Questions tagged [pac-learning]

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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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What is an example of a hard learning problem in the noisily realizable setting?

Suppose $f_{\vec{w}} : \mathbb{R}^n \rightarrow \mathbb{R}$ is a function parameterized by some parameter vector $\vec{w}$. Now for some true parameters $\vec{w}^*$, some noise function $\xi : \mathbb{...
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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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1answer
197 views

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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177 views

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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63 views

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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78 views

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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1answer
87 views

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 ...
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2answers
156 views

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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266 views

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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1answer
105 views

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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1answer
70 views

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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493 views

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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61 views

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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2answers
432 views

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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1answer
275 views

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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78 views

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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734 views

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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1answer
404 views

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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141 views

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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1answer
156 views

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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1answer
144 views

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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163 views

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 ...
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137 views

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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1answer
188 views

Is this variant of PAC learning known?

Here is a problem I've never seen, in a model similar to the PAC model. It asks a similar question to PAC learning, but wishes to optimize, rather than learn. I wonder if this problem is known, has ...
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1answer
338 views

Statistical query model with Gaussian noise?

Kearns' statistical query model is a well-known learning model with noise tolerance. The statistical query oracle takes as input a statistical query of the form $\{\chi, \tau\}$. Here $\chi$ is any ...
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1answer
627 views

What does PAC-learnability say about the learner runtime?

I am new to PAC-learnability. Assume a class $\mathcal{H}$ of hypotheses is PAC-learnable. Then all we know that if we draw polynomial number of examples (in $\delta$ and $\epsilon$), we can return a ...