Questions tagged [machine-learning]

Theoretical questions about Machine learning, especially Computational Learning Theory, including Algorithmic Learning Theory, PAC learning, and Bayesian Inference

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215 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 ...
7
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0answers
154 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 ...
7
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0answers
180 views

Testing the degree of a vertex

Let's say we have a graph $G$ with $n$ vertices. Given $\epsilon>0$ and a specific vertex $v$, consider the problem of deciding whether $\mathrm{deg}(v) < \frac{\epsilon}{3}n$ or $\mathrm{deg}(v)...
7
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0answers
186 views

Sample complexity of PAC learning all k-DNFs over the uniform distribution

Is sample complexity of PAC learning all $k$-DNFs over the uniform distribution known (that is all DNFs with all terms of size at most $k$ and without restriction on the number of terms)? The only ...
6
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90 views

Looking for an easy/pedantic exposition of Renegar's famous result on polynomial optimization

In September $1989$, Renegar had this famous sequence of 3 papers titled, "On the Computational Complexity and Geometry of the First-order Theory of the Reals, Part I/II/III". I was wondering if ...
6
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159 views

Machine learning algorithms on hypergrap models

Graphical models are a very useful tool with many applications, whereby a joint distribution of a set of random variables is modeled using only pairwise dependencies between the variables, and two ...
6
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0answers
124 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 ...
6
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0answers
114 views

Branching Boosting Algorithms

Long/Servedio showed AdaBoost/etc doesn't perform well under noisy environments, but that branching forms of boosting do. Can any point me to a list of branching boosting algorithms, or a reference ...
6
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0answers
180 views

Occam razor for exact learning with membership/equivalence queries

In PAC learning, there is the "Occam razor" principle, which says that learning is qualitatively equivalent finding a succinct hypothesis that is consistent with the training samples. My question is ...
5
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246 views

What precisely is the extra power afforded by using deeper nets?

For any choice of activation function (fix the choice for all the hidden nodes for both the following DNNs) do we know of functions which some $k$ (hidden layer) DNN can compute but a $(k-1)-$DNN can'...
5
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0answers
117 views

extracting/ exploiting similarity of SAT instances by solver

suppose that two SAT formulas on different variables $F_1, F_2$ are given on the input that are known to be true and the problem is to build an algorithm that finds a solution to each. the formulas ...
5
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176 views

Complexity of the Transductive SVM training problem

The Transductive Support Vector Machine training problem is a non-convex mixed integer programming problem: Transductive Support Vector Machine training problem. $$ \begin{align} \mathop{\text{...
4
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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 ...
4
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0answers
1k 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, ...
4
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97 views

Adversarial distributions for PAC lower bounds

The various PAC lower bounds (realizable, agnostic, bounded noise) construct distributions supported on $d$ points, where $d$ is the VC-dimension of the hypothesis class in question. Does anyone ...
4
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0answers
488 views

Practical Implications of Kolmogorov's Result on the Universal Approximation Theorem with Neural Networks

After having read matus's beautiful answer in this thread explaining (among other things) Kolmogorov's result regarding the Universal Approximation Theorem with Neural Networks, I wonder: if just $\...
3
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0answers
93 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 ...
3
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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 ...
3
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0answers
163 views

How useful is program search in the field of programming-language theory?

I've been thinking: computing systems such as the Lambda Calculus and its variations are usually very simple and can be implemented in as few as ~80 lines of Haskell code. There is a self-interpreter ...
3
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68 views

Lower Bound on Zero-order Regret

Here is a brief summary of the experts framework: Given $n$ experts who either give correct or wrong advice for each round $t\in [T]$, an algorithm is required to give a best prediction for each round ...
3
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0answers
258 views

PAC learning and computation over real numbers

I became familiar with the BSS model of computation recently. I find it to be a better model of computation to study complexity of numerical analysis methods (cf. Complexity and Real Computation; ...
3
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0answers
119 views

Finding most informative feature subsets given dataset, clustering algorithm and gold standard partition

I have an $n \times m$ matrix of data $\mathbf{D}$ as well as a $k$-partition $P$ of $n$ indices each representing a row in a dataset. Assuming an arbitrary clustering algorithm $A$, I would like to ...
3
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0answers
80 views

Efficiently Detecting “edges” in the time frequency plane

Given a signal $y(t)\in\mathbb{R}$ I wish to detect edge patterns. $s(f,t)$ is a time-frequency decomposition of $y(t)$ in some window $(t-n,t+n)$ so that $f$ loosely corresponds to a local frequency....
2
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67 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 ...
2
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0answers
68 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 ...
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 ...
2
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0answers
62 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
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0answers
84 views

How does white noise on the output channels influence the training of a neural network?

Let $\rho(\sigma): \mathbb{R} \rightarrow \mathbb{R}$ be a probability density that is parametrized by a parameter vector $\sigma \in \mathbb{R}^s$ (for example the normal distribution where $s = 1$ ...
2
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0answers
160 views

About lower bounding the sample complexity of a distribution

Given a joint probability distribution over a finite number of random variables (each with a finite range space) of which only a certain subset is observable, is there a notion of "sample complexity" ...
2
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0answers
67 views

Impossibility result on metric learning?

Are there any fundamental limitations (impossibility results) known for metric learning? Are there any direct connection reduction from/to that I can use results in clustering? (e.g. this: 2 ) 2 ...
2
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0answers
54 views

Question about the definition of projecting concepts in learning

I am self-studying in the area of query learning and having a difficulty in understanding the definition of closed under projection for concept classes discussed in several papers (for example, here (...
2
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0answers
182 views

Convergence of online convex optimization methods

I am new to this subject so this question might seem a bit trivial Assume that in each round $t\in{{1,...T}}$ we choose $x_t\in K $ where $K$ is a compact and convex set, The common methods for ...
2
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0answers
634 views

Is there a closed form equation for the back-propagation equation update in Neural Networks?

I was trying to understand if there was a way to express the back-propagation equations from neural networks in a better way as to understand them better. I believe the equations can be written in a ...
2
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0answers
47 views

second order regularisation on a neurofuzzy network with Bernstein basis functions

We're trying to build a neural network that uses a neurofuzzy approach. Our reference is the book Adaptive Modelling, Estimation and Fusion from Data: A Neurofuzzy Approach by Chris Harris, Xia Hong, ...
2
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219 views

Dynamical systems analysis of deep learning

I am interested in finding out references that apply dynamical systems analysis to develop the "theory" of deep learning, specifically (say) feedforward deep neural nets. The only paper I seem to have ...
2
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0answers
109 views

Johnson Lindenstrauss for Random variables?

Does the Johnson-Lindenstrauss Lemma apply to any finite-dimensional Hilbert Space? In particular, I am interested in the space of random variables $X = (X_1,...,X_N)$ over $N$ uncertain states. If $\...
2
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0answers
99 views

What mathematical models can analyze and optimize such message passing system?

I look for a mathematical model that can accommodate, analyze and suggest optimizations for a system that can be humanly described as message passing black box programs to which where optimal message ...
2
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0answers
126 views

Cellular neural networks

I'm a new at the cellular neural network, which is a special case of the recurrent neural network: my questions are: I understand the implementation of Chua circuit for the CNN, but I still need to ...
2
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0answers
96 views

Do Genetic Algorithms Expect a Independent Search Space

Genetic Algorithms seem like multiple simulated annealing instances, augmented with a crossover genetic operator. The crossover operator selects predefined genes from two different parent solutions to ...
2
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0answers
90 views

Fast rates — cleanest proof

Fast rates generally refers to generalization bounds interpolating between the $1/n$ consistent rate and the $1/\sqrt n$ agnostic rate. I am aware of two basic approaches for obtaining these: (1) ...
2
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0answers
191 views

Expected probability of error in Vapnik's book

In Vapnik's book "Statistical Learning Theory", Theorem 10.5 states that - for a Support Vector Machine - the expected probability of error (of the optimal hyperplane) is upper bounded by $1/(l+1)$ ...
2
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779 views

How is the complexity of PCA $O(\min(p^3,n^3))$?

I've been reading a paper on Sparse PCA, which is: http://stats.stanford.edu/~imj/WEBLIST/AsYetUnpub/sparse.pdf And it states that, if you have $n$ data points, each represented with $p$ features, ...
2
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0answers
359 views

VC dimension for ellipsoidal classifiers

What is the VC dimension of $g: \mathbb{R}^n \times (\mathbb{R}^{n \times n} \times \mathbb{R}^n \times \mathbb{R}) \rightarrow \{-1,1\}$ defined as $$ g( x, (P_1,p_2,p_3), ) := \text{sgn} \left( x^\...
2
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0answers
751 views

Ultimate Jedi Challenge - Multiarmed Bandit / Reinforcment Learning / advanced AI problem

(This is an attempt to reformulate this question more concisely.) Context You are a Jedi master who wants to prepare a training program (online-algorithm) for his apprentice, "Luke". Luke needs to ...
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0answers
67 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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0answers
75 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 $$ \sum_{j}(V)_{ij}=1, (V)_{ij}\ge 0 $$...
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0answers
38 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 ...
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0answers
55 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 ...
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0answers
49 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 ...
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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 ...