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Aiming to recover the data from several concurrent node failures, linear $r$-LRC codes with locality $r$ were extended into $(r, \delta)$-LRC codes with locality $(r, \delta)$ which can enable the local recovery of a failed node in case of more than one node failure. Optimal LRC codes are those whose parameters achieve the generalized Singleton bound with equality. In the present paper, we are interested in studying optimal LRC codes over small fields and, more precisely, over $\mathbb{F}_4$. We shall adopt an approach by investigating optimal quaternary $(r,\delta)$-LRC codes through their parity-check matrices. Our study includes determining the structural properties of optimal $(r,\delta)$-LRC codes, their constructions, and their complete classification over $\F_4$ by browsing all possible parameters. We emphasize that the precise structure of optimal quaternary $(r,\delta)$-LRC codes and their classification are obtained via the parity-check matrix approach use proofs-techniques different from those used recently for optimal binary and ternary $(r,\delta)$-LRC codes obtained by Hao et al. in [IEEE Trans. Inf. Theory, 2020, 66(12): 7465-7474].

Inspired by a mathematical riddle involving fuses, we define the "fusible numbers" as follows: $0$ is fusible, and whenever $x,y$ are fusible with $|y-x|<1$, the number $(x+y+1)/2$ is also fusible. We prove that the set of fusible numbers, ordered by the usual order on $\mathbb R$, is well-ordered, with order type $\varepsilon_0$. Furthermore, we prove that the density of the fusible numbers along the real line grows at an incredibly fast rate: Letting $g(n)$ be the largest gap between consecutive fusible numbers in the interval $[n,\infty)$, we have $g(n)^{-1} \ge F_{\varepsilon_0}(n-c)$ for some constant $c$, where $F_\alpha$ denotes the fast-growing hierarchy. Finally, we derive some true statements that can be formulated but not proven in Peano Arithmetic, of a different flavor than previously known such statements: PA cannot prove the true statement "For every natural number $n$ there exists a smallest fusible number larger than $n$." Also, consider the algorithm "$M(x)$: if $x<0$ return $-x$, else return $M(x-M(x-1))/2$." Then $M$ terminates on real inputs, although PA cannot prove the statement "$M$ terminates on all natural inputs."

We study the problem of transfer-learning in the setting of stochastic linear bandit tasks. We consider that a low dimensional linear representation is shared across the tasks, and study the benefit of learning this representation in the multi-task learning setting. Following recent results to design stochastic bandit policies, we propose an efficient greedy policy based on trace norm regularization. It implicitly learns a low dimensional representation by encouraging the matrix formed by the task regression vectors to be of low rank. Unlike previous work in the literature, our policy does not need to know the rank of the underlying matrix. We derive an upper bound on the multi-task regret of our policy, which is, up to logarithmic factors, of order $\sqrt{NdT(T+d)r}$, where $T$ is the number of tasks, $r$ the rank, $d$ the number of variables and $N$ the number of rounds per task. We show the benefit of our strategy compared to the baseline $Td\sqrt{N}$ obtained by solving each task independently. We also provide a lower bound to the multi-task regret. Finally, we corroborate our theoretical findings with preliminary experiments on synthetic data.

In this paper we study the problem of efficiently factorizing polynomials in the free noncommutative ring F<x_1,x_2,...,x_n> of polynomials in noncommuting variables x_1,x_2,..., x_n over the field F. We obtain the following result: Given a noncommutative arithmetic formula of size s computing a noncommutative polynomial f in F<x_1,x_2,...,x_n> as input, where F=F_q is a finite field, we give a randomized algorithm that runs in time polynomial in s, n and log q that computes a factorization of f as a product f=f_1f_2\cdots f_r, where each f_i is an irreducible polynomial that is output as a noncommutative algebraic branching program. The algorithm works by first transforming f into a linear matrix L using Higman's linearization of polynomials. We then factorize the linear matrix L and recover the factorization of f. We use basic elements from Cohn's theory of free ideals rings combined with Ronyai's randomized polynomial-time algorithm for computing invariant subspaces of a collection of matrices over finite fields.

In 2017, Bauer, Johnson, Osborne, Riehl, and Tebbe (BJORT) showed that the Abelian functor calculus provides an example of a Cartesian differential category. The definition of a Cartesian differential category is based on a differential combinator which directly formalizes the total derivative from multivariable calculus. However, in the aforementioned work the authors used techniques from Goodwillie's functor calculus to establish a linearization process from which they then derived a differential combinator. This raised the question of what the precise relationship between linearization and having a differential combinator might be. In this paper, we introduce the notion of a linearizing combinator which abstracts linearization in the Abelian functor calculus. We then use it to provide an alternative axiomatization of a Cartesian differential category. Every Cartesian differential category comes equipped with a canonical linearizing combinator obtained by differentiation at zero. Conversely, a differential combinator can be constructed \`a la BJORT when one has a system of partial linearizing combinators in each context. Thus, while linearizing combinators do provide an alternative axiomatization of Cartesian differential categories, an explicit notion of partial linearization is required. This is in contrast to the situation for differential combinators where partial differentiation is automatic in the presence of total differentiation. The ability to form a system of partial linearizing combinators from a total linearizing combinator, while not being possible in general, is possible when the setting is Cartesian closed.

Given a graph function, defined on an arbitrary set of edge weights and node features, does there exist a Graph Neural Network (GNN) whose output is identical to the graph function? In this paper, we fully answer this question and characterize the class of graph problems that can be represented by GNNs. We identify an algebraic condition, in terms of the permutation of edge weights and node features, which proves to be necessary and sufficient for a graph problem to lie within the reach of GNNs. Moreover, we show that this condition can be efficiently verified by checking quadratically many constraints. Note that our refined characterization on the expressive power of GNNs are orthogonal to those theoretical results showing equivalence between GNNs and Weisfeiler-Lehman graph isomorphism heuristic. For instance, our characterization implies that many natural graph problems, such as min-cut value, max-flow value, and max-clique size, can be represented by a GNN. In contrast, and rather surprisingly, there exist very simple graphs for which no GNN can correctly find the length of the shortest paths between all nodes. Note that finding shortest paths is one of the most classical problems in Dynamic Programming (DP). Thus, the aforementioned negative example highlights the misalignment between DP and GNN, even though (conceptually) they follow very similar iterative procedures. Finally, we support our theoretical results by experimental simulations.

The problem of testing low-degree polynomials has received significant attention over the years due to its importance in theoretical computer science, and in particular in complexity theory. The problem is specified by three parameters: field size $q$, degree $d$ and proximity parameter $\delta$, and the goal is to design a tester making as few as possible queries to a given function, which is able to distinguish between the case the given function has degree at most $d$, and the case the given function is $\delta$-far from any degree $d$ function. A tester is called optimal if it makes $O(q^d+1/\delta)$ queries (which are known to be necessary). For the field of size $q$, the natural $t$-flat tester was shown to be optimal first by Bhattacharyya et al. for $q=2$, and later by Haramaty et al. for all prime powers $q$. The dependency on the field size, however, is a tower-type function. We improve the results above, showing that the dependency on the field size is polynomial. Our approach also applies in the more general setting of lifted affine invariant codes, and is based on studying the structure of the collection of erroneous subspaces. i.e. subspaces $A$ such that $f|_{A}$ has degree greater than $d$. Towards this end, we observe that these sets are poorly expanding in the affine version of the Grassmann graph and use that to establish structural results on them via global hypercontractivity. We then use this structure to perform local correction on $f$.

The generalized coloring numbers of Kierstead and Yang offer an algorithmically useful characterization of graph classes with bounded expansion. In this work, we consider the hardness and approximability of these parameters. First, we complete the work of Grohe et al. by showing that computing the weak 2-coloring number is NP-hard. Our approach further establishes that determining the weak $r$-coloring number is APX-hard for all $r \geq 2$. We adapt this to the $r$-coloring number as well, proving APX-hardness for all $r \geq 2$. Our reductions also imply that for every fixed $r \geq 2$, no XP algorithm (runtime $O(n^{f(k)})$) exists for testing if either generalized coloring number is at most $k$. Finally, we give an approximation algorithm for the $r$-coloring number which improves both the runtime and approximation factor of the existing approach of Dvo\v{r}\'{a}k. Our algorithm greedily orders vertices with small enough $\ell$-reach for every $\ell \leq r$ and achieves an $O(C_{r-1} k^{r-1})$-approximation, where $C_i$ is the $i$th Catalan number.

Multi-label text classification refers to the problem of assigning each given document its most relevant labels from the label set. Commonly, the metadata of the given documents and the hierarchy of the labels are available in real-world applications. However, most existing studies focus on only modeling the text information, with a few attempts to utilize either metadata or hierarchy signals, but not both of them. In this paper, we bridge the gap by formalizing the problem of metadata-aware text classification in a large label hierarchy (e.g., with tens of thousands of labels). To address this problem, we present the MATCH solution -- an end-to-end framework that leverages both metadata and hierarchy information. To incorporate metadata, we pre-train the embeddings of text and metadata in the same space and also leverage the fully-connected attentions to capture the interrelations between them. To leverage the label hierarchy, we propose different ways to regularize the parameters and output probability of each child label by its parents. Extensive experiments on two massive text datasets with large-scale label hierarchies demonstrate the effectiveness of MATCH over state-of-the-art deep learning baselines.

This work considers the problem of provably optimal reinforcement learning for episodic finite horizon MDPs, i.e. how an agent learns to maximize his/her long term reward in an uncertain environment. The main contribution is in providing a novel algorithm --- Variance-reduced Upper Confidence Q-learning (vUCQ) --- which enjoys a regret bound of $\widetilde{O}(\sqrt{HSAT} + H^5SA)$, where the $T$ is the number of time steps the agent acts in the MDP, $S$ is the number of states, $A$ is the number of actions, and $H$ is the (episodic) horizon time. This is the first regret bound that is both sub-linear in the model size and asymptotically optimal. The algorithm is sub-linear in that the time to achieve $\epsilon$-average regret for any constant $\epsilon$ is $O(SA)$, which is a number of samples that is far less than that required to learn any non-trivial estimate of the transition model (the transition model is specified by $O(S^2A)$ parameters). The importance of sub-linear algorithms is largely the motivation for algorithms such as $Q$-learning and other "model free" approaches. vUCQ algorithm also enjoys minimax optimal regret in the long run, matching the $\Omega(\sqrt{HSAT})$ lower bound. Variance-reduced Upper Confidence Q-learning (vUCQ) is a successive refinement method in which the algorithm reduces the variance in $Q$-value estimates and couples this estimation scheme with an upper confidence based algorithm. Technically, the coupling of both of these techniques is what leads to the algorithm enjoying both the sub-linear regret property and the asymptotically optimal regret.