4,306 research outputs found
The real projective spaces in homotopy type theory
Homotopy type theory is a version of Martin-L\"of type theory taking
advantage of its homotopical models. In particular, we can use and construct
objects of homotopy theory and reason about them using higher inductive types.
In this article, we construct the real projective spaces, key players in
homotopy theory, as certain higher inductive types in homotopy type theory. The
classical definition of RP(n), as the quotient space identifying antipodal
points of the n-sphere, does not translate directly to homotopy type theory.
Instead, we define RP(n) by induction on n simultaneously with its tautological
bundle of 2-element sets. As the base case, we take RP(-1) to be the empty
type. In the inductive step, we take RP(n+1) to be the mapping cone of the
projection map of the tautological bundle of RP(n), and we use its universal
property and the univalence axiom to define the tautological bundle on RP(n+1).
By showing that the total space of the tautological bundle of RP(n) is the
n-sphere, we retrieve the classical description of RP(n+1) as RP(n) with an
(n+1)-cell attached to it. The infinite dimensional real projective space,
defined as the sequential colimit of the RP(n) with the canonical inclusion
maps, is equivalent to the Eilenberg-MacLane space K(Z/2Z,1), which here arises
as the subtype of the universe consisting of 2-element types. Indeed, the
infinite dimensional projective space classifies the 0-sphere bundles, which
one can think of as synthetic line bundles.
These constructions in homotopy type theory further illustrate the utility of
homotopy type theory, including the interplay of type theoretic and homotopy
theoretic ideas.Comment: 8 pages, to appear in proceedings of LICS 201
Sets in homotopy type theory
Homotopy Type Theory may be seen as an internal language for the
-category of weak -groupoids which in particular models the
univalence axiom. Voevodsky proposes this language for weak -groupoids
as a new foundation for mathematics called the Univalent Foundations of
Mathematics. It includes the sets as weak -groupoids with contractible
connected components, and thereby it includes (much of) the traditional set
theoretical foundations as a special case. We thus wonder whether those
`discrete' groupoids do in fact form a (predicative) topos. More generally,
homotopy type theory is conjectured to be the internal language of `elementary'
-toposes. We prove that sets in homotopy type theory form a -pretopos. This is similar to the fact that the -truncation of an
-topos is a topos. We show that both a subobject classifier and a
-object classifier are available for the type theoretical universe of sets.
However, both of these are large and moreover, the -object classifier for
sets is a function between -types (i.e. groupoids) rather than between sets.
Assuming an impredicative propositional resizing rule we may render the
subobject classifier small and then we actually obtain a topos of sets
Incremental Sparse Bayesian Ordinal Regression
Ordinal Regression (OR) aims to model the ordering information between
different data categories, which is a crucial topic in multi-label learning. An
important class of approaches to OR models the problem as a linear combination
of basis functions that map features to a high dimensional non-linear space.
However, most of the basis function-based algorithms are time consuming. We
propose an incremental sparse Bayesian approach to OR tasks and introduce an
algorithm to sequentially learn the relevant basis functions in the ordinal
scenario. Our method, called Incremental Sparse Bayesian Ordinal Regression
(ISBOR), automatically optimizes the hyper-parameters via the type-II maximum
likelihood method. By exploiting fast marginal likelihood optimization, ISBOR
can avoid big matrix inverses, which is the main bottleneck in applying basis
function-based algorithms to OR tasks on large-scale datasets. We show that
ISBOR can make accurate predictions with parsimonious basis functions while
offering automatic estimates of the prediction uncertainty. Extensive
experiments on synthetic and real word datasets demonstrate the efficiency and
effectiveness of ISBOR compared to other basis function-based OR approaches
Automatic construction of known-item finding test beds
This work is an initial study on the utility of automatically generated queries for evaluating known-item retrieval and how such queries compare to real queries. The main advantage of automatically generating queries is that for any given test collection numerous queries can be produced at minimal cost. For evaluation, this has huge ramifications as state-of-the-art algorithms can be tested on different types of generated queries which mimic particular querying styles that a user may adopt. Our approach draws upon previous research in IR which has probabilistically generated simulated queries for other purposes [2, 3]
Differentiable Unbiased Online Learning to Rank
Online Learning to Rank (OLTR) methods optimize rankers based on user
interactions. State-of-the-art OLTR methods are built specifically for linear
models. Their approaches do not extend well to non-linear models such as neural
networks. We introduce an entirely novel approach to OLTR that constructs a
weighted differentiable pairwise loss after each interaction: Pairwise
Differentiable Gradient Descent (PDGD). PDGD breaks away from the traditional
approach that relies on interleaving or multileaving and extensive sampling of
models to estimate gradients. Instead, its gradient is based on inferring
preferences between document pairs from user clicks and can optimize any
differentiable model. We prove that the gradient of PDGD is unbiased w.r.t.
user document pair preferences. Our experiments on the largest publicly
available Learning to Rank (LTR) datasets show considerable and significant
improvements under all levels of interaction noise. PDGD outperforms existing
OLTR methods both in terms of learning speed as well as final convergence.
Furthermore, unlike previous OLTR methods, PDGD also allows for non-linear
models to be optimized effectively. Our results show that using a neural
network leads to even better performance at convergence than a linear model. In
summary, PDGD is an efficient and unbiased OLTR approach that provides a better
user experience than previously possible.Comment: Conference on Information and Knowledge Management 201
Optimizing Ranking Models in an Online Setting
Online Learning to Rank (OLTR) methods optimize ranking models by directly
interacting with users, which allows them to be very efficient and responsive.
All OLTR methods introduced during the past decade have extended on the
original OLTR method: Dueling Bandit Gradient Descent (DBGD). Recently, a
fundamentally different approach was introduced with the Pairwise
Differentiable Gradient Descent (PDGD) algorithm. To date the only comparisons
of the two approaches are limited to simulations with cascading click models
and low levels of noise. The main outcome so far is that PDGD converges at
higher levels of performance and learns considerably faster than DBGD-based
methods. However, the PDGD algorithm assumes cascading user behavior,
potentially giving it an unfair advantage. Furthermore, the robustness of both
methods to high levels of noise has not been investigated. Therefore, it is
unclear whether the reported advantages of PDGD over DBGD generalize to
different experimental conditions. In this paper, we investigate whether the
previous conclusions about the PDGD and DBGD comparison generalize from ideal
to worst-case circumstances. We do so in two ways. First, we compare the
theoretical properties of PDGD and DBGD, by taking a critical look at
previously proven properties in the context of ranking. Second, we estimate an
upper and lower bound on the performance of methods by simulating both ideal
user behavior and extremely difficult behavior, i.e., almost-random
non-cascading user models. Our findings show that the theoretical bounds of
DBGD do not apply to any common ranking model and, furthermore, that the
performance of DBGD is substantially worse than PDGD in both ideal and
worst-case circumstances. These results reproduce previously published findings
about the relative performance of PDGD vs. DBGD and generalize them to
extremely noisy and non-cascading circumstances.Comment: European Conference on Information Retrieval (ECIR) 201
Balancing Speed and Quality in Online Learning to Rank for Information Retrieval
In Online Learning to Rank (OLTR) the aim is to find an optimal ranking model
by interacting with users. When learning from user behavior, systems must
interact with users while simultaneously learning from those interactions.
Unlike other Learning to Rank (LTR) settings, existing research in this field
has been limited to linear models. This is due to the speed-quality tradeoff
that arises when selecting models: complex models are more expressive and can
find the best rankings but need more user interactions to do so, a requirement
that risks frustrating users during training. Conversely, simpler models can be
optimized on fewer interactions and thus provide a better user experience, but
they will converge towards suboptimal rankings. This tradeoff creates a
deadlock, since novel models will not be able to improve either the user
experience or the final convergence point, without sacrificing the other. Our
contribution is twofold. First, we introduce a fast OLTR model called Sim-MGD
that addresses the speed aspect of the speed-quality tradeoff. Sim-MGD ranks
documents based on similarities with reference documents. It converges rapidly
and, hence, gives a better user experience but it does not converge towards the
optimal rankings. Second, we contribute Cascading Multileave Gradient Descent
(C-MGD) for OLTR that directly addresses the speed-quality tradeoff by using a
cascade that enables combinations of the best of two worlds: fast learning and
high quality final convergence. C-MGD can provide the better user experience of
Sim-MGD while maintaining the same convergence as the state-of-the-art MGD
model. This opens the door for future work to design new models for OLTR
without having to deal with the speed-quality tradeoff.Comment: CIKM 2017, Proceedings of the 2017 ACM on Conference on Information
and Knowledge Managemen
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