arXiv subjects

Marcin Michalski

Publications and source records attributed to Marcin Michalski.

At least 19 recordsLinked to original sources

arXiv2025

An Ideal Zoo in the Baire Space

In this paper, we study the translations into the Baire space of several well-known $\sigma$-ideals and families originally defined on the Cantor space, using their combinatorial characterizations. These include the ideals of null sets, small sets, those generated by closed measure-zero sets, and the meager sets, leading to their "fake" analogues in the Baire space. We also parametrize families related to null sets by functions from $\omega^\omega$. Several structural properties and relations between these families are investigated, including whether they form ideals, the existence of large chains and antichains, orthogonality, the $\kappa$-chain condition, and the determination of certain cardinal invariants.

math.GN↗

arXiv2024

On algebraic sums, trees and ideals in the Baire space

We work in the Baire space $\mathbb{Z}^\omega$ equipped with the coordinate-wise addition $+$. Consider a $\sigma-$ideal $\mathcal{I}$ and a family $\mathbb{T}$ of some kind of perfect trees. We are interested in results of the form: for every $A\in \mathcal{I}$ and a tree $T\in\mathbb{T}$ there exists $T'\in \mathbb{T}, T'\subseteq T$ such that $A+\underbrace{[T']+[T']+\dots +[T']}_{\text{n--times}}\in \mathcal{I}$ for each $n\in\omega$. Explored tree types include perfect trees, uniformly perfect trees, Miller trees, Laver trees and $\omega-$Silver trees. The latter kind of trees is an analogue of Silver trees from the Cantor space. Besides the standard $\sigma$-ideal $\mathcal{M}$ of meager sets, we also analyze $\mathcal{M}_-$ and fake null sets $\mathcal{N}$. The latter two are born out of the characterizations of their respective analogues in the Cantor space. The key ingredient in proofs were combinatorial characterizations of these ideals in the Baire space.

math.GN↗

arXiv2024

On algebraic sums, trees and ideals in the Cantor space

We work in the Cantor space $2^ω$. The results of the paper adhere the following pattern. Let $\mathcal{I}\in \{\mathcal{M}, \mathcal{N}, \mathcal{M}\cap \mathcal{N}, \mathcal{E}\}$ and $T$ be a perfect, uniformly perfect or Silver tree. Then for every $A\in \mathcal{I}$ there exists $T'\subseteq T$ of the same kind as $T$ such that $A+\underbrace{[T']+[T']+\dots +[T']}_{\text{n--times}}\in \mathcal{I}$ for each $n\inω$. We also prove weaker statements for splitting trees. For the case $\mathcal{E}$ we also provide a simple characterization of basis of $\mathcal{E}$. We use these results to prove that the algebraic sum of a generalized Luzin set and a generalized Sierpiński set belongs to $u_0$ and $v_0$, provided that $\mathfrak{c}$ is a regular cardinal.

math.LO↗

arXiv2024

Around Eggleston Theorem

The motivation of this work are the two classical theorems on inscribing rectangles and squares into large subsets of the plane, namely Eggleston Theorem and Mycielski Theorem. Using Shoenfield Absoluteness Theorem we prove that for every Borel subset of the plane with uncountably many positive (with respect to measure or category) vertical section contains a rectangle $P\times B$ where $P$ is perfect and $B$ is Borel and positive. We also obtained a variant of Eggleston Theorem regarding the $σ$-ideal $\mathcal(E)$ generated by closed sets of measure zero. Furthermore we proved that every comeager (resp. conull) subset of the plane contains a rectangle $[T]\times H$, where $T$ is a Spinas tree containing a Silver tree and $H$ is comeager (resp. conull). Moreover we obtained a common generalization of Eggleston Theorem and Mycielski Theorem stating that every comeager (resp. conull) subset of the plane contains a rectangle $[T]\times H$ modulo diagonal, where $T$ is a uniformly perfect tree, $H$ is comeager (resp. conull) and $[T]\subseteq H$.

math.LO↗

arXiv2021

Ideals with Smital properties

A $σ$-ideal $\mathcal{I}$ on a Polish group $(X,+)$ has Smital Property if for every dense set $D$ and a Borel $\mathcal{I}$-positive set $B$ the algebraic sum $D+B$ is a complement of a set from $\mathcal{I}$. We consider several variants of this property and study their connections with countable chain condition, maximality and how well they are preserved via Fubini products.

math.GN↗

arXiv2020

What Matters In On-Policy Reinforcement Learning? A Large-Scale Empirical Study

In recent years, on-policy reinforcement learning (RL) has been successfully applied to many different continuous control tasks. While RL algorithms are often conceptually simple, their state-of-the-art implementations take numerous low- and high-level design decisions that strongly affect the performance of the resulting agents. Those choices are usually not extensively discussed in the literature, leading to discrepancy between published descriptions of algorithms and their implementations. This makes it hard to attribute progress in RL and slows down overall progress [Engstrom'20]. As a step towards filling that gap, we implement >50 such ``choices'' in a unified on-policy RL framework, allowing us to investigate their impact in a large-scale empirical study. We train over 250'000 agents in five continuous control environments of different complexity and provide insights and practical recommendations for on-policy training of RL agents.

cs.LG↗

arXiv2020

Google Research Football: A Novel Reinforcement Learning Environment

Recent progress in the field of reinforcement learning has been accelerated by virtual learning environments such as video games, where novel algorithms and ideas can be quickly tested in a safe and reproducible manner. We introduce the Google Research Football Environment, a new reinforcement learning environment where agents are trained to play football in an advanced, physics-based 3D simulator. The resulting environment is challenging, easy to use and customize, and it is available under a permissive open-source license. In addition, it provides support for multiplayer and multi-agent experiments. We propose three full-game scenarios of varying difficulty with the Football Benchmarks and report baseline results for three commonly used reinforcement algorithms (IMPALA, PPO, and Ape-X DQN). We also provide a diverse set of simpler scenarios with the Football Academy and showcase several promising research directions.

cs.LG↗

arXiv2020

A Large-scale Study of Representation Learning with the Visual Task Adaptation Benchmark

Representation learning promises to unlock deep learning for the long tail of vision tasks without expensive labelled datasets. Yet, the absence of a unified evaluation for general visual representations hinders progress. Popular protocols are often too constrained (linear classification), limited in diversity (ImageNet, CIFAR, Pascal-VOC), or only weakly related to representation quality (ELBO, reconstruction error). We present the Visual Task Adaptation Benchmark (VTAB), which defines good representations as those that adapt to diverse, unseen tasks with few examples. With VTAB, we conduct a large-scale study of many popular publicly-available representation learning algorithms. We carefully control confounders such as architecture and tuning budget. We address questions like: How effective are ImageNet representations beyond standard natural datasets? How do representations trained via generative and discriminative models compare? To what extent can self-supervision replace labels? And, how close are we to general visual representations?

cs.CV↗

arXiv2020

SEED RL: Scalable and Efficient Deep-RL with Accelerated Central Inference

We present a modern scalable reinforcement learning agent called SEED (Scalable, Efficient Deep-RL). By effectively utilizing modern accelerators, we show that it is not only possible to train on millions of frames per second but also to lower the cost of experiments compared to current methods. We achieve this with a simple architecture that features centralized inference and an optimized communication layer. SEED adopts two state of the art distributed algorithms, IMPALA/V-trace (policy gradients) and R2D2 (Q-learning), and is evaluated on Atari-57, DeepMind Lab and Google Research Football. We improve the state of the art on Football and are able to reach state of the art on Atari-57 three times faster in wall-time. For the scenarios we consider, a 40% to 80% cost reduction for running experiments is achieved. The implementation along with experiments is open-sourced so results can be reproduced and novel ideas tried out.

cs.LG↗

arXiv2019

Universal sets for ideals

In this paper we consider a notion of universal sets for ideals. We show that there exist universal sets of minimal Borel complexity for classic ideals like null subsets of $2^ω$ and meager subsets of any Polish space, and demonstrate that the existence of such sets is helpful in establishing some facts about the real line in generic extensions. We also construct universal sets for $\mathcal{E}$ - the $σ$-ideal generated by closed null subsets of $2^ω$, and for some ideals connected with forcing notions: $\mathcal{K}_σ$ subsets of $ω^ω$ and the Laver ideal. We also consider Fubini products of ideals and show that there are $Σ^0_3$ universal sets for $\mathcal{N}\otimes\mathcal{M}$ and $\mathcal{M}\otimes\mathcal{N}$.

math.GN↗

arXiv2019

Rediscovered theorem of Luzin

In 1934 N. N. Luzin proved in his short (but dense) paper \textit{Sur la decomposition des ensembles} that every set $X\subseteq \mathbb{R}$ can be decomposed into two full, with respect to Lebesgue measure or category, subsets. We will try to (at least partially) decipher the reasoning of Luzin and prove this result following his idea.

math.LO↗

arXiv2019

On some relations between properties of invariant $σ$-ideals in Polish spaces

In this paper we shall consider a couple of properties of $σ$-ideals and study relations between them. Namely we will prove that $\mathfrak{c}$-cc $σ$-ideals are tall and that the Weaker Smital Property implies that every Borel $\mathcal{I}$-positive set contains a witness for non($\mathcal{I}$) as well, as satisfying ccc and Fubini Property. We give also a characterization of nonmeasurability of $\mathcal{I}$-Luzin sets and prove that the ideal $[\mathbb{R}]^{\leqω}$ does not posses the Fubini Property using some interesting lemma about perfect sets.

math.GN↗

arXiv2019

A note on sets avoiding rational distances

In this paper we shall give a short proof of the result originally obtained by Ashutosh Kumar that for each $A\subset \mathbb{R}$ there exists $B\subset A$ full in $A$ such that no distance between two distinct points from $B$ is rational. We will construct a Bernstein subset of $\mathbb{R}$ which also avoids rational distances. We will show some cases in which the former result may be extended to subsets of $\mathbb{R}^2$, i. e. it remains true for measurable subsets of the plane and if $non(\mathcal{N})=cof(\mathcal{N})$ then for a given set of positive outer measure we may find its full subset which is a partial bijection and avoids rational distances.

math.GN↗

arXiv2019

Mycielski among trees

Two-dimensional version of the classical Mycielski theorem says that for every comeager or conull set $X\subseteq [0,1]^2$ there exists a perfect set $P\subseteq [0,1]$ such that $P\times P\subseteq X\cup Δ$. We consider generalizations of this theorem by replacing a perfect square with a rectangle $A\times B$, where $A$ and $B$ are bodies of other types of trees with $A\subseteq B$. In particular, we show that for every comeager $G_δ$ set $G\subseteq ω^ω\times ω^ω$ there exist a Miller tree $M$ and a uniformly perfect tree $P\subseteq M$ such that $[P]\times [M]\subseteq G\cupΔ$ and that $P$ cannot be a Miller tree. In the case of measure we show that for every subset $F$ of $2^ω\times 2^ω$ of full measure there exists a uniformly perfect tree $P\subseteq 2^{<ω}$ such that $[P]\times[P]\subseteq F\cupΔ$ and no side of such a rectangle can be a body of a Silver tree or a Miller tree. We also show some properties of forcing extensions of the real line from which we derive nonstandard proofs of Mycielski-like theorems via Shoenfield Absoluteness Theorem.

math.GN↗

arXiv2019

A Large-Scale Study on Regularization and Normalization in GANs

Generative adversarial networks (GANs) are a class of deep generative models which aim to learn a target distribution in an unsupervised fashion. While they were successfully applied to many problems, training a GAN is a notoriously challenging task and requires a significant number of hyperparameter tuning, neural architecture engineering, and a non-trivial amount of "tricks". The success in many practical applications coupled with the lack of a measure to quantify the failure modes of GANs resulted in a plethora of proposed losses, regularization and normalization schemes, as well as neural architectures. In this work we take a sober view of the current state of GANs from a practical perspective. We discuss and evaluate common pitfalls and reproducibility issues, open-source our code on Github, and provide pre-trained models on TensorFlow Hub.

cs.LG↗

arXiv2019

Towards Accurate Generative Models of Video: A New Metric & Challenges

Recent advances in deep generative models have lead to remarkable progress in synthesizing high quality images. Following their successful application in image processing and representation learning, an important next step is to consider videos. Learning generative models of video is a much harder task, requiring a model to capture the temporal dynamics of a scene, in addition to the visual presentation of objects. While recent attempts at formulating generative models of video have had some success, current progress is hampered by (1) the lack of qualitative metrics that consider visual quality, temporal coherence, and diversity of samples, and (2) the wide gap between purely synthetic video data sets and challenging real-world data sets in terms of complexity. To this extent we propose Fréchet Video Distance (FVD), a new metric for generative models of video, and StarCraft 2 Videos (SCV), a benchmark of game play from custom starcraft 2 scenarios that challenge the current capabilities of generative models of video. We contribute a large-scale human study, which confirms that FVD correlates well with qualitative human judgment of generated videos, and provide initial benchmark results on SCV.

cs.CV↗

arXiv2018

Are GANs Created Equal? A Large-Scale Study

Generative adversarial networks (GAN) are a powerful subclass of generative models. Despite a very rich research activity leading to numerous interesting GAN algorithms, it is still very hard to assess which algorithm(s) perform better than others. We conduct a neutral, multi-faceted large-scale empirical study on state-of-the art models and evaluation measures. We find that most models can reach similar scores with enough hyperparameter optimization and random restarts. This suggests that improvements can arise from a higher computational budget and tuning more than fundamental algorithmic changes. To overcome some limitations of the current metrics, we also propose several data sets on which precision and recall can be computed. Our experimental results suggest that future GAN research should be based on more systematic and objective evaluation procedures. Finally, we did not find evidence that any of the tested algorithms consistently outperforms the non-saturating GAN introduced in \cite{goodfellow2014generative}.

stat.ML↗

arXiv2018

MemGEN: Memory is All You Need

We propose a new learning paradigm called Deep Memory. It has the potential to completely revolutionize the Machine Learning field. Surprisingly, this paradigm has not been reinvented yet, unlike Deep Learning. At the core of this approach is the \textit{Learning By Heart} principle, well studied in primary schools all over the world. Inspired by poem recitation, or by $π$ decimal memorization, we propose a concrete algorithm that mimics human behavior. We implement this paradigm on the task of generative modeling, and apply to images, natural language and even the $π$ decimals as long as one can print them as text. The proposed algorithm even generated this paper, in a one-shot learning setting. In carefully designed experiments, we show that the generated samples are indistinguishable from the training examples, as measured by any statistical tests or metrics.

cs.LG↗