The following is from Understanding Machine Learning: Theory to Algorithm textbook: Definition of PAC Learnability: A hypothesis class $\mathcal H$ is PAC learnable if there exist a function $m_H : (0, 1)^2 \rightarrow \mathbb{N}$ and a learning algorithm with the following property: For every $\epsilon, \delta \in (0, 1)$, for every distribution $D$ over $X$, and for every labeling function $f : X \rightarrow \{0,1\}$, if the realizable assumption holds with respect to $\mathcal H,D,f$ then when running the learning …

The below is a picture which denotes the error of an ensemble classifier. Can someone help me understand the notation What does it mean to have (25 and i) in brackets and what is ε^1 is it error of first classifier or the error rate raised to power i. Can someone explain this formulae.

I'm looking for the right notation for features from different types. Let us say that my samples as $m$ features that can be modeled with $X_1,...,X_m$. The features Don't share the same distribution (i.e. some categorical, some numerical, etc.). Therefore, while $X_i$ might be a continuous random variable, $X_j$ could be a discrete random variable. Now, given a data sample $x=(x_1,...,x_m)$, I want to talk about the probability, for example, $P(X_k=x_k)<c$. But $X_k$ might be a continuous variable (i.e. the …

The function in question is (from Appendix B, Proof of proposition 2.1 from Posterior Regularization for Structured Latent Variable Models): $$q(\textbf{Z}) = \frac{p_{\theta}(\textbf{Z}|\textbf{X})exp(\lambda^T \cdot \Phi(\textbf{Z}|\textbf{X}))}{H}$$ The $q(\textbf{Z})$ is a probability distribution of latent variable $z$, such that $\textbf{Z} \in \mathbb{R}^{N \times 1}$, $\textbf{X}$ is a vector of $N$ datapoints such that $\textbf{X} \in \mathbb{R}^{N \times 2}$. The $\lambda$ is a dual variable, such that $\lambda \in \mathbb{R}^{N \times 1}$ and is a vector function $\Phi(\textbf{Z}|\textbf{X})$. $H$ is a constant that …

When reading about convolutional neural networks (CNNs), I often come across a special notation used in the community and in scientific papers, describing the architecture of the network in terms of layers. However, I was not able to find a paper or resource describing this notation in detail. Could someone explain to me the details or point to where it is described or "standardized"? Examples: input−100C3−MP2−200C2−MP2−300C2−MP2−400C2−MP2−500C2−output (source) input−(300nC2−300nC2−MP2)_5−C2−C1−output (source) A good guess seems that xCy are convolution layers (x is …

I'm looking at equations for neural networks and backpropagation and I see this symbol in the equations, ⊙. I thought matrix multiplication of neural networks always involved matrices that matched dimensions on both sides, such as... [3, 3]@[3, 2]. (This is what is happening in the animated gif). What part of a neural net uses a Hadamard product and which uses the Kronecker product? Because I see this notation for the Hadamard product (⊙) in papers and deep learning learning …

I'm struggling to write my algorithm in a concise and correct way. The following is an explanation for an optimizer's update step of part of a vector of weights (not a matrix in my case). I have a vector $\alpha \in \mathbb{R}^d$, and a set $S$ that includes some indices $1\leq i \leq d$ ($S \subseteq \{1,\dots, d\}$). Now, I want to denote that $\alpha$ is 0 for every index $i\in S$, and otherwise it's the value as in $\alpha_i$. …

Apologies in advance, for I have a fairly rudimentary question on the notations for studying Feed-Forward Neural Networks. Here is a nice schematic taken from this blog-post. Here $x_i = f_i(W_i \cdot x_{i-1})$ where $f_i$ is the activation function. Let us denote the number of nodes in the $i^{\text{th}}$ layer by $n_i$ and each example of the training set being $d-$dimensional (i.e., having $d$ features). Which of the following do the nodes in the above graph represent? Each one of …

I am learning from this post. $\alpha$ is the ratio of the first subset, $$\alpha=\frac{\left|D_{1}\right|}{\left|D\right|}$$ according to the context and code of the post, $\left|D\right|$ means the number of samples? What are the pair of vertical lines called? Is it the L1 norm symbol? It does not seem to be the absolute value symbol.

So the first equation above refers to a CNN (rather a committee of CNNs) for image classification. I am unable to understand exactly what the author is trying to do in the first equation. So far, I think they're calculating the index of max likehlihood probabilities for all committees, then adding up the probabilities for all committees for those indices, and finally taking the maximum index. But this seems overly convoluted and I'm not really sure. Could someone clarify this?

When reading papers on neural networks, I occasionally stumble upon the following notation with a semicolon: $$ \text{tanh}(\mathbf{W_c}[\mathbf{c}_t;\mathbf{h}_t]) $$ Unless otherwise noted, does this by default mean the following: vector $\mathbf{c}_t$ is appended to vector $\mathbf{h}_t$ the resulting long vector is dot-producted with a Weight matrix $\mathbf{W}_t$ Finally, the resulting vector is component-wise activated by a hyperbolic tangent function The first bullet point is my main question. Googling for "Vector concatenation notation" doesn't return answers that would resemble the image …

From Sutton and Barto, Reinforcement Learning: An Introduction (second edition draft), in equation 3.4 of page 38. The probabilities given by the four-argument function p completely characterize the dynamics of a finite MDP. From it, one can compute anything else one might want to know about the environment, such as the state-transition probabilities (which we denote, with a slight abuse of notation, as a threeargument function $p(s^{'} | s, a) \dot{=}Pr\{S_t=s^{'} | S_{t-1} = s, A_{t-1}=a\} = \sum_{r\in{R}}{p(s^{'},r|s,a)}$ The author …

In many cases an activation function is notated as g (e.g. Andrew Ng's Course courses), especially if it doesn't refer to any specific activation function such as sigmoid. However, where does this convention come from? And for what reason did g start to be used?

I was going through a paper comparing glove and word2vec. I came across the pound notation shown below. What does it mean when used like this? The link for paper is here