# 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
To determine which equations are incorrect, I will go through each one and check if the condition `C(x+δ) = t` if and only if `f(x+δ) ≤ 0` holds true. 1. `f_1(x') = -loss_{F,t}(x') + 1`: - If `C(x+δ) = t`, then the cross-entropy loss `loss_{F,t}(x+δ)` should be minimized, making `-loss_{F,t}(x+δ)` positive and `f_1(x+δ) > 0`. - If `f_1(x+δ) ≤ 0`, then `-loss_{F,t}(x+δ) ≤ -1`, which implies a high cross-entropy loss and does not necessarily mean `C(x+δ) = t`. - Therefore, this equation is incorrect. 2. `f_2(x') = (max_{i ≠ t}(F(x')_i) - F(x')_t)^+`: - If `C(x+δ) = t`, then `F(x+δ)_t` should be the maximum value, making `max_{i ≠ t}(F(x+δ)_i) - F(x+δ)_t ≤ 0` and `f_2(x+δ) ≤ 0`. - If `f_2(x+δ) ≤ 0`, then `max_{i ≠ t}(F(x+δ)_i) - F(x+δ)_t ≤ 0`, which implies that `F(x+δ)_t` is the maximum value and `C(x+δ) = t`. - Therefore, this equation is correct. 3. `f_3(x') = softplus(max_{i ≠ t}(F(x')_i) - F(x')_t) - log(2)`: - Similar to `f_2`, if `C(x+δ) = t`, then `max_{i ≠ t}(F(x+δ)_i) - F(x+δ)_t ≤ 0`, making `softplus(max_{i ≠ t}(F(x+δ)_i) - F(x+δ)_t) ≤ log(2)` and `f_3(x+δ) ≤ 0`. - If `f_3(x+δ) ≤ 0`, then `softplus(max_{i ≠ t}(F(x+δ)_i) - F(x+δ)_t) ≤ log(2)`, which implies that `max_{i ≠ t}(F(x+δ)_i) - F(x+δ)_t ≤ 0` and `C(x+δ) = t`. - Therefore, this equation is correct. 4. `f_4(x') = (0.5 - F(x')_t)^+`: - If `C(x+δ) = t`, then `F(x+δ)_t` should be greater than 0.5, making `0.5 - F(x+δ)_t ≤ 0` and `f_4(x+δ) ≤ 0`. - If `f_4(x+δ) ≤ 0`, then `0.5 - F(x+δ)_t ≤ 0`, which implies that `F(x+δ)_t ≥ 0.5` but does not necessarily mean `C(x+δ) = t`. - Therefore, this equation is incorrect. 5. `f_5(x') = -log(2F(x')_t - 2)`: - If `C(x+δ) = t`, then `F(x+δ)_t` should be greater than 0.5, making `2F(x+δ)_t - 2 > 0` and `f_5(x+δ) < 0`. - If `f_5(x+δ) ≤ 0`, then `log(2F(x+δ)_t - 2) ≥ 0`, which implies that `2F(x+δ)_t - 2 > 0` and `F(x+δ)_t > 0.5`, but this does not necessarily mean `C(x+δ) = t`. - Therefore, this equation is incorrect. 6. `f_6(x') = (max_{i ≠ t}(Z(x')_i) - Z(x')_t)^+`: - This equation is similar to `f_2`, but it operates on the logits `Z(x')` instead of the softmax output `F(x')`. - If `C(x+δ) = t`, then `Z(x+δ)_t` should be the maximum logit value, making `max_{i ≠ t}(Z(x+δ)_i) - Z(x+δ)_t ≤ 0` and `f_6(x+δ) ≤ 0`. - If `f_6(x+δ) ≤ 0`, then `max_{i ≠ t}(Z(x+δ)_i) - Z(x+δ)_t ≤ 0`, which implies that `Z(x+δ)_t` is the maximum logit value, and after applying the softmax function, `F(x+δ)_t` should be the maximum value, leading to `C(x+δ) = t`. - Therefore, this equation is correct. 7. `f_7(x') = softplus(max_{i ≠ t}(Z(x')_i) - Z(x')_t) - log(2)`: - Similar to `f_6`, but it applies the `softplus` function to the logit difference. - If `C(x+δ) = t`, then `max_{i ≠ t}(Z(x+δ)_i) - Z(x+δ)_t ≤ 0`, making `softplus(max_{i ≠ t}(Z(x+δ)_i) - Z(x+δ)_t) ≤ log(2)` and `f_7(x+δ) ≤ 0`. - If `f_7(x+δ) ≤ 0`, then `softplus(max_{i ≠ t}(Z(x+δ)_i) - Z(x+δ)_t) ≤ log(2)`, which implies that `max_{i ≠ t}(Z(x+δ)_i) - Z(x+δ)_t ≤ 0`, and after applying the softmax function, `F(x+δ)_t` should be the maximum value, leading to `C(x+δ) = t`. - Therefore, this equation is correct. In summary, the incorrect equations are `f_1`, `f_4`, and `f_5`. The correct equations are `f_2`, `f_3`, `f_6`, and `f_7`. The Python list of incorrect equation numbers is: [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. 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
To determine which equations are incorrect according to the specification, we need to analyze each equation and check if it satisfies the condition: $C(x+\delta) = t$ if and only if $f(x+\delta) \le 0$. 1. $f_{1}(x') = -\loss_{F,t}(x') + 1$: This equation is correct. If $C(x+\delta) = t$, then the cross-entropy loss $\loss_{F,t}(x+\delta)$ will be minimized, making $-\loss_{F,t}(x+\delta)$ positive and greater than 0. Adding 1 will make $f_{1}(x+\delta) > 0$. If $C(x+\delta) \neq t$, then $\loss_{F,t}(x+\delta)$ will be larger, making $-\loss_{F,t}(x+\delta) < 0$, and $f_{1}(x+\delta) \le 0$. 2. $f_{2}(x') = (\max_{i \ne t}(F(x')_i) - F(x')_t)^+$: This equation is correct. If $C(x+\delta) = t$, then $F(x+\delta)_t$ will be the maximum value in the softmax output, and $\max_{i \ne t}(F(x+\delta)_i) - F(x+\delta)_t \le 0$, making $f_{2}(x+\delta) \le 0$. If $C(x+\delta) \neq t$, then there exists some $i \neq t$ such that $F(x+\delta)_i > F(x+\delta)_t$, and $\max_{i \ne t}(F(x+\delta)_i) - F(x+\delta)_t > 0$, making $f_{2}(x+\delta) > 0$. 3. $f_{3}(x') = \text{softplus}(\max_{i \ne t}(F(x')_i) - F(x')_t)-\log(2)$: This equation is incorrect. The softplus function is always non-negative, so $\text{softplus}(\max_{i \ne t}(F(x')_i) - F(x')_t) \ge 0$. Subtracting $\log(2)$ from a non-negative value does not guarantee that $f_{3}(x+\delta) \le 0$ when $C(x+\delta) = t$. 4. $f_{4}(x') = (0.5 - F(x')_t)^+$: This equation is incorrect. When $C(x+\delta) = t$, we want $F(x+\delta)_t$ to be as close to 1 as possible, but this equation penalizes $F(x+\delta)_t$ being greater than 0.5, which is the opposite of what we want. 5. $f_{5}(x') = -\log(2 F(x')_t - 2)$: This equation is incorrect. When $C(x+\delta) = t$, we want $F(x+\delta)_t$ to be as close to 1 as possible. However, $-\log(2 F(x+\delta)_t - 2)$ is undefined when $F(x+\delta)_t \ge 1$, which is possible and even desirable when $C(x+\delta) = t$. 6. $f_{6}(x') = (\max_{i \ne t}(Z(x')_i) - Z(x')_t)^+$: This equation is correct. It is similar to $f_{2}$, but operates on the logits $Z(x')$ instead of the softmax output $F(x')$. The analysis is the same as for $f_{2}$. 7. $f_{7}(x') = \text{softplus}(\max_{i \ne t}(Z(x')_i) - Z(x')_t)-\log(2)$: This equation is incorrect for the same reason as $f_{3}$. The softplus function is always non-negative, so subtracting $\log(2)$ does not guarantee that $f_{7}(x+\delta) \le 0$ when $C(x+\delta) = t$. Therefore, the incorrect equations are 3, 4, 5, and 7. The Python list representing these numbers is: [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. 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
To determine which equations are incorrect according to the specification, we need to analyze each equation and ensure that `C(x+delta) = t` if and only if `f(x+delta) <= 0`. 1. `f_1(x') = -loss_{F,t}(x') + 1`: - This equation is correct because `loss_{F,t}(x')` represents the cross-entropy loss for the target class `t`, and when `loss_{F,t}(x') < 1`, it implies that `f_1(x') < 0`, which means `C(x+delta) = t`. 2. `f_2(x') = (max_{i != t}(F(x')_i) - F(x')_t)^+`: - This equation is correct because it calculates the difference between the maximum logit of the non-target classes and the logit of the target class. When `f_2(x') <= 0`, it means that the logit of the target class is greater than or equal to the maximum logit of the non-target classes, implying `C(x+delta) = t`. 3. `f_3(x') = softplus(max_{i != t}(F(x')_i) - F(x')_t) - log(2)`: - This equation is incorrect because the subtraction of `log(2)` does not ensure that `C(x+delta) = t` if and only if `f_3(x+delta) <= 0`. 4. `f_4(x') = (0.5 - F(x')_t)^+`: - This equation is incorrect because it does not consider the logits of the non-target classes. When `F(x')_t > 0.5`, it does not necessarily imply `C(x+delta) = t`. 5. `f_5(x') = -log(2 F(x')_t - 2)`: - This equation is incorrect because it does not correctly represent the condition for `C(x+delta) = t`. The relationship between `f_5(x+delta)` and `C(x+delta) = t` is not clear. 6. `f_6(x') = (max_{i != t}(Z(x')_i) - Z(x')_t)^+`: - This equation is correct because it calculates the difference between the maximum logit of the non-target classes and the logit of the target class before the softmax function. When `f_6(x') <= 0`, it means that the logit of the target class is greater than or equal to the maximum logit of the non-target classes, implying `C(x+delta) = t`. 7. `f_7(x') = softplus(max_{i != t}(Z(x')_i) - Z(x')_t) - log(2)`: - This equation is incorrect for the same reason as `f_3(x')`. The subtraction of `log(2)` does not ensure that `C(x+delta) = t` if and only if `f_7(x+delta) <= 0`. Therefore, the incorrect equations are `f_3(x')`, `f_4(x')`, `f_5(x')`, and `f_7(x')`. The Python list representing the incorrect equations would be: [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. 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
To check if each equation satisfies the condition `C(x+δ) = t if and only if f(x+δ) ≤ 0`, I will go through them one by one: 1. `f_1(x') = -loss_{F,t}(x') + 1`: - If `C(x+δ) = t`, then the loss `loss_{F,t}(x+δ)` should be close to 0 (since the correct label has a high probability). So, `-loss_{F,t}(x+δ) + 1` will be close to 1, which is greater than 0. - If `C(x+δ) != t`, then the loss `loss_{F,t}(x+δ)` will be higher, and `-loss_{F,t}(x+δ) + 1` can be negative or positive. - Therefore, this equation does not satisfy the condition. 2. `f_2(x') = (max_{i != t}(F(x')_i) - F(x')_t)^+`: - If `C(x+δ) = t`, then `F(x+δ)_t` should be the maximum value in `F(x+δ)`, making `max_{i != t}(F(x+δ)_i) - F(x+δ)_t` non-positive, and `f_2(x+δ) = 0`. - If `C(x+δ) != t`, then `F(x+δ)_t` is not the maximum value, so `max_{i != t}(F(x+δ)_i) - F(x+δ)_t` will be positive, and `f_2(x+δ) > 0`. - Therefore, this equation satisfies the condition. 3. `f_3(x') = softplus(max_{i != t}(F(x')_i) - F(x')_t) - log(2)`: - This is similar to `f_2`, but with a constant subtraction and the `softplus` function applied. - The `softplus` function is always positive, so `f_3(x+δ) ≤ 0` if and only if `max_{i != t}(F(x+δ)_i) - F(x+δ)_t ≤ 0`, which is the same condition as `f_2`. - Therefore, this equation satisfies the condition. 4. `f_4(x') = (0.5 - F(x')_t)^+`: - If `C(x+δ) = t`, then `F(x+δ)_t` should be close to 1, making `0.5 - F(x+δ)_t` negative, and `f_4(x+δ) = 0`. - If `C(x+δ) != t`, then `F(x+δ)_t` will be lower than 0.5, making `0.5 - F(x+δ)_t` positive, and `f_4(x+δ) > 0`. - Therefore, this equation satisfies the condition. 5. `f_5(x') = -log(2 F(x')_t - 2)`: - This equation is incorrect because `2 F(x')_t - 2` can be negative, and the logarithm of a negative value is undefined. - Therefore, this equation does not satisfy the condition. 6. `f_6(x') = (max_{i != t}(Z(x')_i) - Z(x')_t)^+`: - This is similar to `f_2`, but with the logits `Z(x')` instead of the softmax output `F(x')`. - The condition `C(x+δ) = t` depends on the softmax output `F(x+δ)`, not the logits `Z(x+δ)`. - Therefore, this equation does not satisfy the condition. 7. `f_7(x') = softplus(max_{i != t}(Z(x')_i) - Z(x')_t) - log(2)`: - This is similar to `f_6`, but with the `softplus` function applied and a constant subtraction. - The issue with `f_6` still applies here, as the condition depends on `F(x+δ)`, not `Z(x+δ)`. - Therefore, this equation does not satisfy the condition. In summary, the equations that are incorrect according to the specification are: `f_1`, `f_5`, `f_6`, and `f_7`. The answer as a Python list: [1, 5, 6, 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. 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 consider each equation one by one: 1. $f_{1}(x') = -\loss_{F,t}(x') + 1$: This equation is correct. When $f_{1}(x') \leq 0$, it means that the loss for the target class $t$ is greater than or equal to 1, implying that the model's prediction is not the target class. Hence, $C(x+\delta) \neq t$. Conversely, when $f_{1}(x') > 0$, the loss for the target class $t$ is less than 1, implying that the model's prediction is the target class. Hence, $C(x+\delta) = t$. 2. $f_{2}(x') = (\max_{i \ne t}( F(x')_i) - F(x')_t)^+$: This equation is correct. When $f_{2}(x') \leq 0$, it means that the maximum probability of the non-target classes is less than or equal to the probability of the target class, implying that the model's prediction is the target class. Hence, $C(x+\delta) = t$. Conversely, when $f_{2}(x') > 0$, the maximum probability of the non-target classes is greater than the probability of the target class, implying that the model's prediction is not the target class. Hence, $C(x+\delta) \neq t$. 3. $f_{3}(x') = \text{softplus}(\max_{i \ne t}(F(x')_i) - F(x')_t)-\log(2)$: This equation is correct. The term $\text{softplus}(\max_{i \ne t}(F(x')_i) - F(x')_t)$ is similar to $f_{2}$, but with a smooth approximation of the maximum function. When $f_{3}(x') \leq 0$, it implies that the maximum probability of the non-target classes is less than or equal to the probability of the target class plus $\log(2)$, which corresponds to a decision boundary for the target class. Hence, $C(x+\delta) = t$. Conversely, when $f_{3}(x') > 0$, the maximum probability of the non-target classes is greater than the probability of the target class plus $\log(2)$, implying that the model's prediction is not the target class. Hence, $C(x+\delta) \neq t$. 4. $f_{4}(x') = (0.5 - F(x')_t)^+$: This equation is incorrect. When $f_{4}(x') \leq 0$, it means that the probability of the target class is greater than or equal to 0.5, but this does not necessarily imply that the model's prediction is the target class, as there could be another class with a higher probability. Conversely, when $f_{4}(x') > 0$, it means that the probability of the target class is less than 0.5, but this does not necessarily imply that the model's prediction is not the target class, as there could be other classes with even lower probabilities. 5. $f_{5}(x') = -\log(2 F(x')_t - 2)$: This equation is incorrect. The condition $f_{5}(x') \leq 0$ does not have a clear interpretation in terms of the model's prediction, as it involves a complex function of the target class probability. Similarly, the condition $f_{5}(x') > 0$ does not have a clear interpretation. 6. $f_{6}(x') = (\max_{i \ne t}( Z(x')_i) - Z(x')_t)^+$: This equation is correct. It is similar to $f_{2}$, but it operates on the logits $Z(x')$ instead of the probabilities $F(x')$. When $f_{6}(x') \leq 0$, it means that the maximum logit of the non-target classes is less than or equal to the logit of the target class, implying that the model's prediction is the target class. Hence, $C(x+\delta) = t$. Conversely, when $f_{6}(x') > 0$, the maximum logit of the non-target classes is greater than the logit of the target class, implying that the model's prediction is not the target class. Hence, $C(x+\delta) \neq t$. 7. $f_{7}(x') = \text{softplus}(\max_{i \ne t}(Z(x')_i) - Z(x')_t)-\log(2)$: This equation is correct. It is similar to $f_{3}$, but it operates on the logits $Z(x')$ instead of the probabilities $F(x')$. The interpretation is similar to $f_{3}$, with the decision boundary for the target class being determined by the logit values. Therefore, the incorrect equations are $f_{4}$ and $f_{5}$, so the answer is: [4, 5]
Resulting in output: False