# 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. Specifically, make sure that $C(x+\delta) = t$ if and only if $f(x+\delta) \le 0$. Think step by step about if this is true for each equation and then give your answer as a python list
# 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. Specifically, make sure that $C(x+\delta) = t$ if and only if $f(x+\delta) \le 0$. Think step by step about if this is true for each equation and then give your answer as a python list
Let's evaluate each of the provided objective functions \( f_1 \) through \( f_7 \) to determine whether they satisfy the condition: \[ C(x+\delta) = t \quad \text{if and only if} \quad f(x+\delta) \leq 0 \] where \( C(x') = \arg\max_i F(x')_i \) assigns label \( t \) to input \( x' = x + \delta \). ### Evaluation of Each Function 1. **\( f_{1}(x') = -\loss_{F,t}(x') + 1 \)** - **Cross-Entropy Loss:** \( \loss_{F,t}(x') = -\log(F(x')_t) \) - **Substitution:** \( f_1(x') = -(-\log(F(x')_t)) + 1 = \log(F(x')_t) + 1 \) - **Condition:** \( f_1(x') \leq 0 \) implies \( \log(F(x')_t) \leq -1 \), which leads to \( F(x')_t \leq e^{-1} \approx 0.3679 \) - **Issue:** This condition indicates that the probability of class \( t \) is low, which contradicts \( C(x') = t \). Therefore, \( f_1 \) does **not** satisfy the required condition. 2. **\( f_{2}(x') = (\max_{i \ne t}( F(x')_i) - F(x')_t)^+ \)** - **Condition for \( f_2(x') \leq 0 \):** \( \max_{i \ne t}(F(x')_i) - F(x')_t \leq 0 \) - **Implication:** \( F(x')_t \geq \max_{i \ne t}(F(x')_i) \), which means class \( t \) has the highest probability. - **Conclusion:** Satisfies \( C(x') = t \) if and only if \( f_2(x') \leq 0 \). **Correct.** 3. **\( f_{3}(x') = \text{softplus}(\max_{i \ne t}(F(x')_i) - F(x')_t) - \log(2)) \)** - **Softplus Function:** Always non-negative. - **Condition for \( f_3(x') \leq 0 \):** Requires the argument of softplus to approach negative infinity, which is impossible. - **Implication:** \( f_3(x') > 0 \) always, so it never satisfies \( f_3(x') \leq 0 \). - **Issue:** Does **not** satisfy the required condition. 4. **\( f_{4}(x') = (0.5 - F(x')_t)^+ \)** - **Condition for \( f_4(x') \leq 0 \):** \( 0.5 - F(x')_t \leq 0 \) implies \( F(x')_t \geq 0.5 \) - **Implication:** While \( F(x')_t \geq 0.5 \) suggests a strong confidence in class \( t \), it doesn't necessarily ensure that \( F(x')_t \) is the maximum among all classes. - **Issue:** Possible instances where \( F(x')_t \geq 0.5 \) but another class has a higher probability. Does **not** strictly enforce \( C(x') = t \). 5. **\( f_{5}(x') = -\log(2 F(x')_t - 2) \)** - **Domain Issue:** The argument of the logarithm, \( 2F(x')_t - 2 \), must be positive. This requires \( F(x')_t > 1 \), which is impossible since \( \sum_{i} F(x')_i = 1 \) and \( F(x')_t \leq 1 \). - **Implication:** Undefined for all valid \( F(x')_t \). - **Issue:** **Incorrect** formulation. 6. **\( f_{6}(x') = (\max_{i \ne t}( Z(x')_i) - Z(x')_t)^+ \)** - **Logits Comparison:** Similar to \( f_2 \), but operates on logits \( Z(x') \) instead of probabilities. - **Condition for \( f_6(x') \leq 0 \):** \( Z(x')_t \geq \max_{i \ne t}(Z(x')_i) \) - **Implication:** \( F(x')_t \) will be the highest probability if logits satisfy this condition. - **Conclusion:** Satisfies \( C(x') = t \) if and only if \( f_6(x') \leq 0 \). **Correct.** 7. **\( f_{7}(x') = \text{softplus}(\max_{i \ne t}(Z(x')_i) - Z(x')_t) - \log(2)) \)** - **Softplus Function:** Always non-negative. - **Condition for \( f_7(x') \leq 0 \):** Requires the argument of softplus to approach negative infinity, which is impossible. - **Implication:** \( f_7(x') > 0 \) always, so it never satisfies \( f_7(x') \leq 0 \). - **Issue:** Does **not** satisfy the required condition. ### Summary of Incorrect Equations The following objective functions do **not** satisfy the condition \( C(x') = t \) if and only if \( f(x') \leq 0 \): - \( f_1 \) - \( f_3 \) - \( f_4 \) - \( f_5 \) - \( f_7 \) ### Answer
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. Specifically, make sure that $C(x+\delta) = t$ if and only if $f(x+\delta) \le 0$. Think step by step about if this is true for each equation and then give your answer as a python list
# 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. Specifically, make sure that $C(x+\delta) = t$ if and only if $f(x+\delta) \le 0$. Think step by step about if this is true for each equation and then give your answer as a python list
Let's analyze each equation to determine whether it satisfies the condition \( C(x') = t \) if and only if \( f(x') \leq 0 \). ### Equation 1: \[ f_{1}(x') = -\loss_{F,t}(x') + 1 \] Assuming the cross-entropy loss \( \loss_{F,t}(x') = -\log(F(x')_t) \), the equation simplifies to: \[ f_{1}(x') = \log(F(x')_t) + 1 \] - **When \( C(x') = t \)**: \( F(x')_t \) is the maximum probability, but \( \log(F(x')_t) + 1 \) does not necessarily satisfy \( \leq 0 \). - **When \( C(x') \neq t \)**: The condition might not hold as expected. **Conclusion**: **Incorrect** ### Equation 2: \[ f_{2}(x') = (\max_{i \ne t}( F(x')_i) - F(x')_t)^+ \] - **When \( C(x') = t \)**: \( \max_{i \ne t}( F(x')_i) - F(x')_t \leq 0 \), so \( f_{2}(x') = 0 \). - **When \( C(x') \neq t \)**: \( f_{2}(x') > 0 \). **Conclusion**: **Correct** ### Equation 3: \[ f_{3}(x') = \text{softplus}\left(\max_{i \ne t}(F(x')_i) - F(x')_t\right) - \log(2) \] - **When \( C(x') = t \)**: \( \max_{i \ne t}(F(x')_i) - F(x')_t \leq 0 \), so \( f_{3}(x') \leq 0 \). - **When \( C(x') \neq t \)**: \( f_{3}(x') > 0 \). **Conclusion**: **Correct** ### Equation 4: \[ f_{4}(x') = (0.5 - F(x')_t)^+ \] - **When \( C(x') = t \)**: For multi-class classification \( m > 2 \), \( F(x')_t \) can be less than 0.5 while still being the highest probability. - **When \( C(x') \neq t \)**: The condition does not align correctly. **Conclusion**: **Incorrect** ### Equation 5: \[ f_{5}(x') = -\log(2 F(x')_t - 2) \] - The term inside the logarithm becomes \( 2F(x')_t - 2 = 2(F(x')_t - 1) \), which is non-positive since \( F(x')_t \leq 1 \). - **Logarithm of a non-positive number is undefined**. **Conclusion**: **Incorrect** ### Equation 6: \[ f_{6}(x') = (\max_{i \ne t}( Z(x')_i) - Z(x')_t)^+ \] - **When \( C(x') = t \)**: \( Z(x')_t \) is the highest logit, so \( f_{6}(x') = 0 \). - **When \( C(x') \neq t \)**: \( f_{6}(x') > 0 \). **Conclusion**: **Correct** ### Equation 7: \[ f_{7}(x') = \text{softplus}\left(\max_{i \ne t}(Z(x')_i) - Z(x')_t\right) - \log(2) \] - **When \( C(x') = t \)**: \( \max_{i \ne t}(Z(x')_i) - Z(x')_t \leq 0 \), so \( f_{7}(x') \leq 0 \). - **When \( C(x') \neq t \)**: \( f_{7}(x') > 0 \). **Conclusion**: **Correct** ### Summary of Incorrect Equations: - **1, 4, 5** **Answer**:
1 | [1, 4, 5] |
Resulting in output: False