Questions tagged [learning-theory]

Questions about the design and analysis of machine learning algorithms.

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What is the VC dimension of the hypothesis class $H=\left\{f_{\theta_{1}, \theta_{2}}: R^{2} \rightarrow\{0,1\} \mid 0<\theta_{1}<\theta_{2}\right\}$?

I would like to know what is the VC dimension of the following hypothesis class. $$H=\left\{f_{\theta_{1}, \theta_{2}}: R^{2} \rightarrow\{0,1\} \mid 0<\theta_{1}<\theta_{2}\right\}$$ where $f_{\...
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46 views

Method for combining derivative free optimization results of different data inputs

I am working on an algorithm that has multiple fixed parameters. The algorithm analyzes time series data and spits out a number. The fixed parameters need to be such that this number is as small as ...
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29 views

Is class of threshold functions Agnostic PAC Learnable?

In "Understanding Machine Learning, From Theory to Algorithms" by Shalev and Ben-David, on page 44 example 6.1, it is proved that the class of threshold functions are PAC learnable. on the other hand, ...
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Labeled points in $\{0,1\}^n$ such that every linear separator requires exponential weights

I want to find labeled samples in $\{0,1\}^n$ such that the Perceptron algorithm takes $2^{\Omega(n)}$ steps to converge. One way to do this would be to find a sequence of labeled examples that are ...
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85 views

PAC learning vs. learning on uniform distribution

The class of function $\mathcal{F}$ is PAC-learnable if there exists an algorithm $A$ such that for any distribution $D$, any unknown function $f$ and any $\epsilon, \delta$ it holds that there exists ...
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24 views

Uniform Convergence and VC Theroy

I've started reading more about statistical learning theory, specifically this paper right here and I cannot understand the following part: It turns out the conditions required to render empirical ...
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What does “smooth control” here mean?

I came across the following statement in this pdf. This allows the learner to have smooth control over the bias-complexity tradeoff. What does "smooth control" here mean? How do you understand ...
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41 views

What are the basics of CS i should know,before I start my journey into machine learning

I am myself a non-cs graduate and would love to be a machine learning engineer. I have learned to code and know the basics of <...
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80 views

VC dimension of the class of polynomial classifiers of degree $n$

I came across this statement on page 85 of the book "understanding machine learning: from theory to algorithms" The general idea is as follows: Consider a binary classification problem with the ...
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30 views

How can I understand the multi-class version of “shattering” intuitively? [closed]

I'm learning machine learning. VC dimension is a good way to measure the complexity of hypothesis class for binary classifier and has a very good intuitive explanation from shattering. When we discuss ...
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Covering numbers to show that H is agnostically PAC-learnable

Suppose $X=[0,1]$ and $Y=[0,1]$, and we use the squared loss Let's define the hypothesis class $H = {h(x) = (x-a)^2 : a \in [0,1]}$. Question: How can covering numbers be used to show that this ...
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66 views

How do you prove the Natarajan's Lemma intuitively?

Let $H$ be a hypothesis class of multiclass predictors; namely, each $h\in H$ is a function from $X$ to $[k]$. Denote the Natarajan dimension of $H$ by $Ndim(H)$. Hope you can give me an intuitive ...
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Hypothesis space in AdaBoost or general Machine learning

I was curious about the following: in most learning algorithms, when an algorithm is said to learn a concept class $C$ then the algorithm outputs a function from the hypothesis space $H$ and often ...
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1answer
64 views

How can the VC-dimension of Turing machine be finite?

The VC-dimension of a hypothesis class $\mathcal{H}$ is defined to be the size of the maximal set $C$ such that $\mathcal{H}$ cannot shutter. This paper shows that the VC-dimension of the set of all ...
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91 views

Dana Angluin's L* algorithm - Hypothesis inconsistent

is it possible for the Dana Angluin's L* algorithm that a hypothesis is inconsistent? So assume we have a closded observation table for a regular language L. Now after creating the hypothesis we will ...
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90 views

Why is Agnostic PAC learnability a stronger criterion as compared to the PAC learnability?

I am trying to understand the mentioned question. According to me, it should be other way around as it is easier to find hypothesis which is approximately correct compared to the optimal hypothesis in ...
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64 views

Weighting function for Non Uniform Learning

Consider a hypothesis class $H = \cup_{n=1}^{\infty} H_n$, where for every $n\in N$, $H_n$ is finite. Find a weighting function $w : H ->[0, 1]$ such that $\sum_{h \in H} w(h) ≤ 1$ and so that for ...
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26 views

what does this phrase mean: “train a policy network”

I am familiar with the basics (and perhaps a substantial amount of basics) of imitation learning and reinforcement learning. In IL (imitation), we take demonstrations from an assumed expert, which we ...
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Query complexity of exact learning and combinatorial parameter

When defining the query complexity of exact learning for a concept $c$ (considered as a function from $\{0,1\}^n \mapsto \{0,1\}$) in a concept class $\mathcal{C}$, we often come across the following ...
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What are good examples of computational theories for A.I. according to David Marr's Definition?

I was reading David Marr's "Artificial Intelligence-A Personal View" and he talks about "computational theory of AI" or what he laters labels as "Type 1" Theory. He provides the example of Chomsky's ...
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339 views

VC dimension of finite unions of one-sided intervals

What is the VC dimension of $k$ finite unions of one-sided intervals: If we take 3 one-sided intervals like $(-\infty, a_1] $, $(-\infty, a_2] $ and $(-\infty, a_3] $, I think union of these ...
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36 views

Does selecting the same arm has the same reward?

In multi-armed bandit problem, we have a set of $K$ arms. In each round $t$, a bandit selects an arm $k$ and receives a reward $r_{kt}$. The objective is to maximize the rewards after $T$ rounds. My ...
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Automatic learning/discovery of logics

Are there efforts to automatically discover new logics? Logics are simple structures - they have formal language, deduction rules, semantics and certain properties that are proved or discarded for ...
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VC dimensions: Let ${x_1, \ldots, x_N}$ be $N$ labelled points on $\mathbb{R}$, then there exists a sinusoid that separates these points

(Proposition, pg 26): Let ${x_1, \ldots, x_N}$ be $N$ points on $\mathbb{R}$, $N \in \mathbb{Z}$, labelled either $+1$ or $-1$ , then there exists a function from the set $\{t \mapsto \sin(\omega t)| \...
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300 views

Geometric intuition behind VC-dimension

Recently, I learnt about VC-dimension and how its boundedness assures PAC learnability on uncountable range spaces (let's assume that hypothesis class is the same as the family of concepts we want to ...
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Neural Network | What is the purpose of hidden layers and how many should I use?

I am pretty new to Neural Networks and I have two questions about hidden layers: 1. What is the purpose of hidden layers? I was wondering this because obviously you can get every result you want with ...
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142 views

Learning curve for model selection [closed]

I'm a newbie in machine learning field. And I need to choose the best model for my classification model, so i use "learning curve" from sk-learn to make selection. I train and plot learning curve on ...
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1answer
40 views

What does it mean the norm symbol applied to a concept?

I'm reading this paper: http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.150.14 or http://www.jmlr.org/papers/volume10/kontorovich09a/kontorovich09a.pdf On page 1101 are introduced 2 functions ...
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Is it true that PAC is a subset of agnostic PAC?

I would like to see the proof or a refernce to it. I feel it is obvious but my tutor insists the other way (agnostic PAC is a subset of PAC, and there are problems in PAC that are not angostic PAC).
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275 views

Faster RCNN: how to translate coordinates

I'm trying to understand and use the Faster R-CNN algorithm on my own data. My question is about ROI coordinates: what we have as labels, and what we want in the end, are ROI coordinates in the input ...
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Detecting the damaged regions in cars

Detecting the regions where a car has been damaged and the extent to which it has been damaged is a very interesting problem. It has potential applications in automatic auto insurance claims. ...
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Is feasible learning using just one algorithm and just one class of functions?

Suppose that in an enterprise there is a section specialized in data prediction problems, and to make easier the software maintainance, the next decision is taken: it will be used only one algorithm ...
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268 views

Random forests on monotone training set yields a monotone classifier?

Suppose I train a random forests classifier on a monotone training set. Is the resulting classifier guaranteed to be a monotone function? Suppose I apply the ID3 algorithm (the greedy algorithm) to ...
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Supervised learning with no prior information

I am now reading two modern books on machine learning theory. Both of them emphasize that, in order to succeed in supervised learning, one must choose a good hypothesis class. "Good" means that it is ...
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VC dimension and binary operations

There are two classes of binary functions, $F_1,F_2$. The VC-dimension of $F_i$ is $d_i$. Is there any simple formula for the VC-dimension of the following classes? $F_\lor := \{ f_1(x) \lor f_2(x) |...
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any hope for a universal automatic parser?

Say you are a program, and you are given some source code but you don't know in what language, it can be C++/Java/Python/Lisp/... all you know is that it is highly structured and LR(1) parse-able, and ...
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1answer
416 views

The VC dimension when the samples are fixed

The VC dimension is usually used in the following way. There is a space of hypotheses. There is an unknown probability distribution. We sample some training-samples from this distribution. We find the ...
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In the learning theory version of Occam's razor, why can't I just declare whatever hypothesis I want to be “shortest”?

Occam's razor states that shorter explanations (formally speaking, hypotheses) are more likely to be correct. Indeed this can be formalized: for a hypothesis class $\mathcal H$ one may ascribe ...
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Is SRM necessary to prove that a countable union of agnostic PAC learnable classes is nonuniformly learnable?

The following I believe is a direct proof of this fact. If a learner is tasked to be $\epsilon$-competitive with a hypothesis $h \in \mathcal H_n$, where $\mathcal H_n$ is agnostic PAC learnable, it ...
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Is there a non-linear version of ICA?

"Independent Component Analysis" is this : someone is sampling a random vector $s \in \mathbb{R}^d$ such that all its components $s_i$ are mutually independent and $\mathbb{E}[s_i^4] < 3$ and the ...
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Why is it NP-hard to learn a disjunction of k variables as a disjunction of fewer than k log n variables?

I'm looking at the claim in An algorithmic theory of learning: Robust concepts and random projection by R. I. Arriaga and S. Vempala (2006): Further, it is NP-hard to learn a disjunction of k ...
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333 views

VC dimension of monotone disjunctions of length k over n variables?

There are of course $n \choose k$ monotone disjunctions which bounds the VC dimension at $\log_2 {n \choose k}$. I'm wondering if this is bound at $k \log_2 n$? (Possibly follows from combinatorial ...
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1answer
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single agent vs multiple agent reinforcement learning

I am confused about 'single' vs 'multiple' agent reinforcement learning. Let's say that I have 1 hunter who I am training to hunt 1 static prey, so that only the hunter is moving around. This is ...
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1answer
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Learning a small disjunction using an input distribution of our choice

I have a boolean function $f: \{0,1\}^n \to \{0,1\}$ that I know takes the form $$f(x_1,\dots,x_n) = x_{i_1} \lor x_{i_2} \lor \dots \lor x_{i_k}.$$ I don't know the values of $i_1,\dots,i_k$, but I ...
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1answer
114 views

Learning a small disjunction

I have a boolean function $f: \{0,1\}^n \to \{0,1\}$ that I know takes the form $$f(x_1,\dots,x_n) = x_{i_1} \lor x_{i_2} \lor \dots \lor x_{i_k},$$ but I don't know the values of $i_1,\dots,i_k$. ...
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106 views

A clarification on the taxonomy of Evolutionary Algorithms

A rather basic question but I am confused about the characterization of a certain local search method which I want to describe in the framework of EAs. In particular, consider an EA which in every ...
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PAC learning axis parallel rectangles

I am trying to understand the proof that the axis parallel rectangles are PAC learnable in the realizable case. This means that given $\epsilon, \delta$ with enough data we can find a function $h$ ...
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1answer
938 views

How do I continue learning programming, beyond the basics? [closed]

DISCLAIMER: I understand that I might not be posting in the right part of StackExchange, or this question might have been asked before (I haven't found it). If this offends anybody, I apologize. I'm ...
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174 views

Sample Complexity for Real-Valued PAC-Learnable Functions

Can anyone shed some light on how the VC Dimension affects the sample complexity bounds of infinite hypothesis classes with real-valued outputs in PAC learning, or how to calculate the sample ...
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1answer
2k views

How to implement the regret matching algorithm?

My question is the following: How to calculate the regret in practice? I am trying to implement the regret matching algorithm but I do not understand how to do it. First, I have $n$ players with the ...