text stringlengths 128 2.05k |
|---|
Remark 4.13 Note that at the extinction steady state the stability condition ( 4.69 ) reads [EQUATION] This is a biologically relevant and natural condition, as it simply says that if mortality and cell division together is higher than recruitment of new cells into the population, then the population dies out. Note the... |
Finally, we establish an instability result, for the case when the semigroup governing the linearised equation is positive and irreducible. |
Proposition 4.14 Assume that condition ( 3.18 ) holds. Then [MATH] implies that the positive steady state [MATH] is unstable. Proof. |
We define the operators [MATH] and [MATH] as follows [EQUATION] Note that if ( 3.18 ) holds, then [MATH] generates a positive irreducible semigroup; moreover, its spectrum is determined by the eigenvalues of its generator [MATH] , which are of finite algebraic multiplicity. Also note that the existence of a (strictly) ... |
, we obtain that [EQUATION] hence the steady state [MATH] is unstable. 5. Examples and simulation results We present three examples to illustrate the asymptotic behaviour of solutions of the CE model. The first example, which is linear, assumes no self-renewal (i.e. telomere restoring capacity) of any cell, and shows a... |
[EQUATION] (Figure ). In all three examples the division modulus [MATH] is [EQUATION] where [MATH] in Example 1, and [MATH] in Examples 2 and 3. (Figure ). Cells which have telomere length below the critical value [MATH] have greatly reduced capacity to divide. Note that [MATH] for [MATH] , so [MATH] does not vanish in... |
[MATH] in Example 1, and [MATH] in Examples 2 and 3. Example 1. No restoration of telomeres occurs in Example 1. The rule governing the telomere length of a daughter cell of length [MATH] from a mother cell with length [MATH] is |
[EQUATION] where [MATH] is a Gaussian distribution in [MATH] with mean [MATH] and standard deviation [MATH] , and [MATH] is a normalization factor (Figure ). Note that [MATH] satisfies ( 3.18 ). The interpretation of this rule is that all daughter cells have telomere length strictly less than their mother cells. The es... |
Example 2. In this example restoration of telomeres occurs in cells with larger telomere lengths. The rule governing the telomere length of a daughter cell of length [MATH] from a mother cell with length [MATH] is |
[EQUATION] where [MATH] is a Gaussian distribution in [MATH] with mean [MATH] and standard deviation [MATH] ; and [MATH] is a normalization factor (Figure and Figure ). Note that [MATH] satisfies ( 3.18 ). The interpretation of this rule is that some daughter cells have telomere length equal or greater than their mothe... |
Example 3. Example 3 is the nonlinear version ( 4.56 )-( 4.58 ) of Example 2, with the same parameters. Additionally, the crowding term [MATH] in Example 3 is defined as |
[EQUATION] with [MATH] . The population stabilizes both in structuring variables [MATH] and [MATH] (see Figure 11 ), as well as in time (Figure 12 ). The self-renewal properties of the longest telomere length cells in Example 3, combined with the nonlinear crowding effect, result in convergence to equilibrium . As in E... |
6. Discussion In this work we have developed a mathematical formulation of cancer cell self-renewal for the clonal evolution model of tumour growth based on telomere restoration. The model allows for a continuum of telomere lengths, and thus contrasts to mathematical treatments of the cancer stem cell model, which inco... |
The cancer stem cell model formulates a hierarchal array of length classes with self-renewing cells in one longest telomere class. The clonal evolution model allows multiple classes of telomere length cells to have self-renewal capacity, corresponding to clonal structuring. |
Our model is thus more tractable for analysis and simulations, which we have provided here. In particular, in this work we focused on the effect of telomere restoring capacity of cancer cells. In particular we showed that the asymptotic behaviour of the linear model is determined by the spectral radius of an integral o... |
Naturally, many issues remain in the mathematical investigation of the clonal evolution model of tumour growth. Important questions which can be addressed in the framework of a mathematical model include the following. |
How do sequential mutations enter into the model formulation? How can quiescent cells be incorporated into the model? How can spatial heterogeneity be formulated in the equations? |
How can more complex nonlinearities be incorporated? How can the model be implemented with actual experimental data? These issues and the development of a general mathematical framework for analysing physiologically structured models with additional distributed structuring variables remain important subjects forf furth... |
# Source: arxiv 1511.05641 # Title: Net2Net: Accelerating Learning via Knowledge Transfer # Sections: all # Downloaded: 2026-03-03T04:14:15.738666+00:00 |
Net2Net : Accelerating Learning via Knowledge Transfer Abstract We introduce techniques for rapidly transferring the information stored in one neural net into another neural net. The main purpose is to accelerate the training of a significantly larger neural net. During real-world workflows, one often trains very many ... |
Introduction We propose a new kind of operation to perform on large neural networks: rapidly transfering knowledge contained in one neural network to another neural network. We call this the Net2Net family of operations. We use Net2Net as a general term describing any process of training a student network significantly... |
Traditionally, machine learning algorithms have been designed to receive a fixed dataset as input, initialize a new model with no knowledge, and train that model to convergence on that dataset. Real workflows are considerably more complicated than this idealized scenario. We advocate Net2Net operations as a useful tool... |
One way that real workflows deviate from the idealized scenario is that machine learning practitioners usually do not train only a single model on each dataset. Instead, one typically trains multiple models, with each model designed to improve upon the previous model in some way. Each step in the iterative design proce... |
Net2Net operations accelerate these workflows by rapidly transferring knowledge from the previous best model into each new model that an experimenter proposes. Instead of training each considered design of model for as much as a month, the experimenter can use Net2Net to train the model for a shorter period of time beg... |
More ambitiously, real machine learning systems will eventually become lifelong learning systems (Thrun, 1995 ; Silver et al., 2013 ; Mitchell et al., 2015 These machine learning systems need to continue to function for long periods of time and continually experience new training examples as these examples become avail... |
Net2Net operations allow us to smoothly instantiate a significantly larger model and immediately begin using it in our lifelong learning system, rather than needing to spend weeks or months re-train a larger model from scratch on the latest, largest version of the training set. |
Methodology In this section, we describe our new Net2Net operations and how we applied them on real deep neural nets. 2.1 Feature prediction |
We briefly experimented with a method that proved not to offer a significant advantage: training a large student network beginning from a random initialization, and introducing a set of extra “teacher prediction” layers into the student network. Specifically, several convolutional hidden layers of the student network w... |
The goal was that the teacher would provide a good internal representation for the task that the student could quickly copy and then begin to refine. The approach resembles the FitNets (Romero et al., 2014 strategy for training very thin networks of moderate depth. Unfortunately, we did not find that this method offere... |
Mahayri et al. ( 2015 independently observed that the benefits of the FitNets training strategy were eliminated after changing the model to use batch normalization. |
The FitNets-style approach to Net2Net learning is very general, in the sense that, if successful, it would allow any architecture of student network to learn from any architecture of teacher network. Though we were not able to make this general approach work, we encourage other researchers to attempt to design fully ge... |
Net2Net strategies in the future. We instead turned to different Net2Net strategies that were limited in scope but more effective. |
2.2 Function-preserving initializations We introduce two effective Net2Net strategies. Both are based on initializing the student network to represent the same function as the teacher, then continuing to train the student network by normal means. Specifically, suppose that a teacher network is represented by a function |
[MATH] where [MATH] is the input to the network, [MATH] is the output of the network, and [MATH] is the parameters of the network. In our experiments, [MATH] is defined by a convolutional network, |
[MATH] is an input image, and [MATH] is a vector of probabilities representing the conditional distribution over object categories given [MATH] |
Our strategy is to choose a new set of parameters [MATH] for a student network [MATH] such that [EQUATION] In our description of the method, we assume for simplicity that both the teacher and student networks contain compositions of standard linear neural network layers of the form |
[MATH] where [MATH] is the activations of hidden layer [MATH] [MATH] is the matrix of incoming weights for this layer, and [MATH] is an activation function, such as as the rectified linear activation function (Jarrett et al., 2009 ; Glorot et al., 2011 |
or the maxout activation function (Goodfellow et al., 2013 We use this layer formalization for clarity, but our method generalizes to arbitrary composition structure of these layers, such as Inception (Szegedy et al., 2014 All of our experiments are in fact performed using Inception networks. |
Though we describe the method in terms of matrix multiplication for simplicity, it readily extends to convolution (by observing that convolution is multiplication by a doubly block circulant matrix) and to the addition of biases (by treating biases as a row of |
[MATH] that are multiplied by a constant input of [MATH] ). We will also give a specific description for the convolutional case in subsequent discussions of the details of the method. Function-preserving initialization carries many advantages over other approaches to growing the network: |
The new, larger network immediately performs as well as the original network, rather than spending time passing through a period of low performance. |
Any change made to the network after initialization is guaranteed to be an improvement, so long as each local step is an improvement. Previous methods could fail to improve over the baseline even if each local step was guaranteed to be an improvement, because the initial change to the larger model size could worsen per... |
It is always “safe” to optimize all parameters in the network. There is never a stage where some layers receive harmful gradients and must be frozen. This is in contrast to approaches such as cascade correlation, where the old units are frozen in order to avoid making poor adaptations while attempting to influence the ... |
2.3 Net2WiderNet Input: [MATH] , the weight matrix of teacher net Use forward inference to generate a consistent random mapping [MATH] |
for [MATH] do [MATH] for [MATH] do [MATH] end for for [MATH] do [MATH] end for end for Output: [MATH] : the transformed weight matrix for wider net. |
Algorithm 1 Net2WiderNet Our first proposed transformation is Net2WiderNet transformation. This allows a layer to be replaced with a wider layer, meaning a layer that has more units. For convolution architectures, this means the layers will have more convolution channels. |
Let us start with a simple special case to illustrate the operation. Suppose that layer [MATH] and layer [MATH] are both fully connected layers, and layer [MATH] uses an elementwise non-linearity. To widen layer [MATH] , we replace [MATH] and [MATH] If layer [MATH] has [MATH] inputs and [MATH] outputs, and layer [MATH]... |
[MATH] Net2WiderNet allows us to replace layer [MATH] with a layer that has [MATH] outputs, with [MATH] We will introduce a random mapping function [MATH] , that satisfies |
[EQUATION] We introduce a new weight matrix [MATH] and [MATH] representing the weights for these layers in the new student network. Then the new weights are given by |
[EQUATION] Here, the first [MATH] columns of [MATH] are copied directly into [MATH] Columns [MATH] through [MATH] of [MATH] are created by choosing a random as defined in [MATH] The random selection is performed with replacement, so each column of [MATH] is copied potentially many times. For weights in [MATH] , we must... |
This description can be generalized to making multiple layers wider, with the layers composed as described by an arbitrary directed acyclic computation graph. This general procedure is illustrated by Fig. So far we have only discussed the use of a single random mapping function to expand one layer. We can in fact intro... |
To explain, we provide examples of two computational graph structures that impose specific constraints on the random remapping functions. |
One example is the layer structure used by batch normalization (Ioffe & Szegedy, 2015 The layer involves both a standard linear transformation, but also involves elementwise multiplication by learned parameters that allow the layer to represent any range of outputs despite the normalization operation. The random remapp... |
Another example is concatenation. If we concatenate the output of layer 1 and layer 2, then pass this concatenated output to layer 3, the remapping function for layer 3 needs to take the concatenation into account. The width of the output of layer 1 will determine the offset of the coordinates of units originating in l... |
To make Net2WiderNet a fully general algorithm, we would need a remapping inference algorithm that makes a forward pass through the graph, querying each operation in the graph about how to make the remapping functions consistent. For our experiments, we manually designed all of the inference necessary rules for the Inc... |
After we get the random mapping, we can copy the weight over and divide by the replication factor, which is formally given by the following equation. |
[EQUATION] It is essential that each unit be replicated at least once, hence the constraint that the resulting layer be wider. This operator can be applied arbitrarily many times; we can expand only one layer of the network, or we can expand all non-output layers. |
In the setting where several units need to share the same weights, for example convolution operations. We can add such constraint to the random mapping generation, such that source of weight is consistent. This corresponds to make a random mapping on the channel level, instead of unit level, the rest procedure remains ... |
When training with certain forms of randomization on the widened layer, such as dropout (Srivastava et al., 2014 it is acceptable to use perfectly transformation preserving Net2WiderNet , as we have described so far. When using a training algorithm that does not use randomization to encourage identical units to learn d... |
2.4 Net2DeeperNet We also introduce a second function-preserving transformation, Net2DeeperNet This allows us to transform an existing net into a deeper one. Specifically, the Net2DeeperNet replaces a layer |
[MATH] with two layers [MATH] The new matrix [MATH] is initialized to an identity matrix, but remains free to learn to take on any value later. This operation is only applicable when [MATH] is chosen such that |
[MATH] for all vectors [MATH] This property holds for the rectified linear activation. To obtain Net2DeeperNet for maxout units, one must use a matrix that is similar to identity, but with replicated columns. Unfortunately, for some popular activation functions, such as the logistic sigmoid, it is not possible to inser... |
In some cases, to build an identity layer requires additional work. For example, when using batch normalization, we must set the output scale and output bias of the normalization layer to undo the normalization of the layer’s statistics. This requires running forward propagation through the network on training data in ... |
The approach we take is a specific case of a more general approach, that is to build a multiple layer network that factorizes the original layer. Making a deeper but equivalent representation. However it is hard to do general factorization of layers which non-linear transformation units, such as rectified linear and ba... |
When using Net2DeeperNet , there is no need to add noise. We can begin training with a student network that represents exactly the same function as the teacher network. Nevertheless, a small amount of noise can be added to break the symmetry more rapidly. At first glance, it appears that Net2Net can only add a new laye... |
Experiments 3.1 Experimental Setup We evaluated our Net2Net operators in three different settings. In all cases we used an Inception-BN network (Ioffe & Szegedy, 2015 trained on ImageNet. In the first setting, we demonstrate that Net2WiderNet |
can be used to accelerate the training of a standard Inception network by initializing it with a smaller network. In the second setting, we demonstrate that Net2DeeperNet allows us to increase the depth of the Inception modules. Finally, we use our Net2Net operators in a realistic setting, where we make more dramatic c... |
We will be comparing our method to some baseline methods: “Random pad”: This is a baseline for Net2WiderNet We widen the network by adding new units with random weights, rather than by replicating units to perform function-preserving initialization. This operation is implemented by padding the pre-existing weight matri... |
“Random initialization” : This is a baseline for Net2DeeperNet We compare the training of a deep network accelerated by initialization from a shallow one against the training of an identical deep network initialized randomly. |
3.2 Net2WiderNet We start by evaluating the method of Net2WiderNet . We began by constructing a teacher network that was narrower than the standard Inception. We reduced the number of convolution channels at each layer within all Inception modules by a factor of [MATH] Because both the input and output number of channe... |
Fig. shows the comparison of different approaches. We can find that the proposed approach gives faster convergence than the baseline approaches. Importantly, |
Net2WiderNet gives the same level of final accuracy as the model trained from random initialization. This indicates that the true size of the model governs the accuracy that the training process can attain. There is no loss in accuracy that comes from initializing the model to mimic a smaller one. |
Net2WiderNet can thus be safely used to get to the same accuracy quicker, reducing the time required to run new experiments. 3.3 Net2DeeperNet |
We conducted experiments with using Net2DeeperNet to make the network deeper. For these experiments, we used a standard Inception model as the teacher network, and increased the depth of each inception module. The convolutional layers in Inception modules use rectangular kernels. The convolutional layers are arranged i... |
obtains good accuracy much faster than training from random initialization, both in terms of training and validation accuracy. 3.4 Exploring Model Design Space with Net2Net |
One of important property of Net2Net is that it enables quick exploration of modelp space, by transforming an existing state-of-art architecture. In this experiment, we made an ambitious exploration of model design space in both wider and deeper directions. Specifically, we enlarged the width of an Inception model to [... |
The results are shown in Fig. . This last approach paid off, yielding a model that sets a new state of the art of 78.5% on our ImageNet validation set. We did not train these larger models from scratch, due to resource and time constraints. However, we reported the convergence curve of the original inception model for ... |
Discussion Our Net2Net operators have demonstrated that it is possible to rapidly transfer knowledge from a small neural network to a significantly larger neural network under some architectural constraints. We have demonstrated that we can train larger neural networks to improve performance on ImageNet recognition usi... |
Net2Net may also now be used as a technique for exploring model families more rapidly, reducing the amount of time needed for typical machine learning workflows. We hope that future research will uncover new ways of transferring knowledge between neural networks. In particular, we hope future research will reveal more ... |
Acknowledgments We would like to thank Jeff Dean and George Dahl for helpful discussions. We also thank the developers of TensorFlow (Abadi et al., 2015 which we used for all of our experiments. We would like to thank Conrado Miranda for helpful feedback that we used to improve the clarity and comprehensiveness of this... |
Appendix A Choice of Hyperparameters for Fine Tuning Net2Net Transformed Models Net2Net involves training neural networks in a different setting than usual, so it is natural to wonder whether the hyperparameters must be changed in some way. We found that, fortunately, nearly all of the hyperparameters that are typicall... |
Appendix B Related work Cascade-correlation (Fahlman & Lebiere, 1990 includes a strategy for learning by enlarging a pre-existing model architecture. This approach uses a very specific architecture where each new hidden unit receives input from all previous hidden units. Also, the pre-existing hidden units do not conti... |
Some work on knowledge transfer between neural networks is motivated by a desire to train deeper networks than would otherwise be possible, by incrementally training deeper networks with knowledge provided by smaller ones (Romero et al., 2014 ; Simonyan & Zisserman, 2015 These works trained convolutional networks with ... |
networks with batch normalization (Ioffe & Szegedy, 2015 trained using RMSProp (Tieleman & Hinton, 2012 The models we refer to as “standard size” have 25 weight layers on the shortest path from input to output, and 47 weight layers on the longest path (3 convolution layers + 11 Inception modules, each Inception module ... |
Model compression (Buciluǎ et al., 2006 ; Hinton et al., 2015 is a technique for transferring knowledge from many models to a single model. It serves a different purpose than our Net2Net technique. Model compression aims to regularize the final model by causing it to learn a function that is similar to the average func... |
Our Net2DeeperNet operator involves inserting layers initialized to represent identity functions into deep networks. Networks with identity weights at initialization have been used before by, for example, Socher et al. ( 2013 , but to our knowledge that have not previously been used to design function-preserving transf... |
The specific operators we propose are both based on the idea of transformations of a neural network that preserve the function it represents. This is a closely related concept to the justification for greedy layerwise pretraining of deep belief networks (Hinton et al., 2006 . A deep belief network is a generative model... |
# Source: arxiv 1511.06987 # Title: Evolutionary algorithms # Sections: all # Downloaded: 2026-03-03T01:56:28.941790+00:00 \ssp Конспект лекций по курсу <<Эволюционные алгоритмы>> |
Еремеев А.В. Abstract. This manuscript contains an outline of lectures on "Evolutionary Algorithms" course read by the author in Omsk State University n.a. F.M.Dostoevsky. The course covers Canonic Genetic Algorithm and various other genetic algorithms as well as evolutioanry algorithms in general. Some facts, such as ... |
Введение Эволюционные алгоритмы (ЭА), к котоpым можно отнести эволюционные стратегии, генетические алгоpитмы, эволюционное программирование, беpут свое начало в pаботах А.Г. Ивахненко |
, Л.А. Растригина , Дж.Холланда И.Реченбеpга , Л. Фогеля, А. Оуэнса, М. Уолша , и дpугих автоpов, вышедших в 60-70-х годах двадцатого века. Основная идея ЭА состоит в компьютеpном моделиpовании пpоцесса эволюции. Пpи этом моделиpование пpедназначено не для исследования биологических популяций, а для pешения пpактически... |
Области применения ЭА: Экономика, управление, инженерные задачи, переработка информации, космос, медицина и т.д. Научные дисциплины, на стыке которых возникла область эволюционных вычислений: |
1. Биология (генетика), 2. Прикладная математика (искусственный интеллект, методы оптимизации, теория вероятности). Постановка задачи безусловной оптимизации в общем виде: |
[EQUATION] где [MATH] - пространство решений (мощности континуума, счетное или конечное). Принцип работы ГА. Один из типичных представителей эволюционных алгоритмов в оптимизации – генетический алгоритм (ГА). При запуске ГА создается <<виpтуальная>> популяция особей (как пpавило достаточно хpанить только их генотипы), ... |
Пpиспособленность особей к условиям окpужающей сpеды выpажается некотоpой монотонной функцией от значения целевой функции задачи. Чем лучше pешение - тем выше приспособленность особей с соответствующим генотипом. Популяция pазвивается за счет отбоpа более пpигодных особей и пpименения к ним случайных опеpатоpов, имитиp... |
Отбоp может осуществляться по-pазному. Особенно распpостpаненными являются опеpатоpы пpопоpциональной селекции (веpоятность выбоpа пpопоpциональна пpигодности), сpезающей селекции (задается pавномеpным pаспpеделением на множестве из [MATH] % лучших генотипов популяции) и туpниpной селекции [MATH] особей извлекаются с п... |
ГА представляет собой универсальную схему для решения самых разных задач. Достаточно определить представление решений в виде генотипа, выбрать операторы селекции и кроссинговера, и алгоритм можно применять. Более того, при достаточно общих предположениях можно доказать, что с вероятностью единица алгоритм находит оптим... |
Основные принципы эволюционных алгоритмов. В области эволюционных алгоритмов находит себе пpименение бионический подход, состоящий в заимствовании пpинципов организации систем из живой пpиpоды. В данном случае имеет место использование принципа постепенных адаптивных преобразований в пределах популяции или вида в ходе,... |
, § 6.4), более <<масштабная>> межвидовая изменчивость (макроэволюция) требует скачкообразных перестроек генотипа и не может быть выведена непосредственно из постепенной внутривидовой изменчивости. В связи с недостаточной изученностью таких скачкообразных перестроек, привлечение известной теории происхождения видов для... |
, § 1.3). Среди специалистов по методам поиска распространено мнение, что хотя эволюционные алгоритмы и подобны природным генетическим <<механизмам>>, биологические принципы не следует рассматривать, как ограничения при построении оптимизационных алгоритмов. Л.А. Растригин: <<… стремление к моделям механизмов биологиче... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.