the following two theorems.
 
 Theorem 3. For $\epsilon \in ( 0 , 0 . 5 )$ , the following statements hold for feature adaptation and self-training: