Questions tagged [machine-learning]
Theoretical questions about Machine learning, especially Computational Learning Theory, including Algorithmic Learning Theory, PAC learning, and Bayesian Inference
0
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0answers
94 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 ...
4
votes
1answer
141 views
What was the significance of Leslie Valiant's, “A Theory of the Learnable?”
It seems like two of the main takeaways were that there is a natural limit to what computers can learn, and learning is bounded by polynomial algorithms. Why was his paper significant in the broader ...
-1
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2answers
68 views
Machine Learning Algorithm To Fill Data Holes
I'm having trouble finding a good place to begin with this. I'm just looking for a name or point to start researching a
Let's say I have 1000 records. 10 of these records are only 90% complete. The ...
2
votes
0answers
66 views
References on generalization bounds
I'm looking for references (books, papers, lecture notes etc) on generalization bounds and their proofs. Specifically, I'm looking to fully understand the technique of defining a hypothesis class (or ...
0
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1answer
23 views
Classic Neural Network Layout for regression: is the score function derivable?
I need to solve an optimization problem based on a function f(X). This function is not known, but it can be estimated from a training set. So first I train a model, then I get the score function f(X), ...
2
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0answers
67 views
how is time complexity defined in computational learning theory
In general, when we say an algorithm $A$ PAC learns $C$ in time $t$, we say $A$ takes time $t$ before outputting a hypothesis $h$, and the hypothesis can be evaluated (on every $x$) in time $t$.
Now ...
6
votes
1answer
204 views
How to generalize VC dimension?
Let's try to generalize the $VC$-dimension (of the class of hyperplanes) to include accuracy/error. Let $S$ be a set of points in $R^d$ and $t$ in $[0,1]$. We say that the class of hyperplanes $t$-...
0
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1answer
76 views
Confusion about covering number
Problem
I do not understand why larger $p$ will give a larger covering number.
Since when $p\geq q$, the corresponding hypercube is also larger (by $\| x \| _ { q } \leq n ^ { ( 1 / q - 1 / p ) } \|...
0
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1answer
37 views
Does fixed hyperparameters perform well regardless the number of training examples?
I'm new in this community and I don't know whether my question is proper for this community. I will delete this post if it is not proper.
I'm interested in deep learning network models and have a ...
1
vote
1answer
76 views
About learning a single Gaussian in total-variation distance
I am looking for the proof of this following result which I saw as being claimed as a "folklore" in a paper. It would be helpful if someone can share a reference where this has been shown!
Let $G$ ...
0
votes
1answer
83 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 ...
3
votes
1answer
90 views
Oncina-Garcia RPNI algorithm for learning DFAs
The question refers to this paper:
ftp://altea.dlsi.ua.es/people/oncina/articulos/asspr1992.pdf
Given a sample of $p$ positive and $n$ negative strings, RPNI constructs a consistent DFA in time $O((p+...
0
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0answers
13 views
Is there a stochastic/online version of the GLM-Tron algorithm?
The GLM-Tron algorithm appeared in Theorem $1$ in this paper, https://arxiv.org/pdf/1104.2018.pdf Is there a stochastic version of this? (...essentially something that will randomly sample a few ...
1
vote
0answers
74 views
Complexity of low-rank matrix factorizations with rows in a simplex and outliers
Our goal is to obtain a matrix factorization in form of $M = U V'$, where $U\in\mathbb{R}^{d\times r}, V \in\mathbb{R}^{N\times r}$ and each row of $V$ satisfies
$$ \sum_{j}(V)_{ij}=1, (V)_{ij}\ge 0 $$...
1
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0answers
37 views
Back-propagation for computing derivative of certain line integral
Consider a function F (think of neural networks) with two sets of parameters: (1) model parameters $\mathbf{w}$, and (2) input data ${\bf x} \in {\mathbb R}^d$. Fix $i \in [d]$, consider the following ...
4
votes
2answers
136 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 $\...
2
votes
1answer
113 views
Is there a gap between weak learning and PAC-learning?
For concreteness lets use the definitions of PAC and weak-learning as in the notes of Avrim Blum (http://www.cs.cmu.edu/~avrim/ML12/lect0208.txt) and also his notes on SQ-Learning (http://www.cs.cmu....
3
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0answers
91 views
Do features always induce a metric?
It is well-known in functional analysis that an inner product always induces a norm and a norm always induces a metric, and the reverse directions do not hold in general. I am wondering if a similar ...
2
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0answers
127 views
Crime prevention using graph theory and machine learning
I am looking for a way to the model the incidence of crime among a network of individuals. Part of it will use machine learning, and part of it will have to resort to some graph theoretic ...
7
votes
1answer
112 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 ...
2
votes
1answer
98 views
Stochastic gradient methods and risk of neural nets
Under many situations it is currently provable that we can minimize the risk of neural nets using stochastic gradient based algorithms. For example : https://arxiv.org/abs/1811.03804, https://arxiv....
2
votes
1answer
76 views
Empirical Rademacher averages versus Hoeffdings bound
Let $M$ be finite set with $n$ distinct elements. I want to probalistically approximate the relative counts $\frac{|P(Q)|}{|M|}$ of $Q \subseteq M$, where $P(Q) = |P \cap M|$.
An upper-bound for ...
2
votes
0answers
56 views
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 ...
2
votes
1answer
101 views
Minimization version of matrix p-norms?
I considered a minimization version of matrix p-norms, defined for a matrix $A$ by
$$
f_p(A)= \min_{x\neq 0} \frac{||Ax||_p}{||x||_p}.
$$
Notice that $f_p(A) = 0$ if and only if $A$'s columns are ...
1
vote
0answers
53 views
Average margin bounds for separable SVM
Suppose we're training a linear separator in the realizable PAC setting. Given $m$ labeled examples $(x_i,y_i)$ in $\mathbb R^d\times\{-1,1\}$, a (consistent) linear separator is a vector $w\in\mathbb ...
1
vote
1answer
59 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 ...
0
votes
0answers
42 views
Function that maps non-linear distribution to normal distribution while maintaining distance
I have a collection $X$ of 10 million $(x,y,z)$ 3-tuples, where $x$, $y$, and $z$ are all numbers between 0 and 1. The distribution of $x$, $y$, and $z$ values are complex, and the distributions of $...
3
votes
2answers
181 views
How can AIC converge in the limit when even 2 parameter models can have infinite VC dimension?
AIC-based model-selection converges to zero error in the limit, and also has finite-sample convergence that is rate-optimal with respect to worst case minimax error [1]. (Note that AIC refers to ...
1
vote
0answers
48 views
Is there a theoretical guarantee that an autoencoder $g$ has $I(x;g(x)) \approx H(x)$?
I know that in general, a function $g$ can be a good auto-encoder (i.e., $g(x) \approx x$ for $x \sim D$) and on the same time $I(g(x);x)$ is small. This is the case when $g$ forms a good correlation ...
3
votes
1answer
81 views
Examples of nontrivial non-discriminatory functions
I am reading Cybenko's "Approximation by Superpositions of a Sigmoidal Function".
The paper defines a discriminatory function as:
$\sigma$ is discriminatory if for a measure $\mu$,
\begin{align}
\int ...
8
votes
1answer
481 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?
1
vote
0answers
31 views
Variability of gradient estimates and convergence rate in stochastic gradient descent/ascent
I am aware that convergence in stochastic gradient problems is very sensitive to the variance of your gradient estimator. One issue I'm running into is that the gradient is a random vector and so ...
1
vote
0answers
50 views
Ranking problem (single instance before others)
Suppose I am trying to learn function $f$ for a ranking-like objective:
The mapping: ranking one element at the top and the rest at the bottom.
More formally, suppose each input instance $I_i$ ...
4
votes
2answers
1k views
What is the best place to get BibTeX entries for ICLR and other machine learning papers?
dblp seems the best i found so far (as also mentioned in the comments and in What is the best place to get BibTeX entries for computer science articles ?) but even there some papers from NIPS ...
4
votes
1answer
108 views
“Learning” when test and train distributions don't match
We know that the theory of PAC-learning is distribution-free, i.e. assuming that the test and train distributions are the same, we have guarantees on learning the hypothesis.
Question: what if the ...
1
vote
0answers
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 ...
1
vote
0answers
57 views
To what extent supervised learning ERM learn first-order knowledge
Suppose I have a collection of (hidden) first-order rules:
$$
\mathcal{R}: \{ Q_i(x) => P_i(x) \}_{i=1}^{k}
$$
all defined over $x \in \mathcal{X}$.
I can use these rules and (automatically) ...
3
votes
1answer
130 views
Tighter Probability Bounds
Let $\mathcal{F}$ be a class of binary functions on a probability space $\Omega$. For $f \in \mathcal{F}$, let $P(f) =\mathbb{E}(f(Z))$ and $P_n(f) = \frac{1}{n} \sum_{i=1}^n f(Z_i)$ where $Z_i$'s are ...
5
votes
1answer
274 views
Universal Approximation - Are ReLUs discriminatory?
In Cybenko's elegant proof of the Universal Approximation Theorem (UAT) he proves that single hidden layer neural networks (with linear output layer) are universal approximators whenever their ...
3
votes
1answer
270 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:
$$\epsilon^2 < \frac{\log{|H_\epsilon|} + \log{1/\delta}}{2m}$$
...
0
votes
1answer
48 views
About using smoothness of the Hessian for getting to approximate criticality of a non-convex objective
Is there any algorithm which shows that under the assumption of Lipschitz smoothness of the Hessian of a non-convex function one can get to its critical point faster?
5
votes
2answers
345 views
Complexity of finding a consistent hyperplane
Given $m$ binary labeled points in $\mathbb{R}^d$, it is well-known that in general it's NP-hard to find a hyperplane that minimizes sample error. A brute-force search considers all $O(m^d)$ sample ...
1
vote
1answer
657 views
Why semi-gradient is used instead of the true gradient in Q-learning?
In reinforcement learning, with function approximation, a popular cost function is the Mean value error.
This involves a target value V_pi and a current value estimate V_hat.
When deriving the update ...
2
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1answer
92 views
Other Uniform Bound
In theoretical machine learning, VC-dimension (VCD) and Rademacher average (RA) are two frequently used uniform bounds, providing better sample complexity than bounds such as Chernoff bound and ...
4
votes
1answer
213 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 ...
1
vote
1answer
181 views
What is the VC dimension of Turing machines with specified maximum size?
Note by "maximum size" in the question I'm referring to the size of the Turing machine's state machine.
I chose Turing machines in the question to make the question concrete, but I'm also more ...
6
votes
2answers
364 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 $\...
5
votes
0answers
105 views
Learning hidden variable distribution
Consider a set of $k$ continuous variables. Each variable $x_k$ is associated with a hidden distribution from which its value is sampled independently of other variables. I am given a set of ...
7
votes
0answers
151 views
Universal approximation theorem of second order
The universal approximation theorem (https://en.wikipedia.org/wiki/Universal_approximation_theorem)
informally states that up to several conditions, any function can be approximated by a shallow ...
-1
votes
1answer
94 views
What is the name of the category of problems that can only be solved with machine learning? [closed]
Wikipedia defines machine learning as the "field of computer science that gives computers the ability to learn without being explicitly programmed". A common example of a problem which machine ...
