Questions tagged [pac-learning]
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Confusion about lower bounds and upper bounds in learning theory
In computer science, lower bounds and upper bounds are defined as follow:
$$m \geq g(n) \implies m = \Omega(g(n))$$
$$m \leq g(n) \implies m = \mathcal{O}(g(n))$$
However, in proving lower bounds and ...
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Learning with zero inductive bias
I want to understand the intuition behind the classic setting of learning theory, we always assume that the model belongs to some known class. Was there a formal proof that we can or can not learn a ...
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What is the condition under which the estimation error increases (logarithmically) with hypothesis class size for a finite hypothesis class
In section 5.2 error decomposition (p.404) from the online book "Shai et al., Understanding Machine Learning: From Theory to Applications", the authors wrote:
As we have shown, for a finite ...
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Why the measure of information complexities for passive and active learning are increasing in research communities?
I am a PhD student working on the theory of active learning.
Over the years, accepted papers in COLT and ALT for active learning are focused on approaches that almost all of them define new ...
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Unable to understand the Sample complexity of PAC learning
I have been studying from the book "Understanding Machine Learning - From Theory to Algorithms" by Shai Shalev-Shwartz and Shai Ben-David
I am struck at corollary 3.2 which states that
Every ...
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Does PAC learnable imply agnostic PAC learnable for binary classification tasks?
The Fundamental Theorem of Statistical Learning from the book "Shai et al., Understanding Machine Learning: From Theory to Algorithms, Cambridge Press University", is written as follows:
...
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67
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The hardness of active learning with fixed budget
I have been looking for theoretical papers studying this question of the hardness of PAC active learning algorithms. I found a few papers studying the problem from a fixed perspective (particular ...
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Learning a PAC-lernable using agnostic-PAC framework
given H a family of functions which is PAC lernable such that for $\epsilon$ error and $\delta $ confidence interval it required $m(\epsilon,\delta)$ samples.
I understood that if we learn H under ...
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Proving existence of efficient PAC learning algorithm without noise info given poly-time algorithm with noise upper bound
How would I prove that if there is an efficient algorithm for PAC learning in the presence of classification noise by an algorithm that is given a noise rate upper bound $\eta_0$ ($1/2 > \eta_0 \...
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PAC learning in the convex case with bounded expected risk
In the context of computational learning theory:
Let $f(w,z): \mathcal{W} \rightarrow \mathbb{R}$ be convex in $w \in \mathcal{W}$ where $\mathcal{W}$ bounded by R. Over $\mathcal{W}$, $|f| \leq C$.
...
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Differing definitions of a weak learner
I've been reading about boosting and have come across basically two definitions of a weak learner. Basically for hypothesis $h$ and target $c$, some definitions says that $h$ is a weak learner if $E[h(...
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PAC learning over continuous functions
I'm wondering if it's possible to use PAC learning to learn a continuous function. For example, if we wanted to learn a probability distribution or a CDF, is it valid to train on some set of m ...
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Learning boolean functions with input-ouput examples and side-information
The Kushilevitz-Mansour, "low-degree", and Goldreich-Levin algorithms aim to learn a function $f: \{0,1\}^n \rightarrow \{0,1\}$ from a sufficiently large set of input-output examples $(x_i, ...
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Relationship between statistical query lower bounds and "traditional" iid sampling lower bounds
Coming from a more statistical background, it is not clear to me if or how lower bounds in the statistical query (SQ) model imply anything useful about traditional learning problems with iid samples (...
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Fat Shattering / VC dimension / Statistical Complexity of piecewise linear functions
I am trying to establish a bound on the VC dimension of piecewise linear continuous functions with $k$ pieces. I am aware of an earlier question which tackles this problem in the case of convex ...
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PAC guarantees for linear prediction under the squared loss
I am looking for generalisation bounds under the squared loss, specifically for the class $\mathcal{F}_{\text{lin}} = \{f(x) = \langle w, x \rangle : \|w\| \leq C\}$ of bounded linear predictors. I am ...
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VC dimension of the class of all polygons with k vertices
VC dimension of the class of convex polygons with $ k $ vertices is known to be $ 2k + 1$.
For the general case I was able to derive a bound of the type $ O(k^2log(k)) $ (probably can be easily ...
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Non-(PAC)-Learnable Classes
I'm learning about PAC-learnability. I've figured out how to show that a class of classifiers is PAC-learnable, but what about if I want to show that a class of classifiers is not PAC-learnable? How ...
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Some issues with proof of Fundamental Theorem of Statistical learning
I am reading the book "Understanding Machine Learning" by Shai Shalev-Shwartz and Shai Ben-David.
The theorem 6.7 has several equivalent statements for a class of functions $H$. The first ...
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An (unusual?) risk bound
I am told that that a bound on the generalization error of the following form exists in terms of something called the ``shattering coefficient" - but I am not able to reference this quantity in ...
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Generalization bound for parameters rather than loss functions
I was wondering if it is possible to obtain high probability bounds (provided finite sample size of the training data) for the distance (say in the l-1 or l-2 norm) between the best parameter set and ...
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No free lunch theorem and finite hypothesis classes
I have read the no free lunch theorem(NFLT) section 5.1 of Understanding machine learning by Shai Shalev-Shwartz. There is also this Corollary 4.6 which states any finite hypothesis class is PAC ...
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Generalisations of the Fundamental Theorem of Statistical Learning to different tasks and losses
The fundamental theorem of statistical learning gives an equivalence between uniform convergence of the empirical risk to learning in the PAC framework.
I have only seen this stated in the case of ...
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Is statistical query learning equivalent to correlational statistical query learning given a fixed distribution?
As title, I saw some paper mentioned they are equivalent, but I'd wonder how to prove they are? is it something to do with PAC as SQ is a restricted version of PAC?
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Logic of learning
Does Robust logic (Leslie Valiant), Default logic (Raymond Reiter) and Circumscription logic (John McCarthy) have any relation?
I was Mathematician and Computer Science (dual degree undergraduate) ...
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Is there a notion of Probably Approximately Correctness in Unsupervised Learning? [closed]
I've been learning a little bit about computational learning theory, but most of what I've seen so far is related to supervised learning. Perhaps dimensionality reduction will be touched on, but not ...
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Why non-uniform learnability does not imply PAC learnability?
PAC guarantees provide us a a learning algorithm $A_n(\cdot)$ and sample complexity bound $n_{\mathcal{F}}(\epsilon,\sigma)$ that ensures
$
P\left[L_P(A(\mathcal{D}^n))-L_P(f^*)\leq \epsilon\right]\...
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Status of the Junta Problem (soft question)
Does the learning theory community in general believe that juntas can be learned in polynomial time?
The naive algorithm works in quasi-polynomial time. MOS's paper shows how to solve the junta ...
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Latest word on cross validation?
It's a standard result leave-one-out cross-validation is an unbiased estimator of the risk (see, e.g., Lemma 4.1 in Mohri, Rostamizadeh, Talwalkar). Are there any "better" results? Such as, say, with ...
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Is this a known learning problem?
Let $(\mathcal{X},\rho)$ be a metric space (say, $\mathcal{X}=[0,1]$ with the Euclidean metric). Let $\alpha:\mathcal{X}\to[0,1]$ be unknown. Suppose that $\mathcal{X}$ is endowed with a distribution $...
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representation of concept classes and pac learning
I was reading the book of Kearns and Vazirani and I didn't completely understand the following:
Let C be a concept class and suppose we want to PAC learn C, they say first consider a larger ...
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Agnostic query learning of decision trees
Gopalan, Kalai, Klivans gave an algorithm
https://dl.acm.org/citation.cfm?id=1374376.1374451
for agnostically learning decision trees $h:\{0,1\}^n\to\{0,1\}$ under the uniform distribution given ...
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Lower bound of real valued bounded function
Is well known that the lower bound on number of example necessary to reach a given error for concept classes $\Omega(d/\varepsilon)$ (cf. also
Agnostic PAC sampling lower bound
)
I am looking for the ...
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Are there hypothesis classes that are hard to learn but easy to test?
Let $H$ be a binary hypothesis class, it is easy to see that if $H$ is (efficiently) properly PAC learnable then it is also (efficiently) testable (here we use the standard notion of within or $\...
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About assumptions needed to get convergence of stochastic gradient methods on non-convex objectives
What are the minimal conditions we know of under which we can prove that a stochastic gradient based algorithm can convergence to criticality on a non-convex objective?
Are there any necessary ...
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Lower bounds for SRM?
This question is about structural risk minimization and model selection. Let $H_n$ be the collection of all binary classifiers on some fixed set with an $n$-bit description length in some fixed ...
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Sample Complexity for Order Statistics
I have a sample complexity question which seems fairly basic, but for which I'm having trouble finding a reference.
Let $F$ be an unknown distribution over $[0,1]$. Denote by $X_{k:n}$ the $k$th of $...
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Sample complexity for learning Boltzmann Distribution parameters
I am trying to think through the number of samples that I would need to estimate the parameters of a Boltzmann partition function to a desirable precision.
Suppose that there are $N$ possible states ...
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Difficulty of "learning" rare instances
Is there any result showing that models (say SVM, Neural-Net, kNN, etc) will have difficulty in learning "rare" instances/tail phenomena?
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Learnability of under some characteristics of the distribution
TLDR; is there any results showing that more concentrated (or easier) distributions are easier to learn?
In PAC-learning, the guarantee is given for any underlying distributions. But in reality, we ...
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PAC-learning bound with epsilon-cover of hypothesis class
In this video at 43:00, a version of the PAC bound for generalization error $\epsilon$, which I hadn't seen before, is quoted:
$$\epsilon^2 < \frac{\log{|H_\epsilon|} + \log{1/\delta}}{2m}$$
...
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Rademacher complexity beyond the agnostic setting
The way I know of to bound generalization error by Rademacher complexity is Theorem 2.4 in this lecture notes, http://ttic.uchicago.edu/~tewari/lectures/lecture9.pdf. Here the quantity on the LHS that ...
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Reference request for the relationship between approximating degree of Boolean functions and learning algorithms
This paper (http://www.cs.columbia.edu/~rocco/Public/stoc01.pdf) from STOC 2001 is possibly the first paper to show how to convert upperbounds on the $\frac{1}{3}-$approximation degree of a Boolean ...
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Proper PAC learning VC dimension bounds
It is well known that for a concept class $\mathcal{C}$ with VC dimension $d$, it suffices to obtain $O\left(\frac{d}{\varepsilon}\log\frac{1}{\varepsilon}\right)$ labelled examples to PAC learn $\...
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Rademacher complexity and lowerbounds in learning theory
Is there any function class known whose Rademacher complexity has a non-trivial lowerbound?
Can the Rademacher complexity be used to lowerbound the generalization error in any learning situation?
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Boolean functions with high query complexity for PAC learning
The most general theorem for PAC learning of Boolean functions that I am aware of is the theorem in section 3.4 of Ryan O'Donnel's book where its basically shown that Boolean functions whose Fourier ...
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Learning from derivative data
In many machine learning algorithm, it is often assumed that outputs of unknown function and their corresponding inputs are given to estimate the unknown function. However, I wonder whether there ...
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Does MCMC belong to the statistical query model?
It is known that a wide range of algorithms fall into the statistical query (SQ) learning model by Michael Kearns. Examples include k-means, logistic regression, naive Bayes (NB), SVM, ICA, PCA, ...
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The Complexity of Properly Learning Decision Trees
Where does this paper prove the middle bullet point of its abstract?
I have looked through that paper fairly thoroughly.
There are three things I want to read how they're getting around.
Reductions ...
user6973
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Learning read-once branching programs with membership queries
Let $B=\{0,1\}$. A read-once branching program of width $n$ and size $w$ is given by a graph with layers $0,\ldots, n$, where the first layer has just the starting node, the last layer has nodes ...
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