Questions tagged [computational-learning-theory]

For questions related to computational learning theory (or, in short, learning theory), which is a research subfield of artificial intelligence devoted to studying the design and mathematical analysis of machine learning algorithms. Computational learning theory (COLT) is largely concerned with computational and data efficiency. A seminal paper in COLT is Valiant's "A theory of the learnable" (1984).

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12
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1answer
339 views

What are the state-of-the-art results on the generalization ability of deep learning methods?

I've read a few classic papers on different architectures of deep CNNs used to solve varied image-related problems. I'm aware there's some paradox in how deep networks generalize well despite ...
12
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0answers
186 views

What is the number of neurons required to approximate a polynomial of degree n?

I learned about the universal approximation theorem from this guide. It states that a network even with a single hidden layer can approximate any function within some bound, given a sufficient number ...
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2answers
612 views

What are the learning limitations of neural networks trained with backpropagation?

In 1969, Seymour Papert and Marvin Minsky showed that Perceptrons could not learn the XOR function. This was solved by the backpropagation network with at least one hidden layer. This type of network ...
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3answers
402 views

Are there any rules of thumb for having some idea of what capacity a neural network needs to have for a given problem?

To give an example. Let's just consider the MNIST dataset of handwritten digits. Here are some things which might have an impact on the optimum model capacity: There are 10 output classes The inputs ...
8
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1answer
685 views

Can neural networks with a sigmoid as the activation function of the output layer approximate continuous functions?

Neural networks are commonly used for classification tasks, in fact from this post it seems like that's where they shine brightest. However, when we want to classify using neural networks, we often ...
8
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3answers
259 views

Why can neural networks generalize at all?

Neural networks are incredibly good at learning functions. We know by the universal approximation theorem that, theoretically, they can take the form of almost any function - and in practice, they ...
7
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3answers
2k views

What is the difference between hypothesis space and representational capacity?

I am reading Goodfellow et al Deeplearning Book. I found it difficult to understand the difference between the definition of the hypothesis space and representation capacity of a model. In Chapter 5,...
7
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1answer
266 views

Are PAC learning and VC dimension relevant to machine learning in practice?

Are PAC learning and VC dimension relevant to machine learning in practice? If yes, what is their practical value? To my understanding, there are two hits against these theories. The first is that ...
7
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2answers
145 views

Can we teach an artificial intelligence through sentences?

Could we teach an AI with sentences such as "ants are small" and "the sky is blue"? Is there any research work that attempts to do this?
6
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1answer
225 views

What does "hard for AI" look like?

In theoretical computer science, there is a massive categorization of the difficulty of various computational problems in terms of their asymptotic worst-time computational complexity. There doesn't ...
6
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2answers
1k views

What are other examples of theoretical machine learning books?

I am looking for a book about machine learning that would suit my physics background. I am more or less familiar with classical and complex analysis, theory of probability, сcalculus of variations, ...
6
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1answer
164 views

Is there a way of converting a neural network to another one that represents the same function?

I have read the paper Neural Turing Machines and the paper On the Computational Power of Neural Nets about the computational power of neural networks. However, it isn't still clear to me one thing. ...
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4answers
1k views

How does size of the dataset depend on VC dimension of the hypothesis class?

This might be a little broad question, but I have been watching Caltech youtube videos on Machine Learning, and in this video prof. is trying to explain how we should interpret the VC dimension in ...
5
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1answer
854 views

What are some resources on computational learning theory?

Pretty soon I will be finishing up Understanding Machine Learning: From Theory to Algorithms by Shai Ben-David and Shai Shalev-Shwartz. I absolutely love the subject and want to learn more, the only ...
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2answers
413 views

Are PAC learnability and the No Free Lunch theorem contradictory?

I am reading the Understanding Machine Learning book by Shalev-Shwartz and Ben-David and based on the definitions of PAC learnability and No Free Lunch Theorem, and my understanding of them it seems ...
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2answers
579 views

How to estimate the capacity of a neural network?

Is it possible to estimate the capacity of a neural network model? If so, what are the techniques involved?
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2answers
136 views

Is it possible to control asymptotic behaviour of neural network models?

Is it possible to specify what the asymptotic behaviour of a Neural Networks (NN) model should be? I am thinking of a NN which tries to learn a mapping $\vec y=f(\vec x)$ with $\vec x$ a vector of ...
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3answers
4k views

How can the generalization error be estimated?

How would you estimate the generalization error? What are the methods of achieving this?
4
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1answer
51 views

How can I estimate how many photos I need to train ResNet-50 for image classification?

I am working on a project where I have to classify around 1000 unique objects. I'm trying to plan how much training data I will need to collect. I was planning on using ResNet-50. Is there anyway I ...
4
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2answers
181 views

Mathematical foundations of the ability to learn

I am an undergraduate student in applied mathematics with an interest in artificial intelligence. I am currently exploring topics where I could do research. Coming from a mathematical background I am ...
4
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1answer
106 views

In deep learning, do we learn a continuous distribution based on the training dataset?

At least at some level, maybe not end-to-end always, but deep learning always learns a function, essentially a mapping from a domain to a range. The domain and range, at least in most cases, would be ...
4
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2answers
112 views

Why does estimation error increase with $|H|$ and decrease with $m$ in PAC learning?

Why does estimation error increase with $|H|$ and decrease with $m$ in PAC learning? I came across this statement in the section 5.2 of the book "understanding machine learning: from theory to ...
4
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1answer
138 views

Does a neural network exist that can learn every possible training data?

The universal approximation theorem states, that a feed-forward network with a single hidden layer containing a finite number of neurons can approximate continuous functions on compact subsets of $R^n$...
3
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2answers
181 views

Will a neural network always predict the correct label if it sees the exact same input during training and testing?

If I'm performing a text classification task using a model built in Keras, and, for example, I am attempting to predict the appropriate tag for a given Stack Overflow question: How do I subtract 1 ...
3
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1answer
225 views

What's the difference between estimation and approximation error?

I'm unable to find online, or understand from context - the difference between estimation error and approximation error in the context of machine learning (and, specifically, reinforcement learning). ...
3
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2answers
110 views

In classification, how does the number of classes affect the model size and amount of data needed to train?

When solving a classification problem with neural nets, be it text or images, how does the number of classes affect the model size and amount of data needed to train? Are there any soft or hard ...
3
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1answer
310 views

What is the maximum number of dichotomies in a square?

I am new to machine learning. I am reading this blog post on the VC dimension. $\mathcal H$ consists of all hypotheses in two dimensions $h: R^2 → \{−1, +1 \}$, positive inside some square boxes and ...
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0answers
126 views

How to show Sauer's Lemma when the inequalities are strict or they are equalities?

I have the following homework. We proved Sauer's lemma by proving that for every class $H$ of finite VC-dimension $d$, and every subset $A$ of the domain, $$ \left|\mathcal{H}_{A}\right| \leq |\...
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0answers
34 views

Batch PTA stopping condition

I am reviewing my Neural Network lectures and I have a doubt: My book's (Haykin) batch PTA describes a cost function which is defined over the set of the misclassified inputs. I have always been ...
3
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0answers
222 views

What is the relation between the definition of learnability of Vapnik and Gold and learnability of neural networks?

Gold showed that a language can be learned only if it contains a finite set of sentences. We know that deep neural networks can implement any function. Does this contradict the Gold's result? What ...
2
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4answers
1k views

How do you distinguish between a complex and a simple model in machine learning?

How do you distinguish between a complex and a simple model in machine learning? Which parameters control the complexity or simplicity of a model? Is it the number of inputs, or maybe the number of ...
2
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2answers
447 views

What is the difference between a learning algorithm and a hypothesis?

What's the distinction between a learning algorithm $A$ and a hypothesis $f$? I'm looking for a few concrete examples, if possible. For example, would the decision tree and random forest be considered ...
2
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3answers
122 views

How would you intuitively but rigorously explain what the VC dimension is?

The VC dimension is a very important concept in computational/statistical learning theory. However, the first time you read its definition, you may not immediately understand what it really represents ...
2
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1answer
282 views

How do I prove that $\mathcal{H}$, with $\mathcal{VC}$ dimension $d$, shatters all subsets with size less than $d-1$?

If a certain hypothesis class $\mathcal{H}$ has a $\mathcal{VC}$ dimension $d$ over a domain $X$, how can I prove that $H$ will shatter all subsets of $X$ with size less than $d$, i.e. $\mathcal{H}$ ...
2
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1answer
38 views

What does the notation $[m]=\{1, \ldots, m\}$ mean in the equation of the empirical error?

The empirical error equation given in the book Understanding Machine Learning: From Theory to Algorithms is My intuition for this equation is: total wrong predictions divided by the total number of ...
2
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1answer
50 views

What is meant by "stable training" of a deep learning model?

I have read it said that the "stable training" of a deep learning model is important. What is meant by "stable training" of a deep learning model?
2
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1answer
92 views

Is there any practical application of knowing whether a concept class is PAC-learnable?

A concept class $C$ is PAC-learnable if there exists an algorithm that can output a hypothesis with probability at least $(1-\delta)$ (the "probably" part), and an error that is less than $\epsilon$ (...
2
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1answer
162 views

How does the number of stacked LSTM layers or units in each layer affect the model complexity?

I playing around sequence modeling to forecast the weather using LSTM. How does the number of layers or units in each layer exactly affect the model complexity (in an LSTM)? For example, if I ...
2
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1answer
84 views

Is there anything theoretically revolutionary about Deep Neural Networks?

In recent years, we have seen quite a lot of impressive display of Deep Neural Network (DNN), as demonstrated most famously by AlphaGo and its cousin programs. But if I understand correctly, deep ...
2
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1answer
88 views

How Dempster-Shafer theory work in AI?

How does Dempster-Shafer theory work in representing ignorance in AI field?
2
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1answer
251 views

What is the minimum number of neurons and hidden layers needed to learn a Boolean function that maps $N$ bits to $1$ bit?

Suppose I have a Boolean function that maps $N$ bits to $1$ bit. If I understand correctly, this function will have $2^{2^N}$ possible configurations of its truth table. What is the minimum number of ...
2
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0answers
56 views

What is the effect of K in K-NN on the VC dimension?

What is the effect of K in K-NN on the VC dimension? When K increases, is the VC dimension decreased or increased, or we can't say anything about this? Is there a reference book that discusses this?
2
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0answers
23 views

Given a dataset and a neural network, is there some heuristic or theorem to determine whether this neural network has enough capacity? [duplicate]

What is the consensus regarding NN "capacity" or expressive power? I remember reading somewhere that expressive power grows exponentially with depth, but I cannot seem to find that exact ...
2
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1answer
53 views

A model for each sub-problem vs one model for the whole problem

Let's say one wants to use a neural net to learn some function $g(x)$. Let's say that we know that $g$ is a combination of two functions (or two sub-problems), $g(x)=f_2(f_1(x))$, and that we have two ...
2
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0answers
64 views

What is the relationship between PAC learning and classic parameter estimation theorems?

What are the differences and similarities between PAC learning and classic parameter estimation theorems (e.g. consistency results when estimating parameters, e.g. with MLE)?
2
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1answer
101 views

An infinite VC dimensional space vs using hierarchical subspaces of finite but growing VC dimensions

I have the following scenario. I have a binary classification problem, whose underlying function is a step function. The probability distribution of feature vectors is a uniform over the domain. Case ...
2
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1answer
189 views

How can neural networks approximate any continuous function but have $\mathcal{VC}$ dimension only proportional to their number of parameters?

Neural networks typically have $\mathcal{VC}$ dimension that is proportional to their number of parameters and inputs. For example, see the papers Vapnik-Chervonenkis dimension of recurrent neural ...
2
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0answers
73 views

Are No Free Lunch theorem and Universal Approximation theorem contradictory in the context of neural networks?

To my understanding NFL states that, we cannot have an hypothesis (let's assume it is an approximator like NN in this case) class that can't achieve certain accuracy parameters $\leq \epsilon$ with ...
2
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0answers
99 views

How can I show that the VC dimension of the set of all closed balls in $\mathbb{R}^n$ is at most $n+3$?

How can I show that the VC dimension of the set of all closed balls in $\mathbb{R}^n$ is at most $n+3$? For this problem, I only try the case $n=2$ for 1. When $n=2$, consider 4 points $A,B,C,D$ and ...
2
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0answers
438 views

A problem about the relation between 1-oracle and 2-oracle PAC model

This problem is about two-oracle variant of the PAC model. Assume that positive and negative examples are now drawn from two separate distributions $\mathcal{D}_{+}$ and $\mathcal{D}_{-} .$ For an ...