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I do not want to sound condescending, but the math you are studying at the undergraduate and even graduate level courses is not advanced. It is the basics. The title of your question should be: Is "basic" math needed/useful in AI research? So, gobble up as much as you can, I have never met a computer scientist who complained about knowing too much math, ...
39
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., here http://www.no-free-lunch.org/ ). Sure enough, deep learning can be "broken" without much difficulty:
http://www.evolvingai.org/fooling
Hence, to be ...
27
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 problem in high dimensions. Current training algorithms are all based on gradient descent, which only guarantees convergence to a critical point (local minimum or ...
20
The two approaches provide very different guarantees.
The JL Lemma says essentially "you give me the error you want, and I'll give you a low dimensional space that captures the distances upto that error". It's also a worst-case pairwise guarantee: for each pair of points, etc etc
The SVD essentially promises "you tell me what dimension you want to live in,...
19
The question is, in my opinion, quite vague and involves some misunderstanding, so this answer attempts only to provide the right vocabulary and point you in the right direction.
There are two fields of computer science that directly study such problems. Inductive inference and computational learning theory. The two fields are very closely related and the ...
17
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 result would be the big breakthrough more so than the conclusion itself.
For Question 2 I want to share some background and a thought. Pretty much all the ...
14
You ask why database aggregations have monoidal structure.
Say we want to combine data values $a$ and $b$, but want to keep things general -- these may be integers, strings, floating point numbers, vectors, matrices, probability distributions, sets, or anything else we want to store and manipulate. So we denote the "aggregation" of $a$ and $b$ by $a.b$.
...
12
The success of a learning algorithm depends critically on the representation. How do you present the input to the algorithm? In an extreme case, suppose you present the numbers as sequences of prime factors -- in this case, learning is quite trivial. In another extreme, consider representing the numbers as binary strings. All the standard learning algorithms ...
12
BP and most of its variants are proved to converge on graphs without cycles. When you have cycles they show very strange behavior sometimes. For these cases people have tried different approximations schemes, for example Sherali-Adams, Lovasz-Schrijver, and Lasserre Hierarchies.
See [1] for a comprehensive review of these approximations. Also (Wainwright ...
11
I'll take a shot at your first question:
Are there examples of natural function families that quantum computers can learn faster than classical computers given cryptographic assumptions?
Well, it depends on the exact model and the resource being minimized. One option is to compare the sample complexity (for distribution-independent PAC learning) of the ...
11
If the learning algorithm is proper (i.e. it always produces a hypothesis from the class $F_n$), then it also gives a testing algorithm -- simply run the learning algorithm, and see whether the hypothesis it produced has error rate $<\epsilon$, which can be done with only $\approx 1/\epsilon^2$ samples. If it does, since the hypothesis is in $F_n$, this ...
11
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 finding is a linear separator in a (possibly-high dimensional) Hilbert space.
For neural networks in general, we don't have any such clear description or even an ...
11
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 $n$ states or fewer.
In general, for $x\in\Sigma^*$ and $f\in C_n$, the problem of evaluating
$f(x)$ is undecidable.
However, suppose we have access to a (...
10
There is no way to choose the parameters A, B, C, D properly;
as it is the case for most heuristics, the parameters are chosen ''by experience''.
Worse, there is no guarantee that the solution (the array of the outputs)
of this heuristic is indeed a feasible TSP tour!
In this context, perhaps interesting to read: Wilson and Pawley,
On the stability of ...
10
PAC comes in two flavors -- "information theoretic PAC" and "efficient PAC." The latter asks for computational efficiency whereas the former cares only about sample size. One usually understands which is referred to from context.
Indeed, it is not known whether (efficient) PAC learning is NP-hard in general, but results on the cryptographic hardness of ...
10
There are many open algorithmic problems. All problems below (other than the last bullet) are NP-hard, so we are interested in the best approximation ratio we can achieve in polynomial time. The following are just a sample:
Given a non-negative submodular function on a universe $U$, find a set $A$ of size at most $k$ maximizing $f(A)$. The best known ...
10
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 $O((d/\varepsilon)\log(1/\varepsilon))$ sample complexity (see Vapnik and Chervonenkis, 1974; Blumer, Ehrenfeucht, Haussler, and Warmuth, 1989).
As for (2), ...
10
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 data. Thus, if a region of space is so sparse as to be poorly represented in the training sample, its probability of appearing during the test phase is low.
You'...
9
We know something close to what you want.
If you look at Ke Yang's "Honest Statistical Queries" -- there is no noise at all, but only "sampling error". In this model, you pass in a parameter $t$, and the Oracle takes $t$ samples, honestly evaluates the passed-in function (onto {0,1}), and returns the average value of the function on the samples.
In ...
9
The approximate Caratheodory theorem goes back to the 60s, and probably way earlier than that (it follows for example from the mistake bound of the preceptron algorithm analysis). As for the dimensionality reduction, the answer is no - the supporting subsets are different subsets, and their number is too large. In particular, the number of possible subsets ...
8
What you describe is a non-stochastic version of the "functional multi-arm bandit problem": you know you have an unknown function from some class C (does not have to be randomly selected), and you have query access to this function. The goal is to find the element which maximizes the function. As you say, depending on the class C, this may or may not require ...
8
If your FSM is a DFA, then this is the Minimum Consistent DFA Problem, which is well known in the machine learning community. This problem is NP complete
Gold1978 - Complexity of Automaton Identification from Given Data.
The problem is also known to be hard to approximate within any polynomial factor. Indeed it is even hard to find an NFA whose number of ...
8
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 to have similar BibTex-ready websites: see http://proceedings.mlr.press/. For instance, going to the abstract of a randomly chosen paper from ICML'15:
ICLR ...
8
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 discussion of lower bounds or no-free-lunch results in this paper. Two is that surely statisticians had been aware of such fundamental limitations on learnability ...
7
(wow! after three years of time passing, this is now easy to answer. funny how that goes! --Daniel)
This "Learning with (Signed) Errors" (LWSE) problem, as invented-and-stated above by me (three years ago), trivially reduces from the Extended Learning with Errors (eLWE) problem first introduced in the work Bi-Deniable Public-Key Encryption by O'Neill, ...
7
For some material more recent than Kearns and Vazirani, you could check out Rocco Servedio's lecture notes for Advanced Topics in Computational Learning Theory, or the notes from Sasha Rakhlin's class.
7
Kearns and Vazirani is maybe a bit old, but good introduction.
7
The LPN problem is indeed believed to be hard, but like most problems we believe are hard, the main reason for it is that many smart people have tried to find an efficient algorithm and failed.
The best "evidence" for LPN's hardness comes from the high statistical query dimension of the parity problem. Statistical queries capture most known learning ...
7
Sorry, but there is lots of evidence that not only is this problem a pain in the univariate case, but moreover that having few dimensions actually makes things worse.
For evidence of the difficulty, please see Proposition 15 in Section 6 ("Exponential dependence on $k$ is inevitable") and the related Figure 1 of the following paper, the follow-up to Ankur ...
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