30,578 research outputs found
Improper Learning by Refuting
The sample complexity of learning a Boolean-valued function class is precisely characterized by its Rademacher complexity. This has little bearing, however, on the sample complexity of efficient agnostic learning.
We introduce refutation complexity, a natural computational analog of Rademacher complexity of a Boolean concept class and show that it exactly characterizes the sample complexity of efficient agnostic learning. Informally, refutation complexity of a class C is the minimum number of example-label pairs required to efficiently distinguish between the case that the labels correlate with the evaluation of some member of C (structure) and the case where the labels are i.i.d. Rademacher random variables (noise). The easy direction of this relationship was implicitly used in the recent framework for improper PAC learning lower bounds of Daniely and co-authors via connections to the hardness of refuting random constraint satisfaction problems. Our work can be seen as making the relationship between agnostic learning and refutation implicit in their work into an explicit equivalence.
In a recent, independent work, Salil Vadhan discovered a similar relationship between refutation and PAC-learning in the realizable (i.e. noiseless) case
More data speeds up training time in learning halfspaces over sparse vectors
The increased availability of data in recent years has led several authors to
ask whether it is possible to use data as a {\em computational} resource. That
is, if more data is available, beyond the sample complexity limit, is it
possible to use the extra examples to speed up the computation time required to
perform the learning task?
We give the first positive answer to this question for a {\em natural
supervised learning problem} --- we consider agnostic PAC learning of
halfspaces over -sparse vectors in . This class is
inefficiently learnable using examples. Our main
contribution is a novel, non-cryptographic, methodology for establishing
computational-statistical gaps, which allows us to show that, under a widely
believed assumption that refuting random formulas is hard, it
is impossible to efficiently learn this class using only
examples. We further show that under stronger
hardness assumptions, even examples do not
suffice. On the other hand, we show a new algorithm that learns this class
efficiently using examples. This
formally establishes the tradeoff between sample and computational complexity
for a natural supervised learning problem.Comment: 13 page
Sum of squares lower bounds for refuting any CSP
Let be a nontrivial -ary predicate. Consider a
random instance of the constraint satisfaction problem on
variables with constraints, each being applied to randomly
chosen literals. Provided the constraint density satisfies , such
an instance is unsatisfiable with high probability. The \emph{refutation}
problem is to efficiently find a proof of unsatisfiability.
We show that whenever the predicate supports a -\emph{wise uniform}
probability distribution on its satisfying assignments, the sum of squares
(SOS) algorithm of degree
(which runs in time ) \emph{cannot} refute a random instance of
. In particular, the polynomial-time SOS algorithm requires
constraints to refute random instances of
CSP when supports a -wise uniform distribution on its satisfying
assignments. Together with recent work of Lee et al. [LRS15], our result also
implies that \emph{any} polynomial-size semidefinite programming relaxation for
refutation requires at least constraints.
Our results (which also extend with no change to CSPs over larger alphabets)
subsume all previously known lower bounds for semialgebraic refutation of
random CSPs. For every constraint predicate~, they give a three-way hardness
tradeoff between the density of constraints, the SOS degree (hence running
time), and the strength of the refutation. By recent algorithmic results of
Allen et al. [AOW15] and Raghavendra et al. [RRS16], this full three-way
tradeoff is \emph{tight}, up to lower-order factors.Comment: 39 pages, 1 figur
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On Learning vs. Refutation
Building on the work of Daniely et al. (STOC 2014, COLT 2016), we study the connection between computationally efficient PAC learning and refutation of constraint satisfaction problems. Specifically, we prove that for every concept class P, PAC-learning P is polynomially equivalent to ârandom-right-hand-side-refutingâ (âRRHS-refutingâ) a dual class P â , where RRHS-refutation of a class Q refers to refuting systems of equations where the constraints are (worst-case) functions from the class Q but the right-hand-sides of the equations are uniform and independent random bits. The reduction from refutation to PAC learning can be viewed as an abstraction of (part of) the work of Daniely, Linial, and Shalev-Schwartz (STOC 2014). The converse, however, is new, and is based on a combination of techniques from pseudorandomness (Yao â82) with boosting (Schapire â90). In addition, we show that PAC-learning the class of DNF formulas is polynomially equivalent to PAC-learning its dual class DNFâ , and thus PAC-learning DNF is equivalent to RRHS-refutation of DNF, suggesting an avenue to obtain stronger lower bounds for PAC-learning DNF than the quasipolynomial lower bound that was obtained by Daniely and Shalev-Schwartz (COLT 2016) assuming the hardness of refuting k-SAT.Engineering and Applied Science
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