Zeyuan Allen Zhu

Mechanism design with approximate valuations

We study single-good auctions when each player knows his own valuation only within a constant multiplicative factor Δ ε (0, 1), known to the mechanism designer. The classical notions of implementation in dominant strategies and implementation in undominated strategies are naturally extended to this setting, but their power is vastly different. On the negative side, we prove that no dominant-strategy mechanism can guarantee social welfare that is significantly better than that achievable by assigning the good to a random player. On the positive side, we provide tight upper and lower bounds for the fraction of the maximum social welfare achievable in undominated strategies, whether deterministically or probabilistically.

Johnson-Lindenstrauss Compression with Neuroscience-Based Constraints.

Significance Significant biological evidence indicates that the brain may perform some form of compression. To be meaningful, such compression should preserve pairwise correlation of the input data. It is mathematically well known that multiplying the input vectors by a sparse and fixed random matrix A achieves the desired compression. But, to implement such an approach in the brain via a synaptic-connectivity matrix, A should also to be sign consistent: that is, all entries in a single column must be either all nonnegative or all nonpositive. This is so because most neurons are either excitatory or inhibitory. We prove that sparse sign-consistent matrices can deliver the desired compression, lending credibility to the hypothesis that correlation-preserving compression occurs in the brain via synaptic-connectivity matrices. Johnson–Lindenstrauss (JL) matrices implemented by sparse random synaptic connections are thought to be a prime candidate for how convergent pathways in the brain compress information. However, to date, there is no complete mathematical support for such implementations given the constraints of real neural tissue. The fact that neurons are either excitatory or inhibitory implies that every so implementable JL matrix must be sign consistent (i.e., all entries in a single column must be either all nonnegative or all nonpositive), and the fact that any given neuron connects to a relatively small subset of other neurons implies that the JL matrix should be sparse. We construct sparse JL matrices that are sign consistent and prove that our construction is essentially optimal. Our work answers a mathematical question that was triggered by earlier work and is necessary to justify the existence of JL compression in the brain and emphasizes that inhibition is crucial if neurons are to perform efficient, correlation-preserving compression.Johnson-Lindenstrauss (JL) matrices implemented by sparse random synaptic connections are thought to be a prime candidate for how convergent pathways in the brain compress information. However, to date, there is no complete mathematical support for such implementations given the constraints of real neural tissue. The fact that neurons are either excitatory or inhibitory implies that every so implementable JL matrix must be sign-consistent (i.e., all entries in a single column must be either all non-negative or all non-positive), and the fact that any given neuron connects to a relatively small subset of other neurons implies that the JL matrix had better be sparse. We construct sparse JL matrices that are sign-consistent, and prove that our construction is essentially optimal. Our work answers a mathematical question that was triggered by earlier work and is necessary to justify the existence of JL compression in the brain, and emphasizes that inhibition is crucial if neurons are to perform efficient, correlation-preserving compression.

Knightian self uncertainty in the vcg mechanism for unrestricted combinatorial auctions

We study the social welfare performance of the VCG mechanism in the well-known and challenging model of self uncertainty initially put forward by Frank H. Knight and later formalized by Truman F. Bewley. Namely, the only information that each player i has about his own true valuation consists of a set of distributions, from one of which is valuation has been drawn. We assume that each player knows his true valuation up to an additive inaccuracy δ, and study the social welfare performance of the VCG mechanism relative to δ > 0. Denoting by MSW the maximum social welfare, we have already shown in [Chiesa, Micali and Zhu 2012] that, even in single-good auctions, no mechanism can guarantee any social welfare greater than MSW / n in dominant strategies or ex-post Nash equilibrium strategies, where n is the number of players. In a separate paper [CMZ14], we have proved that for multi-unit auctions, where it coincides with the Vickrey mechanism, the VCG mechanism performs very well in (Knightian) undominated strategies. Namely, in an n-player m-unit auction, the Vickrey mechanism guarantees a social welfare ≥ - MSW - 2mδ, when each Knightian player chooses an arbitrary undominated strategy to bid in the auction. In this paper we focus on the social welfare performance of the VCG mechanism in unrestricted combinatorial auctions, both in undominated strategies and regret-minimizing strategies. (Indeed, both solution concepts naturally extend to the Knightian setting with player self uncertainty.) Our first theorem proves that, in an n-player m-good combinatorial auction, the VCG mechanism may produce outcomes whose social welfare is ≤ - MSW - ω(2m δ), even when n=2 and each player chooses an undominated strategy. We also geometrically characterize the set of undominated strategies in this setting. Our second theorem shows that the VCG mechanism performs well in regret-minimizing strategies: the guaranteed social welfare is ≥-MSW - 2min{m,n}δ if each player chooses a pure regret-minimizing strategy, and ≥- MSW - O(n2 δ) if mixed strategies are allowed. Finally, we prove a lemma bridging two standard models of rationality: utility maximization and regret minimization. A special case of our lemma implies that, in any game (Knightian or not), every implementation for regret-minimizing players also applies to utility-maximizing players who use regret ONLY to break ties among their undominated strategies. This bridging lemma thus implies that the VCG mechanism continues to perform very well also for the latter players.

Knightian analysis of the vickrey mechanism

We analyze the Vickrey mechanism for auctions of multiple identical goods when the players have both Knightian uncertainty over their own valuations and incomplete preferences. In this model, the Vickrey mechanism is no longer dominant‐strategy, and we prove that all dominant‐strategy mechanisms are inadequate. However, we also prove that, in undominated strategies, the social welfare produced by the Vickrey mechanism in the worst case is not only very good, but also essentially optimal.

Shorter arithmetization of nondeterministic computations

Arithmetizing computation is a crucial component of many fundamental results in complexity theory, including results that gave insight into the power of interactive proofs, multi-prover interactive proofs, and probabilistically-checkable proofs. Informally, an arithmetization is a way to encode a machines computation so that its correctness can be easily verified via few probabilistic algebraic checks.We study the problem of arithmetizing nondeterministic computations for the purpose of constructing short probabilistically-checkable proofs (PCPs) with polylogarithmic query complexity. In such a setting, a PCPs proof length depends (at least!) linearly on the length, in bits, of the encoded computation. Thus, minimizing the number of bits in the encoding is crucial for minimizing PCP proof length.In this paper we show how to arithmetize any T-step computation on a nondeterministic Turing machine by using a polynomial encoding of length O ( T ? ( log ? T ) 2 ) . Previously, the best known length was ? ( T ? ( log ? T ) 4 ) . For nondeterministic random-access machines, our length is O ( T ? ( log ? T ) 2 + o ( 1 ) ) , while prior work only achieved ? ( T ? ( log ? T ) 5 ) .The polynomial encoding that we use is the Reed-Solomon code. When combined with the best PCPs of proximity for this code, our result yields quasilinear-size PCPs with polylogarithmic query complexity that are shorter, by at least two logarithmic factors, than in all prior work.Our arithmetization also enjoys additional properties. First, it is succinct, i.e., the encoding of the computation can be probabilistically checked in ( log ? T ) O ( 1 ) time; this property is necessary for constructing short PCPs with a polylogarithmic-time verifier. Furthermore, our techniques extend, in a certain well-defined sense, to the arithmetization of yet other NEXP-complete languages.

Reconstructing Markov processes from independent and anonymous experiments

We investigate the problem of exactly reconstructing, with high confidence and up to isomorphism, the ball of radius r centered at the starting state of a Markov process from independent and anonymous experiments. In an anonymous experiment, the states are visited according to the underlying transition probabilities, but no global state names are known: one can only recognize whether two states, reached within the same experiment, are the same.We prove quite tight bounds for such exact reconstruction in terms of both the number of experiments and their lengths.