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In recent years, many connections have been made between minimal codes, a classical object in coding theory, and other remarkable structures in finite geometry and combinatorics. One of the main problems related to minimal codes is to give lower and upper bounds on the length $m(k,q)$ of the shortest minimal codes of a given dimension $k$ over the finite field $\mathbb{F}_q$. It has been recently proved that $m(k, q) \geq (q+1)(k-1)$. In this note, we prove that $\liminf_{k \rightarrow \infty} \frac{m(k, q)}{k} \geq (q+ \varepsilon(q) )$, where $\varepsilon$ is an increasing function such that $1.52 <\varepsilon(2)\leq \varepsilon(q) \leq \sqrt{2} + \frac{1}{2}$. Hence, the previously known lower bound is not tight for large enough $k$. We then focus on the binary case and prove some structural results on minimal codes of length $3(k-1)$. As a byproduct, we are able to show that, if $k = 5 \pmod 8$ and for other small values of $k$, the bound is not tight.

In this work we consider stochastic gradient descent (SGD) for solving linear inverse problems in Banach spaces. SGD and its variants have been established as one of the most successful optimisation methods in machine learning, imaging and signal processing, etc. At each iteration SGD uses a single datum, or a small subset of data, resulting in highly scalable methods that are very attractive for large-scale inverse problems. Nonetheless, the theoretical analysis of SGD-based approaches for inverse problems has thus far been largely limited to Euclidean and Hilbert spaces. In this work we present a novel convergence analysis of SGD for linear inverse problems in general Banach spaces: we show the almost sure convergence of the iterates to the minimum norm solution and establish the regularising property for suitable a priori stopping criteria. Numerical results are also presented to illustrate features of the approach.

Reliability of machine learning evaluation -- the consistency of observed evaluation scores across replicated model training runs -- is affected by several sources of nondeterminism which can be regarded as measurement noise. Current tendencies to remove noise in order to enforce reproducibility of research results neglect inherent nondeterminism at the implementation level and disregard crucial interaction effects between algorithmic noise factors and data properties. This limits the scope of conclusions that can be drawn from such experiments. Instead of removing noise, we propose to incorporate several sources of variance, including their interaction with data properties, into an analysis of significance and reliability of machine learning evaluation, with the aim to draw inferences beyond particular instances of trained models. We show how to use linear mixed effects models (LMEMs) to analyze performance evaluation scores, and to conduct statistical inference with a generalized likelihood ratio test (GLRT). This allows us to incorporate arbitrary sources of noise like meta-parameter variations into statistical significance testing, and to assess performance differences conditional on data properties. Furthermore, a variance component analysis (VCA) enables the analysis of the contribution of noise sources to overall variance and the computation of a reliability coefficient by the ratio of substantial to total variance.

In this paper, we study error bounds for {\em Bayesian quadrature} (BQ), with an emphasis on noisy settings, randomized algorithms, and average-case performance measures. We seek to approximate the integral of functions in a {\em Reproducing Kernel Hilbert Space} (RKHS), particularly focusing on the Mat\'ern-$\nu$ and squared exponential (SE) kernels, with samples from the function potentially being corrupted by Gaussian noise. We provide a two-step meta-algorithm that serves as a general tool for relating the average-case quadrature error with the $L^2$-function approximation error. When specialized to the Mat\'ern kernel, we recover an existing near-optimal error rate while avoiding the existing method of repeatedly sampling points. When specialized to other settings, we obtain new average-case results for settings including the SE kernel with noise and the Mat\'ern kernel with misspecification. Finally, we present algorithm-independent lower bounds that have greater generality and/or give distinct proofs compared to existing ones.

Zero-shot coordination in cooperative artificial intelligence (AI) remains a significant challenge, which means effectively coordinating with a wide range of unseen partners. Previous algorithms have attempted to address this challenge by optimizing fixed objectives within a population to improve strategy or behavior diversity. However, these approaches can result in a loss of learning and an inability to cooperate with certain strategies within the population, known as cooperative incompatibility. To address this issue, we propose the Cooperative Open-ended LEarning (COLE) framework, which constructs open-ended objectives in cooperative games with two players from the perspective of graph theory to assess and identify the cooperative ability of each strategy. We further specify the framework and propose a practical algorithm that leverages knowledge from game theory and graph theory. Furthermore, an analysis of the learning process of the algorithm shows that it can efficiently overcome cooperative incompatibility. The experimental results in the Overcooked game environment demonstrate that our method outperforms current state-of-the-art methods when coordinating with different-level partners. Our code and demo are available at //sites.google.com/view/cole-2023.

Motivated by a real-world application, we model and solve a complex staff scheduling problem. Tasks are to be assigned to workers for supervision. Multiple tasks can be covered in parallel by a single worker, with worker shifts being flexible within availabilities. Each worker has a different skill set, enabling them to cover different tasks. Tasks require assignment according to priority and skill requirements. The objective is to maximize the number of assigned tasks weighted by their priorities, while minimizing assignment penalties. We develop an adaptive large neighborhood search (ALNS) algorithm, relying on tailored destroy and repair operators. It is tested on benchmark instances derived from real-world data and compared to optimal results obtained by means of a commercial MIP-solver. Furthermore, we analyze the impact of considering three additional alternative objective functions. When applied to large-scale company data, the developed ALNS outperforms the previously applied solution approach.

We consider the Sobolev embedding operator $E_s : H^s(\Omega) \to L_2(\Omega)$ and its role in the solution of inverse problems. In particular, we collect various properties and investigate different characterizations of its adjoint operator $E_s^*$, which is a common component in both iterative and variational regularization methods. These include variational representations and connections to boundary value problems, Fourier and wavelet representations, as well as connections to spatial filters. Moreover, we consider characterizations in terms of Fourier series, singular value decompositions and frame decompositions, as well as representations in finite dimensional settings. While many of these results are already known to researchers from different fields, a detailed and general overview or reference work containing rigorous mathematical proofs is still missing. Hence, in this paper we aim to fill this gap by collecting, introducing and generalizing a large number of characterizations of $E_s^*$ and discuss their use in regularization methods for solving inverse problems. The resulting compilation can serve both as a reference as well as a useful guide for its efficient numerical implementation in practice.

While most methods for solving mixed-integer optimization problems compute a single optimal solution, a diverse set of near-optimal solutions can often lead to improved outcomes. We present a new method for finding a set of diverse solutions by emphasizing diversity within the search for near-optimal solutions. Specifically, within a branch-and-bound framework, we investigated parameterized node selection rules that explicitly consider diversity. Our results indicate that our approach significantly increases the diversity of the final solution set. When compared with two existing methods, our method runs with similar runtime as regular node selection methods and gives a diversity improvement between 12% and 190%. In contrast, popular node selection rules, such as best-first search, in some instances performed worse than state-of-the-art methods by more than 35% and gave an improvement of no more than 130%. Further, we find that our method is most effective when diversity in node selection is continuously emphasized after reaching a minimal depth in the tree and when the solution set has grown sufficiently large. Our method can be easily incorporated into integer programming solvers and has the potential to significantly increase the diversity of solution sets.

State of the art reinforcement learning has enabled training agents on tasks of ever increasing complexity. However, the current paradigm tends to favor training agents from scratch on every new task or on collections of tasks with a view towards generalizing to novel task configurations. The former suffers from poor data efficiency while the latter is difficult when test tasks are out-of-distribution. Agents that can effectively transfer their knowledge about the world pose a potential solution to these issues. In this paper, we investigate transfer learning in the context of model-based agents. Specifically, we aim to understand when exactly environment models have an advantage and why. We find that a model-based approach outperforms controlled model-free baselines for transfer learning. Through ablations, we show that both the policy and dynamics model learnt through exploration matter for successful transfer. We demonstrate our results across three domains which vary in their requirements for transfer: in-distribution procedural (Crafter), in-distribution identical (RoboDesk), and out-of-distribution (Meta-World). Our results show that intrinsic exploration combined with environment models present a viable direction towards agents that are self-supervised and able to generalize to novel reward functions.

Learning causal relationships between variables is a fundamental task in causal inference and directed acyclic graphs (DAGs) are a popular choice to represent the causal relationships. As one can recover a causal graph only up to its Markov equivalence class from observations, interventions are often used for the recovery task. Interventions are costly in general and it is important to design algorithms that minimize the number of interventions performed. In this work, we study the problem of identifying the smallest set of interventions required to learn the causal relationships between a subset of edges (target edges). Under the assumptions of faithfulness, causal sufficiency, and ideal interventions, we study this problem in two settings: when the underlying ground truth causal graph is known (subset verification) and when it is unknown (subset search). For the subset verification problem, we provide an efficient algorithm to compute a minimum sized interventional set; we further extend these results to bounded size non-atomic interventions and node-dependent interventional costs. For the subset search problem, in the worst case, we show that no algorithm (even with adaptivity or randomization) can achieve an approximation ratio that is asymptotically better than the vertex cover of the target edges when compared with the subset verification number. This result is surprising as there exists a logarithmic approximation algorithm for the search problem when we wish to recover the whole causal graph. To obtain our results, we prove several interesting structural properties of interventional causal graphs that we believe have applications beyond the subset verification/search problems studied here.