7 research outputs found
Towards Stronger Counterexamples to the Log-Approximate-Rank Conjecture
We give improved separations for the query complexity analogue of the
log-approximate-rank conjecture i.e. we show that there are a plethora of total
Boolean functions on input bits, each of which has approximate Fourier
sparsity at most and randomized parity decision tree complexity
. This improves upon the recent work of Chattopadhyay, Mande and
Sherif (JACM '20) both qualitatively (in terms of designing a large number of
examples) and quantitatively (improving the gap from quartic to cubic). We
leave open the problem of proving a randomized communication complexity lower
bound for XOR compositions of our examples. A linear lower bound would lead to
new and improved refutations of the log-approximate-rank conjecture. Moreover,
if any of these compositions had even a sub-linear cost randomized
communication protocol, it would demonstrate that randomized parity decision
tree complexity does not lift to randomized communication complexity in general
(with the XOR gadget)
No Quantum Speedup over Gradient Descent for Non-Smooth Convex Optimization
We study the first-order convex optimization problem, where we have black-box
access to a (not necessarily smooth) function
and its (sub)gradient. Our goal is to find an -approximate minimum of
starting from a point that is distance at most from the true minimum.
If is -Lipschitz, then the classic gradient descent algorithm solves
this problem with queries. Importantly, the number of
queries is independent of the dimension and gradient descent is optimal in
this regard: No deterministic or randomized algorithm can achieve better
complexity that is still independent of the dimension .
In this paper we reprove the randomized lower bound of
using a simpler argument than previous lower
bounds. We then show that although the function family used in the lower bound
is hard for randomized algorithms, it can be solved using
quantum queries. We then show an improved lower bound against quantum
algorithms using a different set of instances and establish our main result
that in general even quantum algorithms need queries
to solve the problem. Hence there is no quantum speedup over gradient descent
for black-box first-order convex optimization without further assumptions on
the function family.Comment: 25 page
Lifting to Parity Decision Trees via Stifling
We show that the deterministic decision tree complexity of a (partial) function or relation f lifts to the deterministic parity decision tree (PDT) size complexity of the composed function/relation f ◦ g as long as the gadget g satisfies a property that we call stifling. We observe that several simple gadgets of constant size, like Indexing on 3 input bits, Inner Product on 4 input bits, Majority on 3 input bits and random functions, satisfy this property. It can be shown that existing randomized communication lifting theorems ([Göös, Pitassi, Watson. SICOMP'20], [Chattopadhyay et al. SICOMP'21]) imply PDT-size lifting. However there are two shortcomings of this approach: first they lift randomized decision tree complexity of f, which could be exponentially smaller than its deterministic counterpart when either f is a partial function or even a total search problem. Second, the size of the gadgets in such lifting theorems are as large as logarithmic in the size of the input to f. Reducing the gadget size to a constant is an important open problem at the frontier of current research. Our result shows that even a random constant-size gadget does enable lifting to PDT size. Further, it also yields the first systematic way of turning lower bounds on the width of tree-like resolution proofs of the unsatisfiability of constant-width CNF formulas to lower bounds on the size of tree-like proofs in the resolution with parity system, i.e., Res(☉), of the unsatisfiability of closely related constant-width CNF formulas
One-way communication complexity and non-adaptive decision trees
We study the relationship between various one-way communication complexity measures of a composed function with the analogous decision tree complexity of the outer function. We consider two gadgets: the AND function on 2 inputs, and the Inner Product on a constant number of inputs. More generally, we show the following when the gadget is Inner Product on 2b input bits for all b ≥ 2, denoted IP. If f is a total Boolean function that depends on all of its n input bits, then the bounded-error one-way quantum communication complexity of f ◦ IP equals Ω(n(b - 1)). If f is a partial Boolean function, then the deterministic one-way communication complexity of f ◦ IP is at least Ω(b · D→dt (f)), where D→dt (f) denotes non-adaptive decision tree complexity of f. To prove our quantum lower bound, we first show a lower bound on the VC-dimension of f ◦ IP. We then appeal to a result of Klauck [STOC’00], which immediately yields our quantum lower bound. Our deterministic lower bound relies on a combinatorial result independently proven by Ahlswede and Khachatrian [Adv. Appl. Math.’98], and Frankl and Tokushige [Comb.’99]. It is known due to a result of Montanaro and Osborne [arXiv’09] that the deterministic one-way communication complexity of f ◦ XOR equals the non-adaptive parity decision tree complexity of f. In contrast, we show the following when the inner gadget is the AND function on 2 input bits. There exists a function for which even the quantum non-adaptive AND decision tree complexity of f is exponentially large in the deterministic one-way communication complexity of f ◦ AND. However, for symmetric functions f, the non-adaptive AND decision tree complexity of f is at most quadratic in the (even two-way) communication complexity of f ◦ AND. In view of the first bullet, a lower bound on non-adaptive AND decision tree complexity of f does not lift to a lower bound on one-way communication complexity of f ◦ AND. The proof of the first bullet above uses the well-studied Odd-Max-Bit function. For the second bullet, we first observe a connection between the one-way communication complexity of f and the Möbius sparsity of f, and then give a lower bound on the Möbius sparsity of symmetric functions. An upper bound on the non-adaptive AND decision tree complexity of symmetric functions follows implicitly from prior work on combinatorial group testing; for the sake of completeness, we include a proof of this result. It is well known that the rank of the communication matrix of a function F is an upper bound on its deterministic one-way communication complexity. This bound is known to be tight for some F. However, in our final result we show that this is not the case when F = f ◦ AND. More precisely we show that for all f, the deterministic one-way communication complexity of F = f ◦ AND is at most (rank(MF))(1 - Ω(1)), where MF denotes the communication matrix of F
Lifting to parity decision trees via Stifling
We show that the deterministic decision tree complexity of a (partial) function or relation f lifts to the deterministic parity decision tree (PDT) size complexity of the composed function/relation f ◦ g as long as the gadget g satisfies a property that we call stifling. We observe that several simple gadgets of constant size, like Indexing on 3 input bits, Inner Product on 4 input bits, Majority on 3 input bits and random functions, satisfy this property. It can be shown that existing randomized communication lifting theorems ([Göös, Pitassi, Watson. SICOMP'20], [Chattopadhyay et al. SICOMP'21]) imply PDT-size lifting. However there are two shortcomings of this approach: first they lift randomized decision tree complexity of f, which could be exponentially smaller than its deterministic counterpart when either f is a partial function or even a total search problem. Second, the size of the gadgets in such lifting theorems are as large as logarithmic in the size of the input to f. Reducing the gadget size to a constant is an important open problem at the frontier of current research. Our result shows that even a random constant-size gadget does enable lifting to PDT size. Further, it also yields the first systematic way of turning lower bounds on the width of tree-like resolution proofs of the unsatisfiability of constant-width CNF formulas to lower bounds on the size of tree-like proofs in the resolution with parity system, i.e., Res(☉), of the unsatisfiability of closely related constant-width CNF formulas
An improved protocol for ExactlyN with more than 3 players
The ExactlyN problem in the number-on-forehead (NOF) communication setting
asks players, each of whom can see every input but their own, if the
input numbers add up to . Introduced by Chandra, Furst and Lipton in 1983,
ExactlyN is important for its role in understanding the strength of randomness
in communication complexity with many players. It is also tightly connected to
the field of combinatorics: its -party NOF communication complexity is
related to the size of the largest corner-free subset in .
In 2021, Linial and Shraibman gave more efficient protocols for ExactlyN for
3 players. As an immediate consequence, this also gave a new construction of
larger corner-free subsets in . Later that year Green gave a further
refinement to their argument. These results represent the first improvements to
the highest-order term for since the famous work of Behrend in 1946. In
this paper we give a corresponding improvement to the highest-order term for
all , the first since Rankin in 1961. That is, we give a more efficient
protocol for ExactlyN as well as larger corner-free sets in higher dimensions.
Nearly all previous results in this line of research approached the problem
from the combinatorics perspective, implicitly resulting in non-constructive
protocols for ExactlyN. Approaching the problem from the communication
complexity point of view and constructing explicit protocols for ExactlyN was
key to the improvements in the setting. As a further contribution we
provide explicit protocols for ExactlyN for any number of players which serves
as a base for our improvement