AI/ML News & Innovations Hub

Linear Regression - Data Subset with Lowest Mean-Squared Error?

Short version: given a linear regression dataset and an integer $K$ , what data subset of size $K$ results in linear parameters with the lowest mean squared error on the entire dataset? Long version: Suppose I have a supervised dataset $D_n := \{x_n, y_n\}_{n=1}^N$ and I want to perform ordinary linear regression by minimizing $$ L_{OLS, N}(w) = ||Y_N - X_N w||_2^2$$ where $Y_N$ is the matrix formed by stacking $y_n$ as row vectors and $X_N$ is the matrix formed by stacking $x_n$ as row vectors. I use the subscript $N$ to remind us that we used all the data to compute the parameter estimates. Alternatively, I might want to perform ridge linear regression by minimizing $$ L_{Ridge, N}(w) = ||Y_N - X_N w||_2^2 + c ||w||_2^2$$ We know that the parameters that minimize the above losses are given by $$w_{OLS, N} = (X_N^T X_N)^{-1} X_N^T Y_N$$ $$w_{Ridge, N} = (X_N^T X_N + c I)^{-1} X_N^T Y_N$$ What I'm curious to know is: if we're forced to choose a subset of only $K data $D_K := \{(x_k, y_k)\}_{k=1}^K \subset D_N$ and we fit parameters $$w_{OLS, K} = (X_K^T X_K)^{-1} X_K^T Y_K$$ $$w_{Ridge, K} = (X_K^T X_K + c I)^{-1} X_K^T Y_K$$ what subset of $K$ data results in parameters $w_{OLS, K}$ and $w_{Ridge, K}$ such that $L_{OLS, N}(w)$ and $L_{Ridge, N}(w)$ (respectively) are minimized?

How to handle missing data in a time serie knowing that the mean will be used in further modelling?

I need to run a survival analysis and use environmental data containing missing values. The environmental data are per day, and I actually want to use the mean of each variable during (let's say) the 10 days prior an event before running a cox model. So that I can take into account the environmental situation and assign one value to each individual. How can I handle missing values in such a case? From many readings, here (such as 1 , 2 or 3 ) and on several stat websites on missing data, multiple imputation would be a suitable method. It is also mentioned in previous articles in my field, but without details on how they implemented that before running a cox model. I am also open to other ideas. The MICE package in R seems a good fit for that. However, it is my understanding that such methods generates several data sets with imputed values, that all the imputed data should then be used in the model, and that the results can then be pooled. Here, how can I "pool" or build one dataset on which I can perform the mean and then use that data set into the cox model? Should I calculate the mean for each imputed data set, and use them in the cox model? Is there a way to "extract" one data set based on the imputed data sets and then take the mean? Other options/best practice ideas/tips on the coding part as well? Thanks!

Considerations for Deploying Machine Learning Models in Production

A common grumble among data science or machine learning researchers or practitioners is that putting a model in production is difficult. As a result, some claim that a large percentage, 87%, of models never see the light of the day in production. “I have a model, I spent considerable time developing it on my laptop. What’s next? How do I get it into our production environment? What should I consider for my ML stack and tooling?”

Why is there a 1 in complexity formula of uniform-cost search?

I am reading the book titled Artificial Intelligence: A Modern Approach 4th ed by Stuart Russell and Peter Norvig. According to the book, the complexity of uniform-cost search is as $$ O(b^{1+\lfloor{C^*/\epsilon}\rfloor}), $$ where $b$ is the branching factor (i.e. the number of available actions in each state), $C^*$ is the cost of the optimal solution, and $\epsilon > 0$ is a lower bound of the cost of each action. My question is: Why is there is a 1 in the formula? For example, suppose in the following tree, the red node is the initial state and the green one is the goal state, and two actions are needed to reach the goal state from the initial state. If the cost of both actions is equal to $\epsilon = 1$ , so, $C^*$ will be $2$ . Therefore, the complexity will be $O(b^{2})$ . But, from the above formula, the complexity will be $O(b^{3})$ . PS. I know there is a similar question in stackoverflow and have read the answer. But there is a disagreement between the answers about the 1.

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Fixed effects in a cross-sectional data

I'm working with research that has cross-sectional data. I have collected information about publicly-listed banks in many countries. For example, for each bank I collected the following information: I have 619 banks in 58 different countries. In my research, I want to test how variations in bank characteristics (Tier 1, Tangible Equity, etc.) affected bank stock returns during crisis time. My equation is as follows: BPb,c = 𝑎 + 𝛽1 RETURNS_2019b + 𝛽2 TIER_1b + 𝛽3 DEPb,+ 𝛽4 NPLb + 𝛽5 NONIIb + 𝛽6 LIQASSb + 𝛽8 SIZEb + 𝛽9 DENb + 𝛽10 ROAEb + 𝛽11 LOANSb + 𝛽3*TANEQb + 𝛾c + ub,c Where BPb,c is the performance of a bank b in country c. The coefficients 𝑎, 𝛽, represent vectors of coefficient estimates and ub.c is the error term. 𝛾c - country fixed effects. In all the literature I have read, fixed effects are applied to panel data models. However, following Beltrati and Stulz (2012), which to my understanding, has cross-sectional data as well, they apply fixed effects and use standard errors clustered by country. Is this approach using country-fixed effects and clustering error by country (with cross-sectional data) logical? Also, perhaps someone could advise how to implement this model in Stata.

Performing Linear Regression using Stochastic Gradient Descent, by batches

I am presented with a data set, where I am supposed to perform linear regression on this using SGD. My first instinct would be to train each data point there is until I reach the last one. Only then will I get my estimate $\hat{y}$ . I understand there are some drawbacks to this: It will take so long to finish, since you have to train all data points. Convergence of parameters is still not guaranteed. Thus, the idea of batching comes into mind. For example, I have a set of 100 data points. I have decided to group them in to 25 batches (or 4 data points each). My questions are: How does this batching work? Do I randomly pick one data point to train from each batch? Meaning, after the end of first run, I will have one estimate. Is it possible that I will have 4 different estimates after having 4 different runs? Should I choose whichever gives the smallest error?

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Stanford AI Lab Papers at CoRL 2021

The Conference on Robot Learning (CoRL 2021) will take place next week. We’re excited to share all the work from SAIL that will be presented, and you’ll find links to papers, videos and blogs below. Feel free to reach out to the contact authors directly to learn more about the work that’s happening at Stanford! List of Accepted Papers LILA: Language-Informed Latent Actions Authors : Siddharth Karamcheti*, Megha Srivastava*, Percy Liang, Dorsa Sadigh Contact : skaramcheti@cs.stanford.edu, megha@cs.stanford.edu Keywords : natural language, shared autonomy, human-robot interaction BEHAVIOR: Benchmark for Everyday Household Activities in Virtual, Interactive, and Ecological Environments Authors : Sanjana Srivastava*, Chengshu Li*, Michael Lingelbach*, Roberto Martín-Martín*, Fei Xia, Kent Vainio, Zheng Lian, Cem Gokmen, Shyamal Buch, C. Karen Liu, Silvio Savarese, Hyowon Gweon, Jiajun Wu, Li Fei-Fei Contact : sanjana2@stanford.edu Links: Paper | Website Keywords : embodied ai, benchmarking, household activities Co-GAIL: Learning Diverse Strategies for Human-Robot Collaboration Authors : Chen Wang, Claudia Pérez-D’Arpino, Danfei Xu, Li Fei-Fei, C. Karen Liu, Silvio Savarese Contact : chenwj@stanford.edu Links: Paper | Website Keywords : learning for human-robot collaboration, imitation learning DiffImpact: Differentiable Rendering and Identification of Impact Sounds Authors : Samuel Clarke, Negin Heravi, Mark Rau, Ruohan Gao, Jiajun Wu, Doug James, Jeannette Bohg Contact : spclar…

Stanford AI Lab Papers at EMNLP/CoNLL 2021

The 2021 Conference on Empirical Methods in Natural Language Processing (EMNLP 2021) will take place next week, colocated with CoNLL 2021. We’re excited to share all the work from SAIL that will be presented, and you’ll find links to papers, videos and blogs below. Feel free to reach out to the contact authors directly to learn more about the work that’s happening at Stanford! List of Accepted Papers Calibrate your listeners! Robust communication-based training for pragmatic speakers Authors : Rose E. Wang, Julia White, Jesse Mu, Noah D. Goodman Contact : rewang@stanford.edu Links: Paper | Video Keywords : language generation, pragmatics, communication-based training, calibration, uncertainty Cross-Domain Data Integration for Named Entity Disambiguation in Biomedical Text Authors : Maya Varma, Laurel Orr, Sen Wu, Megan Leszczynski, Xiao Ling, Christopher Ré Contact : mvarma2@stanford.edu Links: Paper | Video Keywords : named entity disambiguation, biomedical text, rare entities, data integration ContractNLI: A Dataset for Document-level Natural Language Inference for Contracts Authors : Yuta Koreeda, Christopher D. Manning Contact : koreeda@stanford.edu Links: Paper | Website Keywords : natural language inference, contract, law, legal, dataset Venue : The Findings of EMNLP 2021 The Emergence of the Shape Bias Results from Communicative Efficiency Authors : Eva Portelance, Michael C. Frank, Dan Jurafsky, Alessandro Sordoni, Romain Laroche Contact : portelan@stanford.edu Links…

Is there a way to update the neural network to fit the new data without the time required for retraining?

I built a basic neural network in MATLAB. The neural network classifies points on the X-Y axis system into two classes (0 and 1). (I try to get the function that represents a shape from this photo) Every so often the values ​​of the points change slightly and some of the points defined in class 1 become class 0, like in this photo. Is there a way to update the neural network to fit the new data without the time required for retraining?

Ray version 1.8 has been released

Ray version 1.8 has been released! Release highlights include: Ray SGD has been renamed to Ray Train, Ray Datasets is now in beta, and experimental support for Ray on Apple Silicon (M1 Macs)

What's the right statistical analysis for temperature effects on turtle hibernation over time?

I have an assignment in which I am supposed to create a theoretical study design within a biological field. Currently, I have data from a group of seven turtles. For the assignment, I thought it would be interesting to study the effect that temperature has on the hibernation cycle of box turtles; however, I’m not sure what statistical analysis is best My null hypothesis is that temperature does not have a significant effect on the duration of the hibernation cycle of box turtles in my area from 2015-2020. However, I do not have enough experience with statistics to know how best to analyze the data. For the data themselves, each of the seven turtles was tracked roughly once a week for each of the years that I am examining. The start of hibernation is considered two consecutive weeks (two data points) in which the turtle did not move. The end of hibernation is marked as the week in which the turtle begins moving again in the spring. I also have the air and ground temperatures for each day that the turtles were successfully tracked. So, below is a simplified chart of what my data looks like for each turtle. There is some variation in the start and end dates of hibernation for individual turtles between years. Turtle A Date 2015 2016 2017 2018 2019 2020 Day 1 Ground Temp, Air Temp Ground Temp, Air Temp Ground Temp, Air Temp Ground Temp, Air Temp Ground Temp, Air Temp Ground Temp, Air Temp Day 2 Ground Temp, Air Temp Ground Temp, Air Temp Ground Temp, Air Temp Ground Temp, Air Te…

Obtain a between-class similarity. And is the way to do it through PCA valid?

Context: I have a dataset containing instances labeled into different classes, and for all the classes, I have the same set of features. My research question is to identify classes that are more similar to each other. My initial thought was to compare these classes by estimating the pairwise similarity. And, by pairwise similarity, I mean the similarity matrix between all the classes considered. As bellow: Similarity matrix for classes A, B, C, D : A B C D A 1.0, 0.3, 0.7, 0.8 B 0.3, 1.0, 0.2, 0.4 C 0.7, 0.2, 1.0, 0.9 D 0.8, 0.4, 0.9, 1.0 Example: For simplicity, let's consider the iris dataset . And my goal is to find if iris setosa is more similar to iris virginica or to iris versicolor. I want to compute the similarity for each possible pair (a,b) for a,b in (setosa, virginica, and versicolor) . Assume that I have standardized all the features between 0 and 1 universally. Only after standardizing, I separated the iris labeled instances into 3 subsets ( X_setosa , X_virginica , X_versicolor ), according to their classes. Then, I have generated 3 PCs ( PC_setosa , PC_virginica , and PC_versicolor ), one for each set s as bellow: pca_s = PCA(n_components=2) pca.fit_transform(X_s) PC_s = pca_s.components_ My questions are: Does that idea of comparing the PCs (eigenvectors) as a proxy for classes similarity make sense? How could I compare the PCs structures using the cosine similarity? After some googling, I don't know if its better to compare the loadings or the eigenvectors…

Why is coherence of this wavelet transform almost always near 1?

I'm trying to understand the different aspects of a wavelet transform. Wavelet power has made enough sense to me as an analogy of the covariance. However, the wavelet coherence does not make sense to me. The notes in the R package 'WaveletComp' states that coherence is analogous to the coefficient of correlation. In my intuition, that would mean coherence between independent time series should be reasonably low. However, this is not the case, as is shown by the R code below. library(tidyverse) library(WaveletComp) complete_noise % mutate( sample_1 = rnorm(x), sample_2 = rnorm(x) ) wc_noise The plot shows the majority of the area as red, which is up near a coherence of 1.0. Why do independent time series have a high coherence in most of these points on the period-time plot? Edit: I turned off smoothing (it is on by default with WaveletComp) and now the plot shows a coherence of 1 everywhere. I'm taking this to mean the drops in coherence are due to smoothing, which also doesn't make sense to me.

Looking for multilclass classifier that can handle repeated measures

I am looking for a multiclass classifier that can handle repeated measures. Specifically, each of my subjects appears multiple times with the same number of n classes. Now I would like to fit a classifier to this dataset to get an intuition for the feature-label relationship (that is, which features are the most associated with which classes?). Is there a multiclass classifier that can take this into account? EDIT: My problem is a multi-class problem, not a multi-label problem. I did not know that 'multilabel' and 'multiclass' describe different problems, so I changed the wording.

Selective Classification Can Magnify Disparities Across Groups

Selective classification, where models are allowed to “abstain” when they are uncertain about a prediction, is a useful approach for deploying models in settings where errors are costly. For example, in medicine, model errors can have life-or-death ramifications, but abstentions can be easily handled by backing off to a doctor, who then makes a diagnosis. Across a range of applications from vision 1 2 3 and NLP 4 5 , even simple selective classifiers, relying only on model logits, routinely and often dramatically improve accuracy by abstaining. This makes selective classification a compelling tool for ML practitioners 6 7 . However, in our recent ICLR paper, we find that despite reliably improving average accuracy, selective classification can fail to improve and even hurt the accuracy over certain subpopulations of the data . As a motivating example, consider the task of diagnosing pleural effusion, or fluid in the lungs, from chest X-rays. Pleural effusion is often treated with a chest drain, so many pleural effusion cases also have chest drains, while most cases without pleural effusion do not have chest drains 8 . While selective classification improves average accuracy for this task, we find that it does not appreciably improve accuracy on the most clinically relevant subgroup, or subpopulation, of the data: those that have pleural effusion but don’t yet have a chest drain, i.e. those that have pleural effusion but have not yet been treated for it. Practitioners, thus,…

Duke Computer Scientist Wins $1 Million Artificial Intelligence Prize, A ‘New Nobel’

Duke professor Cynthia Rudin is the second recipient of the AAAI Squirrel AI Award for pioneering socially responsible AI. She is being cited for “pioneering scientific work in the area of interpretable and transparent AI systems in real-world deployments, the advocacy for these features in highly sensitive areas such as social justice and medical diagnosis, and serving as a role model for researchers and practitioners.” The post Duke Computer Scientist Wins $1 Million Artificial Intelligence Prize, A ‘New Nobel’ appeared first on AAAI .

What kind of algorithm or approach can I use to find a specific type of object in an image?

What kind of algorithm or approach can I use to find a specific type of object in an image? In particular, I am interested in finding an object like a windmill in an image taken, for example, from Google Maps. The image could be something like this

Choosing the best model for Hidden Markov Model

How do I determine the best model from the Baum-Welch algorithm if I know the number of hidden states? Every time I insert a different initial estimation matrix, I will get a different model from the Baum Welch. Data = 211112211111211222211211221221112122211121122221121212111121111121121122122212 222222222211211121112 Result is the below the code and the actual transition and emission is the same as the initial estimates.

Closed networks vs Networks with a removed delay to predict new data

I've come across two types of neural networks to predict, both from Matlab, the closed structure and the net that removes one delay to find new data. From Matlab's app generated scripts we see: % Closed Loop Network % Use this network to do multi-step prediction. % The function CLOSELOOP replaces the feedback input with a direct % connection from the output layer. netc = closeloop(net); netc.name = [net.name ' - Closed Loop']; view(netc) [xc,xic,aic,tc] = preparets(netc,{},{},T); yc = netc(xc,xic,aic); closedLoopPerformance = perform(net,tc,yc) % Step-Ahead Prediction Network % For some applications it helps to get the prediction a timestep early. % The original network returns predicted y(t+1) at the same time it is % given y(t+1). For some applications such as decision making, it would % help to have predicted y(t+1) once y(t) is available, but before the % actual y(t+1) occurs. The network can be made to return its output a % timestep early by removing one delay so that its minimal tap delay is now % 0 instead of 1. The new network returns the same outputs as the original % network, but outputs are shifted left one timestep. nets = removedelay(net); nets.name = [net.name ' - Predict One Step Ahead']; view(nets) [xs,xis,ais,ts] = preparets(nets,{},{},T); ys = nets(xs,xis,ais); stepAheadPerformance = perform(nets,ts,ys) My question is: What is the real difference between them? Can one uses them equivalently? If yes, why? I mean, even tho the structure or how they are equipp…

Ray version 1.7 has been released

Ray version 1.7 is here. Highlights include: Ray SGD v2 and is in alpha, Ray Workflows is in alpha, and major enhancements to the C++ API

Stanford AI Lab Papers at ICCV 2021

The International Conference on Computer Vision (ICCV 2021) will be hosted virtually next week. We’re excited to share all the work from SAIL that will be presented, and you’ll find links to papers, videos and blogs below. Feel free to reach out to the contact authors directly to learn more about the work that’s happening at Stanford! List of Accepted Papers GLoRIA: A Multimodal Global-Local Representation Learning Framework for Label-efficient Medical Image Recognition Authors : Mars Huang Contact : mschuang@stanford.edu Keywords : medical image, self-supervised learning, multimodal fusion 3D Shape Generation and Completion Through Point-Voxel Diffusion Authors : Linqi Zhou, Yilun Du, Jiajun Wu Contact : linqizhou@stanford.edu Links: Paper | Video | Website Keywords : diffusion, shape generation CAPTRA: CAtegory-level Pose Tracking for Rigid and Articulated Objects from Point Clouds Authors : Yijia Weng*, He Wang*, Qiang Zhou, Yuzhe Qin, Yueqi Duan, Qingnan Fan, Baoquan Chen, Hao Su, Leonidas J. Guibas Contact : yijiaw@stanford.edu Award nominations: Oral Presentation Links: Paper | Video | Website Keywords : category-level object pose tracking, articulated objects Detecting Human-Object Relationships in Videos Authors : Jingwei Ji, Rishi Desai, Juan Carlos Niebles Contact : jingweij@cs.stanford.edu Links: Paper Keywords : human-object relationships, video, detection, transformer, spatio-temporal reasoning Geography-Aware Self-Supervised Learning Authors : Kumar Ayush, Bura…

Decide whether a discrete random variable is usual or unusual using Probability distribution?

We know that in frequency distribution we decide whether a value is usual or unusual depending on how many standard deviations it's away from mean. Moving forward to Probability distribution my teacher explained that we calculate the mean using the following formula $\sum xP(x)$ . Later he said that we can decide whether the discrete random variable $x$ is usual or unusual by calculating how many standard deviations is the $x$ away from 'mean. This doesn't make sense to me since we calculated the mean by multiplying 'the value of the discrete variable' by 'the probability of that variable', so the probability is involved in this formula, as well as the standard deviation formula. Shouldn't we decide whether the $x$ is usual or unusual by looking at its probability?

Dividing a data set into segments with consistent inner behavior, using segmentation algorithms and metrics for consistency

Context of the problem: I have signal data which was recorded in a software system and which shows the runtime of multiple processes over time. In total there are more than 900 processes each having roughly 400 reading points. The data is stored in an m x n matrix with roughly 900 rows (one process per row) and 400 columns (reading points/runtime values). The signal has been divided into five segments with two segments containing values between 50 and 60, and the three remaining segments containing values between 0 and 30. Intuitively, dividing the signal into five segments seems to be the ideal segmentation and guarantees that each segment has relative consistent inner behavior (in relation to the other segments, the data, etc.). Coming to the question: What are appropriate methods for dividing a data set with multiple processes into an optimal number of segments? For single and simple signals/processes the segmentation seems quite intuitive and easy, for multiple processes which have a lot of noise and different behaviors not. My first approach for solving above problem was using variance. The idea is to start with an arbitrary number of segments with the segmentation lines being equally spaced to each other. Now, if needed, each segmentation line can be moved with the goal of minimizing the variance of two neighboring segments that are affected by moving respective line. Doing this for all segments iteratively will minimize the variance of the segments. Ideally, this will…

R package to solve Gaussian MLE under conditional independence constraints

Is there any R package or function to solve Gaussian MLE under conditional independence constraints ? Suppose we have $y_i\overset{i.i.d}{\sim}\mathcal{N}(0,\Sigma_{p\times p})$ , $i = 1,2,\ldots,n$ . We know that $(\Sigma^{-1})_{ij} = 0$ , for some $i,j\in\{1,2,\ldots,p\}$ , i.e. $X_i$ and $X_j$ are conditionally independent given the rest of the variables. We would like to find MLE of $\Sigma$ under the conditional independence constraints. I also tried to implement it according to section 5.1 of the above paper, but I couldn't do it successfully. I wonder if there is any R implementation to find Gaussian MLE under conditional independence constraints?

Serving ML Models in Production: Common Patterns

Over the past couple years, we've listened to ML practitioners across many different industries to learn and improve the tooling around ML production use cases. Through this, we've seen __4 common patterns__ of machine learning in production: 1. Pipeline 2. Ensemble 3. Business Logic 4. Online Learning In the ML serving space, implementing these patterns typically involves a tradeoff between ease of development and production readiness. Ray Serve was built to support these patterns by being both easy to develop and production ready.

Mode for equal consecutive frequencies

I'd like to calculate the mode for the frequency distribution table: .5 - 4 9 4 - 7.5 11 7.5 - 11 11 11 - 14.5 7 14.5 - 18 2 Thanks!

How to Train Really Large Models on Many GPUs?

[Updated on 2022-03-13: add expert choice routing .] [Updated on 2022-06-10]: Greg and I wrote a shorted and upgraded version of this post, published on OpenAI Blog: “Techniques for Training Large Neural Networks”

Regressor contribution in OLS regression

Assume I have the following model, estimated using OLS: $Y_{it}=β0+β1∗X1_{it}+β2∗X2_{it}+β3∗X3_{it}+ϵ_{it}$ I know that some methods exist to compute the relative contribution of each variable to the overall goodness of fit (Shapley decomposition for instance). But is there any easy way to measure a time-varying contribution? I.e. the evolution over time of the contribution of each factor? Probably silly question: what does $(β1∗X1_{it})/Y_{it}$ represent? Can this represent some kind of contribution of $X_1$ to $Y$ ? Many thanks

How to understand the common practices followed for writing a "bounding box" for an image in datasets?

For the image datasets, there may be a bounding box for each image at the dataset. It is an annotation for an image. It is a rectangular box intended for focusing on something inside the image. I read about the following two types of representations for a bounding box. using two points $(x_1, y_1)$ and $(x_2, y_2)$ . $$ $$ Using a point $(x_1, y_1)$ , width, and height. $$ $$ How do understand both the representations, Specifically, does the point $(x_1, y_1)$ used to denote the top right or top left or bottom right or bottom left in both cases?

Which sample size equation to use for testing two onboard computers on a fleet of planes?

Summary: We have both an old tried and true computer (type #1) and brand new (type #2) on 100 airplanes processing the same raw data and want to know if they are equivalent based on the telematics output of those computers. Should we use the sample size equation for two independent samples continuous outcome $$n_i = 2\left(\frac{Z\sigma}{ES}\right)^2$$ or sample size for matched continuous outcome $$n = \left(\frac{Z\sigma_d}{E}\right)^2$$ Details: We have a fleet of airplanes with onboard computers (type #1) that process raw telematics data from hardware such as the engine, brakes, GPS, altitude, etc. We are testing a new onboard computer (type #2) and hypothesize type 2’s will process that raw data identically to type #1. We run a 100 plane experiment where both type 1 and type 2 computers are present on each of the 100 planes and process the same raw data. We have flown 1 million miles so far and our CEO wants to know the sample size needed to decide if the devices are similar enough, e.g. accept or reject the null hypothesis. What I’ve tried: We started by using the sample size for two independent samples where the standard deviation is derived from the known standard deviation of our Type 1 computer. However, now I believe that it would be more appropriate if we used the matched sample equation for sample size because we had 100 planes with both type 1 and a type 2 computer onboard and the output are not truly independent but fit the matched testing scenario better. Sin…

What is the derivative of equation 1 in the paper "Conservative Q-Learning for Offline Reinforcement Learning"?

I am looking at the paper Conservative Q-Learning for Offline Reinforcement Learning , but I'm not sure how they proved theorem 3.1. Here is a screenshot of theorem 3.1. In the proof of theorem 3.1 they say By setting the derivative of Equation 1 to 0, we obtain the following expression ... $$\forall \mathbf{s}, \mathbf{a} \in \mathcal{D}, k, \quad \hat{Q}^{k+1}(\mathbf{s}, \mathbf{a})=\hat{\mathcal{B}}^{\pi} \hat{Q}^{k}(\mathbf{s}, \mathbf{a})-\alpha \frac{\mu(\mathbf{a} \mid \mathbf{s})}{\hat{\pi}_{\beta}(\mathbf{a} \mid \mathbf{s})} \tag{11}\label{11}$$ Here's equation 1 from the paper. $$\hat{Q}^{k+1} \leftarrow \arg \min _{Q} \alpha \mathbb{E}_{\mathbf{s} \sim \mathcal{D}, \mathbf{a} \sim \mu(\mathbf{a} \mid \mathbf{s})}[Q(\mathbf{s}, \mathbf{a})]+\frac{1}{2} \mathbb{E}_{\mathbf{s}, \mathbf{a} \sim \mathcal{D}}\left[\left(Q(\mathbf{s}, \mathbf{a})-\hat{\mathcal{B}}^{\pi} \hat{Q}^{k}(\mathbf{s}, \mathbf{a})\right)^{2}\right] \tag{1}\label{1}$$ My question is: what exactly is the derivative of equation (1)? And how does that result in equation (11)? The $\hat{B}^\pi$ is the empirical Bellman operator and is defined as $\hat{B}^\pi \hat{Q}^k (s,a) = r + \gamma \sum_{s'} \hat{T}(s' \mid s,a) \mathbb{E}_{a'\sim \pi(a' \mid s')}\hat{Q}_k(s', a')$ . Since in offline reinforcement learning, the dataset $\mathcal{D}$ typically does not contain all possible transitions $(s, a, s')$ , the policy evaluation step actually uses an empirical Bellman operator that only backs up a singl…

How can I combine the scores of a multilabel classifier?

I have a keras neural network with 8 outputs and it is a multilabel problem, which means that an observation can be classified into more than one target class. Let's suppose I have the following vector of 8 probabilities (events A, B, C, D, E, F, G, H) calculated by the model to a given observation: [0.31, 0.45, 0.89, 0.02, 0.43, 0.21, 0.96, 0.32] So, we can interpret that: P(A) = 0.31, P(B) = 0.45 and so on... But what if I want the probability of an observation belong to classes A or B, for example ( P(A or B) = ? ). Or the probability P(A and B) = ? I imagine there are some things to take in consideration, like: Are the events independent? How can I check the intersection of the events? I would appreciate references of how can I approach this problem.

How to perform ANOVA for data measured with unequal time interval?

So I have a set of patient data divided into 3 groups based upon their vision. I also have data on few variables that I believe to be relevent which are measured pre-surgery, 1 day post-surgery, and 7-day post-surgery. I am trying to figure out if those variables differ from each other statistically within and between those 3 groups, and a simple F-test should do the trick. However, since the measured time period had uneven time interval (pre-surgery - 1 day post-surger had shorter time interval than 1 day to 7 day). Is there any other test I should perform? Also I am using R. I have read somewhere on how to deal with similar problem using SPSS, but the author was vague and didn't put any explanations on the reason behind. So if you guys can provide a simple procedure on how to perfom it using R it would be great ! Thanks a lot in advance !

Why does Q-function training not query the Q-function value at unobserved states?

In the paper Conservative Q-Learning for Offline Reinforcement Learning , it is stated (section 3.1, page 3) that standard Q-function training does not query the Q-function value at unobserved states, but queries the Q-function at unseen actions I don't see how this is true. For every $(s,a)$ pair, we need to update $Q(s,a)$ to reduce the value $|Q(s,a) - R(s,a) - \gamma E[\max_{a'}Q(s',a')]|$ until it converges to zero. We see the existence of both $a'$ and $s'$ , and $s'$ could be unseen, for example, on the very first update, where we are at $s$ , take action $a$ , and could arrive at any state $s'$ . Can someone explain this?

Marginal likelihood for linear model with random effects to do Bayesian model comparison

Suppose I have behavioral data from multiple participants to four different conditions (four observations per participant per condition). The conditions can be characterized in terms of two fixed effects, x1 and x2 ( i.e. , the experiment is a 2-by-2 design). That is to say, condition 1 has properties such that x1 is 0 and x2 is 0, condition 2 has properties such that x1 is 1 and x2 is 0, condition 3 has properties such that x1 is 0 and x2 is 1, and condition 4 has properties such that x1 is 1 and x2 is 1. Here is some sample data for two participants: data I would like to determine, for each participant, whether their data was more likely to have been generated by model 1 vs. model 2, where model 1 only has fixed effects for x1 and x2 , but model 2 additionally has a fixed effect for the interaction between x1 and x2 . I was thinking I could do this using Bayesian model comparison. That is, assuming model 1 and model 2 are equally likely, I could calculate the marginal likelihoods, $P(X_i\mid m_1)$ and $P(X_i\mid m_2)$ for each participant $i$ , and then just divide one by the other to get a Bayes Factor for each participant. I have two questions: Is this a reasonable approach to determining whether a given participant's data was more likely to have been generated by model 1 vs. model 2? If so, how can I go about doing this? Assuming this is a reasonable approach, I'm not sure what to do to calculate the marginal likelihoods. Since the observations are not IID ( i.e. , ther…

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How to classify instances according to its relative feature value within a group of points?

My real data is very complicated, but briefly it can be formalized as this problem below: currently I have many groups of datapoints, each consisting of 5 datapoints, I want to design an algorithm to label each of the 5 points in a group from 0~4 (Input is a group of 5 points, output is their labels 0~4). On a 2d scatter plot, the problem is like: Train Set | name | feature1 | feature2 | group | label | | ---- | -------- | -------- | -------- | -------- | | G1L0 | 1 | 1 | 1 | 0 | | G1L1 | 4 | 3 | 1 | 1 | | G1L2 | 4 | 5 | 1 | 2 | | G1L3 | 2 | 4.5 | 1 | 3 | | G1L4 | 6 | 5 | 1 | 4 | | ---- | -------- | -------- | -------- | -------- | | G2L0 | 2.6 | 0.8 | 2 | 0 | | G2L1 | 3.9 | 3.5 | 2 | 1 | | G2L2 | 3.4 | 4.5 | 2 | 2 | | G2L3 | 2.3 | 5.5 | 2 | 3 | | G2L4 | 5.6 | 4.5 | 2 | 4 | Test Set | ---- | -------- | -------- | -------- | -------- | | G3L0 | 2.1 | 1.1 | 3 | 0 | | G3L1 | 4.3 | 3.7 | 3 | 1 | | G3L2 | 4.9 | 5.4 | 3 | 2 | | G3L3 | 3.6 | 5.3 | 3 | 3 | | G3L4 | 6.5 | 5.5 | 3 | 4 | The key issue is that the classification of each of the 5 datapoints depends on the information of other points within the group , globally the data suffers greatly from variation of the data, specifically, if I train a traditional classifier on my train set it may end up like: But if I simply use this classifier to predict the test set there will be problem: This is because globally the variation across different groups is huge, but within a single group I can easily classify them manually. Experience…

Is there any reference for stopping the sample after some years after the event date (staggered Difference-in-Differences)?

I am using a generalized difference-in-differences (DiD) equation for a staggered treatment. The law (i.e., treatment) is implemented at different times. From this discussion of @Thomas Bilach here , it seems that you can stop the sample around 1, 2, or 3 years after the laws are implemented in each country. Is there any literature supporting this? I am wondering if this understanding is correct? And does it mean that, if we stop the sample after 2 years after the event date in each country, we need to drop all the observations in this country after 2 years?

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Anastasia.ai provides a powerful platform that enables organizations to operate AI capacities at scale with a fraction of the resources and effort traditionally required. This post covers a demand prediction problem we had and how using Ray to solve this problem led to astonishing results. Specifically, compared to our AWS Batch implementation, our Ray implementation is 9x faster and has reduced the cost by 87%. Update June 2024: Anyscale Endpoints (Anyscale's LLM API Offering) and Private Endpoints (self-hosted LLMs) are now available as part of the Anyscale Platform. Click [here](https://console.anyscale.com/?utm_source=anyscale&utm_medium=blog&utm_campaign=blog_callout&utm_content=june2024_product_update_subheading) to get started on the Anyscale platform.

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What are the Calculus books recommended for beginner to advanced researchers in artificial intelligence?

Calculus is a branch of mathematics that primarily deals with the rate of change of outputs of a function w.r.t the inputs. It contains several concepts including limits, first-order derivatives, higher-order derivatives, chain rule, derivatives of special and standard functions, definite integrals, indefinite integrals, derivative tests, gradients, higher-order gradients, and so on... Calculus has been heavily used in optimization and maybe in several other aspects of artificial intelligence. What are the Calculus textbook(s) recommended that cover all the concepts required for a researcher in artificial intelligence?