How Limited Interaction Hinders Real Communication (and What It Means for Proof and Circuit Complexity)

Rezende, Susanna F. de; Nordström, Jakob; Vinyals, Marc

doi:10.1109/focs.2016.40

Cited by 20 publications

(12 citation statements)

References 37 publications

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“…Communication model References deterministic deterministic [28,14,10,17] nondeterministic nondeterministic [13,11] polynomial degree rank [35,34,29,31] conical junta degree nonnegative rank [13,22] Sherali-Adams LP extension complexity [9,22] sum-of-squares SDP extension complexity [24] The lower bound is tight up to the square root, since (1) can be adapted for composed functions to yield…”

Section: Query Modelmentioning

confidence: 99%

Query-to-Communication Lifting for PNP

et al. 2018

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We prove that the P NP -type query complexity (alternatively, decision list width) of any boolean function f is quadratically related to the P NP -type communication complexity of a lifted version of f . As an application, we show that a certain "product" lower bound method of Impagliazzo and Williams (CCC 2010) fails to capture P NP communication complexity up to polynomial factors, which answers a question of Papakonstantinou, Scheder, and Song (CCC 2014).

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Section: Query Modelmentioning

confidence: 99%

Query-to-Communication Lifting for PNP

et al. 2018

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“…We now utilize the real communication model introduced by Kraíček [1998] (see also de Rezende et al [2016]) in order to establish linear size lower bounds for neural networks. Consider a Boolean function f : {0, 1} d → {0, 1} whose input is split between two players, Alice and Bob.…”

Section: ω(D) Lower Bound For Exact Computationmentioning

confidence: 99%

Size and Depth Separation in Approximating Benign Functions with Neural Networks

Vardi,

Reichman,

Pitassi

et al. 2021

Preprint

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When studying the expressive power of neural networks, a main challenge is to understand how the size and depth of the network affect its ability to approximate real functions. However, not all functions are interesting from a practical viewpoint: functions of interest usually have a polynomially-bounded Lipschitz constant, and can be computed efficiently. We call functions that satisfy these conditions "natural", and explore the benefits of size and depth for approximation of natural functions with ReLU networks. As we show, this problem is more challenging than the corresponding problem for non-natural functions. We give complexity-theoretic barriers to showing depth-lower-bounds: Proving existence of a natural function that cannot be approximated by polynomial-sized networks of depth 4 would settle longstanding open problems in computational complexity. It implies that beyond depth 4 there is a barrier to showing depth-separation for natural functions, even between networks of constant depth and networks of nonconstant depth. We also study size-separation, namely, whether there are natural functions that can be approximated with networks of size O(s(d)), but not with networks of size O(s ′ (d)). We show a complexity-theoretic barrier to proving such results beyond size O(d log 2 (d)), but also show an explicit natural function, that can be approximated with networks of size O(d) and not with networks of size o(d/ log d). For approximation in the L ∞ sense we achieve such separation already between size O(d) and size o(d). Moreover, we show superpolynomial size lower bounds and barriers to such lower bounds, depending on the assumptions on the function. Our size-separation results rely on an analysis of size lower bounds for Boolean functions, which is of independent interest: We show linear size lower bounds for computing explicit Boolean functions with neural networks and threshold circuits.

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“…A theorem of this type was conjectured by Beame, Huynh, and Pitassi [5, §6]. We also note that lifting theory for deterministic tree-like protocols-with applications to monotone formula size, tree-like refutation size, and size-space tradeoffs-has been developed in quite some detail [42,28,20,21,13,56,11]. We import techniques from this line of work into the DAG-like setting.…”

Section: Appetizermentioning

confidence: 99%

“…THEORY OF COMPUTING, Volume 16 (13), 2020, pp. 1-30 mKW search problem S f = input: a pair (x, y) ∈ f −1 (1) × f −1 (0) output: a coordinate i ∈ [n] such that x i > y i CNF search problem S F = input: an n-variable truth assignment z ∈ {0, 1} n output: clause D of F unsatisfied by z, i. e., D(z) = 0…”

Section: Dag-like Modelsmentioning

confidence: 99%