# 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 evaluate whether each equation correctly represents the specification "if and only if $f(x+\delta) \le 0$", let's examine each equation one by one. The given specification requires that a change $\delta$ in input $x$ results in the classifier $C$ assigning the label $t$ if and only if the value of the function $f(x+\delta)$ is less than or equal to zero. 1. **$f_{1}(x') = -\loss_{F,t}(x') + 1$** This equation involves the cross-entropy loss with respect to class $t$. For the condition $C(x+\delta) = t$ to hold iff $f(x+\delta) \le 0$, the loss would need to be greater than or equal to 1 when $C(x+\delta) \ne t$ and less than or equal to 0 when $C(x+\delta) = t$. However, cross-entropy loss doesn't directly translate to this requirement, as it's not guaranteed to be strictly above or below 1 depending on the classification correctness. This makes the equation potentially misleading in aligning with the specification. 2. **$f_{2}(x') = (\max_{i \ne t}( F(x')_i) - F(x')_t)^+$** This equation checks the difference between the maximum probability among incorrect classes and the probability of the correct class $t$, ensuring it's non-negative. This seems in line with the specification since $f(x+\delta) \le 0$ would imply $F(x')_t$ is greater than or equal to any other class probability, which is a necessary condition for correct classification. However, because it uses the positive part function, it doesn't directly translate to the specification's requirement that $f(x+\delta) \le 0$ for correct classification. 3. **$f_{3}(x') = \text{softplus}(\max_{i \ne t}(F(x')_i) - F(x')_t)-\log(2)$** The softplus function output is always positive, and subtracting $\log(2)$ adjusts the threshold. However, for $f(x+\delta) \le 0$ to hold, we need the inside of the softplus to be sufficiently negative, which implies the correct class has a higher probability. This equation seems more aligned with the requirement, but its correctness might depend on the relationship between the probabilities and how they interact with the softplus function and the $\log(2)$ subtraction. 4. **$f_{4}(x') = (0.5 - F(x')_t)^+$** This equation checks if the probability of the correct class $t$ is less than 0.5, implying that for $f(x+\delta) \le 0$ to hold, $F(x')_t$ must be greater than or equal to 0.5. This doesn't necessarily align with the specification since a class can be correctly classified with a probability less than 0.5 if it's the highest among all classes. 5. **$f_{5}(x') = -\log(2 F(x')_t - 2)$** This equation involves a logarithm of a modified probability of class $t$. For the function to output a value $\le 0$, it requires $2 F(x')_t - 2 > 1$, which simplifies to $F(x')_t > 1.5$, an impossible condition since probabilities are in the range [0,1]. This equation is therefore incorrect. 6. **$f_{6}(x') = (\max_{i \ne t}( Z(x')_i) - Z(x')_t)^+$** This equation is similar to $f_{2}$ but operates on logits ($Z(x')$). Since logits can be negative and are not constrained to sum to 1, this formulation allows for a more direct control and interpretation in terms of raw model outputs before the softmax. It's somewhat aligned with the specification but still uses the positive part function, which doesn't guarantee $f(x+\delta) \le 0$ strictly for correct classification. 7. **$f_{7}(x') = \text{softplus}(\max_{i \ne t}(Z(x')_i) - Z(x')_t)-\log(2)$** This is the logit-based counterpart to $f_{3}$ and suffers or benefits from similar considerations. The use of softplus and subtraction by $\log(2)$ might align with the specification under certain conditions, especially considering logits can have a wider range of values than probabilities. Given these analyses, the equations that seem clearly incorrect or not perfectly aligned with the specification due to either impossible conditions or misinterpretation of the requirement for $f(x+\delta) \le 0$ leading to correct classification are:
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. 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 evaluate if each equation meets the requirement that $C(x+\delta) = t$ if and only if $f(x+\delta) \le 0$, let's analyze them one by one. 1. **$f_{1}(x') = -\loss_{F,t}(x') + 1$** The cross-entropy loss $\loss_{F,t}(x')$ is minimized (and hence negative) when the prediction for class $t$ is maximized. This equation does not directly ensure that $C(x+\delta) = t$ if and only if $f(x+\delta) \le 0$ because it aims to make the loss negative but does not directly relate to maximizing the probability of class $t$ over others. 2. **$f_{2}(x') = (\max_{i \ne t}( F(x')_i) - F(x')_t)^+$** This equation ensures that $f(x') \le 0$ when $F(x')_t$ is greater than any other class's output, which aligns with making class $t$ the most probable class. However, due to the positive part $(\cdot)^+$, $f(x')$ can only be zero or positive, never negative, making the specification impossible to satisfy as stated. 3. **$f_{3}(x') = \text{softplus}(\max_{i \ne t}(F(x')_i) - F(x')_t)-\log(2)$** Similar to $f_{2}$, this function aims to make $F(x')_t$ the largest output, but due to the properties of softplus, this function will also never be less than $-\log(2)$, which may not strictly satisfy the given requirement as it doesn't ensure $f(x') \le 0$ directly corresponds to $C(x+\delta) = t$. 4. **$f_{4}(x') = (0.5 - F(x')_t)^+$** This function suggests that for $C(x+\delta) = t$, we require $F(x')_t > 0.5$. However, this criterion does not ensure that $t$ is the class with the maximum probability, only that it surpasses a fixed threshold. This does not necessarily align with the requirement. 5. **$f_{5}(x') = -\log(2 F(x')_t - 2)$** This function is problematic because it becomes undefined for $F(x')_t \le 0.5$. Also, it doesn't ensure that $f(x+\delta) \le 0$ directly corresponds to $C(x+\delta) = t$ as required. 6. **$f_{6}(x') = (\max_{i \ne t}( Z(x')_i) - Z(x')_t)^+$** This equation applies the same logic as $f_{2}$ but uses logits instead of softmax outputs. It has the same issue wherein $f(x')$ can only be zero or positive, making the specification as stated not fully satisfied. 7. **$f_{7}(x') = \text{softplus}(\max_{i \ne t}(Z(x')_i) - Z(x')_t)-\log(2)$** Similar to $f_{6}$ but using the softplus function, this equation also does not guarantee that $f(x+\delta) \le 0$ is equivalent to $C(x+\delta) = t$ due to its mathematical properties. In summary, while some equations may be closer to achieving the stated goal under certain interpretations, strictly speaking, none of them perfectly ensures that $C(x+\delta) = t$ if and only if $f(x+\delta) \le 0$ due to various reasons explained above. However, it seems there might have been a misunderstanding in interpreting the specifications directly for this task, as the task was to identify incorrect equations based on a specific logic flaw or misalignment with the specifications, rather than a broader interpretation of their functionality. Given the initial instruction and after a more careful consideration focusing strictly on the requirement that $C(x+\delta) = t$ if and only if $f(x+\delta) \le 0$, it appears I've overanalyzed the equations in terms of their broader mathematical implications rather than specific errors in formulation according to the provided specification. Every equation aims to reflect conditions under which a certain class is more likely or less likely, but without additional context or understanding the specific mechanism of cross-entropy loss in $f_1$ or the exact behavior of softmax outputs and logits in relation to the condition, pinpointing "incorrect" equations as per the initial request may not be straightforward. Given the nature of the task, a correct analysis would require a deeper dive into each equation's alignment with the specific condition that $f(x+\delta) \le 0$ implies $C(x+\delta) = t$, which was not fully provided here. Thus, without further clarification or a more specific definition of "incorrect" in the context of the setup and specification, providing a list of numbers as requested might not accurately reflect the task's requirements.
Error 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, let's analyze them one by one based on the requirement that $C(x+\delta) = t$ if and only if $f(x+\delta) \le 0$. 1. **Equation \(f_{1}(x')\):** This equation uses the cross-entropy loss \(\loss_{F,t}(x')\) subtracted from 1. The cross-entropy loss decreases as the prediction for class \(t\) increases, which indirectly aligns with the requirement. However, setting this expression to be less than or equal to zero does not directly equate to \(C(x+\delta) = t\), as the loss could be less than 1 for incorrect classifications as well. This equation doesn't guarantee that \(C(x+\delta) = t\) if and only if \(f(x+\delta) \le 0\). 2. **Equation \(f_{2}(x')\):** This equation computes the difference between the maximum predicted probability of incorrect classes and the predicted probability of the target class \(t\). The use of \((\cdot)^+\) ensures that this expression is non-negative when the target class is not the most likely. This formulation aligns well with the requirement, as \(f(x+\delta) \le 0\) would imply \(F(x')_t\) is the maximum, thus \(C(x+\delta) = t\). 3. **Equation \(f_{3}(x')\):** Similar to \(f_{2}\), but applies the softplus function and subtracts \(\log(2)\). The softplus function is smoothly increasing, ensuring \(f(x') > 0\) when the target class is not the most probable. While subtle, the objective of making \(f(x+\delta) \le 0\) might not directly translate to having \(C(x+\delta) = t\), given the transformation by softplus and the subtraction of \(\log(2)\), but it's close in spirit to the specification. 4. **Equation \(f_{4}(x')\):** This equation looks for \(F(x')_t\) to be greater than 0.5, which doesn't guarantee that \(t\) is the class with the maximum probability. This does not fully align with the specification. 5. **Equation \(f_{5}(x')\):** This equation attempts to ensure that \(F(x')_t > 0.5\) through a logarithmic function. However, incorrect for several reasons; notably, it becomes undefined for \(F(x')_t \leq 0.5\), and does not guarantee that \(C(x+\delta) = t\) if and only if \(f(x+\delta) \le 0\). 6. **Equation \(f_{6}(x')\):** This equation correctly focuses on logits (pre-softmax values) and computes the difference in a manner similar to \(f_{2}\), but with logits. This is closely aligned with the objective, as it effectively compares the target logit to others. 7. **Equation \(f_{7}(x')\):** This is a logit-based version of \(f_{3}\) and follows similar logic. It uses softplus and is aligned with the objective, given the context of logits. Given the analysis, the equations that do not properly align with the specification (considering the stipulation that $C(x+\delta) = t$ if and only if $f(x+\delta) \le 0$) are mainly equations 1, 4, and 5. Equation 3 is a borderline case, given its close alignment in spirit but potential for misalignment due to the softplus transformation and subtraction of \(\log(2)\). However, strictly speaking, without additional context on the exact behavior and thresholds, it's challenging to discard it outright based on the provided information. Thus, focusing on clearly incorrect equations:
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. 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 equation in turn, considering whether it satisfies the condition that $C(x+\delta) = t$ if and only if $f(x+\delta) \le 0$. 1. **$f_{1}(x') = -\loss_{F,t}(x') + 1$** The cross-entropy loss decreases as the probability of the correct class $t$ increases, approaching $0$ as the probability approaches $1$. For $f_1(x')$ to be $\le 0$, we would need $\loss_{F,t}(x') \ge 1$, which isn't typical for cross-entropy values as they often range between $0$ and $1$ when probabilities are output from a softmax. This equation may not correctly capture the condition for all cases. 2. **$f_{2}(x') = (\max_{i \ne t}( F(x')_i) - F(x')_t)^+$** This formulation checks if the highest probability among the incorrect classes is greater than the probability of the target class $t$. If this is positive, it implies that the target class is not the one with the highest probability, which contradicts the requirement for classification as $t$. This seems to align with the condition. 3. **$f_{3}(x') = \text{softplus}(\max_{i \ne t}(F(x')_i) - F(x')_t)-\log(2)$** The softplus function is always positive, and subtracting $\log(2)$ adjusts its range. This equation is trying to ensure that the difference between the maximum of incorrect classes and the target is non-negative for classification to not be $t$. However, the condition for $f(x+\delta) \le 0$ might not directly translate to correct classification as $t$ due to the nature of the softplus function and the subtraction of $\log(2)$. 4. **$f_{4}(x') = (0.5 - F(x')_t)^+$** This formulation suggests that $t$ is correctly classified if $F(x')_t > 0.5$. However, in a multi-class setting, having $F(x')_t > 0.5$ is not a necessary condition for $t$ to be the class with the maximum probability. Therefore, this condition might not accurately ensure $C(x+\delta) = t$. 5. **$f_{5}(x') = -\log(2 F(x')_t - 2)$** This equation has a domain issue because $2F(x')_t - 2$ can be negative, making its logarithm undefined. Furthermore, even when defined, this formulation does not ensure that $t$ has the maximum probability. 6. **$f_{6}(x') = (\max_{i \ne t}( Z(x')_i) - Z(x')_t)^+$** This equation uses logits ($Z(x')$) instead of softmax probabilities and checks if any logit excluding $t$ is greater than the logit for $t$. It aligns with ensuring $t$ is not the highest probability class (when considering transformation through softmax), which seems to respect the condition indirectly by working on logits. 7. **$f_{7}(x') = \text{softplus}(\max_{i \ne t}(Z(x')_i) - Z(x')_t)-\log(2)$** Similar to $f_{3}$, but operates on logits. This also uses the softplus function to ensure a non-negative output, which indirectly might not align perfectly with ensuring $t$ is the maximum probability class, especially considering the adjustment with $\log(2)$. Given the specifications and the analysis: - $f_{1}$ has a potential issue with ensuring $C(x+\delta) = t$ because of how cross-entropy behaves. - $f_{4}$ misunderstands the condition for multi-class classification. - $f_{5}$ has domain issues and does not directly ensure that $t$ has the highest probability. Thus, the incorrect equations according to the specification are:
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. 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 that $C(x+\delta) = t$ if and only if $f(x+\delta) \le 0$, let's analyze each function: 1. $f_1(x') = -\loss_{F,t}(x') + 1$ Cross-entropy loss is minimized when the predicted probability for the correct class is maximized. However, minimizing this function (to make it $\leq 0$) does not directly ensure $C(x+\delta) = t$, since it involves a constant shift by 1. This equation could be misleading because it doesn't directly relate to making the class $t$ the maximum probability class. 2. $f_2(x') = (\max_{i \ne t}( F(x')_i) - F(x')_t)^+$ This function aims to make the probability of the true class $t$ higher than any other class by penalizing situations where any other class has a higher probability than class $t$. It fits the specification because making $f_2(x+\delta) \leq 0$ would mean $F(x')_t$ is at least as high as any other class's probability, aligning with $C(x+\delta) = t$. 3. $f_3(x') = \text{softplus}(\max_{i \ne t}(F(x')_i) - F(x')_t)-\log(2)$ This uses softplus to create a smooth approximation to the max function involved in $f_2$, with a constant shift. The goal remains to ensure that no class other than $t$ has a higher probability, but the subtraction of $\log(2)$ and the use of softplus might not directly ensure $C(x+\delta) = t$ if $f(x+\delta) \leq 0$ due to the smoothness and the specific numerical adjustments. 4. $f_4(x') = (0.5 - F(x')_t)^+$ This equation effectively requires $F(x')_t > 0.5$ for $f_4(x+\delta) \leq 0$, which suggests that class $t$ should have a probability greater than 0.5. However, it doesn't ensure that $t$ is the class with the maximum probability, just that it's probable to a certain threshold. 5. $f_5(x') = -\log(2 F(x')_t - 2)$ This equation is problematic because for probabilities (which are in the range $[0,1]$), the term $2 F(x')_t - 2$ will always be negative, rendering the log function undefined. This equation doesn't work properly within the given context. 6. $f_6(x') = (\max_{i \ne t}( Z(x')_i) - Z(x')_t)^+$ This function is applied directly to the logits (outputs before the softmax). Ensuring this function is $\leq 0$ would mean that the logit for class $t$ is not lower than for any other class, which aligns with wanting $C(x+\delta) = t$. This equation seems correct under the assumption that logits directly correlate with class probabilities post-softmax. 7. $f_7(x') = \text{softplus}(\max_{i \ne t}(Z(x')_i) - Z(x')_t)-\log(2)$ Similar to $f_3$, but applied to logits. It aims to make the logit for class $t$ higher than any other through a smooth approximation. The subtraction of $\log(2)$ is arbitrary but does not inherently make the equation incorrect regarding the specification. Given the above analyses, the equations that are incorrect according to the specification, particularly because they either do not ensure $C(x+\delta) = t$ if and only if $f(x+\delta) \le 0$ or are mathematically flawed, are:
1 | [1, 3, 4, 5] |
Resulting in output: False