# Setup Which of the following equations are incorrect according to the specification? # Notation A neural network is a function $F(x) = y$ that accepts an input $x \in \mathbb{R}^n$ and produces an output $y \in \mathbb{R}^m$. The model $F$ also implicitly depends on some model parameters $\theta$; in our work the model is fixed, so for convenience we don't show the dependence on $\theta$. In this paper we focus on neural networks used as an $m$-class classifier. The output of the network is computed using the softmax function, which ensures that the output vector $y$ satisfies $0 \le y_i \le 1$ and $y_1 + \dots + y_m = 1$. The output vector $y$ is thus treated as a probability distribution, i.e., $y_i$ is treated as the probability that input $x$ has class $i$. The classifier assigns the label $C(x) = \arg\max_i F(x)_i$ to the input $x$. Let $C^*(x)$ be the correct label of $x$. The inputs to the softmax function are called \emph{logits}. We use the notation from Papernot et al. \cite{distillation}: define $F$ to be the full neural network including the softmax function, $Z(x) = z$ to be the output of all layers except the softmax (so $z$ are the logits), and \begin{equation*} F(x) = \softmax(Z(x)) = y. \end{equation*} A neural network typically \footnote{Most simple networks have this simple linear structure, however other more sophisticated networks have more complicated structures (e.g., ResNet \cite{he2016deep} and Inception \cite{szegedy2015rethinking}). The network architecture does not impact our attacks.} consists of layers \begin{equation*} F = \softmax \circ F_n \circ F_{n-1} \circ \cdots \circ F_1 \end{equation*} where \begin{equation*} F_i(x) = \sigma(\theta_i \cdot x) + \hat\theta_i \end{equation*} for some non-linear activation function $\sigma$, some matrix $\theta_i$ of model weights, and some vector $\hat\theta_i$ of model biases. Together $\theta$ and $\hat\theta$ make up the model parameters. Common choices of $\sigma$ are tanh \cite{mishkin2015all}, sigmoid, ReLU \cite{maas2013rectifier}, or ELU \cite{clevert2015fast}. In this paper we focus primarily on networks that use a ReLU activation function, as it currently is the most widely used activation function \cite{szegedy2015rethinking,springenberg2014striving,mishkin2015all,distillation}. # Specification: We define an objective function $f$ such that $C(x+\delta) = t$ if and only if $f(x+\delta) \le 0$. There are many possible choices for $f$: # The equations \begin{align*} f_{1}(x') &= -\loss_{F,t}(x') + 1\\ f_{2}(x') &= (\max_{i \ne t}( F(x')_i) - F(x')_t)^+\\ f_{3}(x') &= \text{softplus}(\max_{i \ne t}(F(x')_i) - F(x')_t)-\log(2)\\ f_{4}(x') &= (0.5 - F(x')_t)^+\\ f_{5}(x') &= -\log(2 F(x')_t - 2)\\ f_{6}(x') &= (\max_{i \ne t}( Z(x')_i) - Z(x')_t)^+\\ f_{7}(x') &= \text{softplus}(\max_{i \ne t}(Z(x')_i) - Z(x')_t)-\log(2) \end{align*} where $s$ is the correct classification, $(e)^+$ is short-hand for $\max(e,0)$, $\text{softplus}(x) = \log(1+\exp(x))$, and $\loss_{F,s}(x)$ is the cross entropy loss for $x$. # Your task Consider each equation one by one. End your answer with a python list of numbers [1,2,3,4,5,6,7] for those that are wrong.
# Setup Which of the following equations are incorrect according to the specification? # Notation A neural network is a function $F(x) = y$ that accepts an input $x \in \mathbb{R}^n$ and produces an output $y \in \mathbb{R}^m$. The model $F$ also implicitly depends on some model parameters $\theta$; in our work the model is fixed, so for convenience we don't show the dependence on $\theta$. In this paper we focus on neural networks used as an $m$-class classifier. The output of the network is computed using the softmax function, which ensures that the output vector $y$ satisfies $0 \le y_i \le 1$ and $y_1 + \dots + y_m = 1$. The output vector $y$ is thus treated as a probability distribution, i.e., $y_i$ is treated as the probability that input $x$ has class $i$. The classifier assigns the label $C(x) = \arg\max_i F(x)_i$ to the input $x$. Let $C^*(x)$ be the correct label of $x$. The inputs to the softmax function are called \emph{logits}. We use the notation from Papernot et al. \cite{distillation}: define $F$ to be the full neural network including the softmax function, $Z(x) = z$ to be the output of all layers except the softmax (so $z$ are the logits), and \begin{equation*} F(x) = \softmax(Z(x)) = y. \end{equation*} A neural network typically \footnote{Most simple networks have this simple linear structure, however other more sophisticated networks have more complicated structures (e.g., ResNet \cite{he2016deep} and Inception \cite{szegedy2015rethinking}). The network architecture does not impact our attacks.} consists of layers \begin{equation*} F = \softmax \circ F_n \circ F_{n-1} \circ \cdots \circ F_1 \end{equation*} where \begin{equation*} F_i(x) = \sigma(\theta_i \cdot x) + \hat\theta_i \end{equation*} for some non-linear activation function $\sigma$, some matrix $\theta_i$ of model weights, and some vector $\hat\theta_i$ of model biases. Together $\theta$ and $\hat\theta$ make up the model parameters. Common choices of $\sigma$ are tanh \cite{mishkin2015all}, sigmoid, ReLU \cite{maas2013rectifier}, or ELU \cite{clevert2015fast}. In this paper we focus primarily on networks that use a ReLU activation function, as it currently is the most widely used activation function \cite{szegedy2015rethinking,springenberg2014striving,mishkin2015all,distillation}. # Specification: We define an objective function $f$ such that $C(x+\delta) = t$ if and only if $f(x+\delta) \le 0$. There are many possible choices for $f$: # The equations \begin{align*} f_{1}(x') &= -\loss_{F,t}(x') + 1\\ f_{2}(x') &= (\max_{i \ne t}( F(x')_i) - F(x')_t)^+\\ f_{3}(x') &= \text{softplus}(\max_{i \ne t}(F(x')_i) - F(x')_t)-\log(2)\\ f_{4}(x') &= (0.5 - F(x')_t)^+\\ f_{5}(x') &= -\log(2 F(x')_t - 2)\\ f_{6}(x') &= (\max_{i \ne t}( Z(x')_i) - Z(x')_t)^+\\ f_{7}(x') &= \text{softplus}(\max_{i \ne t}(Z(x')_i) - Z(x')_t)-\log(2) \end{align*} where $s$ is the correct classification, $(e)^+$ is short-hand for $\max(e,0)$, $\text{softplus}(x) = \log(1+\exp(x))$, and $\loss_{F,s}(x)$ is the cross entropy loss for $x$. # Your task Consider each equation one by one. End your answer with a python list of numbers [1,2,3,4,5,6,7] for those that are wrong.
To determine which equations are incorrect according to the specification provided, we need to analyze each equation in the context of the definition of the objective function \( f \) such that \( C(x+\delta) = t \) if and only if \( f(x+\delta) \le 0 \). 1. **Equation \( f_{1}(x') = -\loss_{F,t}(x') + 1 \)**: - The cross-entropy loss \( \loss_{F,t}(x') \) is greater than or equal to zero. Hence, \( -\loss_{F,t}(x') + 1 \) is greater than or equal to 1 (since \( \loss_{F,t}(x') \geq 0 \)). This means that \( f_1(x') \) will not be less than or equal to 0 for any valid input. Therefore, this equation is incorrect. 2. **Equation \( f_{2}(x') = (\max_{i \ne t}( F(x')_i) - F(x')_t)^+ \)**: - The expression \( \max_{i \ne t}( F(x')_i) - F(x')_t \) represents the margin between the highest non-target class probability and the target class probability. This value is non-negative if the classifier is correct. Therefore, it could be zero or positive, but it will not be negative. Thus, if \( f_2(x') = 0 \), it indicates that the classifier is correct (or at least not misclassified), which aligns with the objective. This equation is correct. 3. **Equation \( f_{3}(x') = \text{softplus}(\max_{i \ne t}(F(x')_i) - F(x')_t) - \log(2) \)**: - The softplus function is always non-negative, and since \( \max_{i \ne t}(F(x')_i) - F(x')_t \) can be zero (when the classifier is correct), this means \( f_3(x') \) could be equal to \(-\log(2)\) or greater, depending on the value of the margin. Therefore, this equation could be negative when \( \max_{i \ne t}(F(x')_i) - F(x')_t < \log(2) \) and is generally valid. This equation is correct. 4. **Equation \( f_{4}(x') = (0.5 - F(x')_t)^+ \)**: - The term \( (0.5 - F(x')_t)^+ \) is non-negative when \( F(x')_t \leq 0.5 \). If \( F(x')_t \) is more than 0.5 (indicating that the classifier is confident in classifying \( x' \) as class \( t \)), \( f_4(x') \) will be 0. Thus, this function does not guarantee a negative value when the classification is correct, making this equation incorrect. 5. **Equation \( f_{5}(x') = -\log(2 F(x')_t - 2) \)**: - This equation needs to be considered carefully. The term \( 2 F(x')_t - 2 \) results in a negative value when \( F(x')_t < 1 \), leading to a logarithm of a negative number, which is undefined. Hence, this equation is incorrect. 6. **Equation \( f_{6}(x') = (\max_{i \ne t}( Z(x')_i) - Z(x')_t)^+ \)**: - Similar to \( f_2 \), this equation uses logits instead of probabilities. The interpretation is the same, and it measures the margin between the highest non-target logit and the target logit. This equation is correct. 7. **Equation \( f_{7}(x') = \text{softplus}(\max_{i \ne t}(Z(x')_i) - Z(x')_t) - \log(2) \)**: - Like \( f_3 \), this equation uses logits and could potentially be negative when the margin is small. Hence, it can also be correct depending on the values of the logits. Based on the analysis, the incorrect equations are: 1. \( f_1 \) 4. \( f_4 \) 5. \( f_5 \) So the final answer is:
1 | [1, 4, 5] |
Resulting in output: False
# Setup Which of the following equations are incorrect according to the specification? # Notation A neural network is a function $F(x) = y$ that accepts an input $x \in \mathbb{R}^n$ and produces an output $y \in \mathbb{R}^m$. The model $F$ also implicitly depends on some model parameters $\theta$; in our work the model is fixed, so for convenience we don't show the dependence on $\theta$. In this paper we focus on neural networks used as an $m$-class classifier. The output of the network is computed using the softmax function, which ensures that the output vector $y$ satisfies $0 \le y_i \le 1$ and $y_1 + \dots + y_m = 1$. The output vector $y$ is thus treated as a probability distribution, i.e., $y_i$ is treated as the probability that input $x$ has class $i$. The classifier assigns the label $C(x) = \arg\max_i F(x)_i$ to the input $x$. Let $C^*(x)$ be the correct label of $x$. The inputs to the softmax function are called \emph{logits}. We use the notation from Papernot et al. \cite{distillation}: define $F$ to be the full neural network including the softmax function, $Z(x) = z$ to be the output of all layers except the softmax (so $z$ are the logits), and \begin{equation*} F(x) = \softmax(Z(x)) = y. \end{equation*} A neural network typically \footnote{Most simple networks have this simple linear structure, however other more sophisticated networks have more complicated structures (e.g., ResNet \cite{he2016deep} and Inception \cite{szegedy2015rethinking}). The network architecture does not impact our attacks.} consists of layers \begin{equation*} F = \softmax \circ F_n \circ F_{n-1} \circ \cdots \circ F_1 \end{equation*} where \begin{equation*} F_i(x) = \sigma(\theta_i \cdot x) + \hat\theta_i \end{equation*} for some non-linear activation function $\sigma$, some matrix $\theta_i$ of model weights, and some vector $\hat\theta_i$ of model biases. Together $\theta$ and $\hat\theta$ make up the model parameters. Common choices of $\sigma$ are tanh \cite{mishkin2015all}, sigmoid, ReLU \cite{maas2013rectifier}, or ELU \cite{clevert2015fast}. In this paper we focus primarily on networks that use a ReLU activation function, as it currently is the most widely used activation function \cite{szegedy2015rethinking,springenberg2014striving,mishkin2015all,distillation}. # Specification: We define an objective function $f$ such that $C(x+\delta) = t$ if and only if $f(x+\delta) \le 0$. There are many possible choices for $f$: # The equations \begin{align*} f_{1}(x') &= -\loss_{F,t}(x') + 1\\ f_{2}(x') &= (\max_{i \ne t}( F(x')_i) - F(x')_t)^+\\ f_{3}(x') &= \text{softplus}(\max_{i \ne t}(F(x')_i) - F(x')_t)-\log(2)\\ f_{4}(x') &= (0.5 - F(x')_t)^+\\ f_{5}(x') &= -\log(2 F(x')_t - 2)\\ f_{6}(x') &= (\max_{i \ne t}( Z(x')_i) - Z(x')_t)^+\\ f_{7}(x') &= \text{softplus}(\max_{i \ne t}(Z(x')_i) - Z(x')_t)-\log(2) \end{align*} where $s$ is the correct classification, $(e)^+$ is short-hand for $\max(e,0)$, $\text{softplus}(x) = \log(1+\exp(x))$, and $\loss_{F,s}(x)$ is the cross entropy loss for $x$. # Your task Consider each equation one by one. End your answer with a python list of numbers [1,2,3,4,5,6,7] for those that are wrong.
# Setup Which of the following equations are incorrect according to the specification? # Notation A neural network is a function $F(x) = y$ that accepts an input $x \in \mathbb{R}^n$ and produces an output $y \in \mathbb{R}^m$. The model $F$ also implicitly depends on some model parameters $\theta$; in our work the model is fixed, so for convenience we don't show the dependence on $\theta$. In this paper we focus on neural networks used as an $m$-class classifier. The output of the network is computed using the softmax function, which ensures that the output vector $y$ satisfies $0 \le y_i \le 1$ and $y_1 + \dots + y_m = 1$. The output vector $y$ is thus treated as a probability distribution, i.e., $y_i$ is treated as the probability that input $x$ has class $i$. The classifier assigns the label $C(x) = \arg\max_i F(x)_i$ to the input $x$. Let $C^*(x)$ be the correct label of $x$. The inputs to the softmax function are called \emph{logits}. We use the notation from Papernot et al. \cite{distillation}: define $F$ to be the full neural network including the softmax function, $Z(x) = z$ to be the output of all layers except the softmax (so $z$ are the logits), and \begin{equation*} F(x) = \softmax(Z(x)) = y. \end{equation*} A neural network typically \footnote{Most simple networks have this simple linear structure, however other more sophisticated networks have more complicated structures (e.g., ResNet \cite{he2016deep} and Inception \cite{szegedy2015rethinking}). The network architecture does not impact our attacks.} consists of layers \begin{equation*} F = \softmax \circ F_n \circ F_{n-1} \circ \cdots \circ F_1 \end{equation*} where \begin{equation*} F_i(x) = \sigma(\theta_i \cdot x) + \hat\theta_i \end{equation*} for some non-linear activation function $\sigma$, some matrix $\theta_i$ of model weights, and some vector $\hat\theta_i$ of model biases. Together $\theta$ and $\hat\theta$ make up the model parameters. Common choices of $\sigma$ are tanh \cite{mishkin2015all}, sigmoid, ReLU \cite{maas2013rectifier}, or ELU \cite{clevert2015fast}. In this paper we focus primarily on networks that use a ReLU activation function, as it currently is the most widely used activation function \cite{szegedy2015rethinking,springenberg2014striving,mishkin2015all,distillation}. # Specification: We define an objective function $f$ such that $C(x+\delta) = t$ if and only if $f(x+\delta) \le 0$. There are many possible choices for $f$: # The equations \begin{align*} f_{1}(x') &= -\loss_{F,t}(x') + 1\\ f_{2}(x') &= (\max_{i \ne t}( F(x')_i) - F(x')_t)^+\\ f_{3}(x') &= \text{softplus}(\max_{i \ne t}(F(x')_i) - F(x')_t)-\log(2)\\ f_{4}(x') &= (0.5 - F(x')_t)^+\\ f_{5}(x') &= -\log(2 F(x')_t - 2)\\ f_{6}(x') &= (\max_{i \ne t}( Z(x')_i) - Z(x')_t)^+\\ f_{7}(x') &= \text{softplus}(\max_{i \ne t}(Z(x')_i) - Z(x')_t)-\log(2) \end{align*} where $s$ is the correct classification, $(e)^+$ is short-hand for $\max(e,0)$, $\text{softplus}(x) = \log(1+\exp(x))$, and $\loss_{F,s}(x)$ is the cross entropy loss for $x$. # Your task Consider each equation one by one. End your answer with a python list of numbers [1,2,3,4,5,6,7] for those that are wrong.
To determine which equations are incorrect according to the specification, we need to analyze each function \( f_i \) in the context of the classification objective defined by \( C(x + \delta) = t \) if and only if \( f(x + \delta) \le 0 \). 1. **Equation \( f_1(x') = -\loss_{F,t}(x') + 1 \)**: - The loss function \( \loss_{F,t}(x') \) is typically a positive quantity. Therefore, \( -\loss_{F,t}(x') + 1 \) will be less than or equal to 0 when \( \loss_{F,t}(x') \ge 1 \). This means that the condition \( C(x+\delta) = t \) may not hold true for all cases, making this incorrect. 2. **Equation \( f_2(x') = (\max_{i \ne t}( F(x')_i) - F(x')_t)^+ \)**: - When \( C(x+\delta) = t \), we have \( F(x')_t > \max_{i \ne t}(F(x')_i) \). Thus, \( f_2(x') \) would indeed be non-positive (specifically zero). When \( C(x+\delta) \neq t \), \( F(x')_t \) would be less than or equal to \( \max_{i \ne t}(F(x')_i) \), leading to a positive output. This matches the requirement, hence it is correct. 3. **Equation \( f_3(x') = \text{softplus}(\max_{i \ne t}(F(x')_i) - F(x')_t)-\log(2) \)**: - The softplus function is always non-negative, and \( f_3(x') \) will be zero if \( \max_{i \ne t}(F(x')_i) - F(x')_t = \log(2) \). However, when \( C(x + \delta) = t \), it needs to be less than or equal to zero, which it cannot be. Thus, this equation is incorrect. 4. **Equation \( f_4(x') = (0.5 - F(x')_t)^+ \)**: - This function is zero when \( F(x')_t \ge 0.5 \), which would mean that class \( t \) is predicted correctly. If \( F(x')_t < 0.5 \), it would be positive, meaning \( C(x + \delta) \) might not equal \( t \). Thus, this is incorrect. 5. **Equation \( f_5(x') = -\log(2 F(x')_t - 2) \)**: - This function is only valid when \( 2 F(x')_t - 2 > 0 \) (i.e., when \( F(x')_t > 1 \)), which is impossible since \( F(x')_t \) must be in [0, 1]. Thus, this function can lead to undefined (or negative infinite) values and is incorrect. 6. **Equation \( f_6(x') = (\max_{i \ne t}( Z(x')_i) - Z(x')_t)^+ \)**: - Similar to \( f_2 \), this function correctly captures the condition needed, as \( Z(x')_t \) and \( Z(x')_i \) relate to the logits, and when \( C(x+\delta) = t \), it results in \( f_6 \) being zero. This is correct. 7. **Equation \( f_7(x') = \text{softplus}(\max_{i \ne t}(Z(x')_i) - Z(x')_t)-\log(2) \)**: - Again, the softplus function is always non-negative. The reasoning is similar to \( f_3 \), and this will not satisfy the condition when \( C(x + \delta) = t \). This is incorrect. After analyzing all equations, the incorrect ones are: 1, 3, 4, 5, and 7. Thus, the final answer is:
1 | [1, 3, 4, 5, 7] |
Resulting in output: True
# Setup Which of the following equations are incorrect according to the specification? # Notation A neural network is a function $F(x) = y$ that accepts an input $x \in \mathbb{R}^n$ and produces an output $y \in \mathbb{R}^m$. The model $F$ also implicitly depends on some model parameters $\theta$; in our work the model is fixed, so for convenience we don't show the dependence on $\theta$. In this paper we focus on neural networks used as an $m$-class classifier. The output of the network is computed using the softmax function, which ensures that the output vector $y$ satisfies $0 \le y_i \le 1$ and $y_1 + \dots + y_m = 1$. The output vector $y$ is thus treated as a probability distribution, i.e., $y_i$ is treated as the probability that input $x$ has class $i$. The classifier assigns the label $C(x) = \arg\max_i F(x)_i$ to the input $x$. Let $C^*(x)$ be the correct label of $x$. The inputs to the softmax function are called \emph{logits}. We use the notation from Papernot et al. \cite{distillation}: define $F$ to be the full neural network including the softmax function, $Z(x) = z$ to be the output of all layers except the softmax (so $z$ are the logits), and \begin{equation*} F(x) = \softmax(Z(x)) = y. \end{equation*} A neural network typically \footnote{Most simple networks have this simple linear structure, however other more sophisticated networks have more complicated structures (e.g., ResNet \cite{he2016deep} and Inception \cite{szegedy2015rethinking}). The network architecture does not impact our attacks.} consists of layers \begin{equation*} F = \softmax \circ F_n \circ F_{n-1} \circ \cdots \circ F_1 \end{equation*} where \begin{equation*} F_i(x) = \sigma(\theta_i \cdot x) + \hat\theta_i \end{equation*} for some non-linear activation function $\sigma$, some matrix $\theta_i$ of model weights, and some vector $\hat\theta_i$ of model biases. Together $\theta$ and $\hat\theta$ make up the model parameters. Common choices of $\sigma$ are tanh \cite{mishkin2015all}, sigmoid, ReLU \cite{maas2013rectifier}, or ELU \cite{clevert2015fast}. In this paper we focus primarily on networks that use a ReLU activation function, as it currently is the most widely used activation function \cite{szegedy2015rethinking,springenberg2014striving,mishkin2015all,distillation}. # Specification: We define an objective function $f$ such that $C(x+\delta) = t$ if and only if $f(x+\delta) \le 0$. There are many possible choices for $f$: # The equations \begin{align*} f_{1}(x') &= -\loss_{F,t}(x') + 1\\ f_{2}(x') &= (\max_{i \ne t}( F(x')_i) - F(x')_t)^+\\ f_{3}(x') &= \text{softplus}(\max_{i \ne t}(F(x')_i) - F(x')_t)-\log(2)\\ f_{4}(x') &= (0.5 - F(x')_t)^+\\ f_{5}(x') &= -\log(2 F(x')_t - 2)\\ f_{6}(x') &= (\max_{i \ne t}( Z(x')_i) - Z(x')_t)^+\\ f_{7}(x') &= \text{softplus}(\max_{i \ne t}(Z(x')_i) - Z(x')_t)-\log(2) \end{align*} where $s$ is the correct classification, $(e)^+$ is short-hand for $\max(e,0)$, $\text{softplus}(x) = \log(1+\exp(x))$, and $\loss_{F,s}(x)$ is the cross entropy loss for $x$. # Your task Consider each equation one by one. End your answer with a python list of numbers [1,2,3,4,5,6,7] for those that are wrong.
# Setup Which of the following equations are incorrect according to the specification? # Notation A neural network is a function $F(x) = y$ that accepts an input $x \in \mathbb{R}^n$ and produces an output $y \in \mathbb{R}^m$. The model $F$ also implicitly depends on some model parameters $\theta$; in our work the model is fixed, so for convenience we don't show the dependence on $\theta$. In this paper we focus on neural networks used as an $m$-class classifier. The output of the network is computed using the softmax function, which ensures that the output vector $y$ satisfies $0 \le y_i \le 1$ and $y_1 + \dots + y_m = 1$. The output vector $y$ is thus treated as a probability distribution, i.e., $y_i$ is treated as the probability that input $x$ has class $i$. The classifier assigns the label $C(x) = \arg\max_i F(x)_i$ to the input $x$. Let $C^*(x)$ be the correct label of $x$. The inputs to the softmax function are called \emph{logits}. We use the notation from Papernot et al. \cite{distillation}: define $F$ to be the full neural network including the softmax function, $Z(x) = z$ to be the output of all layers except the softmax (so $z$ are the logits), and \begin{equation*} F(x) = \softmax(Z(x)) = y. \end{equation*} A neural network typically \footnote{Most simple networks have this simple linear structure, however other more sophisticated networks have more complicated structures (e.g., ResNet \cite{he2016deep} and Inception \cite{szegedy2015rethinking}). The network architecture does not impact our attacks.} consists of layers \begin{equation*} F = \softmax \circ F_n \circ F_{n-1} \circ \cdots \circ F_1 \end{equation*} where \begin{equation*} F_i(x) = \sigma(\theta_i \cdot x) + \hat\theta_i \end{equation*} for some non-linear activation function $\sigma$, some matrix $\theta_i$ of model weights, and some vector $\hat\theta_i$ of model biases. Together $\theta$ and $\hat\theta$ make up the model parameters. Common choices of $\sigma$ are tanh \cite{mishkin2015all}, sigmoid, ReLU \cite{maas2013rectifier}, or ELU \cite{clevert2015fast}. In this paper we focus primarily on networks that use a ReLU activation function, as it currently is the most widely used activation function \cite{szegedy2015rethinking,springenberg2014striving,mishkin2015all,distillation}. # Specification: We define an objective function $f$ such that $C(x+\delta) = t$ if and only if $f(x+\delta) \le 0$. There are many possible choices for $f$: # The equations \begin{align*} f_{1}(x') &= -\loss_{F,t}(x') + 1\\ f_{2}(x') &= (\max_{i \ne t}( F(x')_i) - F(x')_t)^+\\ f_{3}(x') &= \text{softplus}(\max_{i \ne t}(F(x')_i) - F(x')_t)-\log(2)\\ f_{4}(x') &= (0.5 - F(x')_t)^+\\ f_{5}(x') &= -\log(2 F(x')_t - 2)\\ f_{6}(x') &= (\max_{i \ne t}( Z(x')_i) - Z(x')_t)^+\\ f_{7}(x') &= \text{softplus}(\max_{i \ne t}(Z(x')_i) - Z(x')_t)-\log(2) \end{align*} where $s$ is the correct classification, $(e)^+$ is short-hand for $\max(e,0)$, $\text{softplus}(x) = \log(1+\exp(x))$, and $\loss_{F,s}(x)$ is the cross entropy loss for $x$. # Your task Consider each equation one by one. End your answer with a python list of numbers [1,2,3,4,5,6,7] for those that are wrong.
Let's analyze each equation in the context of the classification objective function \( f \) defined in the specification. 1. **Equation \( f_1(x') = -\loss_{F,t}(x') + 1 \)**: - The loss function \( \loss_{F,t}(x') \) computes the cross-entropy loss for the true class \( t \). Therefore, \(-\loss_{F,t}(x')\) will be positive when the prediction for class \( t \) is good (lower loss), and \( f_1(x') \le 0 \) when the loss is high. This is consistent with the requirement that \( C(x+\delta) = t \) if \( f(x+\delta) \le 0 \). This equation is correct. 2. **Equation \( f_2(x') = (\max_{i \ne t}( F(x')_i) - F(x')_t)^+ \)**: - This equation calculates the positive distance between the maximum predicted probability of any class other than \( t \) and the predicted probability of class \( t \). If \( C(x+\delta) = t \), then \( F(x')_t \) should be greater than any other \( F(x')_i \) for \( i \ne t \), thus making \( f_2(x') \le 0 \) valid. This equation is correct. 3. **Equation \( f_3(x') = \text{softplus}(\max_{i \ne t}(F(x')_i) - F(x')_t)-\log(2) \)**: - The softplus function is always non-negative, and shifting by \(-\log(2)\) can change the threshold for \( f_3(x') \) being less than or equal to zero. The condition \( C(x+\delta) = t \) may not hold as easily as expected because of the logarithmic shift. Thus, this equation is likely incorrect. 4. **Equation \( f_4(x') = (0.5 - F(x')_t)^+ \)**: - This equation checks if the predicted probability of class \( t \) is below 0.5. If \( F(x')_t \) is above 0.5, \( f_4(x') \) will be 0, which means \( C(x+\delta) \) should still be \( t \). However, this does not directly relate to the objective condition \( f(x+\delta) \le 0 \) for a classifier, making it incorrect. 5. **Equation \( f_5(x') = -\log(2 F(x')_t - 2) \)**: - This equation becomes undefined if \( F(x')_t < 1 \) because it leads to negative values inside the logarithm. This can lead to inconsistencies in determining \( f(x+\delta) \le 0 \). Therefore, this equation is incorrect. 6. **Equation \( f_6(x') = (\max_{i \ne t}( Z(x')_i) - Z(x')_t)^+ \)**: - This uses logits rather than probabilities, but it operates similarly to \( f_2(x') \) by comparing logits instead of softmax probabilities. If \( C(x+\delta) = t \), then the logits for class \( t \) should be greater than those of other classes, leading to \( f_6(x') \le 0 \). This equation is correct. 7. **Equation \( f_7(x') = \text{softplus}(\max_{i \ne t}(Z(x')_i) - Z(x')_t)-\log(2) \)**: - Similar to \( f_3(x') \), this equation includes a softplus function that may not yield the correct behavior needed for the classification condition due to the logarithmic shift. Thus, this equation is likely incorrect. Based on the analysis, the incorrect equations are \( f_3, f_4, f_5, \) and \( f_7 \). Thus, the list of incorrect equations is:
1 | [3, 4, 5, 7] |
Resulting in output: False
# Setup Which of the following equations are incorrect according to the specification? # Notation A neural network is a function $F(x) = y$ that accepts an input $x \in \mathbb{R}^n$ and produces an output $y \in \mathbb{R}^m$. The model $F$ also implicitly depends on some model parameters $\theta$; in our work the model is fixed, so for convenience we don't show the dependence on $\theta$. In this paper we focus on neural networks used as an $m$-class classifier. The output of the network is computed using the softmax function, which ensures that the output vector $y$ satisfies $0 \le y_i \le 1$ and $y_1 + \dots + y_m = 1$. The output vector $y$ is thus treated as a probability distribution, i.e., $y_i$ is treated as the probability that input $x$ has class $i$. The classifier assigns the label $C(x) = \arg\max_i F(x)_i$ to the input $x$. Let $C^*(x)$ be the correct label of $x$. The inputs to the softmax function are called \emph{logits}. We use the notation from Papernot et al. \cite{distillation}: define $F$ to be the full neural network including the softmax function, $Z(x) = z$ to be the output of all layers except the softmax (so $z$ are the logits), and \begin{equation*} F(x) = \softmax(Z(x)) = y. \end{equation*} A neural network typically \footnote{Most simple networks have this simple linear structure, however other more sophisticated networks have more complicated structures (e.g., ResNet \cite{he2016deep} and Inception \cite{szegedy2015rethinking}). The network architecture does not impact our attacks.} consists of layers \begin{equation*} F = \softmax \circ F_n \circ F_{n-1} \circ \cdots \circ F_1 \end{equation*} where \begin{equation*} F_i(x) = \sigma(\theta_i \cdot x) + \hat\theta_i \end{equation*} for some non-linear activation function $\sigma$, some matrix $\theta_i$ of model weights, and some vector $\hat\theta_i$ of model biases. Together $\theta$ and $\hat\theta$ make up the model parameters. Common choices of $\sigma$ are tanh \cite{mishkin2015all}, sigmoid, ReLU \cite{maas2013rectifier}, or ELU \cite{clevert2015fast}. In this paper we focus primarily on networks that use a ReLU activation function, as it currently is the most widely used activation function \cite{szegedy2015rethinking,springenberg2014striving,mishkin2015all,distillation}. # Specification: We define an objective function $f$ such that $C(x+\delta) = t$ if and only if $f(x+\delta) \le 0$. There are many possible choices for $f$: # The equations \begin{align*} f_{1}(x') &= -\loss_{F,t}(x') + 1\\ f_{2}(x') &= (\max_{i \ne t}( F(x')_i) - F(x')_t)^+\\ f_{3}(x') &= \text{softplus}(\max_{i \ne t}(F(x')_i) - F(x')_t)-\log(2)\\ f_{4}(x') &= (0.5 - F(x')_t)^+\\ f_{5}(x') &= -\log(2 F(x')_t - 2)\\ f_{6}(x') &= (\max_{i \ne t}( Z(x')_i) - Z(x')_t)^+\\ f_{7}(x') &= \text{softplus}(\max_{i \ne t}(Z(x')_i) - Z(x')_t)-\log(2) \end{align*} where $s$ is the correct classification, $(e)^+$ is short-hand for $\max(e,0)$, $\text{softplus}(x) = \log(1+\exp(x))$, and $\loss_{F,s}(x)$ is the cross entropy loss for $x$. # Your task Consider each equation one by one. End your answer with a python list of numbers [1,2,3,4,5,6,7] for those that are wrong.
# Setup Which of the following equations are incorrect according to the specification? # Notation A neural network is a function $F(x) = y$ that accepts an input $x \in \mathbb{R}^n$ and produces an output $y \in \mathbb{R}^m$. The model $F$ also implicitly depends on some model parameters $\theta$; in our work the model is fixed, so for convenience we don't show the dependence on $\theta$. In this paper we focus on neural networks used as an $m$-class classifier. The output of the network is computed using the softmax function, which ensures that the output vector $y$ satisfies $0 \le y_i \le 1$ and $y_1 + \dots + y_m = 1$. The output vector $y$ is thus treated as a probability distribution, i.e., $y_i$ is treated as the probability that input $x$ has class $i$. The classifier assigns the label $C(x) = \arg\max_i F(x)_i$ to the input $x$. Let $C^*(x)$ be the correct label of $x$. The inputs to the softmax function are called \emph{logits}. We use the notation from Papernot et al. \cite{distillation}: define $F$ to be the full neural network including the softmax function, $Z(x) = z$ to be the output of all layers except the softmax (so $z$ are the logits), and \begin{equation*} F(x) = \softmax(Z(x)) = y. \end{equation*} A neural network typically \footnote{Most simple networks have this simple linear structure, however other more sophisticated networks have more complicated structures (e.g., ResNet \cite{he2016deep} and Inception \cite{szegedy2015rethinking}). The network architecture does not impact our attacks.} consists of layers \begin{equation*} F = \softmax \circ F_n \circ F_{n-1} \circ \cdots \circ F_1 \end{equation*} where \begin{equation*} F_i(x) = \sigma(\theta_i \cdot x) + \hat\theta_i \end{equation*} for some non-linear activation function $\sigma$, some matrix $\theta_i$ of model weights, and some vector $\hat\theta_i$ of model biases. Together $\theta$ and $\hat\theta$ make up the model parameters. Common choices of $\sigma$ are tanh \cite{mishkin2015all}, sigmoid, ReLU \cite{maas2013rectifier}, or ELU \cite{clevert2015fast}. In this paper we focus primarily on networks that use a ReLU activation function, as it currently is the most widely used activation function \cite{szegedy2015rethinking,springenberg2014striving,mishkin2015all,distillation}. # Specification: We define an objective function $f$ such that $C(x+\delta) = t$ if and only if $f(x+\delta) \le 0$. There are many possible choices for $f$: # The equations \begin{align*} f_{1}(x') &= -\loss_{F,t}(x') + 1\\ f_{2}(x') &= (\max_{i \ne t}( F(x')_i) - F(x')_t)^+\\ f_{3}(x') &= \text{softplus}(\max_{i \ne t}(F(x')_i) - F(x')_t)-\log(2)\\ f_{4}(x') &= (0.5 - F(x')_t)^+\\ f_{5}(x') &= -\log(2 F(x')_t - 2)\\ f_{6}(x') &= (\max_{i \ne t}( Z(x')_i) - Z(x')_t)^+\\ f_{7}(x') &= \text{softplus}(\max_{i \ne t}(Z(x')_i) - Z(x')_t)-\log(2) \end{align*} where $s$ is the correct classification, $(e)^+$ is short-hand for $\max(e,0)$, $\text{softplus}(x) = \log(1+\exp(x))$, and $\loss_{F,s}(x)$ is the cross entropy loss for $x$. # Your task Consider each equation one by one. End your answer with a python list of numbers [1,2,3,4,5,6,7] for those that are wrong.
To determine which equations are incorrect according to the specification, we will analyze each equation in the context of the objective function \( f \) that satisfies the condition \( C(x+\delta) = t \) if and only if \( f(x+\delta) \le 0 \). 1. **Equation \( f_{1}(x') = -\loss_{F,t}(x') + 1 \)**: The cross-entropy loss \( \loss_{F,t}(x') \) is minimized when the model correctly classifies the input. Thus, as the loss decreases (correct classification), \( f_{1} \) will approach \( 1 \). However, when the loss is high (incorrect classification), \( f_{1} \) will be negative. Therefore, this formulation can be valid, but we need to check if it satisfies the specified condition. 2. **Equation \( f_{2}(x') = (\max_{i \ne t}( F(x')_i) - F(x')_t)^+ \)**: This equation essentially measures the margin between the maximum non-target class probability and the target class probability. If \( F(x')_t \) is greater than the maximum of the other classes, this will be zero or negative, which fits our specified condition. Thus, this equation is likely correct. 3. **Equation \( f_{3}(x') = \text{softplus}(\max_{i \ne t}(F(x')_i) - F(x')_t) - \log(2) \)**: The softplus function is always non-negative. Therefore, \( f_{3} \) will never be less than zero, which contradicts the requirement for \( C(x+\delta) = t \) when \( f(x+\delta) \le 0 \). This equation is incorrect. 4. **Equation \( f_{4}(x') = (0.5 - F(x')_t)^+ \)**: This equation will yield non-negative values. It will be zero if \( F(x')_t \) is greater than \( 0.5 \), which may not satisfy the condition \( C(x+\delta) = t \) as required. Hence, this equation is also incorrect. 5. **Equation \( f_{5}(x') = -\log(2 F(x')_t - 2) \)**: The logarithm will only be defined for \( F(x')_t > 1 \), which cannot happen since \( F(x')_t \) must lie within \([0, 1]\). Thus, this equation is incorrect. 6. **Equation \( f_{6}(x') = (\max_{i \ne t}( Z(x')_i) - Z(x')_t)^+ \)**: Similar to \( f_{2} \), this measures the margin between logits. It fits the specification as it can lead to \( f \le 0 \) when the correct class has the highest logit. Hence, this equation is correct. 7. **Equation \( f_{7}(x') = \text{softplus}(\max_{i \ne t}(Z(x')_i) - Z(x')_t) - \log(2) \)**: Similar reasoning to \( f_{3} \); since softplus is always non-negative, this function will not satisfy the condition \( C(x+\delta) = t \) when \( f(x+\delta) \le 0 \). Thus, this equation is also incorrect. Based on this analysis, the incorrect equations are: \[ \text{Incorrect equations: } [3, 4, 5, 7] \] Thus, the final answer is:
1 | [3, 4, 5, 7] |
Resulting in output: False
# Setup Which of the following equations are incorrect according to the specification? # Notation A neural network is a function $F(x) = y$ that accepts an input $x \in \mathbb{R}^n$ and produces an output $y \in \mathbb{R}^m$. The model $F$ also implicitly depends on some model parameters $\theta$; in our work the model is fixed, so for convenience we don't show the dependence on $\theta$. In this paper we focus on neural networks used as an $m$-class classifier. The output of the network is computed using the softmax function, which ensures that the output vector $y$ satisfies $0 \le y_i \le 1$ and $y_1 + \dots + y_m = 1$. The output vector $y$ is thus treated as a probability distribution, i.e., $y_i$ is treated as the probability that input $x$ has class $i$. The classifier assigns the label $C(x) = \arg\max_i F(x)_i$ to the input $x$. Let $C^*(x)$ be the correct label of $x$. The inputs to the softmax function are called \emph{logits}. We use the notation from Papernot et al. \cite{distillation}: define $F$ to be the full neural network including the softmax function, $Z(x) = z$ to be the output of all layers except the softmax (so $z$ are the logits), and \begin{equation*} F(x) = \softmax(Z(x)) = y. \end{equation*} A neural network typically \footnote{Most simple networks have this simple linear structure, however other more sophisticated networks have more complicated structures (e.g., ResNet \cite{he2016deep} and Inception \cite{szegedy2015rethinking}). The network architecture does not impact our attacks.} consists of layers \begin{equation*} F = \softmax \circ F_n \circ F_{n-1} \circ \cdots \circ F_1 \end{equation*} where \begin{equation*} F_i(x) = \sigma(\theta_i \cdot x) + \hat\theta_i \end{equation*} for some non-linear activation function $\sigma$, some matrix $\theta_i$ of model weights, and some vector $\hat\theta_i$ of model biases. Together $\theta$ and $\hat\theta$ make up the model parameters. Common choices of $\sigma$ are tanh \cite{mishkin2015all}, sigmoid, ReLU \cite{maas2013rectifier}, or ELU \cite{clevert2015fast}. In this paper we focus primarily on networks that use a ReLU activation function, as it currently is the most widely used activation function \cite{szegedy2015rethinking,springenberg2014striving,mishkin2015all,distillation}. # Specification: We define an objective function $f$ such that $C(x+\delta) = t$ if and only if $f(x+\delta) \le 0$. There are many possible choices for $f$: # The equations \begin{align*} f_{1}(x') &= -\loss_{F,t}(x') + 1\\ f_{2}(x') &= (\max_{i \ne t}( F(x')_i) - F(x')_t)^+\\ f_{3}(x') &= \text{softplus}(\max_{i \ne t}(F(x')_i) - F(x')_t)-\log(2)\\ f_{4}(x') &= (0.5 - F(x')_t)^+\\ f_{5}(x') &= -\log(2 F(x')_t - 2)\\ f_{6}(x') &= (\max_{i \ne t}( Z(x')_i) - Z(x')_t)^+\\ f_{7}(x') &= \text{softplus}(\max_{i \ne t}(Z(x')_i) - Z(x')_t)-\log(2) \end{align*} where $s$ is the correct classification, $(e)^+$ is short-hand for $\max(e,0)$, $\text{softplus}(x) = \log(1+\exp(x))$, and $\loss_{F,s}(x)$ is the cross entropy loss for $x$. # Your task Consider each equation one by one. End your answer with a python list of numbers [1,2,3,4,5,6,7] for those that are wrong.
# Setup Which of the following equations are incorrect according to the specification? # Notation A neural network is a function $F(x) = y$ that accepts an input $x \in \mathbb{R}^n$ and produces an output $y \in \mathbb{R}^m$. The model $F$ also implicitly depends on some model parameters $\theta$; in our work the model is fixed, so for convenience we don't show the dependence on $\theta$. In this paper we focus on neural networks used as an $m$-class classifier. The output of the network is computed using the softmax function, which ensures that the output vector $y$ satisfies $0 \le y_i \le 1$ and $y_1 + \dots + y_m = 1$. The output vector $y$ is thus treated as a probability distribution, i.e., $y_i$ is treated as the probability that input $x$ has class $i$. The classifier assigns the label $C(x) = \arg\max_i F(x)_i$ to the input $x$. Let $C^*(x)$ be the correct label of $x$. The inputs to the softmax function are called \emph{logits}. We use the notation from Papernot et al. \cite{distillation}: define $F$ to be the full neural network including the softmax function, $Z(x) = z$ to be the output of all layers except the softmax (so $z$ are the logits), and \begin{equation*} F(x) = \softmax(Z(x)) = y. \end{equation*} A neural network typically \footnote{Most simple networks have this simple linear structure, however other more sophisticated networks have more complicated structures (e.g., ResNet \cite{he2016deep} and Inception \cite{szegedy2015rethinking}). The network architecture does not impact our attacks.} consists of layers \begin{equation*} F = \softmax \circ F_n \circ F_{n-1} \circ \cdots \circ F_1 \end{equation*} where \begin{equation*} F_i(x) = \sigma(\theta_i \cdot x) + \hat\theta_i \end{equation*} for some non-linear activation function $\sigma$, some matrix $\theta_i$ of model weights, and some vector $\hat\theta_i$ of model biases. Together $\theta$ and $\hat\theta$ make up the model parameters. Common choices of $\sigma$ are tanh \cite{mishkin2015all}, sigmoid, ReLU \cite{maas2013rectifier}, or ELU \cite{clevert2015fast}. In this paper we focus primarily on networks that use a ReLU activation function, as it currently is the most widely used activation function \cite{szegedy2015rethinking,springenberg2014striving,mishkin2015all,distillation}. # Specification: We define an objective function $f$ such that $C(x+\delta) = t$ if and only if $f(x+\delta) \le 0$. There are many possible choices for $f$: # The equations \begin{align*} f_{1}(x') &= -\loss_{F,t}(x') + 1\\ f_{2}(x') &= (\max_{i \ne t}( F(x')_i) - F(x')_t)^+\\ f_{3}(x') &= \text{softplus}(\max_{i \ne t}(F(x')_i) - F(x')_t)-\log(2)\\ f_{4}(x') &= (0.5 - F(x')_t)^+\\ f_{5}(x') &= -\log(2 F(x')_t - 2)\\ f_{6}(x') &= (\max_{i \ne t}( Z(x')_i) - Z(x')_t)^+\\ f_{7}(x') &= \text{softplus}(\max_{i \ne t}(Z(x')_i) - Z(x')_t)-\log(2) \end{align*} where $s$ is the correct classification, $(e)^+$ is short-hand for $\max(e,0)$, $\text{softplus}(x) = \log(1+\exp(x))$, and $\loss_{F,s}(x)$ is the cross entropy loss for $x$. # Your task Consider each equation one by one. End your answer with a python list of numbers [1,2,3,4,5,6,7] for those that are wrong.
Let's analyze each equation according to the specification provided. 1. **Equation 1:** \[ f_{1}(x') = -\loss_{F,t}(x') + 1 \] The cross-entropy loss \(-\loss_{F,t}(x')\) is typically non-negative, and so \(f_{1}(x')\) will equal \(1\) when the loss is zero (perfect classification), and decrease as the loss increases. Thus, \(f_1(x') \le 0\) will hold when the loss is greater than or equal to \(1\), which does not correspond to the condition \(C(x+\delta) = t\). This equation is incorrect. 2. **Equation 2:** \[ f_{2}(x') = (\max_{i \ne t}( F(x')_i) - F(x')_t)^+ \] \(f_2(x') \le 0\) if \(F(x')_t \ge \max_{i \ne t}(F(x')_i)\), which means \(C(x') = t\). This equation is correct. 3. **Equation 3:** \[ f_{3}(x') = \text{softplus}(\max_{i \ne t}(F(x')_i) - F(x')_t)-\log(2) \] The softplus function is non-negative, and \(f_3(x') \le 0\) would hold when \(\max_{i \ne t}(F(x')_i) - F(x')_t \le \log(2)\). This condition can be satisfied when \(C(x') = t\) does not hold. Thus, this equation is incorrect. 4. **Equation 4:** \[ f_{4}(x') = (0.5 - F(x')_t)^+ \] This function will be less than or equal to zero when \(F(x')_t \ge 0.5\). This corresponds with \(C(x') = t\) if \(F(x')_t\) is the highest probability. This equation is correct. 5. **Equation 5:** \[ f_{5}(x') = -\log(2 F(x')_t - 2) \] The argument \(2 F(x')_t - 2\) must be positive for the log to be defined, which is only true if \(F(x')_t > 1\), which is impossible since \(F(x')_t\) is a probability. Thus, \(f_5(x')\) cannot be correctly defined, making this equation incorrect. 6. **Equation 6:** \[ f_{6}(x') = (\max_{i \ne t}( Z(x')_i) - Z(x')_t)^+ \] This equation mirrors \(f_2\) but operates on logits \(Z(x')\). The same reasoning applies: \(f_6(x') \le 0\) if \(Z(x')_t \ge \max_{i \ne t}(Z(x')_i)\), which corresponds to \(C(x') = t\). This equation is correct. 7. **Equation 7:** \[ f_{7}(x') = \text{softplus}(\max_{i \ne t}(Z(x')_i) - Z(x')_t)-\log(2) \] Similar to \(f_3\), the softplus function will be non-negative, and \(f_7(x') \le 0\) can hold under conditions that do not correspond to \(C(x') = t\). This equation is incorrect. Based on this analysis, the incorrect equations are: 1, 3, 5, and 7. Therefore, the final answer is:
1 | [1, 3, 5, 7] |
Resulting in output: False