Sigmoid is a Leading Data Solutions company offering best-in-class services in Data Engineering and Data Science

Sigmoid

Animate your web applications with a customizable ease-out. Ease-out animations are great for creating animations that feel like they respond instantly but still cruise to a gentle halt. Most easings specify one of a small number of easing curves: easeOutQuad, easeOutSine, easeOutCubic, etc. However, the sharpness of that curve is not configurable. Here I show how to create a configurable ease-out function that will work for animating any property you desire. How to Use It The easing function takes a single parameter , that specifies the steepness of the curve. The larger is, the steeper the curve, and the higher the velocity in the middle section relative to the beginning and ending. k k The will produce a function that takes a parameter that represents the percentage of the animation completed as a number between 0 and 1. It’s important to note that this system only produces numbers that you can use within an animation framework (that you might have to build). An example is included in the . easeOutFactory t framework jsFiddle Here is an example of the usage: // LISTING 1 (Javascript)// easeOutFactory produces a function representing an ease-out. var easing = easeOutFactory(4); // choose k = 4, see Fig. 2 easing(0.00); // 0easing(0.10); // 0.2047403251776069easing(0.25); // 0.4793609299265755easing(0.50); // 0.7900128291929867easing(0.75); // 0.9389236079466523easing(0.90); // 0.9821358147987899easing(1.00); // 1 You can your easing curve visually using this . Modify to adjust the steepness. design OS X Grapher file k A demonstration of four different Rienzi Gokea. k values implemented in the engine of mobile game Epsilon Jump ( iOS , Android ) graciously provided by The Theory This section is not required for use. Read on to understand the techniques underlying the code so that you will be able to develop your own easing curves. The basic idea is to take the ease-in-out curve developed in and use the right half of it to get our adjustable ease-out. Ease In, Out: The Sigmoid Factory At the conclusion of The Sigmoid Factory, we had this equation developed to describe a sigmoid centered at the origin: Equation 1: An instrumented sigmoid centered at x = 0. α is our knob to tune the steepness, and t is our animation progression from 0 to 1. Figure 1: The centered sigmoid function from Eqn. 2. For our animations, we constrain the useful region of the function to be between 0 and 1 on the time ( )and distance ( ) axes. Notice how the plotted function is substantially less than 1 at = 1 in Fig. 1. We can apply the same scaling trick as we used in The Sigmoid Factory to bump it up to 1. That is to say, we normalize the function by its value at = 1. x y x x Because we’re not shifting the function around, just scaling it, the math works out much more simply than before. Equation 2: The ease out equation. s(t, k) is as specified in Eqn. 1. t is time, k is a steepness constant. Figure 2: The Ease Out Function from Eqn. 2 plotted as k = 4. You can find the . grapher file here I’ll leave Ease-In as an exercise to the reader. Think about how to flip things around. 😘 Hint: Versus Traditional Ease-Outs There is a slight deficiency with this method versus easeOutQuad and easeOutCubic. Figure 3: easeOutCubic (black) and easeOutQuad (gray) You’ll notice that at = 1, both of these eases have zero vertical velocity. That is to say, their d_y_/d_x_ at 1 is 0. The adjustable sigmoid ease described here gets quite close to zero starting around = 6 or 7. If you’re using a below this range, it might look a bit nicer to use one of the traditional eases. x k k Table 1: Approximate Corresponding Eases EASING           EQN            K APPROX. sin(2πt/4)     3.25 -t^2 + 2t      3.50 (t-1)^3 + 1    5.50 (t-1)^5 + 1    8.00 easeOutSine easeOutQuad easeOutCubic easeOutQuint If you enjoyed this article, please follow me on Medium. You might also like The Sigmoid Factory , The Bounce Factory and The Spring Factory . References — Where the theory for the ease-in-out curve used in this article was developed. Ease In, Out: The Sigmoid Factory — A collection of many commonly used easing curves. http://easings.net/ — A Grapher file I developed to design my ease out functions. Ease Out Grapher File

Ease Out: The Half Sigmoid

Animate your web applications with a customizable ease-in-out. It’s very common for animations to be specified as ease-in-out. It’s a very pleasing sensation to witness an object speed up, cruise, and slow to a halt. Most easings specify one of a small number of easing curves: easeInOutQuad, easeInOutSine, easeInOutCubic, etc. However, the sharpness of that curve is not configurable. Here I show how to create a configurable ease-in-out function that will work for animating any property you desire. How to Use It The easing function takes a single parameter , that specifies the steepness of the curve. The larger is, the steeper the curve, and the higher the velocity in the middle section relative to the beginning and ending. k k The will produce a function that takes a parameter that represents the percentage of the animation completed as a number between 0 and 1. It’s important to note that this system only produces numbers that you can use within an animation framework (that you might have to build). An example framework is included in the . sigmoidFactory t jsFiddle Here is an example of the usage: var easing = sigmoidFactory(7); easing(0); // nearly 0 (5.551115123125783e-17)easing(1); // 1easing(0.5); // 0.5easing(0.25); // 0.028453023879735584easing(0.75); // 0.9715469761202644 You can your easing curve visually using this . Modify to adjust the steepness. design OS X Grapher file k A demonstration of four different Rienzi Gokea. k values implemented in the engine of mobile game Epsilon Jump ( iOS , Android ) graciously provided by The Theory This section is not required for use. Read on to understand the techniques underlying the code so that you will be able to develop your own easing curves. Firstly, we need a function that looks like . There are two that come to mind that fit the bill: (Fig. 1, which we will use) and [1] (which we will not use). easeInOut sigmoid arctangent Figure 1. The Sigmoid Function However, these curves both have some defects we’ll have to remedy. We want our animations to start at 0 and end at 1. Both functions begin in the middle of the at 0, and asymptotically approach their final values (0 and 1 in the case of sigmoid, -π/2 to π/2 in the case of arctangent). animation We’ll need to apply a number of fixes to our chosen function in order to make it ready for service. Move the center point to 0.5 along the x axis. Instrument a steepness knob. Force the function to cross 0 at 0, and 1 at 1. I’ve selected the sigmoid curve for our fixes because it’s based on which is generally easier to analyze than arctangent, though it’s nearly as easy to work with in this case. That exercise will be left to the reader. e Equation 1. The Sigmoid Function Moving the sigmoid is quite easy. Using an algebraic manipulation, if our easing function is , then moving the center point by 0.5 is accomplished by . f(t) f(t-0.5) Similarly, if we multiply by a chosen knob, , we can make the function “accelerate and condense” if > 1, or make it “slow and spread out” if < 1. t f(kt) k k Equation 2. Sigmoid augmented with a knob k and centered at t = 0.5. In Eqn. 2, we move the manipulations of into the function in order to make the function easier to use. This solves our first two problems, but take a look at the graph below where Eqn. 2 has been plotted with = 6. t k Figure 2. Eqn. 2 plotted with k = 6. Note that the ends are significantly above 0 and below 1. This is pretty close! However, the beginning and ending of the animation is respectively above 0 and below 1. This will cause an animation to exhibit a “jump” at the ends. If k is set sufficiently high (above about 11 or 12), you won’t even notice the problem. However, this restricts our range of expression as we can’t use any shallow curves, but it might be good enough for you and you can stop here if you so choose. While your first intuition, as it was mine, might be to find some way to make the sigmoid pass though zero by doing some trick like multiplying by or using some crazy conditional statements or reflections, there is a simpler way that’s kind of a hack. Notice that the error at the ends of the sigmoid is symmetric — the distance from 0 is the same as the distance from 1. What if we could just scale the result so that 0 passes through 0 and 1 passes through 1? t There are several steps to get this strategy to work, but in brief, you start with a basic sigmoid centered at 0 (so the range is -1/2 to 1/2), multiply by the correction factor (0.5 / f(1)), then figure out how to translate it up (+0.5) and to the right (t-0.5). You then condense the range of the function by 2 (it reaches -0.5 at -1 and +0.5 at 1). Equation 3. Evolved from Eqn. 2, this is our instrumented sigmoid centered at 0. Equation 4. The fully developed ease in out equation. s(t, α) comes from Eqn. 3. The first term is the correction factor, the second term is our sigmoid, and the third term moves the curve into position. Figure 3. Equation 4 graphed at k = 3. You can find the . grapher file here If you enjoyed this article, please follow me on Medium. You might also like The Bounce Factory and The Spring Factory , and others on Analytic Animations . Endnotes Here is a picture of arctangent: Inverse (Arc)tangent 2. In a discussion on , the function was brought to my attention. It has derivatives of 0 at 1 and 0 and is commonly used in shaders. Reddit smoothstep Figure 4. Smoothstep References — A collection of many commonly used easing curves. http://easings.net/ — A Grapher file I developed to design my easing functions. Ease In Out Graph

Ease In, Out: The Sigmoid Factory

Binary Classification: Understanding Activation and Loss Functions with a PyTorch Example

The brain of a human child is spectacularly amazing. Even in any previously unknown situation, the brain makes a decision based on its primal knowledge. Depending on the outcome, it learns and remembers the most optimal choices to be taken in that particular scenario. On a high level, this process of learning can be understood as a ’trial and error’ process, where the brain tries to maximise the occurrence of positive outcomes. What is Reinforcement Learning? Similar is the inception of Reinforcement Learning. An ideal machine is like a child’s brain, that can remember each and every decision taken in given tasks. Likewise, the goal is to try and optimise the results. In Reinforcement Learning, the learner isn’t told which action to take, but is instead made to try and discover actions that would yield maximum reward. In the most interesting and challenging cases, actions may not only affect the immediate reward, but also impact the next situation and all subsequent rewards. These two characteristics: ‘trial and error search’ and ‘delayed reward’ are the most distinguishing features of reinforcement learning. Machines vs Humans Many of us must have heard about the famous Alpha Go, built by Google using Reinforcement Learning. This machine has even beaten the world champion Lee Sudol in the abstract strategy board game of Go! Elon Musk in a famous debate on AI with Jack Ma, explained how machines are becoming smarter than humans. Reinforcement Learning is definitely one of the areas where machines have already proven their capability to outsmart humans. The Video Games Analogy Reinforcement Learning can be understood by an example of video games. A typical video game usually consists of: Fig: A Video Game Analogy of Reinforcement Learning An (player) who moves around doing stuffAn that the agent exists in (map, room). An that the agent takes (moves upward one space, sells cloak). A that the agent acquires (coins, killing other players). A that the agent currently exists in (on a particular square of a map, part of a room). A that the agent may have (level up, getting as many rewards as possible) agent environment action reward state goal The agent basically runs through sequences of state-action pairs in the given environment, observing the rewards that result, to figure out the best path for the agent to take in order to reach the . goal Rules of Reinforcement Learning The Markov decision process lays the foundation stone for Reinforcement Learning and formally describes an observable environment. There are two important parts of Reinforcement Learning: This is a function that maps a given state to probabilities of selecting each possible action from that state. In simple terms, it identifies the best probable set of actions an agent should take in order to maximize their reward. This policy is learnt by taking random decisions and then iterating backwards (updating weights) once there is a positive or negative end result. Policy Learning: These are functions of states, or of state-action pairs, that estimate how good is it for an agent to be in a given state, or how good is it for the agent to perform a given action in a given state. Unlike Policy Learning, Q-Learning takes two inputs: state and action, and returns a value for each pair. If you’re at an intersection, Q-Learning will tell you the expected value of each action that your agent could take (left, right, etc.). Value Functions (Q Learning): Applications of Reinforcement Learning There are numerous application areas of Reinforcement Learning. Starting from robotics and games to self-driving cars, Reinforcement Learning has found applications in many areas. Famous researchers in the likes of Andrew Ng, Andrej Karpathy and David Silverman are betting big on the future of Reinforcement Learning. Final Words It seems till date that the idea of outsmarting humans in every field is farfetched. But the seed has been sown and companies like Google and Tesla have shown that if machines and humans work together, the future has many opportunities to offer. As far as Reinforcement Learning is concerned, we at Sigmoid are excited about its future and its game changing applications. About Abhijeet is a Data Scientist at . He mainly works in the domain of Recommendation Engines, Time Series Forecasting, Reinforcement Learning and Computer Vision. Sigmoid Previously published here .

Reinforcement Learning: A Brief Introduction to Rules and Applications

In this post, we will talk about the Universal approximation theorem and we will also prove the theorem graphically. This is a follow-up post of my previous post on . Sigmoid Neuron Citation Note: The content and the structure of this article is based on the deep learning lectures from One-Fourth Labs — Padhai . Sigmoid Neuron Before we go into the actual discussion of Universal approximation theorem, I will give a brief recap on working of Sigmoid Neuron and how it will handle non-linearly separable data. A sigmoid neuron is similar to the perceptron neuron, for every input it has weight associated with the input. The weights indicate the importance of the input in the decision-making process. The output from the sigmoid is not 0 or 1 like the perceptron model instead it is a real value between 0–1 which can be interpreted as a probability. The most commonly used sigmoid function is the logistic function, which has a characteristic of an “ ” shaped curve. xi wi S Sigmoid Neuron Representation Handling Non-Linear Data From my previous post on , we have seen that even though we have introduced non-linear Sigmoid it is still not able to effectively separate ones and zeroes. The important point is that from a rigid decision boundary in , 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 a deep neural network. Sigmoid Neuron perceptron Modeling Complex Relations In real life, we deal with complex functions where the relationship between input and output might be complex, unlike the simple sigmoid neuron or perceptron. Complex Functions How do we even come up with such complex functions? It’s hard to come up with such functions there could be infinite possibilities for such a function, we need a simple approach. To solve this problem, let's take an analogy of building a house. You can think of the house as a complex function, the way we will build a house is we start with a building a block i.e... brick in this case. First, we will lay the foundation and on top of that we will lay another layer and we will just keep continuing in this manner till we get very complex output i.e… house. In this process, we are combining the very simple building blocks in an effective way so that we can get this house. Modeling Complex Functions In our case instead of bricks and houses, we are interested in complex functions. The building block in building complex functions is sigmoid neuron. The way we are going to create complex functions is that we will combine the sigmoid neurons in an effective way so that there will be an interaction between the neurons present at different layers. So that the predicted output is a complex function of inputs and there will be weights associated across multiple layers. Now the question arises, how do we decide how many layers to use, how many neurons in each layer etc… these are valid questions but the discussion on these questions is out of the scope of this article. Now I need to prove to you that the output from the complex function created using the network of neurons is very close to the true relationship between input and the output. Deep neural network with a certain number of hidden layers should be able to approximate any function that exits between input and output. says:that a feed-forward network with a single hidden layer containing a finite number of neurons can approximate continuous functions on compact subsets of , under mild assumptions on the activation function — Wikipedia Universal approximation theorem R In simple terms, UAT says that — you can always come up with a deep neural network that will approximate any complex relation between input and output. Proof of Universal Approximation Theorem In this section, we will see the illustrative proof of Universal Approximation theorem. For this illustrative proof, we will consider a simple example where we have one-dimensional input and output , the graph shown below represents the true relationship between input and the output. Let’s assume that I don’t know what that function equation between and and I can’t come up with a representation for the equation. x y x y True relationship between x and y To solve this problem, I will break this function into multiple smaller parts so that each part is represented by a simpler function. By combining the series of smaller functions (rectangular bars/towers) I can approximate the relation between and as close to the true relationship possible. x y Combination of simpler functions The key point to note in this case is that I don’t have to worry about coming up with complex equations to represent the relationship between input and output. I can just come up with a simple function and use the combination of these functions to approximate the relationship to my true relationship. The more functions that I choose in this method the better will be my approximation. How do we come up with these rectangles/towers and how it will tie back to the sigmoid neuron? Let’s take two sigmoid functions having a very steep slope and notice that they have a different place at which they peak. The left sigmoid peaks just before zero and right sigmoid peaks just after zero. If I just subtract these two functions the net effect is going to be a tower (rectangular output). If we can get the series of these towers, then we can approximate any true function between input and output. Neural Network Can we come up with a neural network to represent this operation of subtracting one sigmoid function from another?. Neural network for subtraction if we have an input and it is passed through the two sigmoid neurons and the output from these two neurons are combined in another neuron with weights +1 and -1 i.e… same is as subtracting these two outputs, then we will get our tower. Now you can see that we have our building block ready which is a connection of three sigmoid neurons. If we can construct many such building blocks and add all of them up, we can approximate any complex true relationship between input and output. This is called the , to approximate any kind of relationship between input and output. x representational power of deep neural networks Photo by on Sergey Pesterev Unsplash Conclusion In this post, we briefly looked at the working of sigmoid neuron and its ability to handle non-linear data. We then looked at how can we model complex relationships and saw the formal definition of Universal Approximation theorem and then went on to see the illustrative proof of Universal Approximation theorem. Recommended Reading _The building block of the deep neural networks is called the sigmoid neuron._towardsdatascience.com Sigmoid Neuron — Deep Neural Networks _Read writing from Niranjan Kumar in Hacker Noon. Intern at HSBC Analytics. ML and DL Enthusiast. Writer at…_hackernoon.com Niranjan Kumar - Hacker Noon In my next post, we will discuss the feed forward neural works in detail for both regression and classification tasks. Stay tuned for that. is working as an intern at HSBC Analytics division. He is passionate about deep learning and AI. He is one of the top writers at in . Connect with me on or follow me on for updates about upcoming articles on deep learning and Artificial Intelligence. Niranjan Kumar Medium Artificial Intelligence LinkedIn twitter

Illustrative Proof of Universal Approximation Theorem

This paper is available on arxiv under CC 4.0 license.
Authors:
(1) Ghazaleh H. Torbati, Max Planck Institute for Informatics Saarbrucken, Germany & ghazaleh@mpi-inf.mpg.de;
(2) Andrew Yates, University of Amsterdam Amsterdam, Netherlands & a.c.yates@uva.nl;
(3) Anna Tigunova, Max Planck Institute for Informatics Saarbrucken, Germany & tigunova@mpi-inf.mpg.de;
(4) Gerhard Weikum, Max Planck Institute for Informatics Saarbrucken, Germany & weikum@mpi-inf.mpg.de. Table of Links Abstract and Introduction
Related Work
Methodology
Experimental Design
Experimental Results
Conclusion
Ethics Statement and References III. METHODOLOGY A. System Architecture The CUP framework is based on a two-tower architecture for representation learning (one ”tower” for users, the other for items, following the prevalent architecture in neural information retrieval with query and document/passage encodings). The two towers are jointly trained, coupled by the shared loss function and ground truth. Figure 1 shows a pictorial overview. User reviews and items descriptions are fed into BERT followed by a feed-forward network to learn latent representations. Downstream, the vectors are simply compared by a dot product for scores that indicate whether the user likes an item or not. Importantly, unlike prior works, we allow the top-most layer of BERT itself to be fine-tuned as part of the end-to-end training process. The per-item text usually comprises book titles, tags like categories or genre labels, which can be coarse (e.g., “thriller”) or fine-grained (e.g., “Scandinavian crime noir”), and a short description of the book contents. The per-user text can comprise the titles and tags of her training-set books and the entirety of her review texts, which vary widely in length and informativeness, hence the need for smart text selection. In the following, we present the CUP training procedure in Subsection III-B, the test/deployment-time inference in Subsection III-C, the judicious selection of input text snippets in Subsection III-D, and the selection of negative training samples in Subsection III-E. B. Training Another optional mechanism is to process multiple chunks of bounded-length text, which could be of interest for long user reviews. In this case, the per-user representations are learned for each chunk, and there is an additional max-pooling step between the FFN and the dot product, to combine the per-chunk vectors. We use this option for comparison in experiments. As it incurs m times higher computational and energy costs, with m being the number of chunks, we do not favor it. A user-item pair is classified with score calculated by with <, > denoting dot product and σ for the sigmoid function. We use the Adam optimizer to minimize the binary cross entropy loss between predicted labels and the ground truth with sampled negatives. During training we update the topmost layer of BERT, which allows end-to-end training of all components. This is an important difference to earlier works with similar architectures, which use frozen BERT to encode user and item representations. C. Inference Prediction for Ranking. At test time, a prediction is made for user-item pairs. We encode the item description by running it through the trained network, and we compare it to the already learned user vector, which is based on the user’s training-time reviews (none for the given test item). The scores for different test items, computed by the final dot product, yield the ranking of the candidate items. This is a very efficient computation, following established practice in neural information retrieval [37]. Search-based Recommendation. In a deployed system (as opposed to lab experiments with test samples), a typical usage mode would be search-based re-ranking: a user provides context with a tag-based query or an example of a specific liked item, which can be thought of as query-by-example. The user’s expectation is to see a ranked list of recommended items that are similar to her positive sample (as opposed to recommendations from all kinds of categories). The system achieves this by first computing approximate matches to the query item (i.e., similarity-search neighbors), and then re-ranking a shortlist of say top-100 candidates. The CUP framework supports this mode, by using a light-weight BM25 retrieval model, querying all unlabeled items with the category and textual description of the positive point at hand, and keeping the top-100 highest scoring matches. D. Coping with Long and Noisy Texts For constructing user profiles from text, the simplest idea would be to concatenate all available reviews into a long token sequence. Two problems arise, though. User reviews are a noisy mix of descriptive elements (e.g., “the unusual murder weapon”), sentiment expressions (e.g., “it was fun to read”) and personal but irrelevant statements (e.g., “I read only on weekends”). Only the first aspect is helpful for content-based profiling (as the sentiment is already captured by the fact that the user likes the book). Second, the entirety of user-provided text can be too long to be fully digested by the Transformer. Even when it would fit into the token budget, the computational and energy cost is quadratic in the number of input tokens. Therefore, we tightly limit the tokens for each user’s text profile to 128, and devise a suite of light-weight techniques for judiciously selecting the most informative pieces. Variants of our methodology are to create concise profiles by generative language models upfront (e.g., using T5 or ChatGPT), or to digest multiple chunks of text with downstream aggregation by max-pooling or averaging. We include these techniques in our experimental comparisons, but emphasize that both of these alternatives have much higher computational cost and climate footprint. Our techniques for selecting the most informative parts of user reviews to construct concise profiles, are as follows: • Weighted Phrases: selected unigrams, bigrams or trigrams, ordered by descending tf-idf weights, where tf is the frequency of the phrase in all of the given user’s reviews, and idf is pre-computed on Google books n-grams to capture informativeness. • Weighted Sentences: selected sentences, ordered by descending idf weights, where a sentence’s total weight is the sum of the per-word idf weights normalized by sentence length. • Similar Sentences: selected sentences, ordered by descending similarity scores computed via Sentence-BERT [38] for comparing the user-review sentences against the sentences of the item description. To ensure that the selected set is not dominated by a single item, the sentences are picked from different items in a round-robin manner. • ChatGPT-generated Profiles: feeding all reviews of a user, in large chunks, into ChatGPT and instructing it to characterize the user’s book interests with a few short keyphrases. The output size is limited per chunk with a total budget of 128 tokens, to enforce concise profiles. • T5-generated Keywords: using a T5 model fine-tuned for keyword generation, to cast each user’s review text into a set of keywords, concatenated to create the profile. We give examples of selected profile constructions for a Goodreads and an Amazon user in Table VII. An online demo of the CUP framework is accessible on our website[2]. E. Coping with Unlabeled Data A challenge for training in the data-poor regime is how to handle the extreme skew between positive samples and unlabeled data points for sparse users. The crux in many recommender applications is that there are extremely few, if any, explicitly negative samples, such as books rated with low marks. This holds also for the datasets in this work. Therefore, we introduce and experiment with two different techniques to construct negative training samples from unlabeled data: • Uniform random samples. Under the closed world assumption (CWA), aka Selected Completely At Random, negative training points are sample uniformly from all unlabeled data. This is the baseline technique, widely used in prior works. • Weighted pos-neg samples. Prior works on PU learning [39], with positive and unlabeled data and without explicitly negative samples, is largely based on treating unlabeled points as pairs of samples, one positive and one negative with fractional weights. The weights can be based on (learned estimates of) class priors, but the extreme skew in our data renders these techniques ineffective. Instead we leverage the fact that relatedness measures between item pairs can be derived from tagging features or interaction data. We compute item-item relatedness via basic matrix factorization of the user-item matrix for the entire dataset. The relatedness of two items is set to the scalar product between their latent vectors, re-scaled for normalization between 0 and 1. Then, each originally unlabeled sample is cloned, with one instance positive with weight proportional to its average relatedness to the user’s explicitly positive points. The negative clone’s weight is set to the complement. [2] https://sirup.mpi-inf.mpg.de/ This paper is available on arxiv under CC 4.0 license. Authors: (1) Ghazaleh H. Torbati, Max Planck Institute for Informatics Saarbrucken, Germany & ghazaleh@mpi-inf.mpg.de; (2) Andrew Yates, University of Amsterdam Amsterdam, Netherlands & a.c.yates@uva.nl; (3) Anna Tigunova, Max Planck Institute for Informatics Saarbrucken, Germany & tigunova@mpi-inf.mpg.de; (4) Gerhard Weikum, Max Planck Institute for Informatics Saarbrucken, Germany & weikum@mpi-inf.mpg.de. This paper is available on arxiv under CC 4.0 license. Authors: (1) Ghazaleh H. Torbati, Max Planck Institute for Informatics Saarbrucken, Germany & ghazaleh@mpi-inf.mpg.de; (2) Andrew Yates, University of Amsterdam Amsterdam, Netherlands & a.c.yates@uva.nl; (3) Anna Tigunova, Max Planck Institute for Informatics Saarbrucken, Germany & tigunova@mpi-inf.mpg.de; (4) Gerhard Weikum, Max Planck Institute for Informatics Saarbrucken, Germany & weikum@mpi-inf.mpg.de. Table of Links Abstract and Introduction Related Work Methodology Experimental Design Experimental Results Conclusion Ethics Statement and References Abstract and Introduction Abstract and Introduction Related Work Related Work Methodology Methodology Experimental Design Experimental Design Experimental Results Experimental Results Conclusion Conclusion Ethics Statement and References Ethics Statement and References III. METHODOLOGY A. System Architecture A. System Architecture The CUP framework is based on a two-tower architecture for representation learning (one ”tower” for users, the other for items, following the prevalent architecture in neural information retrieval with query and document/passage encodings). The two towers are jointly trained, coupled by the shared loss function and ground truth. Figure 1 shows a pictorial overview. User reviews and items descriptions are fed into BERT followed by a feed-forward network to learn latent representations. Downstream, the vectors are simply compared by a dot product for scores that indicate whether the user likes an item or not. Importantly, unlike prior works, we allow the top-most layer of BERT itself to be fine-tuned as part of the end-to-end training process. The per-item text usually comprises book titles, tags like categories or genre labels, which can be coarse (e.g., “thriller”) or fine-grained (e.g., “Scandinavian crime noir”), and a short description of the book contents. The per-user text can comprise the titles and tags of her training-set books and the entirety of her review texts, which vary widely in length and informativeness, hence the need for smart text selection. In the following, we present the CUP training procedure in Subsection III-B, the test/deployment-time inference in Subsection III-C, the judicious selection of input text snippets in Subsection III-D, and the selection of negative training samples in Subsection III-E. B. Training B. Training Another optional mechanism is to process multiple chunks of bounded-length text, which could be of interest for long user reviews. In this case, the per-user representations are learned for each chunk, and there is an additional max-pooling step between the FFN and the dot product, to combine the per-chunk vectors. We use this option for comparison in experiments. As it incurs m times higher computational and energy costs, with m being the number of chunks, we do not favor it. A user-item pair is classified with score calculated by with <, > denoting dot product and σ for the sigmoid function. We use the Adam optimizer to minimize the binary cross entropy loss between predicted labels and the ground truth with sampled negatives. During training we update the topmost layer of BERT, which allows end-to-end training of all components. This is an important difference to earlier works with similar architectures, which use frozen BERT to encode user and item representations. C. Inference C. Inference Prediction for Ranking. At test time, a prediction is made for user-item pairs. We encode the item description by running it through the trained network, and we compare it to the already learned user vector, which is based on the user’s training-time reviews (none for the given test item). The scores for different test items, computed by the final dot product, yield the ranking of the candidate items. This is a very efficient computation, following established practice in neural information retrieval [37]. Prediction for Ranking. Search-based Recommendation. In a deployed system (as opposed to lab experiments with test samples), a typical usage mode would be search-based re-ranking: a user provides context with a tag-based query or an example of a specific liked item, which can be thought of as query-by-example. The user’s expectation is to see a ranked list of recommended items that are similar to her positive sample (as opposed to recommendations from all kinds of categories). The system achieves this by first computing approximate matches to the query item (i.e., similarity-search neighbors), and then re-ranking a shortlist of say top-100 candidates. The CUP framework supports this mode, by using a light-weight BM25 retrieval model, querying all unlabeled items with the category and textual description of the positive point at hand, and keeping the top-100 highest scoring matches. D. Coping with Long and Noisy Texts D. Coping with Long and Noisy Texts For constructing user profiles from text, the simplest idea would be to concatenate all available reviews into a long token sequence. Two problems arise, though. User reviews are a noisy mix of descriptive elements (e.g., “the unusual murder weapon”), sentiment expressions (e.g., “it was fun to read”) and personal but irrelevant statements (e.g., “I read only on weekends”). Only the first aspect is helpful for content-based profiling (as the sentiment is already captured by the fact that the user likes the book). Second, the entirety of user-provided text can be too long to be fully digested by the Transformer. Even when it would fit into the token budget, the computational and energy cost is quadratic in the number of input tokens. Therefore, we tightly limit the tokens for each user’s text profile to 128, and devise a suite of light-weight techniques for judiciously selecting the most informative pieces. Variants of our methodology are to create concise profiles by generative language models upfront (e.g., using T5 or ChatGPT), or to digest multiple chunks of text with downstream aggregation by max-pooling or averaging. We include these techniques in our experimental comparisons, but emphasize that both of these alternatives have much higher computational cost and climate footprint. Our techniques for selecting the most informative parts of user reviews to construct concise profiles, are as follows: • Weighted Phrases: selected unigrams, bigrams or trigrams, ordered by descending tf-idf weights, where tf is the frequency of the phrase in all of the given user’s reviews, and idf is pre-computed on Google books n-grams to capture informativeness. • Weighted Sentences: selected sentences, ordered by descending idf weights, where a sentence’s total weight is the sum of the per-word idf weights normalized by sentence length. • Similar Sentences: selected sentences, ordered by descending similarity scores computed via Sentence-BERT [38] for comparing the user-review sentences against the sentences of the item description. To ensure that the selected set is not dominated by a single item, the sentences are picked from different items in a round-robin manner. • ChatGPT-generated Profiles: feeding all reviews of a user, in large chunks, into ChatGPT and instructing it to characterize the user’s book interests with a few short keyphrases. The output size is limited per chunk with a total budget of 128 tokens, to enforce concise profiles. • T5-generated Keywords: using a T5 model fine-tuned for keyword generation, to cast each user’s review text into a set of keywords, concatenated to create the profile. We give examples of selected profile constructions for a Goodreads and an Amazon user in Table VII. An online demo of the CUP framework is accessible on our website[2]. E. Coping with Unlabeled Data E. Coping with Unlabeled Data A challenge for training in the data-poor regime is how to handle the extreme skew between positive samples and unlabeled data points for sparse users. The crux in many recommender applications is that there are extremely few, if any, explicitly negative samples, such as books rated with low marks. This holds also for the datasets in this work. Therefore, we introduce and experiment with two different techniques to construct negative training samples from unlabeled data: • Uniform random samples. Under the closed world assumption (CWA), aka Selected Completely At Random, negative training points are sample uniformly from all unlabeled data. This is the baseline technique, widely used in prior works. Uniform random samples. • Weighted pos-neg samples. Prior works on PU learning [39], with positive and unlabeled data and without explicitly negative samples, is largely based on treating unlabeled points as pairs of samples, one positive and one negative with fractional weights. The weights can be based on (learned estimates of) class priors, but the extreme skew in our data renders these techniques ineffective. Instead we leverage the fact that relatedness measures between item pairs can be derived from tagging features or interaction data. We compute item-item relatedness via basic matrix factorization of the user-item matrix for the entire dataset. The relatedness of two items is set to the scalar product between their latent vectors, re-scaled for normalization between 0 and 1. Then, each originally unlabeled sample is cloned, with one instance positive with weight proportional to its average relatedness to the user’s explicitly positive points. The negative clone’s weight is set to the complement. [2] https://sirup.mpi-inf.mpg.de/

Recommendations by Concise User Profiles from Review Text: Methodology

Loss of Amazon Rainforest Resilience: Materials and Methods

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.

SIGMOID

sigmoid.com

-- employees

Since n.d.

COMPANY RANKING

SIGMOID

EVERGREEN INDEX #9509

Representing this company? Click here to claim and customize this page!

Sigmoid WIKI

Apply to get your Tech Company News Page live on HackerNoon.com today!

SIGMOID

sigmoid.com

-- employees

Since n.d.

COMPANY RANKING

SIGMOID

EVERGREEN INDEX #9509

RELATED COMPANIESRANK

STORIES ABOUT SIGMOID

Representing this company? Click here to claim and customize this page!

Sigmoid WIKI

Apply to get your Tech Company News Page live on HackerNoon.com today!