arXiv subjects

Daniel Keren

Publications and source records attributed to Daniel Keren.

5 recordsLinked to original sources

arXiv2026

Stationarity is not enough: tightness of the quantum mechanical bootstrap and the copositive cone

The numerical bootstrap for quantum mechanics tests a candidate state's positivity only against sums of squares, whereas every physical state assigns nonnegative expectation to every pointwise nonnegative polynomial. In one dimension the two coincide; in two or more they do not. Whether this gap is realized depends sharply on which constraint set is imposed. For the stationary bootstrap, which imposes <[H,O]> = 0 and is the relaxation appropriate to thermal and mixed states, we exhibit a two-dimensional quartic double well and a moment vector that satisfies every level-three stationary constraint exactly, has a positive definite moment matrix, and yet assigns a negative expectation to a polynomial nonnegative on R^2; it is therefore the moment sequence of no state. All data are rational, and every step is verified in exact arithmetic. For the eigenstate bootstrap, which additionally imposes = E , the same search finds no violation in any of five settings spanning two, three and five degrees of freedom and truncation levels three and four, tested against complete families of separating polynomials. The single exception occurs at a truncation so low that only one eigenstate constraint survives, and the theory presented here accounts for it. We identify the mechanism: the eigenstate constraints bound the high momentum moments, otherwise unbounded on the feasible set, and it is those unbounded directions that reach the region between the two cones. Finally, under the reflection symmetries of a typical potential the relevant obstruction is copositivity rather than nonnegativity, which, for the quartic witnesses available at the lowest truncation, places the first possible failure at five degrees of freedom. We conjecture that the eigenstate constraints imply an Archimedean-type bound on the momentum moments, and formulate the corresponding tightness statement.

cs.SC↗

arXiv2026

How Much Can a Few Engine Moves Help? Quantifying Limited Cheating in Chess

Cheating in chess, by using advice from powerful software, has become a major problem, reaching the highest levels. As opposed to the large majority of previous work, which concerned {\em detection} of cheating, here we try to evaluate the possible gain in performance, obtained by cheating a limited number of times during a game. We develop threshold-based and Bellman-style intervention policies, and test them in a controlled engine-vs-engine setting using Stockfish. A judicious choice of 1 or 2 cheats yields average scores of 0.71 and 0.82, respectively, compared to 0.51 with no cheats. We also introduce a fast, engine-free simulator that enables hyperparameter optimization without running games, closely matching the engine-based optimum. The goal of this work is not to assist cheaters, but to measure the effectiveness of cheating -- which is crucial as part of the effort to contain and detect it.

cs.AI↗

arXiv2024

A practical, fast method for solving sum-of-squares problems for very large polynomials

Sum of squares (SOS) optimization is a powerful technique for solving problems where the positivity of a polynomials must be enforced. The common approach to solve an SOS problem is by relaxation to a Semidefinite Program (SDP). The main advantage of this transormation is that SDP is a convex problem for which efficient solvers are readily available. However, while considerable progress has been made in recent years, the standard approaches for solving SDPs are still known to scale poorly. Our goal is to devise an approach that can handle larger, more complex problems than is currently possible. The challenge indeed lies in how SDPs are commonly solved. State-Of-The-Art approaches rely on the interior point method, which requires the factorization of large matrices. We instead propose an approach inspired by polynomial neural networks, which exhibit excellent performance when optimized using techniques from the deep learning toolbox. In a somewhat counter-intuitive manner, we replace the convex SDP formulation with a non-convex, unconstrained, and \emph{over parameterized} formulation, and solve it using a first order optimization method. It turns out that this approach can handle very large problems, with polynomials having over four million coefficients, well beyond the range of current SDP-based approaches. Furthermore, we highlight theoretical and practical results supporting the experimental success of our approach in avoiding spurious local minima, which makes it amenable to simple and fast solutions based on gradient descent. In all the experiments, our approach had always converged to a correct global minimum, on general (non-sparse) polynomials, with running time only slightly higher than linear in the number of polynomial coefficients, compared to higher than quadratic in the number of coefficients for SDP-based methods.

math.OC↗

arXiv2023

Geometric Covering using Random Fields

A set of vectors $S \subseteq \mathbb{R}^d$ is $(k_1,\varepsilon)$-clusterable if there are $k_1$ balls of radius $\varepsilon$ that cover $S$. A set of vectors $S \subseteq \mathbb{R}^d$ is $(k_2,δ)$-far from being clusterable if there are at least $k_2$ vectors in $S$, with all pairwise distances at least $δ$. We propose a probabilistic algorithm to distinguish between these two cases. Our algorithm reaches a decision by only looking at the extreme values of a scalar valued hash function, defined by a random field, on $S$; hence, it is especially suitable in distributed and online settings. An important feature of our method is that the algorithm is oblivious to the number of vectors: in the online setting, for example, the algorithm stores only a constant number of scalars, which is independent of the stream length. We introduce random field hash functions, which are a key ingredient in our paradigm. Random field hash functions generalize locality-sensitive hashing (LSH). In addition to the LSH requirement that ``nearby vectors are hashed to similar values", our hash function also guarantees that the ``hash values are (nearly) independent random variables for distant vectors". We formulate necessary conditions for the kernels which define the random fields applied to our problem, as well as a measure of kernel optimality, for which we provide a bound. Then, we propose a method to construct kernels which approximate the optimal one.

cs.DS↗

arXiv2019

Adaptive Communication Bounds for Distributed Online Learning

We consider distributed online learning protocols that control the exchange of information between local learners in a round-based learning scenario. The learning performance of such a protocol is intuitively optimal if approximately the same loss is incurred as in a hypothetical serial setting. If a protocol accomplishes this, it is inherently impossible to achieve a strong communication bound at the same time. In the worst case, every input is essential for the learning performance, even for the serial setting, and thus needs to be exchanged between the local learners. However, it is reasonable to demand a bound that scales well with the hardness of the serialized prediction problem, as measured by the loss received by a serial online learning algorithm. We provide formal criteria based on this intuition and show that they hold for a simplified version of a previously published protocol.

cs.DC↗