Questions tagged [learning-theory]
Questions about the design and analysis of machine learning algorithms.
87
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Transductive Information Maximization vs classification with feature embedding in higher dimensional spaces?
Recent research work has shown that transductive learning/inference outperforms standard methods that were used before, where people embed features in a high dimensional space and then use the ...
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Transductive Learning vs Inductive Learning in Machine Learning
Several recent research work has shown that transductive learning/inference outperforms inductive learning/inference during classification problems. This has been found in few-shot learning, other ...
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Question about the proof for the sample complexity of axis-aligned rectangles
The classical proof for the sample complexity of the hypothesis class of axis-aligned rectangles usually begins by stating that our $A(S) \subset R^*$, where $R^*$ is the target function. My only ...
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Infinite VC Dim not PAC learnable
This is usually shown by an application of the Statistical No Free Lunch Theorem.
But is this possible to show this by working simply with the definition of PAC learnability and the sample complexity ...
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Regarding constant * opt approximation in agnostic learning
In standard agnostic learning, we assume that there is a concept class $H\subseteq \{h:\{0,1\}^n\rightarrow \{0,1\}\}$. Given samples from a distribution $D:\{0,1\}^n\times \{0,1\}\rightarrow [0,1]$, ...
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Multi-class sample complexity for PAC learning using "VC dimension"
VC dimension covers the binary classification case, i.e. when we want to get a predictor $X \to \{0, 1\}$.
Using VC dimension, we can get the upper bound on the sample complexity for PAC-learning.
In ...
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Precise definition of Universal Learner in Machine Learning
It is surprising to me that I cannot find a precise definition of universal learner on the internet. I can guess what it should bebut I don't want to make a mistake, therefore I have come here.
Here's ...
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Generalization error bound in case of collaborative learning
I am reading the paper "Collaborative PAC Learning" by Blum et al. So I will try to setup the problem here as to avoid reading the complete section (personalized setting).
Assume there are $...
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Understanding halving algorithm in online learning
I am working through "Understanding Machine Learning Theory" by Shai Shalev-Schwartz. In the chapter "Online learning" I came across the halving algorithm, the author uses the ...
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42
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pseudo-dimension for knapsack problem
Let $v_i, s_i$ be the value and size of item $i$, let $\rho \in \mathbb{R}$, n be the maximum number of items. Then we add items based on $\frac{v_i}{s_i^{\rho}}$ in decreasing order. I was trying to ...
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Hoeffding's inequality applicability for sample complexity
I am trying to determine some bounds for sample complexity. Suppose we have a bounded loss function $\ell$ and target function $f:\mathcal{X}\to\mathcal{Y}$. Hypothesis $h$ is learned, then the ...
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A confidence interval algorithm for Disagreement coefficient
My question has to do with the disagreement coefficient in active learning. I've been trying to solve the following question, where I need an algorithm to derive a confidence interval for $\theta$, ...
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Applications of derivative only, zeroth-order free optimization
I understand what is derivative-free optimization, and I am thinking a similar problem where the function $f$ we are optimizing is unknown and the only information we can acquire is the derivative. In ...
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Pseudo-dimension of a subset of affine functions
Let's say there are two sets of affine functions.
$\mathcal{A} = \{ax +b \mid a,b \in \mathbb{R}\}$
$\mathcal{H} = \{2x + 1, x, 3x + 4, 4x\}$
I know that the $\mathrm{Pdim}(\mathcal{A}) = 2$. From ...
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Uniform convergence for a class of finite dimension
The following theorem is cited in Balcan, M.F., Sandholm, T. and Vitercik, E., 2019. Estimating approximate incentive compatibility which I am currently reading and it is referenced to David Pollard. ...
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Is memorization necessary in learning computer science?
Backgrounds
Hi guys, I am now trying to teach myself some basic computer science theories. Specifically, I am using the book, CSAPP (computer systems, from a programmer’s perspective) and 15-213 ...
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AdaBoost - why using such alpha function?
I'm reading the paper where AdaBoost was invented (link), and I couldn't understand why they have chosen the formula α_t = 1/2 * ln((1-ε_t) / ε_t).
snippet:
...
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Obtaining a set of $O(\log n)$ classifiers using multiplicative weights algorithms
I'm trying to find an algorithm that uses the multiplicative weights algorithm to obtain a set of $O(\log n)$ classifiers that classify a set $X=\{x_1, x_2, ...,x_n\}$ where the set of labels is $l \...
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Perceptron - Generalization Bounds & Compression Bounds
A distribution $P$ over $\mathbb{R}^{d} \times\{-1,+1\}$ being $(\gamma, R)$ -separable. We now let $\mathcal{P}_{\gamma}$ denote the set of all $(\gamma, 1)$ separable distributions. For a ...
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Compression Bounds - Determine and Visualize for hypothesis vs VC dimension
I want to visualize or calculate the compression bounds for hypothesis classes. I learnt how to figure out the VC dimension. Let's say I define two hypothesis class. For example:
$$
H_k = \{h ∈ (0,1)...
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Sample compression scheme and bounding the VC dimension
There is a compression function takes any sample $S$, for which there exists a function $h ∈ H$ with $L_S(h)$, and compresses it to a subset of $k$ sample points. Similarly, there is a decompression ...
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How is the RKHS norm related to sample complexity or other learning theory properties?
This is a somewhat soft question. Given two reproducing kernel Hilbert spaces (RKHSs) $H_1$ and $H_2$, if their RKHS norms only differ by a constant, i.e., $C_1\|f\|_{H_1}\le \|f\|_{H_2} \le C_2\|f\|_{...
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Density of uniform distribution over two disjoint squares
A probability distribution $P$ over $X \times \{0, 1\}$. $P$ can be defined in term of its marginal distribution over $X$ , which we will denote by $P_X$ and the conditional labeling distribution, ...
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VC dimension of axis-aligned hyperplanes and their complements
This is a problem of VC that I've been trying to solve. Any help is appreciated.
Let's assume hypothesis classes $H_{\mathit{init}}$ of initial segments over domain $X = \mathbb R$ and $H_{\mathit{...
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348
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Empirical Risk and True Risk - Generalization Error Proof
I showed that, over an uncountable domain,learner A and a distribution P, such that for every sample size m and all samples S from $P^m$
$$
: L_S(A(S)) − L_P (A(S))| = 1
$$
Now I wanna prove for ...
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What is the VC dimension of the hypothesis class $H=\left\{f_{\theta_{1}, \theta_{2}}: R^{2} \rightarrow\{0,1\} \mid 0<\theta_{1}<\theta_{2}\right\}$?
I would like to know what is the VC dimension of the following hypothesis class.
$$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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1
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87
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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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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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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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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 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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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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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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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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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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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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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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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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650
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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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1
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39
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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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423
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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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162
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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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