FeedFlo

In this post, we will try to build our own deep learning library in Python and begin to write a simple feedforward neural network. The focus will be on the . The content covering the training of our network will occur in the next post. forward pass Below you will find information on how feedforward neural networks take in input and produce an output from it. Firstly, What is a Neural Network? Neural networks are a machine learning technique that is loosely inspired by the model of the brain. As with all machine learning techniques, it learns from a dataset that contains inputs and their corresponding outputs. Neural networks consist of layers. Each layer is connected to the next layer with and . weights biases These weights and biases are used by the network to calculate the output it will give. They are adjusted when the network trains, so that the network produces the optimal output based on the data it trained on. This diagram shows a 3 layer neural network. The lines connecting the nodes are used to represent the weights and biases of the network. How do They Work? The MATH! Each layer has its own weights and bias. The weights and biases initially start as a matrix of random values. A basic feedforward neural network consists of only . linear layers Linear layers produce their output with the following formula: x @ w + b


Where...
x is the input to the layer
w is the weights of the layer
b is the bias of the layer
(@ means matrix multiply) The output of each layer is fed as an input into the next layer. Note If you are unaware of how matrix multiplication works, this website explains it nicely. here This is all we will cover for now - next post we will get into the mathematics behind how these weights and biases get corrected in training! Activation Functions Layers of neural nets are composed of nodes. Activation functions are applied to layers to determine which nodes should "fire"/"activate". This "firing" is observed in the human brain too, hence why it was introduced in neural networks since they are loosely based on the model of the brain. Activation functions also allow the network to model non-linear data. Without activation functions, the neural network would just be a linear regression model, meaning it would not be able to model most real-world data. There are multiple activation functions, but here are the most common ones used... Sigmoid The sigmoid function maps inputs to a value between 0 and 1, as shown in the graph below. (x is the input vector) Relu (Rectified Linear) The Relu function only allows positive values of the input vector to pass through. Negative values are mapped to 0. For example: [[-5, 10]  
 [15, -10] --> relu --> [[0, 10]
                         [15, 0]] Tanh Tanh is similar to Sigmoid, except it maps inputs to values between -1 and 1. Softmax Softmax takes in an input and maps it out as a probability distribution (meaning all the values in the output sum to 1). (z is the input vector, K is the length of the input vector) Writing the Code We will need NumPy for our matrix operations... import numpy as np First, let's write our linear layer class: class Linear:
    def __init__(self, units):
        #units specify how many nodes are in the layer
        self.units = units
        self.initialized = False

    def __call__(self, x):
        #initialize weights and biases if layer hasn't been called before
        if not self.initialized:
            self.w = np.random.randn(self.input.shape[-1], self.units)
            self.b = np.random.randn(self.units)
            self.initialized = True

        return self.input @ self.w + self.b Example usage... x = np.array([[0, 1]])
layer = Linear(5)
print (layer(x))

# => [[-2.63399933 -1.18289984  0.32129587  0.2903246  -0.2602642 ]] Now let's write all our activation function classes, following the formulae given previously: class Sigmoid:
    def __call__(self, x):
        return 1 / (1 + np.exp(-x))

class Relu:
    def __call__(self, x):
        return np.maximum(0, x)   

class Softmax:
    def __call__(self, x):
        return np.exp(x) / np.sum(np.exp(x))   

class Tanh:
    def __call__(self, x):
        return np.tanh(x) Now let's write a "Model" class, which will act as a container for all our layers / the actual neural network class. class Model:
    def __init__(self, layers):
        self.layers = layers
    
    def __call__(self, x):
        output = x
        for layer in self.layers:
            output = layer(x)
            
        return output Save all of those classes into " " (or any name you wish). layer.py Now we can build a simple neural network, with our tiny library so far: import layers
import numpy as np

#inputs array
x = np.array([[0, 1], [0, 0], [1, 1], [0, 1]])

#network uses all the layers we have designed so far
net = layers.Model([
    layers.Linear(32),
    layers.Sigmoid(),
    layers.Linear(16),
    layers.Softmax(),
    layers.Linear(8),
    layers.Tanh(),
    layers.Linear(4),
    layers.Relu(),
])

print (net(x)) Output:
[[0.         3.87770361 0.17602662 0.        ]
 [0.         3.85640582 0.22373699 0.        ]
 [0.         3.77290517 0.2469388  0.        ]
 [0.         3.87770361 0.17602662 0.        ]] Also published here.

How to Build Feedforward Neural Networks: A Step-by-Step Guide 

Building a Feedforward Neural Network from Scratch in Python

Feedforward neural networks are also known as (MLN). These networks of models are called feedforward because the information only travels forward in the neural network, through the input nodes then through the hidden layers (single or many layers) and finally through the output nodes. In MLN there are no feedback connections such that the output of the network is fed back into itself. These networks are represented by a combination of many simpler models(sigmoid neurons). Multi-layered Network of Neurons Citation Note: The content and the structure of this article is based on the deep learning lectures from One-Fourth Labs — Padhai . Motivation: Non-Linear Data Before we talk about the feedforward neural networks, let’s understand what was the need for such neural networks. Traditional models like perceptron — which takes real inputs and give boolean output only works if the data is linearly separable. That means that the positive points (green) should lie on one side of the boundary and negative points (red) lie another side of the boundary. As you can see from the below figure perceptron is doing a very poor job of finding the best decision boundary to separate positive and negative points. Perceptron Decision Surface for Non-Linear Data Next, we have the sigmoid neuron model this is similar to , but the sigmoid model is slightly modified such that the output from the sigmoid neuron is much smoother than the step functional output from perceptron. Although we have introduced the non-linear sigmoid neuron function, it is still not able to effectively separate red points(negatives) from green points(positives). perceptron Sigmoid Neuron Decision Boundary for Non-Linear Data The important point is that from a rigid decision boundary in perceptron, we have taken our first step in the direction of creating a decision boundary that works well for non-linearly separable data. Hence the sigmoid neuron is the building block of our feedforward neural network. From my previous post on , we have proved that even though single sigmoid neuron can’t deal with non-linear data. If we connect multiple sigmoid neurons in an effective way, we can approximate the combination of neurons to any complex relationship between input and the output, required to deal with non-linear data. Universal Approximation theorem Combination of Sigmoid Neurons for Non-Linear Data Feedforward Neural Networks Multi-layered Network of neurons is composed of many sigmoid neurons. MLNs are capable of handling the non-linearly separable data. The layers present between the input and output layers are called hidden layers. The hidden layers are used to handle the complex non-linearly separable relations between input and the output. Simple Deep Neural Network In this section let’s see how can we use a very simple neural network to solve complex non-linear decision boundaries. Let’s take an example of a mobile phone like/dislike predictor with two variables: screen size, and the cost. It has a complex decision boundary as shown below: Decision Boundary We know that by using a single sigmoid neuron, it is impossible to obtain this kind of non-linear decision boundary. Regardless of how we vary the sigmoid neuron parameters and . Now change the situation and use a simple network of neurons for the same problem and see how it handles. w b Simple Neural Network We have our inputs x₁ — screen size and x₂— price going into the network along with the bias b₁ and b₂. Now let’s break down the model neuron by neuron to understand. We have our first neuron (leftmost) in the first layer which is connected to inputs x₁ and x₂ with the weights w₁₁ and w₁₂ and the bias b₁ and b₂. The output of that neuron represented as h₁, which is a function of x₁ and x₂ with parameters w₁₁ and w₁₂. Output of the first Neuron If we apply the sigmoid function to the inputs x₁ and x₂ with the appropriate weights w₁₁, w₁₂ and bias b₁ we would get an output h₁, which would be some real value between 0 and 1. The sigmoid output for the first neuron h₁ will be given by the following equation, Output Sigmoid for First Neuron Next up, we have another neuron in the first layer which is connected to inputs x₁ and x₂ with the weights w₁₃ and w₁₄ along with the bias b₃ and b₄. The output of the second neuron represented as h₂. Output of the second Neuron Similarly, the sigmoid output for the second neuron h₂ will be given by the following equation, So far we have seen the neurons present in the first layer but we also have another output neuron which takes h₁ and h₂ as the input as opposed to the previous neurons. The output from this neuron will be the final predicted output, which is a function of h₁ and h₂. The predicted output is given by the following equation, Complex looking Equation!! Really?. Comment if you understood it or not. Earlier we can adjust only w₁, w₂ and b — parameters of a single sigmoid neuron. Now we can adjust the 9 parameters (w₁₁, w₁₂, w₁₃, w₁₄, w₂₁, w₂₂, b₁, b₂, b₃), which allows the handling of much complex decision boundary. By trying out the different configurations for these parameters we would be able to find the optimal surface where output for the entire middle area (red points) is one and everywhere else is zero, what we desire. Decision Boundary from Network The important point to note is that even with the simple neural network we were able to model the complex relationship between the input and output. Now the question arises, how do we know in advance this particular configuration is good and why not add few more layers between or add few more neurons in the first layer. All these questions are valid but for now, we will keep things simple take the network as it is. We will discuss these questions and a lot more in detail when we discuss . hyper-parameter tunning Generic Deep Neural Network In the previous, we have seen the neural network for a specific task, now we will talk about the neural network in generic form. Generic Network without Connections Let’s assume we have our network of neurons with two hidden layers (in blue but there can more than 2 layers if needed) and each hidden layer has 3 sigmoid neurons there can be more neurons but for now I am keeping things simple. We have three inputs going into the network (for simplicity I used only three inputs but it can take n inputs) and there are two neurons in the output layer. As I said before we will take this network as it is and understand the intricacies of the deep neural network. First, I will explain the terminology then we will go into how these neurons interact with each other. For each of these neurons, two things will happen Pre-activation represented by ‘a’: It is a weighted sum of inputs plus the bias. Activation represented by ‘h’: Activation function is Sigmoid function. Generic Network with Connections Let’s understand the network neuron by neuron. Consider the first neuron present in the first hidden layer. The first neuron is connected to each of the inputs by weight ₁. W Going forward I will be using this format of the indices to represent the weights and biases associated with a particular neuron, W(Layer number)(Neuron number in the layer)(Input number) b(Layer number)(Bias number associated for that input) W₁₁₁— Weight associated with the first neuron present in the first hidden layer connected to the first input. W₁₁₂— Weight associated with the first neuron present in the first hidden layer connected to the second input. b₁₁ — Bias associated with the first neuron present in the first hidden layer. b₁₂ — Bias associated with the second neuron present in the first hidden layer. …. Here W₁ a weight matrix containing the individual weights associated with the respective inputs. The pre-activation at each layer is the weighted sum of the inputs from the previous layer plus bias. The mathematical equation for pre-activation at each layer ‘i’ is given by, Pre-activation Function The activation at each layer is equal to applying the sigmoid function to the output of pre-activation of that layer. The mathematical equation for the activation at each layer ‘i’ is given by, Activation Function Finally, we can get the predicted output of the neural network by applying some kind of activation function (could be softmax depending on the task) to the pre-activation output of the previous layer. The equation for the predicted output is shown below, Output Activation Function Computations in Deep Neural Network We have seen the terminology and functional aspect of the neural network. Now we will see how the computations inside a DNN takes place. Consider that you have 100 inputs and 10 neurons in the first and second hidden layers. Each of the 100 inputs will be connected to the neurons will be The weight matrix of the first neuron W₁ will have a total of 10 x 100 weights. Weight Matrix Remember, we are following a very specific format for the indices of the weight and bias variable as shown below, W(Layer number)(Neuron number in the layer)(Input number) b(Layer number)(Bias number associated for that input) Now let’s see how we can compute the pre-activation for the first neuron of the first layer a₁₁. We know that pre-activation is nothing but the weighted sum of inputs plus bias. In other words, it is the dot product between the first row of the weight matrix W₁ and the input matrix X plus bias b₁₁. Similarly the pre-activation for other 9 neurons in the first layer given by, In short, the overall pre-activation of the first layer is given by, Where, W₁ is a matrix containing the individual weights associated with the corresponding inputs and b₁ is a vector containing(b₁₁, b₁₂, b₁₃,….,b₁₀) the individual bias associated with the sigmoid neurons. The activation for the first layer is given by, Where ‘g’ represents the sigmoid function. Remember that a₁ is a vector of 10 pre-activation values, here we are applying the element-wise sigmoid function on all these 10 values and storing them in another vector represented as h₁. Similarly, we can compute the pre-activation and activation values for ’n’ number of hidden layers present in the network. Output Layer of DNN So far we have talked about the computations in the hidden layer. Now we will talk about the computations in the output layer. Generic Network with Connections We can compute the pre-activation a₃ at the output layer by taking the dot product of weights associated W₃ and activation of the previous layer h₂ plus bias vector b₃. Pre-activation for the output layer To find out the predicted output from the network, we are applying some function (which we don’t know yet) to the pre-activation values. Predicted Values These two outputs will form a probability distribution that means their summation would be equal to 1. The output Activation function is chosen depending on the task at hand, can be softmax or linear. Softmax Function We will use the Softmax function as the output activation function. The most frequently used activation function in deep learning for classification problems. Softmax Function In the Softmax function, the output is always positive, irrespective of the input. Now, let us illustrate the Softmax function on the above-shown network with 4 output neurons. The output of all these 4 neurons is represented in a vector ‘a’. To this vector, we will apply our softmax activation function to get the predicted probability distribution as shown below, Applying the Softmax Function By applying the softmax function we would get a predicted probability distribution and our true output is also a probability distribution, we can compare these two distributions to compute the loss of the network. Loss Function In this section, we will talk about the loss function for binary and multi-class classification. The purpose of the loss function is to tell the model that some correction needs to be done in the learning process. In general, the number of neurons in the output layer would be equal to the number of classes. But in the case of binary classification, we can use only one sigmoid neuron which outputs the probability P(Y=1) therefore we can obtain P(Y=0) = 1-P(Y=1). In the case of classification, We will use the cross-entropy loss to compare the predicted probability distribution and the true probability distribution. Cross-entropy loss for binary classification is given by, Cross-entropy loss for multi-class classification is given by, Learning Algorithm: Non-Mathy Version We will be looking at the non-math version of the learning algorithm using gradient descent. The objective of the learning algorithm is to determine the best possible values for the parameters, such that the overall loss of the deep neural network is minimized as much as possible. The learning algorithm goes like this, We initialize all the weights (w₁₁₁, w₁₁₂,…) and (b₁, b₂,….) randomly. We then iterate over all the observations in the data, for each observation find the corresponding predicted distribution from the neural network and compute the loss using cross-entropy function. Based on the loss value, we will update the weights such that the overall loss of the model at the new parameters will be of the model. w b less than the current loss Photo by on Vasily Koloda Unsplash Conclusion In this post, we briefly looked at the limitations of traditional models like perceptron and sigmoid neuron to handle the non-linear data and then we went on to see how we can use a simple neural network to solve complex decision boundary problem. We then looked at the neural network in a generic sense and the computations behind the neural network in detail. Finally, we have looked at the learning algorithm of the deep neural network. Recommended Reading: Sigmoid Neuron Learning Algorithm Explained With Math In my next post, we will discuss how to implement the feedforward neural network from scratch in python using numpy. So make sure you follow me on medium to get notified as soon as it drops. Until then Peace :) NK. Connect with me on or follow me on for updates on future posts & discussions on the current story. LinkedIn twitter

Deep Learning: Feedforward Neural Networks Explained

Generative Artificial Intelligence for Software Engineering: Background

Backpropagation - The Most Fundamental Training Systems Algorithm in Modern Generative AI

This paper is available on arxiv under CC BY-NC-ND 4.0 DEED license.
Authors:
(1) U Jin Choi, Department of mathematical science, Korea Advanced Institute of Science and Technology & ujchoi@kaist.ac.kr;
(2) Kyung Soo Rim, Department of mathematics, Sogang University & ksrim@sogang.ac.kr. Table of Links Introduction
Organization and notation
Problem Setting and Preliminaries
Generalized Wiener algebra and Fréchet derivative
Characterization of coefficients
Coefficients from an ergodic process
Conclusion & References Conclusion In this mathematical study, we delve into the realm of statistical inference and introduce a novel approach to variational non-Bayesian inference. Most significantly, we propose a new method for uniquely determining the hidden PDF solely from a random sample while leveraging theory. Beyond merely approximating the shape of the unrevealed PDF, we yield results that provide practical assistance in inferring important moments, such as the mean and variance, from the estimated PDF. These days, methods utilizing artificial intelligence are widely employed, but they generally exhibit a drawback of depending on initial conditions, i.e., a prior distribution, and iterative computations, i.e., a backpropagation. The limitations of such approaches lie in their reliance on these conditions to achieve global optimization. We emphasize the significant contribution of the proposed method to predictive and classification models, even without any information on populations, highlighting its potential applications in various domains such as finance, economics, weather forecasting, and machine learning, all of which present unique challenges. The most well-known method for approximating hidden PDFs is the variational approach in a Bayesian context. In contrast, our study extends this problem by seeking to precisely determine unknown PDFs through a system of equations. Our approach includes proving the Fréchet differentiability of entropy to establish the uniqueness of the energy function space in the Wiener algebra. We then derive the unique determination of the energy function through the minimization of KL-divergence. Leveraging the Ergodic theorem, we elucidate that solutions to equations comprising polynomial function series are the coefficients of the energy function and numerically substantiate the convergence of partial sums of the energy function obtained from a finite number of equations. In summary, our mathematical exploration has unveiled the potential of variational Non-Bayesian inference in Wiener space. We anticipate that these mathematical ideas can offer an innovative framework for probability density estimation and predictive modeling. Ultimately, our research emphasizes its potential contribution to reshaping mathematical horizons and expanding the boundaries of knowledge in the field of statistical methodology. REFERENCES [1] K. Bharath A. Saha and S. Kurtek, A geometric variational approach to bayesian inference, Journal of the American Statistical Association 115 (2020), no. 530, 822–835. [2] P. Alquier and J. Ridgway, Concentration of tempered posteriors and of their variational approximations, The Annals of Statistics 48 (2020), no. 3, 1475–1497. [3] A. R. Barron and C. Sheu, Approximation of density functions by sequences of exponential families, The Annals of Statistics 19 (1991), no. 3, 1347–1369. [4] C. M. Bishop and M. E. Tipping, Probabilistic principal component analysis, J. R. Statist. Soc. B 61 (1999), no. 3, 611–622. [5] F. W. Olver, D. Lozier, R. F. Boisvert and C. W. Clark, NIST Handbook of Mathematical Functions, Cambridge University Press, 2010. [6] C Brouder and F Patras, One-particle irreducibility with initial correlations, (2011). [7] I. Csiszar, I-divergence geometry of probability distributions and minimization problems, Ann. Probab. 3 (1975), no. 1, 146–158. [8] G. Cybenko, Approximation by superpositions of a sigmoidal function, Mathematics of Control, Signals and Systems 2 (1989), 303–314. [9] L. Grafakos, Classical and Modern Fourier Analysis, Prentice Hall, 2003. [10] D. E. Rumelhart, G. E. Hinton and R. Williams, Learning representations by back-propagating errors, Nature 323 (1986), 533–536. [11] G. E. Hinton, A Practical Guide to Training Restricted Boltzmann Machines, University of Toronto, 2010. [12] K. Hornik, Approximation capabilities of multilayer feedforward networks, Neural Networks 4 (1991), no. 2, 251–257. [13] Y. Katznelson, An introduction to harmonic analysis (third ed.), New York: Cambridge Mathematical Library, 2004. [14] D. M. Blei, A. Kucukelbir and J. D. McAuliffe, Variational inference: A review for statisticians, Jour. of the American Statistical Association 112 (2017), no. 518, 859–877. [15] S. Kullback and R. A. Leibler, On information and sufficiency, Ann. Math. Statistics 22 (1951), no. 1, 79–86. [16] H. M. Möller and R. Tenberg, Multivariate polynomial system solving using intersections of eigenspaces, Journal of Symbolic Computation 32 (2001), no. 5, 513–531. [17] B. Mourrain, V. Y. Pan and O. Ruatta, Accelerated solution of multivariate polynomial systems of equations, SIAM Journal on Computing 32 (2003), no. 2, 435–454. [18] W. Rudin, Functional Analysis, McGraw-Hill, 1991. [19] D. W. Stroock, Probability theory, an analytic view, Cambridge University Press, 1993. [20] B. Sturmfels, Solving systems of polynomial equations (cbms regional conference series in mathematics), Amer. Math. Soc., 2002. [21] N. Wiener, Tauberian theorems, Annals of Mathematics 33 (1932), no. 1, 1–100. [22] F. Zhang and C. Gao, Convergence rates of variational posterior distributions, The Annals of Statistics 48 (2020), no. 4, 2180–2207. This paper is available on arxiv under CC BY-NC-ND 4.0 DEED license. Authors: (1) U Jin Choi, Department of mathematical science, Korea Advanced Institute of Science and Technology & ujchoi@kaist.ac.kr; (2) Kyung Soo Rim, Department of mathematics, Sogang University & ksrim@sogang.ac.kr. This paper is available on arxiv under CC BY-NC-ND 4.0 DEED license. Authors: Authors: (1) U Jin Choi, Department of mathematical science, Korea Advanced Institute of Science and Technology & ujchoi@kaist.ac.kr; (2) Kyung Soo Rim, Department of mathematics, Sogang University & ksrim@sogang.ac.kr. Table of Links Introduction Organization and notation Problem Setting and Preliminaries Generalized Wiener algebra and Fréchet derivative Characterization of coefficients Coefficients from an ergodic process Conclusion & References Introduction Introduction Organization and notation Organization and notation Problem Setting and Preliminaries Problem Setting and Preliminaries Generalized Wiener algebra and Fréchet derivative Generalized Wiener algebra and Fréchet derivative Characterization of coefficients Characterization of coefficients Coefficients from an ergodic process Coefficients from an ergodic process Conclusion & References Conclusion & References Conclusion In this mathematical study, we delve into the realm of statistical inference and introduce a novel approach to variational non-Bayesian inference. Most significantly, we propose a new method for uniquely determining the hidden PDF solely from a random sample while leveraging theory. Beyond merely approximating the shape of the unrevealed PDF, we yield results that provide practical assistance in inferring important moments, such as the mean and variance, from the estimated PDF. These days, methods utilizing artificial intelligence are widely employed, but they generally exhibit a drawback of depending on initial conditions, i.e., a prior distribution, and iterative computations, i.e., a backpropagation. The limitations of such approaches lie in their reliance on these conditions to achieve global optimization. We emphasize the significant contribution of the proposed method to predictive and classification models, even without any information on populations, highlighting its potential applications in various domains such as finance, economics, weather forecasting, and machine learning, all of which present unique challenges. The most well-known method for approximating hidden PDFs is the variational approach in a Bayesian context. In contrast, our study extends this problem by seeking to precisely determine unknown PDFs through a system of equations. Our approach includes proving the Fréchet differentiability of entropy to establish the uniqueness of the energy function space in the Wiener algebra. We then derive the unique determination of the energy function through the minimization of KL-divergence. Leveraging the Ergodic theorem, we elucidate that solutions to equations comprising polynomial function series are the coefficients of the energy function and numerically substantiate the convergence of partial sums of the energy function obtained from a finite number of equations. In summary, our mathematical exploration has unveiled the potential of variational Non-Bayesian inference in Wiener space. We anticipate that these mathematical ideas can offer an innovative framework for probability density estimation and predictive modeling. Ultimately, our research emphasizes its potential contribution to reshaping mathematical horizons and expanding the boundaries of knowledge in the field of statistical methodology. REFERENCES [1] K. Bharath A. Saha and S. Kurtek, A geometric variational approach to bayesian inference, Journal of the American Statistical Association 115 (2020), no. 530, 822–835. [2] P. Alquier and J. Ridgway, Concentration of tempered posteriors and of their variational approximations, The Annals of Statistics 48 (2020), no. 3, 1475–1497. [3] A. R. Barron and C. Sheu, Approximation of density functions by sequences of exponential families, The Annals of Statistics 19 (1991), no. 3, 1347–1369. [4] C. M. Bishop and M. E. Tipping, Probabilistic principal component analysis, J. R. Statist. Soc. B 61 (1999), no. 3, 611–622. [5] F. W. Olver, D. Lozier, R. F. Boisvert and C. W. Clark, NIST Handbook of Mathematical Functions, Cambridge University Press, 2010. [6] C Brouder and F Patras, One-particle irreducibility with initial correlations, (2011). [7] I. Csiszar, I-divergence geometry of probability distributions and minimization problems, Ann. Probab. 3 (1975), no. 1, 146–158. [8] G. Cybenko, Approximation by superpositions of a sigmoidal function, Mathematics of Control, Signals and Systems 2 (1989), 303–314. [9] L. Grafakos, Classical and Modern Fourier Analysis, Prentice Hall, 2003. [10] D. E. Rumelhart, G. E. Hinton and R. Williams, Learning representations by back-propagating errors, Nature 323 (1986), 533–536. [11] G. E. Hinton, A Practical Guide to Training Restricted Boltzmann Machines, University of Toronto, 2010. [12] K. Hornik, Approximation capabilities of multilayer feedforward networks, Neural Networks 4 (1991), no. 2, 251–257. [13] Y. Katznelson, An introduction to harmonic analysis (third ed.), New York: Cambridge Mathematical Library, 2004. [14] D. M. Blei, A. Kucukelbir and J. D. McAuliffe, Variational inference: A review for statisticians, Jour. of the American Statistical Association 112 (2017), no. 518, 859–877. [15] S. Kullback and R. A. Leibler, On information and sufficiency, Ann. Math. Statistics 22 (1951), no. 1, 79–86. [16] H. M. Möller and R. Tenberg, Multivariate polynomial system solving using intersections of eigenspaces, Journal of Symbolic Computation 32 (2001), no. 5, 513–531. [17] B. Mourrain, V. Y. Pan and O. Ruatta, Accelerated solution of multivariate polynomial systems of equations, SIAM Journal on Computing 32 (2003), no. 2, 435–454. [18] W. Rudin, Functional Analysis, McGraw-Hill, 1991. [19] D. W. Stroock, Probability theory, an analytic view, Cambridge University Press, 1993. [20] B. Sturmfels, Solving systems of polynomial equations (cbms regional conference series in mathematics), Amer. Math. Soc., 2002. [21] N. Wiener, Tauberian theorems, Annals of Mathematics 33 (1932), no. 1, 1–100. [22] F. Zhang and C. Gao, Convergence rates of variational posterior distributions, The Annals of Statistics 48 (2020), no. 4, 2180–2207.

Variational Non-Bayesian Inference of the Probability Density Function: Conclusion & References

This paper is available on arxiv under CC 4.0 license. Authors: (1) Pierre Colombo, Equall, Paris, France & MICS, CentraleSupelec, Universite Paris-Saclay, France; (2) Victor Pellegrain, IRT SystemX Saclay, France & France & MICS, CentraleSupelec, Universite Paris-Saclay, France; (3) Malik Boudiaf, ÉTS Montreal, LIVIA, ILLS, Canada; (4) Victor Storchan, Mozilla.ai, Paris, France; (5) Myriam Tami, MICS, CentraleSupelec, Universite Paris-Saclay, France; (6) Ismail Ben Ayed, ÉTS Montreal, LIVIA, ILLS, Canada; (7) Celine Hudelot, MICS, CentraleSupelec, Universite Paris-Saclay, France; (8) Pablo Piantanida, ILLS, MILA, CNRS, CentraleSupélec, Canada. Table of Links Abstract & Introduction Related Work API-based Few-shot Learning An Enhanced Experimental Setting Experiments Conclusions Limitations, Acknowledgements, & References Appendix A: Proof of Proposition Appendix B: Additional Experimental Results 2 Related Work 2.1 Few-shot learning in NLP Numerous studies have tackled the task of FSL in Natural Language Processing (NLP) by utilizing pre-trained language models (Devlin et al., 2018; Liu et al., 2019b; Radford et al., 2019; Yang et al., 2019). These methods can be classified into three major categories: prompt-based, parameter efficient tuning, and in-context learning. Prompt-based FSL: Prompt-based FSL involves the use of natural language prompts or templates to guide the model to perform a specific task (Ding et al., 2021; Liu et al., 2023). For example, the seminal work (Schick and Schütze, 2020a) proposed a model called PET, which uses a pre-defined set of prompts to perform various NLP tasks as text classification. They also impose a choice of a verbalizer which highly impacts the classification performances (Cui et al., 2022; Hu et al., 2021a). However, recent studies have questioned the benefits of prompt-based learning due to the high variability in performance caused by the choice of prompt (Liu et al., 2022). To address this issue, researchers have proposed prompt tuning which involves a few learnable parameters in addition to the prompt (Lester et al., 2021). Nevertheless, these approaches face limitations when learning from API: (i) encoder access for gradient computation is infeasible (as in R1), (ii) prompting requires to send data and label which raises privacy concerns (as in R3), and (iii) labeling new points is time-consuming (see in R3) and expensive due to the need to send all shots for each input token[1]. These methods, such as adapters (Houlsby et al., 2019; Pfeiffer et al., 2020), keep most of the model’s parameters fixed during training and only update small feedforward networks that are inserted within the larger model architecture. A recent example is T-FEW (Liu et al., 2022), which adds learned vectors that rescale the network’s internal activations. Additionally, it requires a set of manually created prompts for each dataset making it hard to use in practice. Relying on parameter-efficient fine-tuning methods with an API is not possible due to the need to compute gradients of the encoder (as per R1) and the requirement to send both the labeling schema and the labels, which violates R3. Parameter-efficient fine-tuning. In-context learning models are models that utilize input-to-output training examples as prompts to make predictions, without any parameter updates (Wei et al., 2022). These models, such as text-davinci, rely solely on the provided examples to generate predictions, without any additional training. However, a significant drawback of this approach is that the user must supply the input, label examples, and task description, which becomes prohibitively expensive when the number of classes or shots increases, is slow (Liu et al., 2022) (R2) and raises data privacy concerns (as highlighted in R3). Additionally, the inability to reuse text embeddings for new tasks or with new labels without querying the model’s API limits practicality and scalability, making reusable encoding unfeasible for in-context learning models [2]. In Context Learning (ICL). or quite long stood as the de-facto paradigm for FSL ((Snell et al., 2017; Rusu et al., 2019; Sung et al., 2018b; Lee et al., 2019; Raghu et al., 2019; Sun et al., 2019a)). In meta-learning, the objective is to provide the model with the intrinsic ability to learn in a data-efficient manner. For instance, MAML ((Finn et al., 2017a; Antoniou et al., 2018)), arguably the most popular meta-learning method, tries to train a model such that it can be fine-tuned end-to-end using only a few supervised samples while retaining high generalization ability. Unlike the three previous lines of work, meta-learning methods operate by modifying the pre-training procedure and therefore assume access to both the training data and the model, which wholly breaks both R1 and R3. 2.2 Inductive vs transductive learning Learning an inductive classifier on embeddings generated by an API-based model, as proposed by (Snell et al., 2017), is a common baseline for performing FSL. This approach is prevalent in NLP, where a parametric model is trained on data to infer general rules that are applied to label new, unseen data (known as inductive learning (Vapnik, 1999)). However, in FSL scenarios with limited labeled data, this approach can be highly ambiguous and lead to poor generalization. Transduction offers an attractive alternative to inductive learning (Sain, 1996). Unlike inductive learning, which infers general rules from training data, transduction involves finding rules that work specifically for the unlabeled test data. By utilizing more data, such as unlabeled test instances, and aiming for a more localized rule rather than a general one, transductive learning has shown promise and practical benefits in computer vision (Boudiaf et al., 2020, 2021; Ziko et al., 2020). Transductive methods yield substantially better performance than their inductive counterparts by leveraging the statistics of the query set (Dhillon et al., 2019). However, this approach has not yet been explored in the context of textual data. [1] The cost of API queries is determined by the number of input tokens that are transmitted. [2] Furthermore, as the number of considered classes increases, the fixed size of the transformer limits the number of possible shots that can be fed to the model. Previous studies have often neglected this limitation by focusing on a few numbers of labels.

Transductive Learning for Textual Few-Shot Classification: Related Work

This paper is under CC 4.0 license. available on arxiv Authors: (1) Amir Noorizadegan, Department of Civil Engineering, National Taiwan University; (2) D.L. Young, Core Tech System Co. Ltd, Moldex3D, Department of Civil Engineering, National Taiwan University & dlyoung@ntu.edu.tw; (3) Y.C. Hon, Department of Mathematics, City University of Hong Kong; (4) C.S. Chen, Department of Civil Engineering, National Taiwan University & dchen@ntu.edu.tw. Table of Links Abstract & Introduction Neural Networks PINN for Solving Inverse Burgers’ Equation Residual Network Numerical Results Results, Acknowledgments & References 2 Neural Networks In this section, we will explore the utilization of feedforward neural networks for solving interpolation problems, specifically focusing on constructing accurate approximations of functions based on given data points. 2.1 Feedforward Neural Networks The feedforward neural network, also known as a multilayer perceptron (MLP), serves as a foundational architecture in artificial neural networks. Comprising interconnected layers of neurons, the information flow progresses unidirectionally from the input layer through hidden layers to the output layer. This process, termed “feedforward,” entails transforming input data into desired output predictions. The core constituents of a feedforward neural network are its individual neurons. A neuron computes a weighted sum of its inputs, augmented by a bias term, before applying an activation function to the result. It is important to note that while hidden layers utilize activation functions to introduce non-linearity, the last layer (output layer) typically does not apply an activation function to its outputs. 2.2 Neural Networks for Interpolation In the context of interpolation problems, feedforward neural networks can be leveraged to approximate functions based on a given set of data points. The primary objective is to construct a neural network capable of accurately predicting function values at points not explicitly included in the provided dataset. 2.2.1 Training Process The training of the neural network involves adjusting its weights and biases to minimize the disparity between predicted outputs and actual data values. This optimization process is typically driven by algorithms such as gradient descent, which iteratively update network parameters to minimize a chosen loss function. 2.2.2 Loss Function for Interpolation This loss function guides the optimization process, steering the network toward producing accurate predictions.

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