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Safe Policy Improvement (SPI) aims at provable guarantees that a learned policy is at least approximately as good as a given baseline policy. Building on SPI with Soft Baseline Bootstrapping (Soft-SPIBB) by Nadjahi et al., we identify theoretical issues in their approach, provide a corrected theory, and derive a new algorithm that is provably safe on finite Markov Decision Processes (MDP). Additionally, we provide a heuristic algorithm that exhibits the best performance among many state of the art SPI algorithms on two different benchmarks. Furthermore, we introduce a taxonomy of SPI algorithms and empirically show an interesting property of two classes of SPI algorithms: while the mean performance of algorithms that incorporate the uncertainty as a penalty on the action-value is higher, actively restricting the set of policies more consistently produces good policies and is, thus, safer.

Two mechanisms have recently been proposed that can significantly speed up finding distant improving solutions via mutation, namely using a random mutation rate drawn from a heavy-tailed distribution ("fast mutation", Doerr et al. (2017)) and increasing the mutation strength based on stagnation detection (Rajabi and Witt (2020)). Whereas the latter can obtain the asymptotically best probability of finding a single desired solution in a given distance, the former is more robust and performs much better when many improving solutions in some distance exist. In this work, we propose a mutation strategy that combines ideas of both mechanisms. We show that it can also obtain the best possible probability of finding a single distant solution. However, when several improving solutions exist, it can outperform both the stagnation-detection approach and fast mutation. The new operator is more than an interleaving of the two previous mechanisms and it also outperforms any such interleaving.

We introduce a simple and fast method for comparing graphs of different sizes. Existing approaches are often either limited to comparing graphs with the same number of vertices or are computationally unscalable. We propose the Embedded Laplacian Distance (ELD) for comparing graphs of potentially vastly different sizes. Our approach first projects the graphs onto a common, low-dimensional Laplacian embedding space that respects graphical structure. This reduces the problem to that of comparing point clouds in a Euclidean space. A distance can then be computed efficiently via a natural sliced Wasserstein approach. We show that the ELD is a pseudo-metric and is invariant under graph isomorphism. We provide intuitive interpretations of the ELD using tools from spectral graph theory. We test the efficacy of the ELD approach extensively on both simulated and real data. Results obtained are excellent.

In this work, a sequential decoder for convolutional codes over channels that are vulnerable to insertion, deletion, and substitution errors, is described and analyzed. The decoder expands the code trellis by introducing a new channel state variable, called drift state, as proposed by Davey-MacKay. A suitable decoding metric on that trellis for sequential decoding is derived, in a manner that generalizes the original Fano metric. Under low-noise environments, this approach reduces the decoding complexity by a couple orders of magnitude in comparison to Viterbi's algorithm, albeit at relatively higher frame error rates. An analytical method to determine the computational cutoff rate is also suggested. This analysis is supported with numerical evaluations of frame error rates and computational complexity, which are compared with respect to optimal Viterbi decoding.

A toric code, introduced by Hansen to extend the Reed-Solomon code as a $k$-dimensional subspace of $\mathbb{F}_q^n$, is determined by a toric variety or its associated integral convex polytope $P \subseteq [0,q-2]^n$, where $k=|P \cap \mathbb{Z}^n|$ (the number of integer lattice points of $P$). There are two relevant parameters that determine the quality of a code: the information rate, which measures how much information is contained in a single bit of each codeword; and the relative minimum distance, which measures how many errors can be corrected relative to how many bits each codeword has. Soprunov and Soprunova defined a good infinite family of codes to be a sequence of codes of unbounded polytope dimension such that neither the corresponding information rates nor relative minimum distances go to 0 in the limit. We examine different ways of constructing families of codes by considering polytope operations such as the join and direct sum. In doing so, we give conditions under which no good family can exist and strong evidence that there is no such good family of codes.

This paper is concerned with list decoding of $2$-interleaved binary alternant codes. The principle of the proposed algorithm is based on a combination of a list decoding algorithm for (interleaved) Reed-Solomon codes and an algorithm for (non-interleaved) alternant codes. The decoding radius exceeds the binary Johnson radius and the newly derived upper bound on the returned list size scales polynomially in the code parameters. Further, we provide simulation results on the probability of successful decoding by the proposed algorithm.

Ongoing traffic changes, including those triggered by the COVID-19 pandemic, reveal the necessity to adapt our public transport systems to the ever-changing users' needs. This work shows that single and multi objective stances can be synergistically combined to better answer the transit network design problem (TNDP). Single objective formulations are dynamically inferred from the rating of networks in the approximated (multi-objective) Pareto Front, where a regression approach is used to infer the optimal weights of transfer needs, times, distances, coverage, and costs. As a guiding case study, the solution is applied to the multimodal public transport network in the city of Lisbon, Portugal. The system takes individual trip data given by smartcard validations at CARRIS buses and METRO subway stations and uses them to estimate the origin-destination demand in the city. Then, Genetic Algorithms are used, considering both single and multi objective approaches, to redesign the bus network that better fits the observed traffic demand. The proposed TNDP optimization proved to improve results, with reductions in objective functions of up to 28.3%. The system managed to extensively reduce the number of routes, and all passenger related objectives, including travel time and transfers per trip, significantly improve. Grounded on automated fare collection data, the system can incrementally redesign the bus network to dynamically handle ongoing changes to the city traffic.

We introduce a novel framework for optimization based on energy-conserving Hamiltonian dynamics in a strongly mixing (chaotic) regime and establish its key properties analytically and numerically. The prototype is a discretization of Born-Infeld dynamics, with a squared relativistic speed limit depending on the objective function. This class of frictionless, energy-conserving optimizers proceeds unobstructed until slowing naturally near the minimal loss, which dominates the phase space volume of the system. Building from studies of chaotic systems such as dynamical billiards, we formulate a specific algorithm with good performance on machine learning and PDE-solving tasks, including generalization. It cannot stop at a high local minimum and cannot overshoot the global minimum, yielding an advantage in non-convex loss functions, and proceeds faster than GD+momentum in shallow valleys.

Given a graph $G = (V,E)$, a threshold function $t~ :~ V \rightarrow \mathbb{N}$ and an integer $k$, we study the Harmless Set problem, where the goal is to find a subset of vertices $S \subseteq V$ of size at least $k$ such that every vertex $v\in V$ has less than $t(v)$ neighbors in $S$. We enhance our understanding of the problem from the viewpoint of parameterized complexity. Our focus lies on parameters that measure the structural properties of the input instance. We show that the problem is W[1]-hard parameterized by a wide range of fairly restrictive structural parameters such as the feedback vertex set number, pathwidth, treedepth, and even the size of a minimum vertex deletion set into graphs of pathwidth and treedepth at most three. On dense graphs, we show that the problem is W[1]-hard parameterized by cluster vertex deletion number. We also show that the Harmless Set problem with majority thresholds is W[1]-hard when parameterized by the treewidth of the input graph. We prove that the Harmless Set problem can be solved in polynomial time on graph with bounded cliquewidth. On the positive side, we obtain fixed-parameter algorithms for the problem with respect to neighbourhood diversity, twin cover and vertex integrity of the input graph. We show that the problem parameterized by the solution size is fixed parameter tractable on planar graphs. We thereby resolve two open questions stated in C. Bazgan and M. Chopin (2014) concerning the complexity of {\sc Harmless Set} parameterized by the treewidth of the input graph and on planar graphs with respect to the solution size.

The Transformer is widely used in natural language processing tasks. To train a Transformer however, one usually needs a carefully designed learning rate warm-up stage, which is shown to be crucial to the final performance but will slow down the optimization and bring more hyper-parameter tunings. In this paper, we first study theoretically why the learning rate warm-up stage is essential and show that the location of layer normalization matters. Specifically, we prove with mean field theory that at initialization, for the original-designed Post-LN Transformer, which places the layer normalization between the residual blocks, the expected gradients of the parameters near the output layer are large. Therefore, using a large learning rate on those gradients makes the training unstable. The warm-up stage is practically helpful for avoiding this problem. On the other hand, our theory also shows that if the layer normalization is put inside the residual blocks (recently proposed as Pre-LN Transformer), the gradients are well-behaved at initialization. This motivates us to remove the warm-up stage for the training of Pre-LN Transformers. We show in our experiments that Pre-LN Transformers without the warm-up stage can reach comparable results with baselines while requiring significantly less training time and hyper-parameter tuning on a wide range of applications.