362 research outputs found
LIPIcs, Volume 251, ITCS 2023, Complete Volume
LIPIcs, Volume 251, ITCS 2023, Complete Volum
Providing Private and Fast Data Access for Cloud Systems
Cloud storage and computing systems have become the backbone of many applications such as streaming (Netflix, YouTube), storage (Dropbox, Google Drive), and computing (Amazon Elastic Computing, Microsoft Azure). To address the ever growing demand for storage and computing requirements of these applications, cloud services are typically im-plemented over a large-scale distributed data storage system. Cloud systems are expected to provide the following two pivotal services for the users: 1) private content access and 2) fast content access. The goal of this thesis is to understand and address some of the challenges that need to be overcome to provide these two services.
The first part of this thesis focuses on private data access in distributed systems. In particular, we contribute to the areas of Private Information Retrieval (PIR) and Private Computation (PC). In the PIR problem, there is a user who wishes to privately retrieve a subset of files belonging to a database stored on a single or multiple remote server(s). In the PC problem, the user wants to privately compute functions of a subset of files in the database. The PIR and PC problems seek the most efficient solutions with the minimum download cost that enable the user to retrieve or compute what it wants privately.
We establish fundamental bounds on the minimum download cost required for guaran-teeing the privacy requirement in some practical and realistic settings of the PIR and PC problems and develop novel and efficient privacy-preserving algorithms for these settings. In particular, we study the single-server and multi-server settings of PIR in which the user initially has a random linear combination of a subset of files in the database as side in-formation, referred to as PIR with coded side information. We also study the multi-server setting of the PC in which the user wants to privately compute multiple linear combinations of a subset of files in the database, referred to as Private Linear Transformation.
The second part of this thesis focuses on fast content access in distributed systems. In particular, we study the use of erasure coding to handle data access requests in distributed storage and computing systems. Service rate region is an important performance metric for coded distributed systems, which expresses the set of all data access request rates that can be simultaneously served by the system. In this context, two classes of problems arise: 1) characterizing the service rate region of a given storage scheme and finding the optimal request allocation, and 2) designing the underlying erasure code to handle a given desired service rate region.
As contributions along the first class of problems, we characterize the service rate region of systems with some common coding schemes such as Simplex codes and Reed-Muller codes by introducing two novel techniques: 1) fractional matching and vertex cover on graph representation of codes, and 2) geometric representations of codes. Moreover, along the second class of code design, we establish some lower bounds on the minimum storage required to handle a desired service rate region for a coded distributed system and in some regimes, we design efficient storage schemes that provide the desired service rate region while minimizing the storage requirements
Using gadget construction in structural convergence
Structural convergence is a framework for convergence of graphs and relational struc- tures based on the probability of satisfaction of first-order formulas. We consider gadget construction, which is a ubiquitous tool in many areas of mathematics, as a method for constructing convergent sequences of structures. Both for elementary and local conver- gence, we investigate the behavior of a sequence created by gadget construction from convergent sequences of base structures and gadgets. We show that elementary conver- gence is always preserved while additional assumptions are necessary for local convergence as witnessed by several examples. We give various different conditions that ensure local convergence. One of them states that the resulting sequence is local convergent if the replaced edges are dense in the sequence of base structures. The sufficient conditions are partially complemented by inverse theorems. 1Strukturální konvergence je framework konvergence grafů a relačních struktur zalo- žený na počítání pravděpodobnosti splnění formulí predikátové logiky. V práci navrhu- jeme gadgetovou konstrukci, která nalezla uplatnění v mnoha oblastech matematiky, jako metodu výroby konvergentních posloupností relačních struktur. Pro elementární a lokální konvergenci studujeme chování posloupnosti struktur vytvořených gadgetovou konstrukcí z konvergentních poslouností základních struktur a gadgetů. Ukazujeme, že elementární konvergence je vždy zachována, zatímco v případě lokální konvergence je potřeba dalších předpokladů, což ilustrujeme řadou příkladů. Dokazujeme několik postačujících podmínek pro zachování lokální konvergence. Jedna z nich říká, že posloupnost vytvořených struk- tur je lokálně konvergentní, pokud v posloupnosti základních struktur byly nahrazované hrany husté. Představené postačující podmínky částečně komplementujeme inverzními větami. 1Computer Science Institute of Charles UniversityInformatický ústav Univerzity KarlovyFaculty of Mathematics and PhysicsMatematicko-fyzikální fakult
The externally definable Ramsey property and fixed points on type spaces
We discuss the externally definable Ramsey property, a weakening of the
Ramsey property for ultrahomogeneous structures, where the only colourings
considered are those that are externally definable: that is, definable with
parameters in an elementary extension. We show a number of basic results
analogous to the classical Ramsey theory, and show that, for an
ultrahomogeneous structure M, the externally definable Ramsey property is
equivalent to the dynamical statement that, for each natural number n, every
subflow of the space of n-types with parameters in M has a fixed point. We
discuss a range of examples, including results regarding the lexicographic
product of structures.Comment: 42 pages, 1 figur
LIPIcs, Volume 261, ICALP 2023, Complete Volume
LIPIcs, Volume 261, ICALP 2023, Complete Volum
Regularization and Optimal Multiclass Learning
The quintessential learning algorithm of empirical risk minimization (ERM) is
known to fail in various settings for which uniform convergence does not
characterize learning. It is therefore unsurprising that the practice of
machine learning is rife with considerably richer algorithmic techniques for
successfully controlling model capacity. Nevertheless, no such technique or
principle has broken away from the pack to characterize optimal learning in
these more general settings.
The purpose of this work is to characterize the role of regularization in
perhaps the simplest setting for which ERM fails: multiclass learning with
arbitrary label sets. Using one-inclusion graphs (OIGs), we exhibit optimal
learning algorithms that dovetail with tried-and-true algorithmic principles:
Occam's Razor as embodied by structural risk minimization (SRM), the principle
of maximum entropy, and Bayesian reasoning. Most notably, we introduce an
optimal learner which relaxes structural risk minimization on two dimensions:
it allows the regularization function to be "local" to datapoints, and uses an
unsupervised learning stage to learn this regularizer at the outset. We justify
these relaxations by showing that they are necessary: removing either dimension
fails to yield a near-optimal learner. We also extract from OIGs a
combinatorial sequence we term the Hall complexity, which is the first to
characterize a problem's transductive error rate exactly.
Lastly, we introduce a generalization of OIGs and the transductive learning
setting to the agnostic case, where we show that optimal orientations of
Hamming graphs -- judged using nodes' outdegrees minus a system of
node-dependent credits -- characterize optimal learners exactly. We demonstrate
that an agnostic version of the Hall complexity again characterizes error rates
exactly, and exhibit an optimal learner using maximum entropy programs.Comment: 40 pages, 2 figure
Generalised Indiscernibles, Dividing Lines, and Products of Structures
Generalised indiscernibles highlight a strong link between model theory and
structural Ramsey theory. In this paper, we use generalised indiscernibles as
tools to prove results in both these areas. More precisely, we first show that
a reduct of an ultrahomogenous -categorical structure which has
higher arity than the original structure cannot be Ramsey. In particular, the
only nontrivial Ramsey reduct of the generically ordered random -hypergraph
is the linear order. We then turn our attention to model-theoretic dividing
lines that are characterised by collapsing generalised indiscernibles, and
prove, for these dividing lines, several transfer principles in (full and
lexicographic) products of structures. As an application, we construct new
algorithmically tame classes of graphs
String Diagrams for -calculi and Functional Computation
This tutorial gives an advanced introduction to string diagrams and graph
languages for higher-order computation. The subject matter develops in a
principled way, starting from the two dimensional syntax of key categorical
concepts such as functors, adjunctions, and strictification, and leading up to
Cartesian Closed Categories, the core mathematical model of the lambda calculus
and of functional programming languages. This methodology inverts the usual
approach of proceeding from syntax to a categorical interpretation, by
rationally reconstructing a syntax from the categorical model. The result is a
graph syntax -- more precisely, a hierarchical hypergraph syntax -- which in
many ways is shown to be an improvement over the conventional linear term
syntax. The rest of the tutorial focuses on applications of interest to
programming languages: operational semantics, general frameworks for type
inference, and complex whole-program transformations such as closure conversion
and automatic differentiation
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