# 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
# Analysis of Equations Let me examine each equation to determine if it satisfies the requirement: $C(x+\delta) = t$ if and only if $f(x+\delta) \leq 0$. Remember that $C(x) = \arg\max_i F(x)_i$, so $C(x) = t$ means $F(x)_t > F(x)_i$ for all $i \neq t$. ## Equation 1: $f_{1}(x') = -\loss_{F,t}(x') + 1$ The cross-entropy loss for target class $t$ is $\loss_{F,t}(x') = -\log(F(x')_t)$ So $f_1(x') = \log(F(x')_t) + 1$ For $f_1(x') \leq 0$, we need $\log(F(x')_t) \leq -1$, which means $F(x')_t \leq e^{-1} \approx 0.368$ This doesn't ensure $C(x') = t$ - we want $F(x')_t$ to be the maximum probability, not less than some threshold. **Incorrect** ## Equation 2: $f_{2}(x') = (\max_{i \ne t}( F(x')_i) - F(x')_t)^+$ For $f_2(x') \leq 0$, we need $\max_{i \ne t}(F(x')_i) - F(x')_t \leq 0$ This is equivalent to $F(x')_t \geq \max_{i \ne t}(F(x')_i)$, which is exactly the condition for $C(x') = t$. **Correct** ## Equation 3: $f_{3}(x') = \text{softplus}(\max_{i \ne t}(F(x')_i) - F(x')_t)-\log(2)$ For $f_3(x') \leq 0$: $\log(1+\exp(\max_{i \ne t}(F(x')_i) - F(x')_t)) \leq \log(2)$ $1+\exp(\max_{i \ne t}(F(x')_i) - F(x')_t) \leq 2$ $\exp(\max_{i \ne t}(F(x')_i) - F(x')_t) \leq 1$ $\max_{i \ne t}(F(x')_i) - F(x')_t \leq 0$ $F(x')_t \geq \max_{i \ne t}(F(x')_i)$ This is exactly the condition for $C(x') = t$. **Correct** ## Equation 4: $f_{4}(x') = (0.5 - F(x')_t)^+$ For $f_4(x') \leq 0$, we need $0.5 - F(x')_t \leq 0$, so $F(x')_t \geq 0.5$ This is sufficient but not necessary for $C(x') = t$. For example, in a 3-class problem, $F(x')_t$ could be 0.4 with the other classes at 0.3 each, making $t$ the correct class despite $F(x')_t < 0.5$. **Incorrect** ## Equation 5: $f_{5}(x') = -\log(2 F(x')_t - 2)$ For $f_5(x') \leq 0$, we need $-\log(2 F(x')_t - 2) \leq 0$ This means $\log(2 F(x')_t - 2) \geq 0$ $2 F(x')_t - 2 \geq 1$ $F(x')_t \geq 1.5$ Since probabilities must be ≤ 1, this equation can never be satisfied. **Incorrect** ## Equation 6: $f_{6}(x') = (\max_{i \ne t}( Z(x')_i) - Z(x')_t)^+$ For $f_6(x') \leq 0$, we need $Z(x')_t \geq \max_{i \ne t}(Z(x')_i)$ Since softmax preserves the ordering of values, the highest logit corresponds to the highest probability, which is exactly the condition for $C(x') = t$. **Correct** ## Equation 7: $f_{7}(x') = \text{softplus}(\max_{i \ne t}(Z(x')_i) - Z(x')_t)-\log(2)$ Similar to equation 3, this requires $Z(x')_t \geq \max_{i \ne t}(Z(x')_i)$, which is equivalent to $C(x') = t$. **Correct** The incorrect equations are [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
# Evaluating each equation for correctness I'll examine each equation to determine if it satisfies the condition: $C(x+\delta) = t$ if and only if $f(x+\delta) \leq 0$. ## $f_1(x') = -\text{loss}_{F,t}(x') + 1$ The cross-entropy loss is typically $\text{loss}_{F,t}(x') = -\log(F(x')_t)$, so: $f_1(x') = \log(F(x')_t) + 1$ For this to be ≤ 0, we need $F(x')_t \leq e^{-1} \approx 0.368$. However, $C(x') = t$ means $F(x')_t$ is the maximum class probability, which doesn't necessarily imply $F(x')_t \leq 0.368$. In fact, in most cases where $C(x') = t$, we would have $F(x')_t > 0.368$, causing $f_1(x') > 0$. **Incorrect**: The "if and only if" condition is violated. ## $f_2(x') = (\max_{i \neq t}(F(x')_i) - F(x')_t)^+$ This function equals 0 when $F(x')_t \geq \max_{i \neq t}(F(x')_i)$ (i.e., when class $t$ has the highest probability), and is positive otherwise. Therefore, $f_2(x') \leq 0$ if and only if $C(x') = t$. **Correct** ## $f_3(x') = \text{softplus}(\max_{i \neq t}(F(x')_i) - F(x')_t) - \log(2)$ Since $\text{softplus}(x) = \log(1+\exp(x))$: - When $C(x') = t$, we have $\max_{i \neq t}(F(x')_i) - F(x')_t \leq 0$ - The softplus of a non-positive number is ≤ $\log(2)$ - Therefore $f_3(x') \leq 0$ when $C(x') = t$ Conversely, when $C(x') \neq t$, we have $\max_{i \neq t}(F(x')_i) - F(x')_t > 0$, making the softplus > $\log(2)$, so $f_3(x') > 0$. **Correct** ## $f_4(x') = (0.5 - F(x')_t)^+$ This equals 0 when $F(x')_t \geq 0.5$ and is positive when $F(x')_t < 0.5$. However, in a multi-class problem, $C(x') = t$ doesn't imply $F(x')_t \geq 0.5$. For example, with probabilities [0.4, 0.3, 0.3], class 1 would be chosen even though $F(x')_1 < 0.5$. **Incorrect**: The condition is not satisfied. ## $f_5(x') = -\log(2F(x')_t - 2)$ For this to be defined in the real domain, we need $2F(x')_t - 2 > 0$, which means $F(x')_t > 1$. Since $F(x')_t$ is a probability output from softmax, it's always ≤ 1. This means the function is undefined for all valid inputs. **Incorrect**: The function is not well-defined. ## $f_6(x') = (\max_{i \neq t}(Z(x')_i) - Z(x')_t)^+$ This works like $f_2$ but uses logits instead of probabilities. Since softmax preserves the ordering of values, $Z(x')_t > Z(x')_i$ for all $i \neq t$ if and only if $F(x')_t > F(x')_i$ for all $i \neq t$. Therefore, $f_6(x') \leq 0$ if and only if $C(x') = t$. **Correct** ## $f_7(x') = \text{softplus}(\max_{i \neq t}(Z(x')_i) - Z(x')_t) - \log(2)$ Similar to $f_3$, but using logits instead of probabilities. By the same reasoning, $f_7(x') \leq 0$ if and only if $C(x') = t$. **Correct** # Answer The incorrect equations are [1, 4, 5]
Resulting in output: False