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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Characterize the high probability bound for learning algorithm

Suppose we have a dataset $S = (x_1, \dots x_n)$ drawn i.i.d from distribution $D$, a learning algorithm $A$ and error function $err$. The performance of $A$ is therefore defined by the error/...
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Is it possible to overfit a model on infinite amounts of data?

This is a theoretical question. Is it possible to overfit a model on infinite amounts of data? Let me clarify there are no duplicates. Say, we have a generator function that produces data, with the ...
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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?
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Why was the VC dimension not defined for all configurations of $d$ points?

Let's start with a typical definition of the VC dimension (as described in this book) Definition $3.10$ (VC-dimension) The $V C$ -dimension of a hypothesis set $\mathcal{H}$ is the size of the ...
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3answers
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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 ...
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Does distribution of data augmentation parameters matter?

Idea Let's say we have simple pictures dataset containing 40x40 images of digits. We have only one image of each digit. We want to use that as training set, but we need more data, so we use data ...
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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?
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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 ...
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How can I estimate the minimum number of training samples needed to get interesting results with WGAN?

Let's say we have a WGAN where the generator and critic have 8 layers and 5 million parameters each. I know that the greater the number of training samples the better, but is there a way to know the ...
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53 views

Can an ML model sort a random sequence of numbers from 1 to $ 2^{2^{512}} $ in our universe in infinite time?

I am pondering on the question in the title. As a human being, somehow I can sort a random sequence of numbers from 1 to $ 2^{2^{512}} $ in our universe in infinite time (But I am not sure.). Can an ...
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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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Utilizing continuous variables in concept learning algorithms

Suppose I'm using the Spambase dataset, with 57 real continuous variables and one binary label, and I'd like to implement the Find-S or LGG or similar algorithms. How do I represent the dataset for ...
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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 ...
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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). ...
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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, ...
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1answer
70 views

What is the representational capacity of a learning algorithm? [duplicate]

The definition I see for representational capacity is "the family of functions the learning algorithm can choose from when varying the parameters in order to reduce a training objective." (...
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77 views

Does this $\max$ mean that we need to maximize the regret in this regret formula?

I found that the regret in Online Machine Learning is stated as: $$\operatorname{Regret}_{T}(h)=\sum_{t=1}^{T} l\left(p_{t}, y_{t}\right)-\sum_{t=1}^{T} l\left(h(x), y_{t}\right),$$ where $p_t$ is the ...
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131 views

Is the VC Dimension meaningful in the context of Reinforcement Learning?

Is the VC dimension meaningful for reinforcement learning (RL), as a machine learning (ML) method? How?
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How estimate the minimum size of an autoencoder to overfit the training data?

Given e.g. $1$M vectors of $1000$ floating points each, where every point in vectors is sampled from a uniform distribution between $-1$ to $1$, how to estimate the minimum network size required ...
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72 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 $\epsilon$ (...
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52 views

Why is probability that at least one hypothesis out of $k$ being consistent with $m$ training examples $k(1- \epsilon)^m$?

My question is actually related to the addition of probabilities. I am reading on computational learning theory from Tom Mitchell's machine learning book. In chapter 7, when proving the upper bound ...
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1answer
52 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 ...
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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)?
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How can a machine learning problem be reduced as a communication problem?

I once heard that the problem of approximating an unknown function can be modeled as a communication problem. How is this possible?
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What do we mean by saying "VC dimension gives a LOOSE, not TIGHT bound"?

From what I understand VC dimension is what establishes the feasibility of learning for infinite hypothesis sets, the only kind we would use in practice. But, the literature (i.e. Learning from Data)...
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780 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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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 ...
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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 ...
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1answer
129 views

Understanding relation between VC Symmetrization Lemma and Generalization Bounds

I am new in the field of Machine Learning so I wanted to start of by reading more about mathematics and history behind it. I am currently reading, in my opinion, a very good and descriptive paper on ...
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175 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 ...
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1answer
96 views

If a neural network is a universal function approximator, can it have any prior beliefs?

Let us confine ourselves to the case where we have a $n$ dimensional input and a $+1$ or $-1$ output. It can be shown that: For every $n$, there exists a dense NN of depth 2, such that it contains ...
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1answer
57 views

Can feature engineering change the selection of the model according to the minimum description length?

The definition of MDL according to these slides is: The minimum description length (MDL) criteria in machine learning says that the best description of the data is given by the model which ...
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1answer
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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. $\mathcal{H}$ ...
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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 ...
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670 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 ...
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1answer
151 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 ...
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394 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 ...
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1answer
51 views

Why does the discrepancy measure involve a supremum over the hypothesis space?

I am referring specifically to the disc defined by Kuznetsov and Mohri in https://arxiv.org/pdf/1803.05814.pdf This is a kind of worst case path dependent generalization error. But what is the ...
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534 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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172 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 ...
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397 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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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 ...
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428 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 ...
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How to Prove This Inequality, Related to Generalization Error (Not Using Rademacher Complexity)?

This is an inequality on page 36 of the Foundations of Machine Learning by Mohri, but the author only states it without proof. $$ \mathbb{P}\left[\left|R(h)-\widehat{R}_{S}(h)\right|>\epsilon\right]...
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Convert a PAC-learning algorithm into another one which requires no knowledge of the parameter

This is part of the exercise 2.13 in the book Foundations of Machine Learning (page 28). You can refer to chapter 2 for the notations. Consider a family of concept classes $\left\{\mathcal{C}_{s}\...
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248 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 ...
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248 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 ...
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2answers
423 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 ...
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332 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 ...
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1answer
83 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 ...