Theoretical questions about Machine learning, especially Computational Learning Theory, including Algorithmic Learning Theory, PAC learning, and Bayesian Inference
1
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0answers
26 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
43 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 ...
-2
votes
0answers
34 views
Does anybody know worth reading papers on “How computers (software's) can help in eradicating corruption at every place”? [closed]
Paper can be on any of the following topics,
- Governance
- Private organisations
- Defense
- etc.
But the paper(s) must present a practical idea which can be implemented to eradicate ...
0
votes
0answers
40 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
141 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
40 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
48 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
472 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
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0answers
24 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 $I_i$ ...
1
vote
1answer
177 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
102 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
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0answers
60 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
51 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
110 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
145 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
141 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
41 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
303 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
251 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
votes
1answer
77 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
145 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 ...
0
votes
0answers
58 views
How hard is nearly Bayes optimal reinforcement learning?
Consider a set of $n$ MDPs (Markov decision processes). An MDP $M$ is selected from this set according to some probability distribution $\xi$ and then interacts with a fixed policy $\pi$ for time $T$ ...
1
vote
1answer
145 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
248 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
104 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 ...
6
votes
0answers
119 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
93 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 ...
3
votes
0answers
61 views
Is there some research about infinitely many-armed bandit with non-stationary assumption?
Is there some research about infinitely many-armed bandit with non-stationary assumption? I have found the paper about infinitely many-armed bandit under stationary (or stochastic) assumption. And I ...
0
votes
0answers
55 views
Classification time complexity (NOT training time complexity) of different algorithms
I would like to know, what is the classification time complexity of these algorithms given the number of features d
logistic regression (I suspect O(d))
Support Vector Machines (Have no idea, read ...
2
votes
2answers
109 views
Can machine learning algorithms be trained to discard nonsense?
Afaik, the problem with many machine learning algorithms is that they will often label nonsense into some categories.
What measures can one take to discard nonsense results?
Eg. if you have a bot ...
1
vote
1answer
71 views
References for the computational complexity of training neural networks
I'm looking for a good review paper or book chapter that offers an accessible introduction to the computational complexity of training neural networks for classification problems.
In particular, I'm ...
9
votes
5answers
627 views
Can neural networks be used to devise algorithms?
After the newer and newer successes of neural networks in playing board games, one feels that the next goal we set could be something more useful than beating humans in Starcraft. More precisely, I ...
0
votes
1answer
56 views
Learning a discrete distribution in $\ell_r$ norm
Let $P=(p_1,\ldots,p_d)$ be a distribution on $[d]$. Given $n$ iid draws from $P$, we construct some empirical estimate
$\hat P_n=(\hat p_{n,1},\ldots,\hat p_{n,d})$. Let us define the $r$-risk
by
$$ ...
4
votes
1answer
432 views
Understanding the No Free Lunch Theorem
I came across the No Free Lunch Theorem via Jürgen Schmidhuber's paper on Universal Search and there were a couple remarks on NFL which stood out
to me. The first was that we can't define a uniform ...
0
votes
0answers
54 views
Approximation Capabilities of Partially Connected Neural Networks
A well known result (Cybenko 1989) holds that a single-layer feedforward "multilayer perceptron" style network:
$f(x) = \sum_{i=1}^{K}\alpha_{i}\sigma(w_{i}^{T}x + \theta_{i})$
Can approximate any ...
1
vote
1answer
87 views
$L_\mathcal{D}(A(S)) \le 0.1$ with prob at least $0.9$ implies PAC learnability
Suppose we have a hypothesis class $\mathcal{H}$ that is non-uniform learnable via sample compelxity function $m_{\text{NUL}}:[0,1]^2 \times \mathcal{H} \rightarrow \mathbb{N}$.
If we define $\mathcal{...
13
votes
0answers
190 views
Differential privacy and data poisoning
A differentially private algorithm takes datasets containing inputs and produces randomized outputs, such that no small change in the dataset can shift the distribution of outputs by too much. This ...
4
votes
1answer
385 views
Autoencoders and information compression
Disclaimer: I know very (very) little about deep nets, besides what an introductory course on machine learning would teach on neural networks, and skimming some paper abstracts and introductions.
If ...
0
votes
0answers
45 views
Wavelet based Non linear optimization technique
I am outlining a method for solving Non Linear optimization problems.
Consider the system of equations:--------------------------------- 1
f1(a0, a1, a2, a3 ......... an) = 0
f2(a0, a1, a2, a3 ........
12
votes
1answer
468 views
Is BPP vs. P a real problem after we know BPP lies in P/poly?
We know (for now about 40 years, thank Adleman, Bennet and Gill) that the inclusion BPP $\subseteq$ P/poly, and an even stronger BPP/poly $\subseteq$ P/poly hold.
The "/poly" means that we work non-...
1
vote
0answers
135 views
Convergence of Q-learning with non-linear function approximation
Q-learning is a well-known algorithm in Reinforcement learning which enjoys great empirical success but with insufficient theoretical understanding. In the tabular setting, it is known that if each ...
23
votes
1answer
941 views
Functions that are Not Efficiently Computable but Learnable
We know that (see, e.g., Theorems 1 and 3 of [1]), roughly speaking, under suitable conditions, functions that can be efficiently computed by Turing machine in polynomial time ("efficiently computable"...
3
votes
2answers
369 views
Learning a coin's bias (localized)
It's well known that the minimax sample complexity for estimating the bias $p$ of a coin to additive error $\epsilon$ with confidence $\delta$ is
$\Theta(\epsilon^{-2}\log(1/\delta))$. What if we ...
4
votes
1answer
187 views
Adversarial Machine Learning, Learning with (Malicious) noise
I am reading some old papers regarding Learning With Malicious Noise. In one of them, Learning in the presence of Malicious Errors, by Kearns and Li $[1]$ (https://www.cis.upenn.edu/~mkearns/papers/...
7
votes
1answer
302 views
Applications of Takens' theorem to TCS?
My apologies if the question is a tad vague—I did try to search the literature for more, but didn't find anything (the similarity between the keywords "Takens" and "taken" on Google may be partly to ...
4
votes
1answer
191 views
Textbook/resources for a beginning researcher in (Machine) Learning Theory
I'm looking to begin understanding basic concepts, notions, results and definitions in the area of Computational Learning Theory (or the theory of Machine Learning), as is done in the theoretical ...
1
vote
0answers
114 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 ...
4
votes
0answers
747 views
Universal Approximation Theorem for non-sigmoidal activation functions
The most cited Universal Approximation Theories for multi-layer feedforward neural networks by Cybenko (1989) and Hornik (1991) assume the activation functions of the network to be sigmoidal. However, ...
1
vote
0answers
68 views
Off-policy Monte Carlo Control
The off-policy Monte Carlo control algorithm to learn the optimal state-value function $V^*$ is given as follows, which is obtained from Sutton's book.
I have three questions concerning this ...
