30,523 research outputs found
Parametric Polynomial Time Perceptron Rescaling Algorithm
Let us consider a linear feasibility problem with a possibly infinite number of inequality constraints posed in an on-line setting: an algorithm suggests a candidate solution, and the oracle either confirms its feasibility, or outputs a violated constraint vector. This model can be solved by subgradient optimisation algorithms for non-smooth functions, also known as the perceptron algorithms in the machine learning community, and its solvability depends on the problem dimension and the radius of the constraint set. The classical perceptron algorithm may have an exponential complexity in the worst case when the radius is infinitesimal [1]. To overcome this difficulty, the space dilation technique was exploited in the ellipsoid algorithm to make its running time polynomial [3]. A special case of the space dilation, the rescaling procedure is utilised in the perceptron rescaling algorithm [2] with a probabilistic approach to choosing the direction of dilation. A parametric version of the perceptron rescaling algorithm is the focus of this work. It is demonstrated that some fixed parameters of the latter algorithm (the initial estimate of the radius and the relaxation parameter) may be modified and adapted for particular problems. The generalised theoretical framework allows to determine convergence of the algorithm with any chosen set of values of these parameters, and suggests a potential way of decreasing the complexity of the algorithm which remains the subject of current research
Entropy landscape of solutions in the binary perceptron problem
The statistical picture of the solution space for a binary perceptron is
studied. The binary perceptron learns a random classification of input random
patterns by a set of binary synaptic weights. The learning of this network is
difficult especially when the pattern (constraint) density is close to the
capacity, which is supposed to be intimately related to the structure of the
solution space. The geometrical organization is elucidated by the entropy
landscape from a reference configuration and of solution-pairs separated by a
given Hamming distance in the solution space. We evaluate the entropy at the
annealed level as well as replica symmetric level and the mean field result is
confirmed by the numerical simulations on single instances using the proposed
message passing algorithms. From the first landscape (a random configuration as
a reference), we see clearly how the solution space shrinks as more constraints
are added. From the second landscape of solution-pairs, we deduce the
coexistence of clustering and freezing in the solution space.Comment: 21 pages, 6 figures, version accepted by Journal of Physics A:
Mathematical and Theoretica
The impact of architecture on the performance of artificial neural networks
A number of researchers have investigated the impact of network architecture on the performance of artificial neural networks. Particular attention has been paid to the impact on the performance of the multi-layer perceptron of architectural issues, and the use of various strategies to attain an optimal network structure. However, there are still perceived limitations with the multi-layer perceptron and networks that employ a different architecture to the multi-layer perceptron have gained in popularity in recent years, particularly, networks that implement a more localised solution, where the solution in one area of the problem space does not impact, or has a minimal impact, on other areas of the space. In this study, we discuss the major architectural issues affecting the performance of a multi-layer perceptron, before moving on to examine in detail the performance of a new localised network, namely the bumptree. The work presented here examines the impact on the performance of artificial neural networks of employing alternative networks to the long established multi-layer perceptron. In particular, networks that impose a solution where the impact of each parameter in the final network architecture has a localised impact on the problem space being modelled are examined. The alternatives examined are the radial basis function and bumptree neural networks, and the impact of architectural issues on the performance of these networks is examined. Particular attention is paid to the bumptree, with new techniques for both developing the bumptree structure and employing this structure to classify patterns being examined
Constructive Preference Elicitation over Hybrid Combinatorial Spaces
Preference elicitation is the task of suggesting a highly preferred
configuration to a decision maker. The preferences are typically learned by
querying the user for choice feedback over pairs or sets of objects. In its
constructive variant, new objects are synthesized "from scratch" by maximizing
an estimate of the user utility over a combinatorial (possibly infinite) space
of candidates. In the constructive setting, most existing elicitation
techniques fail because they rely on exhaustive enumeration of the candidates.
A previous solution explicitly designed for constructive tasks comes with no
formal performance guarantees, and can be very expensive in (or unapplicable
to) problems with non-Boolean attributes. We propose the Choice Perceptron, a
Perceptron-like algorithm for learning user preferences from set-wise choice
feedback over constructive domains and hybrid Boolean-numeric feature spaces.
We provide a theoretical analysis on the attained regret that holds for a large
class of query selection strategies, and devise a heuristic strategy that aims
at optimizing the regret in practice. Finally, we demonstrate its effectiveness
by empirical evaluation against existing competitors on constructive scenarios
of increasing complexity.Comment: AAAI 2018, computing methodologies, machine learning, learning
paradigms, supervised learning, structured output
Playing Billiard in Version Space
A ray-tracing method inspired by ergodic billiards is used to estimate the
theoretically best decision rule for a set of linear separable examples. While
the Bayes-optimum requires a majority decision over all Perceptrons separating
the example set, the problem considered here corresponds to finding the single
Perceptron with best average generalization probability. For randomly
distributed examples the billiard estimate agrees with known analytic results.
In real-life classification problems the generalization error is consistently
reduced compared to the maximal stability Perceptron.Comment: uuencoded, gzipped PostScript file, 127576 bytes To recover 1) save
file as bayes.uue. Then 2) uudecode bayes.uue and 3) gunzip bayes.ps.g
Multifractality and percolation in the coupling space of perceptrons
The coupling space of perceptrons with continuous as well as with binary
weights gets partitioned into a disordered multifractal by a set of random input patterns. The multifractal spectrum can be
calculated analytically using the replica formalism. The storage capacity and
the generalization behaviour of the perceptron are shown to be related to
properties of which are correctly described within the replica
symmetric ansatz. Replica symmetry breaking is interpreted geometrically as a
transition from percolating to non-percolating cells. The existence of empty
cells gives rise to singularities in the multifractal spectrum. The analytical
results for binary couplings are corroborated by numerical studies.Comment: 13 pages, revtex, 4 eps figures, version accepted for publication in
Phys. Rev.
Stability of the replica symmetric solution in diluted perceptron learning
We study the role played by the dilution in the average behavior of a
perceptron model with continuous coupling with the replica method. We analyze
the stability of the replica symmetric solution as a function of the dilution
field for the generalization and memorization problems. Thanks to a Gardner
like stability analysis we show that at any fixed ratio between the
number of patterns M and the dimension N of the perceptron (),
there exists a critical dilution field above which the replica symmetric
ansatz becomes unstable.Comment: Stability of the solution in arXiv:0907.3241, 13 pages, (some typos
corrected
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