Questions tagged [lg.learning]

Machine learning and learning theory: PAC learning, algorithmic learning theory, and computational aspects of Bayesian inference and graphical models.

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Minimax computation for classification problems with smooth densities functions

Fix $d=1$, $r \in (0,\infty)$ and a neigborhood $\Omega$ of $0$ in $\mathbb R^d$ and let and let $W^{1,\infty}(r)$ be the Sobolev ball continuously differentiable functions $f:\mathbb R^d \to \mathbb ...
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How to read a COLT or other paper related learning theory?

I am a master student right now. And first time met the theoretical computer science, I am really interested in it, and especially the learning theory part. Wish to do research about this part in the ...
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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,\...
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2 answers
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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 ...
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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 ...
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1 answer
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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 ...
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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 ...
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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⁡(...
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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 ...
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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 ...
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understanding generalized coupon collector for distributions or learning mixture of distribution

Lets suppose we have a set $S=\{1,\ldots,n\}$ and $P$ is the uniform distribution over two subsets $T_1,T_2\subseteq S$, each of size $m\leq n/100$. Now, suppose somehow is given uniform samples from ...
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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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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 ...
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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 ...
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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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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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About estimating escape time of gradient Langevin dynamics

I am trying to understand the argument in the proof of Lemmma 6.3 (page 18) of this paper https://arxiv.org/abs/1902.08179. Let me summarize the conceptual crux of the argument here using a slightly ...
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1 answer
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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 ...
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1 answer
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Reconstruction of a sequence generated by a Markov chain - reference request

Let S be a finite sequence of symbols from a finite alphabet, with gaps - that is on some known locations an unknown number of symbols are missing. Assuming that the sequence , including the symbols ...
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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 ...
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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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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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3 votes
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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 ...
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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 ...
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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 ...
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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 ...
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1 answer
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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 ) } \|...
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2 answers
482 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 $G$ ...
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1 answer
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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 the ...
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2 votes
1 answer
330 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+...
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1 vote
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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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1 answer
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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....
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4 votes
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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 ...
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7 votes
2 answers
280 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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4 votes
1 answer
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What is the current state of the art in black-box grammar induction?

Grammar induction of Context Free Languages seems to be a very well researched field. I would like to know the current state of the art in inducing a Context Free Grammar (I am reading up Higuera's ...
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2 votes
1 answer
120 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 ...
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2 votes
0 answers
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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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1 vote
0 answers
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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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3 votes
2 answers
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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 ...
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2 votes
1 answer
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On-policy/Off-policy Offline/Online Evaluation: Which would be an example of Online Off-Policy Evaluation?

In the context of the following question: off-policy and offline policy reinforcement learning , it can be concluded that off-policy/on-policy learning can be orthogonal to an online/offline sampling ...
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5 votes
1 answer
162 views

Is Bayes optimal RL of a finite set of DFAs feasible?

Let $Q$ be a finite set of states, $\Sigma$ a finite alphabet, $q_0\in Q$ the start state and $F\subseteq Q$ the set of accepting sets. Let $\{\delta_k:Q\times\Sigma\rightarrow Q\}_{k=1}^n$ be a set ...
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3 votes
1 answer
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Rademacher complexity for piecewise-linear convex function

Consider a function family $$\ell(x)=\max_{1\leq k\leq K} a_k^\top x + b_k,$$ where $a_k,b_k \in \mathbb{R}^d$ are bounded in the sense of some norm and $K\geq 2$. What is the best upper bound on the ...
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8 votes
1 answer
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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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4 votes
1 answer
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"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 ...
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1 vote
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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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1 vote
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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) ...
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5 votes
2 answers
508 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 ...
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3 votes
1 answer
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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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5 votes
1 answer
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Tight VC bound for agnostic learning

The following result is supposedly known. However, the proofs I am able to find all prove a weaker result with an extra log factor. Where can I find the proof of the tight bound? Theorem. Let $\...
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