42
votes
What kind of answer does TCS want to the question "Why do neural networks work so well?"
There are a bunch of "no free lunch" theorems in machine learning, roughly stating that there can be no one master learning algorithm that performs uniformly better than all other algorithms (see, e.g....
- 10.6k
27
votes
What kind of answer does TCS want to the question "Why do neural networks work so well?"
There are two main gaps in our understanding of neural networks: optimization hardness and generalization performance.
Training a neural network requires solving a highly non-convex optimization ...
22
votes
Accepted
Is BPP vs. P a real problem after we know BPP lies in P/poly?
Not sure how much of an answer this is, I'm just indulging in some rumination.
Question 1 could be equally asked about P $\neq$ NP and with a similar answer -- the techniques/ideas used to prove the ...
- 7,768
15
votes
Accepted
Circuit and Formula Lower Bounds for Separating Sparse Sets of Strings
Yes, any such pair can be separated by a formula of size $O(n)$. More generally, any disjoint pair $P,N\subseteq\{0,1\}^n$ of size $s=|P|+|N|$ can be separated by a decision tree of size $O(s)$, which ...
- 18.2k
12
votes
Functions that are Not Efficiently Computable but Learnable
I will formalize a variant of this question where "efficiency" is replaced by "computability".
Let $C_n$ be the concept class of all languages $L\subseteq\Sigma^*$
recognizable by Turing machines on $...
- 10.6k
11
votes
What kind of answer does TCS want to the question "Why do neural networks work so well?"
Another take on this question, to add to @Aryeh's remarks: For many other models of learning, we know the "shape" of the hypothesis space. SVMs are the best example of this, in that what you're ...
- 32.1k
11
votes
Accepted
Proper PAC learning VC dimension bounds
My thanks to Aryeh for bringing this question to my attention.
As others have mentioned, the answer to (1) is Yes, and the
simple method of Empirical Risk Minimization in $\mathcal{C}$ achieves
the ...
- 126
10
votes
Accepted
Difficulty of "learning" rare instances
In the classic PAC learning (i.e., classification) model, rare instances are not a problem. This is because the learner's test points are assumed to come from the same distribution as the training ...
- 10.6k
9
votes
What is the best place to get BibTeX entries for ICLR and other machine learning papers?
For NeurIPS (previously NIPS), the NeurIPS website itself is a good source: look for a paper, go to its page, and click on the "BibTex" link. Here is a random sample (!):
ICML, COLT, and JMLR appear ...
- 4,481
8
votes
Accepted
What was the significance of Leslie Valiant's, "A Theory of the Learnable?"
I think that "there is a natural limit to what computers can learn", while definitely true, is not one of the main takeaways of Valiant's paper, for two reasons. One is that I could not find any ...
- 10.6k
7
votes
Accepted
Complexity of finding a consistent hyperplane
Second version, hopefully correct.
I claim that solving the feasibility problem $\exists? x: Ax \le b$ reduces in strongly polynomial time to finding a linear separator. Then it's easy to reduce ...
- 18.2k
7
votes
Accepted
What is the best place to get BibTeX entries for ICLR and other machine learning papers?
As an update, I noticed that DBLP has added ICLR to tracking as of today. Now it has ICLR papers and their bibtex available at https://dblp.uni-trier.de/db/conf/iclr/
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6
votes
If machine learning techniques keep improving, what's the role of algorithmics in the future?
This is a question that has been haunting me recently, so I am glad you asked it.
However, I am less interested in classifying the application areas for which machine learning will dominate the ...
- 342
6
votes
Accepted
Learning a coin's bias (localized)
Write $p=p_0=1-q$. We may assume that $\epsilon<\eta \le p\le 1/2$. Then the sample complexity is of order $\log(1/\delta)$ times the reciprocal of the relative entropy $D((p,q)||(p+\epsilon,q-\...
- 451
6
votes
Proper PAC learning VC dimension bounds
Your questions (1) and (2) are related. First, let's talk about proper PAC learning. It is known that there are proper PAC learners that achieve zero sample error, and yet require $\Omega(\frac{d}{\...
- 10.6k
6
votes
Accepted
Latest word on cross validation?
It is not hard to see that without additional stability assumptions one won't be able to get high probability bounds. For example consider predicting unbiased coin using majority label in the sample. ...
- 881
6
votes
Accepted
Reference Request: Computational Learning Theory
Another good introductory book is "Foundations of Machine Learning" by Mohri et al.: https://www.amazon.com/Foundations-Machine-Learning-Mehryar-Mohri/dp/0262039400/. It has a large overlap with the ...
- 12k
6
votes
Accepted
What's the intuition behind Rademacher complexity?
The standard "intuition" is that the Rademacher complexity quantifies the ability of the function class $F$ to fit symmetric random noise: a low value (close to 0) means that this ability is ...
- 10.6k
5
votes
Accepted
Adversarial Machine Learning, Learning with (Malicious) noise
In the proof of the positive result in [2] you are referring to, namely Theorem 2, the argument goes as follows. For every possible concept $L_i$ of the hypothesis class $\mathcal{H} = \{L_1,\dots, ...
- 4,481
5
votes
Textbook/resources for a beginning researcher in (Machine) Learning Theory
People are going to recommend
http://www.cs.nyu.edu/~mohri/mlbook/
and
http://www.cs.huji.ac.il/~shais/UnderstandingMachineLearning/understanding-machine-learning-theory-algorithms.pdf
-- so I ...
- 10.6k
5
votes
Learning a coin's bias (localized)
Yuval Peres gave the answer in terms of the Kullback-Leibler divergence. Another way is to recall that the sample complexity will be captured by the inverse of the squared Hellinger distance between ...
- 4,481
5
votes
Are there distribution properties which are "maximally" hard to test?
Sorry for unearthing this post -- it is quite old, but I figured having it answered may not be that bad an idea.
First, it looks like you performed your Chernoff bound with some slightly odd setting ...
- 4,481
5
votes
Accepted
Understanding the No Free Lunch Theorem
You're asking about optimization and universal search, BUT machine-learning is tagged and you're wondering about "a uniform distribution on an infinite" discrete set so perhaps this will be helpful. ...
- 10.6k
5
votes
Proper PAC learning VC dimension bounds
To add to the currently accepted answer:
Yes. The $$O\left(\frac{d}{\varepsilon}\log\frac{1}{\varepsilon}\right)$$
sample complexity upper bound holds for proper PAC learning as well (although it is ...
- 4,481
5
votes
Accepted
Oncina-Garcia RPNI algorithm for learning DFAs
The algorithm is named RPNI, not RNPI.
Given that the language generating the inputs is regular and that enough examples are given (the characteristic set), the algorithm returns the canonical (i.e., ...
5
votes
What are some good resources for strengthening my theoretical foundation for machine learning?
I suggest
Probability and Computing: Randomized Algorithms and Probabilistic Analysis by Mitzenmacher and Upfal. Probability is at the foundation of machine learning and it's one of the weakest ...
- 10.6k
5
votes
Accepted
Complexity of constructing minimum depth decision trees
I think I can see a fairly easy reduction from 3DM. Let $B=\{0^J\}$, i.e., it is a singleton set with the only zero element. The points of $A$ correspond to the points of the 3DM that are to be ...
- 14.1k
5
votes
Accepted
Is there an equivalent to VC-dimension for density estimation as opposed to classification?
For distributions with finite support of size $d$, when the error metric is the $\ell_1$ distance, the analogue of VC dimension is exactly $d$. (In fact, it's pretty much the VC dimension -- since to ...
- 10.6k
4
votes
Accepted
Learning from derivative data
If your function is $f:\mathbb{R}\to\mathbb{R}$, you can "learn" $f'$ as a standard regression problem (linear, polynomial, etc.) and then recover $f$ up to an additive constant by integrating $f'$. ...
- 10.6k
4
votes
What kind of answer does TCS want to the question "Why do neural networks work so well?"
The principle of Information Bottleneck has been proposed to explain the success of deep nueral networks.
Here is a quote from Quanta magazine
Last month, a YouTube video of a conference talk in ...
- 21.1k
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