Questions tagged [machine-learning]
Theoretical questions about Machine learning, especially Computational Learning Theory, including Algorithmic Learning Theory, PAC learning, and Bayesian Inference
325
questions
-1
votes
1answer
55 views
equivalence between Bayesian prior distribution and regularization metric?
Ridge and LASSO can be interpreted as OLS with priors over the coefficients (respectively, Gaussian and Laplacian). How much does this generalize? Given a prior, does it imply a regularization term ...
0
votes
0answers
18 views
Why can methods like ReSuMe, Chronotron and SPAN only train single-layer spiking neural networks?
ReSuMe, Chronotron and SPAN all use STDP-like local learning rules to implement their training algorithm (though they approach the training differently, e.g. SPAN uses gradient descent via spikes ...
0
votes
0answers
38 views
Is it possible to count the total number of local minima for a scalar, multivariate function?
We can assume the function is differentiable, but it is also non-convex and setting the gradient equal to zero has no analytical solution.
We can also assume that the domain is bounded, namely the ...
2
votes
2answers
305 views
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 ...
4
votes
1answer
117 views
What is tightest known (VC-style) sample complexity bound for uniform convergence of empirical means?
The following result is adapted from Anthony and Bartlett, 1999 (Theorem 4.9).
Theorem There exist positive constants $m_0 \le 400$, $c_1 \le 8$, $c_2 \le 41$, $c_3 \ge 1/576$ such that, if $(\Omega,\...
0
votes
2answers
150 views
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 ...
0
votes
0answers
33 views
Peer Based Machine Learning
Side Note: I originally asked this question on stack overflow but it was closed because apparently it wasn't the right place to ask the question. Please direct me to the right place if you don't think ...
1
vote
1answer
99 views
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 ...
2
votes
1answer
69 views
Bayes-consistent cost-sensitive classification
In cost-sensitive classification,
we have a confusion (or cost) matrix
$C$, where $C(i,j)$ is the cost incurred
for predicting label $i$ when nature
specifies $j$. The costs are non-negative,
but no ...
-1
votes
1answer
87 views
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 ...
3
votes
1answer
64 views
Examples of learning via exactly integrable gradient flows
If $\ell (\vec{w}, \vec{z})$ is the loss function at weights $\vec{w}$ and for data $\vec{z}$ then corresponding to a distribution ${\cal D}$ we can consider doing gradient flow with step-length $\eta ...
7
votes
0answers
157 views
A variant of transfer learning
Suppose we want to train $K$ linear classifiers based on iid samples. Each classifier is of the form
$x\mapsto\mathrm{sign}(w\cdot x+\theta)$, with the constraint that the hyperplane $w$ is the same ...
2
votes
1answer
133 views
An invariance theorem for algorithmically random data in statistical learning
Motivation:
The following invariance theorem for statistical learning in the setting of algorithmically random data occurred to me yesterday. This theorem uses the fact that the property of ...
1
vote
0answers
97 views
On the equivalence of incompressibility and incompleteness in machine learning
Motivation:
Motivated by the question of what problems are suitably addressed using methods in machine learning, I decided to formulate this general problem using Algorithmic Information Theory. My ...
1
vote
2answers
100 views
VC generalization bound extended to other types of target functions
In Y. S. Abu-Mostafa's book "Learning from Data", he mentions on page 55 after deriving the VC generalization bound for a binary target function that "it can be extended to other types ...
2
votes
0answers
44 views
Energy-Based Modeling vs Deep Learning
I am doing some research on machine learning algorithms in the context of a seminar, which focuses on Energy-Based Modeling vs Deep Learning specifically in working with images Modeling. Now I know ...
12
votes
2answers
313 views
Circuit and Formula Lower Bounds for Separating Sparse Sets of Strings
We say that a pair $(P,N)$ of subsets of strings from $\{0,1\}^n$ is an $n$-pair if $|P|=|N|=n$. Intuitively, sucha a pair consists of a set $P$ with $n$ positive $n$-bit strings, and a set $N$ with $...
0
votes
1answer
127 views
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 ...
0
votes
1answer
44 views
Machine Learning: Calibrating SubGroups of Probability Predictions inside a Dataset which should sum to 100%
I am working on an interesting type of problem where I want to make predicitons for individual elements within subgroups- with the knowledge that the sum of the probabilities within a subgroup should ...
1
vote
1answer
125 views
VC-dimension of infinite set of triangle wave
I am searching for the VC-dimension of the following:
What is the VC-dimension of the infinite set of triangle wave functions with
amplitude 1 and period parameter p on points on the line?
2πarcsin(...
1
vote
1answer
615 views
What's the intuition behind Rademacher complexity?
As stated, what exactly is the intuition behind Rademacher Complexity which is defined:
Rademacher complexity captures the richness of a family of functions by measuring the degree to which a ...
-2
votes
1answer
158 views
How to calculate complexity in a high dimensional space?
Edit: 'Fitness landscape analysis' was mentioned as a relevant measure. If you're going to downvote the post, at least leave a comment what is wrong.
For a specific f(), I'm defining a term '...
6
votes
1answer
300 views
Is there an equivalent to VC-dimension for density estimation as opposed to classification?
VC-dimension can be used to quantify the capacity for classifier models and compute generalization bounds, but is there an equivalent concept that can be applied to density estimation, e.g. to compute ...
0
votes
1answer
472 views
VC dimension for balanced binary decision trees
What is the VC dimension of all balanced binary decision tree of
depth $k$ in $\{0,1\}^d$? Does it depend on depth $k$ or dimension $d$?
1
vote
1answer
109 views
Rademacher Complexity of the Composition with an Indicator
Consider the statistical learning setting where you have an arbitrary hypothesis space $\mathcal{H}$, a data space $\mathcal{Z}$, and a bounded loss function $\ell: \mathcal{H}\times \mathcal{Z} \...
7
votes
1answer
275 views
Testing for finite expectation
The mean of a positive random variable $X$ is either finite or infinite; define $J(X)$ to be $0$ in the former case and $1$ in the latter case. Claim: there does not exist a function $J_n$ from the ...
2
votes
0answers
101 views
Polynomial convergence to optimal move of the UCT algorithm. Missing proof?
This is a question regarding the theoretical convergence guarantees of the UCT algorithm, a popular variation of the Monte Carlo Tree Search algorithm (used in games, planning, reinforcement learning, ...
1
vote
0answers
49 views
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 ...
6
votes
3answers
568 views
Reference Request: Computational Learning Theory
Pretty soon I will be finishing up Understanding Machine Learning by Shai Ben-David and Shai Shalev-Shwartz. I absolutely love the subject and want to learn more, the only issue is I'm having trouble ...
2
votes
1answer
141 views
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]\...
3
votes
0answers
167 views
Understanding Dudley Chaining Argument for Rademacher Bound
I follow the proof of the Dudley chaining/metric entropy bound of the (empirical) Rademacher complexity, but I don't have any intuition for why this bound should be true. In particular, I don't know ...
7
votes
1answer
337 views
Complexity of constructing minimum depth decision trees
I am interested in the computational complexity of
Problem 1: Given a finite, non-empty set $J$, given $A, B \subseteq \{0,1\}^J$ such that $A \cap B = \emptyset$, and given $n \in \mathbb{N}$, does ...
0
votes
0answers
77 views
Using martingale arguments to prove convergence of iterative algorithms
Can someone give me typical/educative examples of how martingales can be used to prove convergence of an iterative algorithmS?
The examples I know of can only go so far as to show that there exists ...
4
votes
0answers
107 views
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 ...
5
votes
1answer
208 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 ...
4
votes
1answer
277 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 $...
0
votes
1answer
120 views
Singular Value in Machine Learning
I'm reading the paper (http://proceedings.mlr.press/v9/glorot10a/glorot10a.pdf) from Glorot and Bengio. There is something that I don't understand at the abstract section on page 1.
"Training may be ...
1
vote
1answer
78 views
Characterize a point cloud
Background: I have multiple point clouds (sets of objects) $\{S_i\}_{i\in\mathbb{N}}$ of variable size and "purity" (meaning that some sets contain very similar objects, some show a high diversity; ...
4
votes
2answers
201 views
What are some good resources for strengthening my theoretical foundation for machine learning?
I'm a computer science major and I'm taking a lot of machine learning courses. I'm finding that my theoretical foundation on subjects like calculus and linear algebra are not as strong as I'd like ...
4
votes
1answer
80 views
Terminology and references for a learning model
Let's say we're doing regression over $[0,1]^d$ -- either in the PAC sense with bounded-range agnostic noise or in the more classical-statistics sense with additive Gaussian noise. Suppose further ...
1
vote
0answers
56 views
Average smoothness learning rates
This question is somewhat related to this one. There are many results in statistics where convergence rates (including minimax ones) are given in terms of the smoothness properties of the underlying ...
0
votes
0answers
78 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 ...
1
vote
0answers
56 views
Which algorithms can be used to measure similarity for two very different languages?
recently I have read this paper, A Survey of text similarity approaches, and I discovered that there are a lot of algorithms that can be used to measure similarity.
At present I am applying the ...
1
vote
0answers
70 views
Sentences in what kinds of grammar in the Chomsky hierarchy can be parsed by an LSTM of a given size?
Given
an LSTM $N$ of a given size $A$,
a sentence $S$ with a given number of words $B$,
a Chomsky grammar hierarchy level $C$ in 0-3,
a Chomsky grammar $G$ of level $C$ of size $D$,
A given fixed, ...
1
vote
0answers
217 views
Minimum number of hours of speech needed to train a neural net to recognize speech [closed]
From a theoretical computer science point of view, is there a lower limit on the number of hours of speech needed to train a neural net to translate speech to text? An estimate from CMU is 3000-5000 ...
2
votes
0answers
88 views
Which computational framework lies behind the Chinese “Social Credit System”?
BACKGROUND
The Social Credit System is a data-driven reputation system which draws on several sources to label various entities, namely businesses and individual citizens, with a trustworthiness ...
1
vote
0answers
86 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 ...
2
votes
1answer
172 views
How many samples are needed to reconstruct a path?
Consider an input set of vertices $V$ and vertices $s,t\in V$.
The goal is to learn some unknown shortest path from $s$ to $t$; the set of edges of the graph is hidden at first and there may be ...
3
votes
1answer
90 views
Agnostic query learning for DFAs
Angluin's membership+equivalence query algorithm allows to efficiently and exactly learn a target $n$-state DFA. But what if the target DFA is huge, or the target concept is not even a regular ...
0
votes
0answers
118 views
Lower bound to agnostic learning with membership queries
Setting:
Let $X$ be a finite set and $C = \{0, 1\}^X$ a finite family of classifiers on $X$. Fix an $f \in \{0, 1\}^X$ not in $C$, a (possibly randomized and adaptive) learner $A$ has access to a ...
