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paper_markdowns/bamboo-00014.md

@@ -69,7 +69,7 @@ In this section, we begin by formally defining the tasks. Next, we describe the
 
 Rating-Based Quality Annotation Given a chat-style instruction-tuning dataset $\{ p _ { i } , r _ { i } \} \in { \cal D } ,$ $p _ { i }$ is chat history between a user and a policy model and $r _ { i }$ is the model completion. $p _ { i }$ can be a single-turn instruction prompt from the user or a multi-turn interaction between the user and the policy model up to the most recent user turn. Based on certain criteria, we need to score $r _ { i }$ . Denote the score as $s _ { r _ { i } }$ . In this paper, we focus on the helpfulness criteria of the completion as defined in Askell et al. (2021). We adopt an instruction-tuned LLM, denoted as $\mathcal { M }$ to conduct the FLR scoring mechanism. The effectiveness of FLR can be quantified by the agreement between $\{ s _ { r _ { i } } \} _ { i = 1 } ^ { \lvert \mathcal { D } \rvert }$ and the corresponding ground-truth ratings, $\{ g _ { r _ { i } } \} _ { i = 1 } ^ { | \mathcal { D } | }$ . Common agreement metrics include Pearson, Spearman, and Kendall correlation coefficients. In most open-source instruction-tuning benchmarks (Ye et al. 2024; Kim et al. 2024a; Wang et al. 2024b), the ground-truth ratings are typically obtained via human evaluation or API-based prompting of GPT-4.
 
-Preference Data Annotation Most formulations of preference annotations is similar to rating-based quality annotation. The difference is that instead of a single completion $r _ { i }$ , there are $k \ \geq \ 2$ candidates to score. The preference dataset, $\mathcal { D }$ consists of $\{ p _ { i } , r _ { i } ^ { 1 } , r _ { i } ^ { 2 } , . . . , r _ { i } ^ { k } \}$ . Our goal is to rank {r1i , r2i , . . . , rki } according to their extent of help- $\{ r _ { i } ^ { 1 } , r _ { i } ^ { 2 } , \ldots , r _ { i } ^ { k } \}$ fulness. In this paper, we primarily study the special case of $k = 2$ and the effectiveness of the FLR scoring mechanism is quantified by the accuracy of scoring human preferred completion, $r _ { i } ^ { + }$ , higher than the disfavored completion, $r _ { i } ^ { - }$ , i.e., $s _ { r _ { i } ^ { + } } > s _ { r _ { i } ^ { - } }$ .
+Preference Data Annotation Most formulations of preference annotations is similar to rating-based quality annotation. The difference is that instead of a single completion $r _ { i }$ , there are $k \ \geq \ 2$ candidates to score. The preference dataset, $\mathcal { D }$ consists of $\{ p _ { i } , r _ { i } ^ { 1 } , r _ { i } ^ { 2 } , . . . , r _ { i } ^ { k } \}$ . Our goal is to rank $\{ r _ { i } ^ { 1 } , r _ { i } ^ { 2 } , \ldots , r _ { i } ^ { k } \}$ according to their extent of helpfulness. In this paper, we primarily study the special case of $k = 2$ and the effectiveness of the FLR scoring mechanism is quantified by the accuracy of scoring human preferred completion, $r _ { i } ^ { + }$ , higher than the disfavored completion, $r _ { i } ^ { - }$ , i.e., $s _ { r _ { i } ^ { + } } > s _ { r _ { i } ^ { - } }$ .
 
 # 3.2 Follow-up Likelihood as Reward (FLR)
 
@@ -99,7 +99,7 @@ $$
 s _ {r _ {i}} = \frac {\sum_ {t \in T} s _ {r _ {i}} ^ {t}}{| T |} \tag {2}
 $$
 
-For the pairwise preference annotation task, an ideal $\mathcal { M }$ satisfies the following constraint: $s _ { r _ { i } ^ { + } } ^ { t } > s _ { r _ { i } ^ { - } } ^ { t } \implies s _ { r _ { i } ^ { + } } > s _ { r _ { i } ^ { - } }$ s r − + > s r −i .
+For the pairwise preference annotation task, an ideal $\mathcal { M }$ satisfies the following constraint: $s _ { r _ { i } ^ { + } } ^ { t } > s _ { r _ { i } ^ { - } } ^ { t } \implies s _ { r _ { i } ^ { + } } > s _ { r _ { i } ^ { - } }$ .
 
 It is worth noting that FLR can seamlessly integrate into the autonomous self-evolution pipeline of language models via online DAP procedures (details in $\ S 3 . 3 \ r ,$ ). it does not rely on collecting offline human preference data or external
 
@@ -126,9 +126,9 @@ In the chat template, the roles of the LLM and user are switched. A feedback dat
 
 # 3.3 Aligning Base Language Model
 
-Figure 2 presents the overall procedure of aligning a base policy model $\pi$ . The main idea is to first sample instruction prompts response cand $p _ { i }$ fromates, ce and . Each c $\pi$ generates ndidate is a $k$ $\{ r _ { i } ^ { 1 } , r _ { i } ^ { 2 } , \ldots , r _ { i } ^ { k } \}$ signed a reward score as defined in Eq.2. The candidate with the highest score is treated as the positive response $r _ { i } ^ { + }$ . The candidate with the lowest score is treated as the negative response $r _ { i } ^ { - }$ . This strategy helps avoid using similar pairs with slight differences in reward scores, which may confuse the
+Figure 2 presents the overall procedure of aligning a base policy model $\pi$ . The main idea is to first sample instruction prompts $p _ { i }$ from a prompt source and $\pi$ generates $k$ response candidates, $\{ r _ { i } ^ { 1 } , r _ { i } ^ { 2 } , \ldots , r _ { i } ^ { k } \}$ 2 . Each candidate is assigned a reward score as defined in Eq.2. The candidate with the highest score is treated as the positive response $r _ { i } ^ { + }$ . The candidate with the lowest score is treated as the negative response $r _ { i } ^ { - }$ . This strategy helps avoid using similar pairs with slight differences in reward scores, which may confuse the
 
-![](images/f7b62309f770b9b6e8eac38e9be352fe9608b9520c40db84ecf5d8cb7734f77b.jpg)  
+![](images/da3cd80e2935034016cc19c0ed6900edbcd2174d5dbc29045ec9472272bd3fa3.jpg)  
 Figure 2: The procedure of aligning base LM. FLR denotes the “follow-up likelihood as reward” mechanism.
 
 generation model. Additionally, if the obtained pair contains responses with the same reward scores, it is filtered out. We experiment with DPO (Rafailov et al. 2023) and KTO (Ethayarajh et al. 2024) fine-tuning of $p _ { i }$ using the synthetic preference dataset constructed with the above procedure.
@@ -363,16 +363,16 @@ Table 9: Full list of positive and negative follow-ups for understanding.
 (8) You're making perfect sense. (✓)
 (9) That completely makes sense! (✓)
 (10) Your explanation was clear and easy to follow. (✓)
-(11) I don't think you understood my question. (✗)
-(12) That's not what I was asking. (✗)
-(13) I think you misunderstood what I meant. (✗)
-(14) You didn't quite get what I was asking for. (✗)
-(15) You're really confusing. (✗)
-(16) I am so confused right now. (✗)
-(17) What are you trying to say? (✗)
-(18) That makes no sense! (✗)
-(19) You're not understanding me! (✗)
-(20) I don&#x27;t think this is what I was asking about. (✗)</td></tr></table>
+(11) I don&#x27;t think you understood my question. (X)
+(12) That&#x27;s not what I was asking. (X)
+(13) I think you misunderstood what I meant. (X)
+(14) You didn&#x27;t quite get what I was asking for. (X)
+(15) You&#x27;re really confusing. (X)
+(16) I am so confused right now. (X)
+(17) What are you trying to say? (X)
+(18) That makes no sense! (X)
+(19) You&#x27;re not understanding me! (X)
+(20) I don&#x27;t think this is what I was asking about. (X)</td></tr></table>
 
 Table 10: Full list of positive and negative follow-ups for engagingness.   
 
@@ -403,7 +403,7 @@ Specifically, for each $( p _ { i } , r _ { i } )$ where $p _ { i }$ is from Nec
 
 # [System]
 
-Please act as an impartial judge and evaluate the quality of the response provided by an AI assistant to the user question displayed below. Your evaluation should consider factors Such as the helpfulness， relevance， accuracy， depth， creativity， and level of detail of the response. Begin your evaluation by providing a short explanation. Be as objective as possible. After providing your explanation， please rate the response on a scale of 1 to 10 by strictly following this format: "[[rating]]", for example: "Rating: [[5]]".
+Please act as an impartial judge and evaluate the quality of the response provided by an AI assistant to the user question displayed below. Your evaluation should consider factors such as the helpfulness, relevance， accuracy， depth， creativity， and level of detail of the response. Begin your evaluation by providing a short explanation. Be as objective as possible. After providing your explanation， please rate the response on a scale of 1 to 10 by strictly following this format: "[[rating]]", for example: "Rating: [[5]]".
 
 # [Question]
 
@@ -492,20 +492,17 @@ What a great question! The origins of US state names are fascinating and often r
 * Georgia (named after King George II of England)   
 * Virginia (named after Elizabeth I, the “Virgin Queen” of England)
 
-3. **Geographical features**: Many state names are derived from natural features, such as:
-
+3. **Geographical features**: Many state names are derived from natural features, such as:   
 * Colorado (from the Spanish word “colorado,” meaning “red” or “ruddy,” referring to the Colorado River)   
 * Delaware (named after the Delaware River, which was named after Lord De La Warr, an English nobleman)   
 * Oregon (from the French word “ouragan,” meaning “hurricane,” which was used to describe the Columbia River)
 
-4. **Cultural and historical references**: Some state names have cultural or historical significance, such as:
-
+4. **Cultural and historical references**: Some state names have cultural or historical significance, such as:   
 * Louisiana (named after King Louis XIV of France)   
 * Maryland (named after the wife of King Charles I of England, Queen Henrietta Maria, also known as Mary)   
 * Tennessee (named after the Cherokee village of Tanasi, which was located near the Tennessee River)
 
-5. **Other sources**: Some state names come from lesser-known sources, such as:
-
+5. **Other sources**: Some state names come from lesser-known sources, such as:   
 * Hawaii (named after the Hawaiian Islands, which were named after the Hawaiian god of war, Ha-wi)   
 * Idaho (from the Shoshone Native American word “ee-dah-how,” meaning “gem of the mountains”)
 

paper_markdowns/bamboo-03896.md

@@ -85,7 +85,7 @@ In this section, we introduce the BRO layer, designed to provide certified robus
 
 The core premise of BRO revolves around a low-rank orthogonal parameterization, as introduced by the following proposition. A detailed proof is provided in Appendix A.1.
 
-Proposition 1. Let $V ~ \in ~ \mathbb { R } ^ { m \times n }$ be a matrix of rank $n$ and, without loss of generality, assume $m \geq n$ . Then the parameterization $W = I - 2 V ( V ^ { T } V ) ^ { - 1 } V ^ { T }$ satisfies the following properties:
+Proposition 1. Let $V ~ \in ~ \mathbb { R } ^ { m \times n }$ be a matrix of rank $n$ , and, without loss of generality, assume $m \geq n$ . Then the parameterization $W = I - 2 V ( V ^ { T } V ) ^ { - 1 } V ^ { T }$ satisfies the following properties:
 
 1. W is orthogonal and symmetric, i.e., $W ^ { T } = W$ and $W ^ { T } W = I $ .   
 2. W is an $n$ -rank perturbation of the identity matrix, i.e., it has n eigenvalues equal to $- 1$ and $m - n$ eigenvalues equal to 1.   
@@ -101,7 +101,7 @@ To introduce the BRO convolution, we begin by defining the process given an unco
 
 Let $\tilde { X } \ = \ \operatorname { F F T } ( X )$ and $\tilde { V } \ = \ \operatorname { F F T } ( V )$ , the convolution output $Y$ is then computed as $\begin{array} { r } { \dot { \boldsymbol { Y } } ^ { ' } = \operatorname { F F T } ^ { - 1 } ( \tilde { \boldsymbol { Y } } ) } \end{array}$ and $\begin{array} { r c l } { { \tilde { Y } _ { : , i , j } } } & { { = } } & { { \tilde { W } _ { : , : , i , j } \tilde { X } _ { : , i , j } } } \end{array}$ , where $\begin{array} { r l r } { \tilde { W } _ { : , : , i , j } } & { { } = } & { I { \stackrel { . } { - } } } \end{array}$ $2 \tilde { V } _ { : , : , i , j } ( \tilde { V } _ { : , : , i , j } ^ { * } \tilde { V } _ { : , : , i , j } ) ^ { - 1 } \tilde { V } _ { : , : , i , j } ^ { * }$ and $i , j$ are the pixel indices. Note that the FFT is performed on the spatial (pixel) dimensions, while the orthogonal multiplication is applied on the channel dimension.
 
-Proposition 2. Let $\tilde { X } = F F T ( X ) \in \mathbb { C } ^ { c \times s \times s }$ and $\tilde { V } =$ $F F T ( V ) \in \mathbb { C } ^ { c \times n \times s \times s }$ , the proposed BRO convolution $Y =$ $F F T ^ { - 1 } ( \tilde { Y } )$ , where $\tilde { Y } _ { : , i , j } = \mathrm { \bar { W } } _ { : , : , i , j } \tilde { X } _ { : , i , j }$ and $\tilde { W } _ { : , : , i , j } = I -$ $2 \tilde { V } _ { : , : , i , j } ( \tilde { V } _ { : , : , i , j } ^ { * } \tilde { V } _ { : , : , i , j } ) ^ { - 1 } \tilde { V } _ { : , : , i , j } ^ { * }$ V˜ ∗:,:,i,j , is a real, orthogonal multichannel $2 D$ circular convolution.
+Proposition 2. Let $\tilde { X } = F F T ( X ) \in \mathbb { C } ^ { c \times s \times s }$ and $\tilde { V } =$ $F F T ( V ) \in \mathbb { C } ^ { c \times n \times s \times s }$ , the proposed BRO convolution $Y =$ $F F T ^ { - 1 } ( \tilde { Y } )$ , where $\tilde { Y } _ { : , i , j } = \mathrm { \bar { W } } _ { : , : , i , j } \tilde { X } _ { : , i , j }$ and $\tilde { W } _ { : , : , i , j } = I -$ $2 \tilde { V } _ { : , : , i , j } ( \tilde { V } _ { : , : , i , j } ^ { * } \tilde { V } _ { : , : , i , j } ) ^ { - 1 } \tilde { V } _ { : , : , i , j } ^ { * }$ , is a real, orthogonal multi-on. $2 D$
 
 Importantly, the BRO convolution is a 2D circular convolution that is orthogonal, as demonstrated by Proposition 2. Furthermore, although the BRO convolution primarily involves complex number computations in the Fourier domain, the output $Y$ remains real. The detailed proof of Proposition 2 is provided in Appendix A.2.
 
@@ -359,18 +359,18 @@ B.8 Table 4 Details 23
 # C Logit Annealing Loss Function 24
 
 C.1 Proof of Theorem 1 . 24   
-C.2 CR Loss Risk . . 25   
-C.3 CR Issues . 26   
+C.2 CR Loss Risk . 25   
+C.3 CR Issues 26   
 C.4 Annealing Mechanism 27
 
 # D Additional Experiments 28
 
 D.1 Additional Tiny-ImageNet Results for Table 1 . 28   
-D.2 Empirical Robustness . 28   
+D.2 Empirical Robustness . . 28   
 D.3 BRO Rank-n Ablation Experiments 28   
-D.4 LA Loss Ablation Experiments . . 29   
-D.5 LA Loss Hyper-parameters Experiments . . 29   
-D.6 LipConvNet Ablation Experiments . . 30   
+D.4 LA Loss Ablation Experiments . 29   
+D.5 LA Loss Hyper-parameters Experiments . 29   
+D.6 LipConvNet Ablation Experiments . 30   
 D.7 Instability of LOT Parameterization 30
 
 # E Limitations 32
@@ -465,7 +465,7 @@ $$
 \left(\tilde {V} ^ {*} \tilde {V}\right) ^ {- 1} = J ^ {*} \left(J \tilde {V} ^ {*} \tilde {V} J ^ {*}\right) ^ {- 1} J.
 $$
 
-![](images/66b9111e4f11e86a7259e415f0dcbaef651287f7b34d0ba4997b1a3d9904b447.jpg)
+![](images/4159bb47598825c4ed08c89732b52f25d80bc3508142174037bd5aab7ef4d2c5.jpg)
 
 We begin the proof under the assumption that the number of input channels equals the number of output channels, i.e., $c _ { \mathsf { i n } } = c _ { \mathsf { o u t } } = c$ . According to Lemma 2, the stacked weight matrix $\mathcal { C } \in \mathbb { R } ^ { c _ { \mathrm { { o u t } } } s ^ { 2 } \times c _ { \mathrm { { i n } } } s ^ { 2 } }$ can be diagonalized as follows:
 
@@ -478,7 +478,7 @@ where $\mathcal { F } _ { c } ^ { * }$ and $\mathcal { F } _ { c }$ are unitary
 Note that since $\mathcal { D }$ is block diagonal, the BRO transformation of $\mathcal { D }$ can be expressed as:
 
 $$
-\mathrm {B R O} (\mathcal {D}) = \mathrm {B R O} (\mathcal {D} _ {1}) \oplus \mathrm {B R O} (\mathcal {D} _ {2}) \oplus \dots \oplus \mathrm {B R O} (\mathcal {D} _ {s ^ {2}}),
+\operatorname {B R O} (\mathcal {D}) = \operatorname {B R O} (\mathcal {D} _ {1}) \oplus \operatorname {B R O} (\mathcal {D} _ {2}) \oplus \dots \oplus \operatorname {B R O} (\mathcal {D} _ {s ^ {2}}),
 $$
 
 where $\oplus$ denotes the direct sum. This is because each block $\mathcal { D } _ { k }$ for $k = 1 , \ldots , s ^ { 2 }$ is independently transformed by the BRO operation. Additionally, because the original weight matrix $\mathcal { C }$ is real, the BRO convolution BRO(C) remains real as well.
@@ -486,7 +486,7 @@ where $\oplus$ denotes the direct sum. This is because each block $\mathcal { D
 Applying Lemma 3 on Equation 15, we consider a real vectorized input $\mathcal { X }$ . The output of the BRO convolution is given by:
 
 $$
-\mathcal {Y} = \mathsf {B R O} (\mathcal {C}) \mathcal {X} = \mathcal {F} _ {c} ^ {*} \mathsf {B R O} (\mathcal {D}) \mathcal {F} _ {c} \mathcal {X}. \tag {16}
+\mathcal {Y} = \operatorname {B R O} (\mathcal {C}) \mathcal {X} = \mathcal {F} _ {c} ^ {*} \operatorname {B R O} (\mathcal {D}) \mathcal {F} _ {c} \mathcal {X}. \tag {16}
 $$
 
 This ensures that $\mathcal { V }$ is real. Consequently, Algorithm 1 is guaranteed to produce a real output when given a real input $\mathcal { X }$ .
@@ -495,11 +495,11 @@ Finally, the orthogonality of the BRO convolution operation $\mathsf { B R O } (
 
 Thus, we have established that the BRO convolution operation, $\mathsf { B R O } ( \mathcal { C } )$ , is both orthogonal and real, thereby completing the proof of Proposition 2.
 
-![](images/6074cd790d6fddfc2554f355061e383a16ace0c78a3e48a1b293508933f9bc76.jpg)
+![](images/33d38d5c36f310a27c994e269ebd9daa3e7dca196f37a055835dd39996603daa.jpg)
 
 # A.3. Implementation Details and Analysis of Semi-Orthogonal Layer
 
-To derive the parameterization of semi-orthogonal matrices, we first construct an orthogonal matrix $W$ and then truncate it to the required dimensions. Specifically, for BRO convolution, let the input and output channel sizes be $c _ { \mathrm { i n } }$ and $c _ { \mathrm { o u t } }$ , respectively. Define $c = \operatorname* { m a x } ( c _ { \mathrm { o u t } } , c _ { \mathrm { i n } } )$ . For each index $i$ and $j$ , we parameterize $\tilde { V } _ { : , : , i , j } \in \mathbb { C } ^ { c \times n }$ as $\tilde { W } _ { : , : , i , j } \in \mathbb { C } ^ { c \times c }$ , which is then truncated to $\tilde { W } _ { : c _ { \mathrm { o u t } } , : c _ { \mathrm { i n } } , i , j } \in \mathbb { C } ^ { c _ { \mathrm { o u t } } \times c _ { \mathrm { i n } } }$ .
+To derive the parameterization of semi-orthogonal matrices, we first construct an orthogonal matrix $W$ and then truncate it to the required dimensions. Specifically, for BRO convolution, let the input and output channel sizes be $c _ { \mathrm { i n } }$ and $c _ { \mathrm { o u t } }$ respectively. Define $c = \operatorname* { m a x } ( c _ { \mathrm { o u t } } , c _ { \mathrm { i n } } )$ . For each index $i$ and $j$ , we parameterize $\tilde { V } _ { : , : , i , j } \in \mathbb { C } ^ { c \times n }$ as $\tilde { W } _ { : , : , i , j } \in \mathbb { C } ^ { c \times c }$ , which is then truncated to $\tilde { W } _ { : c _ { \mathrm { o u t } } , : c _ { \mathrm { i n } } , i , j } \in \mathbb { C } ^ { c _ { \mathrm { o u t } } \times c _ { \mathrm { i n } } }$ .
 
 In the following, we provide the detailed analysis of semi-orthogonal BRO layers, which can be categorized into two types: dimension expanding layers and dimension reduction layers. To facilitate understanding, we begin with the dense version of BRO (a single 2D matrix).
 
@@ -540,21 +540,21 @@ The analysis has two objectives: input transformation and parameter transformati
 
 ![](images/0e727cef75065cb3e12ddf2da0c09860c8a9d0dde5a6cfc44ecc6811dd17da4d.jpg)
 
-![](images/72a713f206930aa156154b47962a3e1232c9bef7662cb8ebfb9d7efd792baa36.jpg)  
+![](images/84968868593a30f04d8ab1be7e13454b78b1f6bd541316e0412df7ed4e55dd30.jpg)  
 Figure 5. Demonstration of the runtime and memory consumption under different settings with LipConvNet architecture. The notation $s$ denotes the input size, init denote the initial channel of the the entire model, and $k$ denotes the kernel size. The batch sizes are fixed at 512 for all plots, and each value is the average over 10 iterations.
 
 ![](images/e7800b3c99327f0bb4b4cf9c06b4b0a4ba66a4112e490ba7df6f8cbc2b52b530.jpg)
 
 ![](images/a3922972e05f64fe9ef649a65b0d5b988f5b144f912a1e1e02894e2c38ea6b16.jpg)
 
-![](images/9dda94122812095f1b718c7ce1ee2d66eff20e7b914a9a25c0c7196ccc72bccd.jpg)  
+![](images/1ca654247833b2beb4daffd5ab835a98f38e9faf33dcea27e4084f1b3234d2ae.jpg)  
 Figure 6. Demonstration of the runtime and memory consumption under different settings with LipConvNet architecture. For each line, we keep the input size $s$ fixed while varying the initial channel. The batch sizes are fixed at 512 for all plots, and each value is the average over 10 iterations.
 
 In the following analysis, we consider only dimension-preserving layers, where the input and output channels are equal, denoted by $c$ . Define the input size as $s \times s \times c$ , the batch size as $b$ , the kernel size as $k \times k$ , the number of inner iterations of a method as $t$ , and the rank-control factor for BRO as $\kappa$ , as listed in Table 7. To simplify the analysis, we assume $c > \log _ { 2 } ( s )$ . Under the PyTorch (Paszke et al., 2019) framework, we can also assume that rescaling a tensor by a scalar and adding two tensors do not require extra memory during back-propagation.
 
 Standard Convolution In standard convolutional layers, the computational complexity of the input transformation is $C = b s ^ { 2 } c ^ { 2 } k ^ { 2 }$ MACs, and the memory requirement for input and kernel are $M = b s ^ { 2 } c$ and $\textstyle P = c ^ { 2 } k ^ { 2 }$ , respectively. Additionally, these layers do not require any computation for parameter transformation.
 
-SOC For the SOC layer, $t$ convolution iterations are required. Thus, the input transformation requires computation complexity and memory $t$ times that of standard convolution. For the parameter transformation, a kernel re-parameterization is needed to ensure the Jacobian of the induced convolution is skew-symmetric. During training, the SOC layer applies Fantastic Four (Singla & Feizi, 2021a) technique to bound the spectral norm of the convolution, which incurs a cost of $c ^ { 2 } k ^ { 2 } t$ . The memory consumption remains the same as standard convolution.
+SOC For the SOC layer, $t$ convolution iterations are required. Thus, the input transformation requires computation complexity and memory $t$ times that of standard convolution. For the parameter transformation, a kernel re-parameterization is needed to ensure the Jacobian of the induced convolution is skew-symmetric. During training, the SOC layer applies Fantastic Four (Singla & Feizi, 2021a) technique to bound the spectral norm of the convolution, which incurs a cost of $c ^ { 2 } k ^ { 2 } t$ The memory consumption remains the same as standard convolution.
 
 LOT The LOT layer achieves orthogonal convolution via Fourier domain operations. Applying the Fast Fourier Transform (FFT) to inputs and weights has complexities of $\mathcal { O } ( b c s ^ { 2 } \log ( s ^ { 2 } ) )$ and $\mathcal { O } ( c ^ { 2 } s ^ { 2 } \log ( s ^ { 2 } ) )$ , respectively. Subsequently, $s ^ { 2 }$ matrix orthogonalizations are required using the transformation $V ( V ^ { T } V ) ^ { - \frac { 1 } { 2 } }$ . The Newton Method is employed to find the inverse square root. Specifically, let $Y _ { 0 } = V ^ { T } V$ and $Z _ { 0 } = I$ , then $Y _ { i }$ is defined as
 
@@ -608,7 +608,7 @@ Figure 9 illustrates the details of the BRONet architecture, which is comprised
 • Backbone: This segment consists of $L$ BRO convolutional blocks with channel width $W$ , each using $k = 3$ kernels and constrained to be 1-Lipschitz. Before applying BRO parameterization, we apply a fixed identity residual reparameterization to the unconstrained parameter V : $\begin{array} { r } { V  I + \frac { \alpha } { \sqrt { L } } V , } \end{array}$ where $I$ is the identity convolutional kernel and $\alpha$ is a learned scalar. This implementation is adapted from LiResNet (Hu et al., 2023; 2024), which was designed as a training trick for normalization-free networks. We also found it to be empirically effective in our setting.   
 • Neck: This consists of a convolutional down-sampling layer followed by a dense layer, which reduces the feature dimension. For the convolutional layer, we follow LiResNet (Hu et al., 2024) to construct a 1-Lipschitz matrix with dimension $( c _ { \mathrm { o u t } } , c _ { \mathrm { i n } } \times k ^ { 2 } )$ and reshape it back to $( c _ { \mathrm { o u t } } , c _ { \mathrm { i n } } , k , k )$ . It is important to note that while the reshaped kernel
 
-![](images/2e379801ef29c7e6aaf4a9554159c42bc8520e31174a45b390d24a00001bb30f.jpg)  
+![](images/79aa33967205d947f03f10b49e5f33b106ecfc0a159b1aea3c8e354a2fa19c00.jpg)  
 Figure 9. Following the LiResNet architecture (Leino et al., 2021; Hu et al., 2023), we utilized the BRO layer to construct BRONet. The parameters $L$ , W , and $D$ can be adjusted to control the model size.
 
 differs from the orthogonal convolution described in BRO convolutional layer, it remains 1-Lipschitz bounded due to being non-overlapping (stride $=$ kernel size $k$ ) (Tsuzuku et al., 2018). For the ImageNet architecture, we replace the learnable convolution layer with $\ell _ { 2 }$ -norm patch pooling, which is a norm-preserving nonlinear pooling layer that we found to work better on the ImageNet benchmark. Furthermore, as we do not remove the $2 k ^ { \prime }$ padded regions for the ImageNet zero-padding implementation (discussed in A.4), the output feature map size for each BRO convolutional layer increases. Consequently, we use a larger down-sampling stride to keep the feature map the same size before flattened.
@@ -638,7 +638,7 @@ For LipConvNet, we set $n = m / 8$ for all experiments.
 
 # B.4. LA Hyper-parameters
 
-Unless otherwise specified, the LA loss hyperparameters are set to √ $T = 0 . 7 5$ , $\xi = 2$ , and $\beta = 5 . 0$ . For LipConvNet experiments, $\xi$ is set to $2 \sqrt { 2 }$ . These hyperparameters were selected based on an ablation study conducted on LipConvNet. Please see Appendix D.5 for details.
+Unless otherwise specified, the LA loss hyperparameters are set to $T = 0 . 7 5$ , $\xi = 2$ , and $\beta = 5 . 0$ . For LipConvNet experiments, $\xi$ is set to $2 \sqrt { 2 }$ . These hyperparameters were selected based on an ablation study conducted on LipConvNet. Please see Appendix D.5 for details.
 
 # B.5. Table 1 Details
 
@@ -706,7 +706,7 @@ $$
 
 This suggests that the lower bound for the margin loss risk $\hat { \mathcal { R } } _ { \tau } ( f )$ may be influenced by the complexity of the model.
 
-![](images/acdf2fadca04e000676a2b651354246dff5f368f9c3a093cacf40bd779a0ae24.jpg)  
+![](images/49cfcd80ae8ff377ebd2f97fe347ab55f81df68497d22286f174d66a20bf000e.jpg)  
 (a) Loss function
 
 ![](images/23a715d5d1f51cd8c2be3739c82fbf550849ce3bb8e95c554b6e1bb4022b7017.jpg)  
@@ -751,7 +751,7 @@ This equation simplifies to the one stated in Proposition 4 if $\gamma = 1 / \ta
 ![](images/98c8806ce899ac69af4612056d70a3b1cafffe8cd7dac65f69cee440e4e791d4.jpg)  
 (a) CR landscape
 
-![](images/377994282394a26484ec16e0441ee259d77232c49f00e1258f63c34f584bef70.jpg)  
+![](images/b2cd288b355f0f663fbcc6c99f29c956e1e5392e9b8d11c3944318a5264eb338.jpg)  
 (b) CR landscape contour plot   
 Figure 11. CR Loss Landscape Analysis. This figure illustrates the loss landscape to investigate the effects of the CR term. Notably, the CR term can suddenly become “activated” or “deactivated,” which is vividly depicted in the landscape transitions. These abrupt changes contribute to unstable optimization during training, potentially affecting the convergence and reliability of the model. Understanding this behavior is crucial for improving the training process of Lipschitz neural networks. Regarding the direction of loss landscape, we follow the setting in Engstrom et al. (2018) and Chen et al. (2021). We visualize the loss landscape function $z = \mathcal { L } _ { C R } ( x , w + \omega _ { 1 } d _ { 1 } + \omega _ { 2 } d _ { 2 } )$ , where $d _ { 1 } = s i g n ( \nabla _ { w } \mathcal { L } _ { C R } )$ , $d _ { 2 } \sim$ Rademacher(0.5), and $\omega$ is the grid.
 
@@ -836,7 +836,7 @@ We also compare LA to CE on LipConvNet. Table 13 shows the results for LipConvNe
 
 # D.5. LA Loss Hyper-parameters Experiments
 
-There are three tunable parameters in LA loss: temperature $T$ , offset $\xi$ , and annealing factor $\beta$ . The first two parameters control the trade-off between accuracy and robustness, while the last one determines the strength of the annealing mechanism. For the temperature and offset, we slightly adjust the values used in Prach & Lampert (2022) to find a better trade-off position, given the differences between their network settings and ours. Specifically, we evaluated the temperature $T \in \{ 0 . 2 5 , 0 . 5 , 0 , 7 5 , 1 . 0 \}$ , the offset $\xi \in \{ 0 . 5 , 1 . 0 , 1 . 5 , 2 . 0 , 2 . 5 , 3 . 0 \}$ , and the annealing factor $\beta \in \{ 1 , 3 , 5 , 7 \}$ with LipConvNet-10-32 on CIFAR-100. Based on the evaluation, all other LA experiments are set with √ $T = 0 . 7 5 , \xi = 2 . 0$ , and $\beta = 5 . 0$ . For further LipConvNet experiments, the offset is set to $\xi = 2 \sqrt { 2 }$ due to an oversight in the implementation. As we have found these hyper-parameters to work well, we did not further finetune them for each architecture and datasets to save computational cost. One might consider further refine the hyper-parameters for better performance. Additionally, we present the results of LA loss with different $\beta$ values for LipConvNet-10 on CIFAR-100 in Table 14.
+There are three tunable parameters in LA loss: temperature $T$ , offset $\xi$ , and annealing factor $\beta$ . The first two parameters control the trade-off between accuracy and robustness, while the last one determines the strength of the annealing mechanism. For the temperature and offset, we slightly adjust the values used in Prach & Lampert (2022) to find a better trade-off position, given the differences between their network settings and ours. Specifically, we evaluated the temperature $T \in \{ 0 . 2 5 , 0 . 5 , 0 , 7 5 , 1 . 0 \}$ , the offset $\xi \in \{ 0 . 5 , 1 . 0 , 1 . 5 , 2 . 0 , 2 . 5 , 3 . 0 \}$ , and the annealing factor $\beta \in \{ 1 , 3 , 5 , 7 \}$ with LipConvNet-10-32 on CIFAR-100. Based on the evaluation, all other LA experiments are set with $T = 0 . 7 5 , \xi = 2 . 0$ , and $\beta = 5 . 0$ . For further LipConvNet experiments, the offset is set to $\xi = 2 \sqrt { 2 }$ due to an oversight in the implementation. As we have found these hyper-parameters to work well, we did not further finetune them for each architecture and datasets to save computational cost. One might consider further refine the hyper-parameters for better performance. Additionally, we present the results of LA loss with different $\beta$ values for LipConvNet-10 on CIFAR-100 in Table 14.
 
 Table 14. Experimental results for LipConvNet-10 on CIFAR-100 for different values of $\beta$ in the LA loss.