51 research outputs found
New Sampling Lower Bounds via the Separator
Suppose that a target distribution can be approximately sampled by a low-depth decision tree, or more generally by an efficient cell-probe algorithm. It is shown to be possible to restrict the input to the sampler so that its output distribution is still not too far from the target distribution, and at the same time many output coordinates are almost pairwise independent.
This new tool is then used to obtain several new sampling lower bounds and separations, including a separation between AC0 and low-depth decision trees, and a hierarchy theorem for sampling. It is also used to obtain a new proof of the Patrascu-Viola data-structure lower bound for Rank, thereby unifying sampling and data-structure lower bounds
Algorithms and Lower Bounds in Circuit Complexity
Computational complexity theory aims to understand what problems can be efficiently solved by computation. This thesis studies computational complexity in the model of Boolean circuits. Boolean circuits provide a basic mathematical model for computation and play a central role in complexity theory, with important applications in separations of complexity classes, algorithm design, and pseudorandom constructions. In this thesis, we investigate various types of circuit models such as threshold circuits, Boolean formulas, and their extensions, focusing on obtaining complexity-theoretic lower bounds and algorithmic upper bounds for these circuits. (1) Algorithms and lower bounds for generalized threshold circuits: We extend the study of linear threshold circuits, circuits with gates computing linear threshold functions, to the more powerful model of polynomial threshold circuits where the gates can compute polynomial threshold functions. We obtain hardness and meta-algorithmic results for this circuit model, including strong average-case lower bounds, satisfiability algorithms, and derandomization algorithms for constant-depth polynomial threshold circuits with super-linear wire complexity. (2) Algorithms and lower bounds for enhanced formulas: We investigate the model of Boolean formulas whose leaf gates can compute complex functions. In particular, we study De Morgan formulas whose leaf gates are functions with "low communication complexity". Such gates can capture a broad class of functions including symmetric functions and polynomial threshold functions. We obtain new and improved results in terms of lower bounds and meta-algorithms (satisfiability, derandomization, and learning) for such enhanced formulas. (3) Circuit lower bounds for MCSP: We study circuit lower bounds for the Minimum Circuit Size Problem (MCSP), the fundamental problem of deciding whether a given function (in the form of a truth table) can be computed by small circuits. We get new and improved lower bounds for MCSP that nearly match the best-known lower bounds against several well-studied circuit models such as Boolean formulas and constant-depth circuits
Dynamical subset sampling of quantum error correcting protocols
Quantum error correcting (QEC) stabilizer codes enable protection of quantum
information against errors during storage and processing. Simulation of noisy
QEC codes is used to identify the noise parameters necessary for advantageous
operation of logical qubits in realistic quantum computing architectures.
Typical quantum error correction techniques contain intermediate measurements
and classical feedback that determine the actual noisy circuit sequence in an
instance of performing the protocol. Dynamical subset sampling enables
efficient simulation of such non-deterministic quantum error correcting
protocols for any type of quantum circuit and incoherent noise of low strength.
As an importance sampling technique, dynamical subset sampling allows one to
effectively make use of computational resources to only sample the most
relevant sequences of quantum circuits in order to estimate a protocol's
logical failure rate with well-defined error bars. We demonstrate the
capabilities of dynamical subset sampling with examples from fault-tolerant
(FT) QEC. We show that, in a typical stabilizer simulation with incoherent
Pauli noise of strength , our method can reach a required sampling
accuracy on the logical failure rate with two orders of magnitude fewer samples
than direct Monte Carlo simulation. Furthermore, dynamical subset sampling
naturally allows for efficient simulation of realistic multi-parameter noise
models describing faulty quantum processors. It can be applied not only for QEC
in the circuit model but any noisy quantum computing framework with incoherent
fault operators including measurement-based quantum computation and quantum
networks.Comment: 33 pages, 26 figure
Barriers to Black-Box Constructions of Traitor Tracing Systems
Reducibility between different cryptographic primitives is a fundamental problem in modern cryptography. As one of the primitives, traitor tracing systems help content distributors recover the identities of users that collaborated in the pirate construction by tracing pirate decryption boxes. We present the first negative result on designing efficient traitor tracing systems via black-box constructions from symmetric cryptographic primitives, e.g. one-way functions. More specifically, we show that there is no secure traitor tracing scheme in the random oracle model, such that , where is the length of user key, is the length of ciphertext and is the number of users, under the assumption that the scheme does not access the oracle to generate user keys. To our best knowledge, almost all the practical (non-artificial) cryptographic schemes (not limited to traitor tracing systems) via black-box constructions satisfy this assumption. Thus, our negative results indicate that most of the standard black-box reductions in cryptography cannot help construct a more efficient traitor tracing system.
We prove our results by extending the connection between traitor tracing systems and differentially private database sanitizers to the setting with random oracle access. After that, we prove the lower bound for traitor tracing schemes by constructing a differentially private sanitizer that only queries the random oracle polynomially many times. In order to reduce the query complexity of the sanitizer, we prove a large deviation bound for decision forests, which might be of independent interest
Simplicity Bias in Transformers and their Ability to Learn Sparse Boolean Functions
Despite the widespread success of Transformers on NLP tasks, recent works
have found that they struggle to model several formal languages when compared
to recurrent models. This raises the question of why Transformers perform well
in practice and whether they have any properties that enable them to generalize
better than recurrent models. In this work, we conduct an extensive empirical
study on Boolean functions to demonstrate the following: (i) Random
Transformers are relatively more biased towards functions of low sensitivity.
(ii) When trained on Boolean functions, both Transformers and LSTMs prioritize
learning functions of low sensitivity, with Transformers ultimately converging
to functions of lower sensitivity. (iii) On sparse Boolean functions which have
low sensitivity, we find that Transformers generalize near perfectly even in
the presence of noisy labels whereas LSTMs overfit and achieve poor
generalization accuracy. Overall, our results provide strong quantifiable
evidence that suggests differences in the inductive biases of Transformers and
recurrent models which may help explain Transformer's effective generalization
performance despite relatively limited expressiveness.Comment: Preprin
LIPIcs, Volume 251, ITCS 2023, Complete Volume
LIPIcs, Volume 251, ITCS 2023, Complete Volum
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