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## Testing halfspaces (2009)

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### Other Repositories/Bibliography

Venue: | IN PROC. 20TH ANNUAL SYMPOSIUM ON DISCRETE ALGORITHMS (SODA |

Citations: | 33 - 15 self |

### Citations

2909 |
An introduction to probability theory and its applications
- Feller
- 1971
(Show Context)
Citation Context ...ted in arbitrary θ, hence we prove more general versions of the theorems here. First we state the well-known Berry-Esseen theorem, a version of the Central Limit Theorem with error bounds (see, e.g., =-=[Fel68]-=-): Theorem 29. Let `(x) = c1x1 + · · · + cnxn be a linear form over the random ±1 bits xi. Let τ be such that |ci| ≤ τ for all i, and write σ = √∑ c2i . Write F for the c.d.f. of `(x)/σ; i.e., F (t) =... |

1269 | 2000. An Introduction to Support Vector Machines and other kernel-based learning methods - Cristianini, Shawe-Taylor |

665 | Perceptrons: An Introduction to Computational Geometry - Minsky, Papert - 1969 |

475 | Property testing and its connection to learning and approximation
- Goldreich, Goldwasser, et al.
- 1998
(Show Context)
Citation Context ...arbitrary f and we would like to distinguish whether f is a member of C or ǫ-far from every member of C. Though it is well known that any proper learning algorithm can be used as a testing algorithm1 =-=[GGR98]-=-, testing potentially requires fewer queries than learning (and indeed when this is the case, a query-efficient testing algorithm can be used to check whether f is close to C before bothering to run a... |

422 |
Probability in Banach spaces
- Ledoux, Talagrand
- 1991
(Show Context)
Citation Context ... functions f : {−1, 1}n → {−1, 1} and Hermite analysis of functions f : Rn → {−1, 1}. For more information on Fourier analysis see, e.g., [Šte00]; for more information on Hermite analysis see, e.g., =-=[LT91]-=-. Fourier analysis. Here we consider functions f : {−1, 1}n → R, and we think of the inputs x to f as being distributed according to the uniform probability distribution. The set of such functions for... |

356 | Self-testing/correcting with applications to numerical problems - Blum, Luby, et al. - 1993 |

286 |
Limit Theorems of Probability Theory
- Petrov
- 1995
(Show Context)
Citation Context ... A ⊆ R is any interval then Pr[`(x)/σ ∈ A] τ/σ≈ Pr[X ∈ A]. We will sometimes find it useful to quote a special case of the Berry-Essen theorem (with a sharper constant). The following can be found in =-=[Pet95]-=-: Theorem 30. In the setup of Theorem 29, for any λ ≥ τ and any θ ∈ R it holds that Pr[|`(x)− θ| ≤ λ] ≤ 6λ/σ. 11 We will use the following proposition: Proposition 31. Let f(x) = sgn(c · x− θ) be an L... |

223 | Optimal inapproximability results for Max-Cut and other 2-variable CSPs
- Khot, Kindler, et al.
(Show Context)
Citation Context ...vided by Proposition 2.2 item 3 and Theorem 2.1. Theorem 3.2. Let f ∈ LTFn,τ . Then∣∣∣∑ni=1 f̂(i)2 − 2π ∣∣∣ ≤ O(τ2/3). Proof. Let ρ > 0 be small (chosen later). Using Proposition 7.1 and Theorem 5 of =-=[KKMO07]-=-, we have ∑ S ρ|S|f̂(S)2 = 2 π arcsinρ±O(τ). On the LHS side we have that f̂(S) = 0 for all even |S| since f is an odd function, and therefore, |∑S ρ|S|f̂(S)2 − ρ∑|S|=1 f̂(S)2| ≤ ρ3∑|S|≥3 f̂(S)2 ≤ ρ3.... |

216 | Normal approximation and asymptotic expansions - Bhattacharya, Rao - 1986 |

154 | On convergence proofs on perceptrons - Novikoff - 1962 |

140 | Threshold circuits of bounded depth - Hajnal, Maass, et al. - 1993 |

103 | On ACC and threshold circuits - Yao - 1990 |

85 | The Perceptron: a model for brain functioning - Block - 1962 |

72 | Testing monotonicity - Goldreich, Goldwasser, et al. |

62 | Monotonicity testing over general poset domains - Fischer, Matsliah, et al. - 2002 |

57 | Testing juntas - Fischer, Kindler, et al. |

48 | Property Testing: A Learning Theory Perspective
- Ron
(Show Context)
Citation Context ...Yao90, Blo62, Nov62, MP68, STC00] and related references). The relationship between learning and property testing has been the subject of much recent work (see e.g. the references cited in the survey =-=[Ron07]-=-). In a typical learning setup we are given access (via queries ∗Supported by an NSF Graduate Fellowship, NSF grant 012702-001, NSF grant 0514771, NSF grant 0732334, NSF CAREER award CCF-0347282, NSF ... |

46 | Testing basic boolean formulae
- Parnas, Ron, et al.
(Show Context)
Citation Context ...ing to run a more query-intensive learning algorithm). Several classes of Boolean functions that are of interest in learning theory have recently been studied from a testing perspective. For example, =-=[PRS02]-=- show how to test dictator functions, monomials, and O(1)- term monotone DNFs with query complexity O(1ǫ ), and [FKR+02] show how to test k-juntas with query complexity poly(k, 1ǫ ). Most recently, [D... |

44 | Testing low-degree polynomials over GF(2 - Alon, Kaufman, et al. - 2003 |

35 |
On the characterization of threshold functions
- Chow
(Show Context)
Citation Context ...ish new structural results about LTFs which essentially characterize LTFs in terms of their degree-0 and degree1 Fourier coefficients. For functions mapping {−1, 1}n to {−1, 1} it has long been known =-=[Cho61]-=- that any linear threshold function f is completely specified by the n+1 parameters consisting of its degree-0 and degree-1 Fourier coefficients (also referred to as its Chow parameters). While this s... |

35 | Testing for concise representations - Diakonikolas, Lee, et al. - 2007 |

35 | Every linear threshold function has a low-weight approximator
- Servedio
(Show Context)
Citation Context ...puts an estimate of E[f ] (i.e. the fraction of inputs that satisfy f) to within an additive ±ǫ. This is arguably the simplest explicit open derandomization problem of which we are aware. The work of =-=[Ser06]-=- gives a deterministic algorithm which runs in time polynomial in n but exponential in 1/ǫ. Can the insights into LTFs from the current paper be used to obtain a truly polynomial algorithm? References... |

33 |
How much are increasing sets positively correlated
- Talagrand
- 1996
(Show Context)
Citation Context ...≤ τ1/6. (22) (We assume in this theorem that τ is less than a sufficiently small constant.) Proof. We first dispense with the case that |E[f1]| ≥ 1 − τ1/10. In this case, Proposition 2.2 of Talagrand =-=[Tal96]-=- implies that ∑n i=1 f̂1(i) 2 ≤ O(τ2/10 log(1/τ)), and Proposition 24 (item 3) implies that W (E[f1]) ≤ O(τ2/10 log(1/τ)). Thus∣∣∣∣∣ n∑ i=1 f̂1(i)2 −W (E[f1]) ∣∣∣∣∣ ≤ O(τ1/5 log(1/τ)) ≤ τ1/6, so (21) ... |

32 | Active learning using arbitrary binary valued queries
- Kulkarni, Mitter, et al.
- 1993
(Show Context)
Citation Context ...o testing, any learning algorithm — even one with black-box query access to f — must make at least Ω(nǫ ) queries to learn an unknown LTF to accuracy ǫ (this follows easily from, e.g., the results of =-=[KMT93]-=-). Thus the query complexity of learning is linear in n, while the query complexity of our testing algorithm is independent of n. Note that the assumption that our testing algorithm has query access t... |

21 | On restricted-focus-of-attention learnability of Boolean functions - Birkendorf, Dichterman, et al. - 1998 |

14 |
Theory of majority switching elements
- Muroga, Toda, et al.
- 1961
(Show Context)
Citation Context ...ger vector with entries at most 2O(s log s) in absolute value and θ∗ is also an integer in this range. (Since |J | ≤ s, any LTF on J can be expressed thus by the well-known result of Muroga et al. 36 =-=[MTT61]-=-, which says that any LTF over s Boolean variables may be expressed using weights that are all integers of absolute value at most 2O(s log s).) Since f is 2-close to g, we know that for at least a 1 ... |

13 | A Bound on the Precision Required to Estimate a Boolean Perceptron from its Average Satisfying Assignment - Goldberg |

12 | Fourier Transform in Computer Science - Stefankovic - 2000 |

5 | Fourier transform in computer science - ˇStefankovič - 2000 |

5 | Decision trees: More theoretical justification for practical algorithms
- Fiat, Pechyony
- 2004
(Show Context)
Citation Context ...s intuition is in fact correct, however proving the former statement turns out to be much easier than the latter. To start, we state the following very simple fact (an explicit proof appears in, e.g.,=-=[FP04]-=-). Fact 37. Let f = sgn(w1x1 + · · · + wnxn − θ) be an LTF such that |w1| ≥ |wi| for all i ∈ [n]. Then |Inf1(f)| ≥ |Infi(f)| for all i ∈ [n]. Using this fact together with the Berry-Esseen theorem we ... |

3 | Estimating a Boolean perceptron from its average satisfying assignment: A bound on the precision required - Goldberg - 2001 |

2 | Testing for concise representations. Submitted for publication - Diakonikolas, Lee, et al. - 2007 |

1 |
The Chow Parameters Problem. To appear
- O’Donnell, Servedio
- 2008
(Show Context)
Citation Context ...rent techniques are required. 1 Fourier coefficients. These results have already proved useful in other work; indeed, structural results from this paper play a crucial role in the recent algorithm of =-=[OS08]-=- which efficiently learns any LTF given only its degree-0 and degree-1 Fourier coefficients. As we describe in the conclusion, we are hopeful that our techniques will help resolve other open questions... |

1 | Testing low-degree polynomials over GF(2), booktitle - Alon, Kaufman, et al. |

1 | Monotonicity testing over general poset domains - Fiat, Pechyony - 2002 |