In Linear Logic ($\mathsf{LL}$), the exponential modality $!$ brings forth a distinction between non-linear proofs and linear proofs, where linear means using an argument exactly once. Differential Linear Logic ($\mathsf{DiLL}$) is an extension of Linear Logic which includes additional rules for $!$ which encode differentiation and the ability of linearizing proofs. On the other hand, Graded Linear Logic ($\mathsf{GLL}$) is a variation of Linear Logic in such a way that $!$ is now indexed over a semiring $R$. This $R$-grading allows for non-linear proofs of degree $r \in R$, such that the linear proofs are of degree $1 \in R$. There has been recent interest in combining these two variations of $\mathsf{LL}$ together and developing Graded Differential Linear Logic ($\mathsf{GDiLL}$). In this paper we present a sequent calculus for $\mathsf{GDiLL}$, as well as introduce its categorical semantics, which we call graded differential categories, using both coderelictions and deriving transformations. We prove that symmetric powers always give graded differential categories, and provide other examples of graded differential categories. We also discuss graded versions of (monoidal) coalgebra modalities, additive bialgebra modalities, and the Seely isomorphisms, as well as their implementations in the sequent calculus of $\mathsf{GDiLL}$.
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We prove that the blocklength $n$ of a linear $3$-query locally correctable code (LCC) $\mathcal{L} \colon {\mathbb F}^k \to {\mathbb F}^n$ with distance $\delta$ must be at least $n \geq 2^{\Omega\left(\left(\frac{\delta^2 k}{(|{\mathbb F}|-1)^2}\right)^{1/8}\right)}$. In particular, the blocklength of a linear $3$-query LCC with constant distance over any small field grows exponentially with $k$. This improves on the best prior lower bound of $n \geq \tilde{\Omega}(k^3)$ [AGKM23], which holds even for the weaker setting of $3$-query locally decodable codes (LDCs), and comes close to matching the best-known construction of $3$-query LCCs based on binary Reed-Muller codes, which achieve $n \leq 2^{O(k^{1/2})}$. Because there is a $3$-query LDC with a strictly subexponential blocklength [Yek08, Efr09], as a corollary we obtain the first strong separation between $q$-query LCCs and LDCs for any constant $q \geq 3$. Our proof is based on a new upgrade of the method of spectral refutations via Kikuchi matrices developed in recent works [GKM22, HKM23, AGKM23] that reduces establishing (non-)existence of combinatorial objects to proving unsatisfiability of associated XOR instances. Our key conceptual idea is to apply this method with XOR instances obtained via long-chain derivations, a structured variant of low-width resolution for XOR formulas from proof complexity [Gri01, Sch08].
We bound the smoothed running time of the FLIP algorithm for local Max-Cut as a function of $\alpha$, the arboricity of the input graph. We show that, with high probability, the following holds (where $n$ is the number of nodes and $\phi$ is the smoothing parameter): 1) When $\alpha = O(\sqrt{\log n})$ FLIP terminates in $\phi poly(n)$ iterations. Previous to our results the only graph families for which FLIP was known to achieve a smoothed polynomial running time were complete graphs and graphs with logarithmic maximum degree. 2) For arbitrary values of $\alpha$ we get a running time of $\phi n^{O(\frac{\alpha}{\log n} + \log \alpha)}$. This improves over the best known running time for general graphs of $\phi n^{O(\sqrt{ \log n })}$ for $\alpha = o(\log^{1.5} n)$. Specifically, when $\alpha = O(\log n)$ we get a significantly faster running time of $\phi n^{O(\log \log n)}$.
We present the notion of \emph{reasonable utility} for binary mechanisms, which applies to all utility functions in the literature. This notion induces a partial ordering on the performance of all binary differentially private (DP) mechanisms. DP mechanisms that are maximal elements of this ordering are optimal DP mechanisms for every reasonable utility. By looking at differential privacy as a randomized graph coloring, we characterize these optimal DP in terms of their behavior on a certain subset of the boundary datasets we call a boundary hitting set. In the process of establishing our results, we also introduce a useful notion that generalizes DP conditions for binary-valued queries, which we coin as suitable pairs. Suitable pairs abstract away the algebraic roles of $\varepsilon,\delta$ in the DP framework, making the derivations and understanding of our proofs simpler. Additionally, the notion of a suitable pair can potentially capture privacy conditions in frameworks other than DP and may be of independent interest.
The Weisfeiler-Leman (WL) dimension of a graph parameter $f$ is the minimum $k$ such that, if $G_1$ and $G_2$ are indistinguishable by the $k$-dimensional WL-algorithm then $f(G_1)=f(G_2)$. The WL-dimension of $f$ is $\infty$ if no such $k$ exists. We study the WL-dimension of graph parameters characterised by the number of answers from a fixed conjunctive query to the graph. Given a conjunctive query $\varphi$, we quantify the WL-dimension of the function that maps every graph $G$ to the number of answers of $\varphi$ in $G$. The works of Dvor\'ak (J. Graph Theory 2010), Dell, Grohe, and Rattan (ICALP 2018), and Neuen (ArXiv 2023) have answered this question for full conjunctive queries, which are conjunctive queries without existentially quantified variables. For such queries $\varphi$, the WL-dimension is equal to the treewidth of the Gaifman graph of $\varphi$. In this work, we give a characterisation that applies to all conjunctive qureies. Given any conjunctive query $\varphi$, we prove that its WL-dimension is equal to the semantic extension width $\mathsf{sew}(\varphi)$, a novel width measure that can be thought of as a combination of the treewidth of $\varphi$ and its quantified star size, an invariant introduced by Durand and Mengel (ICDT 2013) describing how the existentially quantified variables of $\varphi$ are connected with the free variables. Using the recently established equivalence between the WL-algorithm and higher-order Graph Neural Networks (GNNs) due to Morris et al. (AAAI 2019), we obtain as a consequence that the function counting answers to a conjunctive query $\varphi$ cannot be computed by GNNs of order smaller than $\mathsf{sew}(\varphi)$.
We present a structure-preserving Eulerian algorithm for solving $L^2$-gradient flows and a structure-preserving Lagrangian algorithm for solving generalized diffusions. Both algorithms employ neural networks as tools for spatial discretization. Unlike most existing methods that construct numerical discretizations based on the strong or weak form of the underlying PDE, the proposed schemes are constructed based on the energy-dissipation law directly. This guarantees the monotonic decay of the system's energy, which avoids unphysical states of solutions and is crucial for the long-term stability of numerical computations. To address challenges arising from nonlinear neural-network discretization, we first perform temporal discretization on these variational systems. This approach is computationally memory-efficient when implementing neural network-based algorithms. The proposed neural-network-based schemes are mesh-free, allowing us to solve gradient flows in high dimensions. Various numerical experiments are presented to demonstrate the accuracy and energy stability of the proposed numerical schemes.
We consider the massively parallel computation (MPC) model, which is a theoretical abstraction of large-scale parallel processing models such as MapReduce. In this model, assuming the widely believed 1-vs-2-cycles conjecture, solving many basic graph problems in $O(1)$ rounds with a strongly sublinear memory size per machine is impossible. We improve on the recent work of Holm and T\v{e}tek [SODA 2023] that bypass this barrier for problems when a planar embedding of the graph is given. In the previous work, on graphs of size $n$ with $O(n/\mathcal{S})$ machines, the memory size per machine needs to be at least $\mathcal{S} = n^{2/3+\Omega(1)}$, whereas we extend their work to the fully scalable regime, where the memory size per machine can be $\mathcal{S} = n^{\delta}$ for any constant $0< \delta < 1$. We give the first constant round fully scalable algorithms for embedded planar graphs for the problems of (i) connectivity and (ii) minimum spanning tree (MST). Moreover, we show that the $\varepsilon$-emulator of Chang, Krauthgamer, and Tan [STOC 2022] can be incorporated into our recursive framework to obtain constant-round $(1+\varepsilon)$-approximation algorithms for the problems of computing (iii) single source shortest path (SSSP), (iv) global min-cut, and (v) $st$-max flow. All previous results on cuts and flows required linear memory in the MPC model. Furthermore, our results give new algorithms for problems that implicitly involve embedded planar graphs. We give as corollaries constant round fully scalable algorithms for (vi) 2D Euclidean MST using $O(n)$ total memory and (vii) $(1+\varepsilon)$-approximate weighted edit distance using $\widetilde{O}(n^{2-\delta})$ memory. Our main technique is a recursive framework combined with novel graph drawing algorithms to compute smaller embedded planar graphs in constant rounds in the fully scalable setting.
This paper studies the prediction of a target $\mathbf{z}$ from a pair of random variables $(\mathbf{x},\mathbf{y})$, where the ground-truth predictor is additive $\mathbb{E}[\mathbf{z} \mid \mathbf{x},\mathbf{y}] = f_\star(\mathbf{x}) +g_{\star}(\mathbf{y})$. We study the performance of empirical risk minimization (ERM) over functions $f+g$, $f \in F$ and $g \in G$, fit on a given training distribution, but evaluated on a test distribution which exhibits covariate shift. We show that, when the class $F$ is "simpler" than $G$ (measured, e.g., in terms of its metric entropy), our predictor is more resilient to heterogeneous covariate shifts} in which the shift in $\mathbf{x}$ is much greater than that in $\mathbf{y}$. Our analysis proceeds by demonstrating that ERM behaves qualitatively similarly to orthogonal machine learning: the rate at which ERM recovers the $f$-component of the predictor has only a lower-order dependence on the complexity of the class $G$, adjusted for partial non-indentifiability introduced by the additive structure. These results rely on a novel H\"older style inequality for the Dudley integral which may be of independent interest. Moreover, we corroborate our theoretical findings with experiments demonstrating improved resilience to shifts in "simpler" features across numerous domains.
For a set of points in $\mathbb{R}^d$, the Euclidean $k$-means problems consists of finding $k$ centers such that the sum of distances squared from each data point to its closest center is minimized. Coresets are one the main tools developed recently to solve this problem in a big data context. They allow to compress the initial dataset while preserving its structure: running any algorithm on the coreset provides a guarantee almost equivalent to running it on the full data. In this work, we study coresets in a fully-dynamic setting: points are added and deleted with the goal to efficiently maintain a coreset with which a k-means solution can be computed. Based on an algorithm from Henzinger and Kale [ESA'20], we present an efficient and practical implementation of a fully dynamic coreset algorithm, that improves the running time by up to a factor of 20 compared to our non-optimized implementation of the algorithm by Henzinger and Kale, without sacrificing more than 7% on the quality of the k-means solution.
We consider the problem of testing and learning quantum $k$-juntas: $n$-qubit unitary matrices which act non-trivially on just $k$ of the $n$ qubits and as the identity on the rest. As our main algorithmic results, we give (a) a $\widetilde{O}(\sqrt{k})$-query quantum algorithm that can distinguish quantum $k$-juntas from unitary matrices that are "far" from every quantum $k$-junta; and (b) a $O(4^k)$-query algorithm to learn quantum $k$-juntas. We complement our upper bounds for testing quantum $k$-juntas and learning quantum $k$-juntas with near-matching lower bounds of $\Omega(\sqrt{k})$ and $\Omega(\frac{4^k}{k})$, respectively. Our techniques are Fourier-analytic and make use of a notion of influence of qubits on unitaries.
We study the recursion-theoretic complexity of Positive Almost-Sure Termination ($\mathsf{PAST}$) in an imperative programming language with rational variables, bounded nondeterministic choice, and discrete probabilistic choice. A program terminates positive almost-surely if, for every scheduler, the program terminates almost-surely and the expected runtime to termination is finite. We show that $\mathsf{PAST}$ for our language is complete for the (lightface) co-analytic sets ($\Pi^1_1$-complete). This is in contrast to the related notions of Almost-Sure Termination ($\mathsf{AST}$) and Bounded Termination ($\mathsf{BAST}$), both of which are arithmetical ($\Pi^0_2$ and $\Sigma^0_2$ complete respectively). Our upper bound implies an effective procedure to reduce reasoning about probabilistic termination to non-probabilistic fair termination in a model with bounded nondeterminism, and to simple program termination in models with unbounded nondeterminism. Our lower bound shows the opposite: for every program with unbounded nondeterministic choice, there is an effectively computable probabilistic program with bounded choice such that the original program is terminating $iff$ the transformed program is $\mathsf{PAST}$. We show that every program has an effectively computable normal form, in which each probabilistic choice either continues or terminates execution immediately, each with probability $1/2$. For normal form programs, we provide a sound and complete proof rule for $\mathsf{PAST}$. Our proof rule uses transfinite ordinals. We show that reasoning about $\mathsf{PAST}$ requires transfinite ordinals up to $\omega^{CK}_1$; thus, existing techniques for probabilistic termination based on ranking supermartingales that map program states to reals do not suffice to reason about $\mathsf{PAST}$.
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