Second, in the non-parametric machine learning setting, we provide an explicit algorithm combining the previous scheme with Nyström projection techniques, andprovethatitachievesoptimal generalization bounds with atime complexity of orderO(ndfλ), a memory complexity of orderO(df2λ) and no dependence on the condition number, generalizing the results known for leastsquaresregression.Here nisthenumberofobservationsand dfλ istheassociated degrees of freedom.

In this paper, we study large-scale convex optimization algorithms based on the Newton method applied to regularized generalized self-concordant losses, which include logistic regression and softmax regression. We first prove that our new simple scheme based on a sequence of problems with decreasing regularization parameters is provably globally convergent, that this convergence is linear with a constant factor which scales only logarithmically with the condition number. In the parametric setting, we obtain an algorithm with the same scaling than regular first-order methods but with an improved behavior, in particular in ill-conditioned problems. Second, in the non parametric machine learning setting, we provide an explicit algorithm combining the previous scheme with Nystr\om projections techniques, and prove that it achieves optimal generalization bounds with a time complexity of order O(n df), a memory complexity of order O(df^2) and no dependence on the condition number, generalizing the results known for least squares regression. Here n is the number of observations and df is the associated degrees of freedom. In particular, this is the first large-scale algorithm to solve logistic and softmax regressions in the non-parametric setting with large condition numbers and theoretical guarantees.

Large language models (LLMs) are known for their exceptional performance in natural language processing, making them highly effective in many human life-related tasks. The attention mechanism in the Transformer architecture is a critical component of LLMs, as it allows the model to selectively focus on specific input parts. The softmax unit, which is a key part of the attention mechanism, normalizes the attention scores. Hence, the performance of LLMs in various NLP tasks depends significantly on the crucial role played by the attention mechanism with the softmax unit.In-context learning is one of the celebrated abilities of recent LLMs. Without further parameter updates, Transformers can learn to predict based on few in-context examples.

In this paper, we study large-scale convex optimization algorithms based on the Newton method applied to regularized generalized self-concordant losses, which include logistic regression and softmax regression. We first prove that our new simple scheme based on a sequence of problems with decreasing regularization parameters is provably globally convergent, that this convergence is linear with a constant factor which scales only logarithmically with the condition number. In the parametric setting, we obtain an algorithm with the same scaling than regular first-order methods but with an improved behavior, in particular in ill-conditioned problems. Second, in the non parametric machine learning setting, we provide an explicit algorithm combining the previous scheme with Nystr\"om projections techniques, and prove that it achieves optimal generalization bounds with a time complexity of order O(n df), a memory complexity of order O(df 2) and no dependence on the condition number, generalizing the results known for least squares regression. Here n is the number of observations and df is the associated degrees of freedom.

Robustness to malicious attacks is of paramount importance for distributed learning. Existing works often consider the classical Byzantine attacks model, which assumes that some workers can send arbitrarily malicious messages to the server and disturb the aggregation steps of the distributed learning process. To defend against such worst-case Byzantine attacks, various robust aggregators have been proven effective and much superior to the often-used mean aggregator. In this paper, we show that robust aggregators are too conservative for a class of weak but practical malicious attacks, as known as label poisoning attacks, where the sample labels of some workers are poisoned. Surprisingly, we are able to show that the mean aggregator is more robust than the state-of-the-art robust aggregators in theory, given that the distributed data are sufficiently heterogeneous. In fact, the learning error of the mean aggregator is proven to be optimal in order. Experimental results corroborate our theoretical findings, demonstrating the superiority of the mean aggregator under label poisoning attacks.

Large language models (LLMs) and generative AI have played a transformative role in computer research and applications. Controversy has arisen as to whether these models output copyrighted data, which can occur if the data the models are trained on is copyrighted. LLMs are built on the transformer neural network architecture, which in turn relies on a mathematical computation called Attention that uses the softmax function. In this paper, we show that large language model training and optimization can be seen as a softmax regression problem. We then establish a method of efficiently performing softmax regression, in a way that prevents the regression function from generating copyright data. This establishes a theoretical method of training large language models in a way that avoids generating copyright data.

Originally published on Towards AI the World's Leading AI and Technology News and Media Company. If you are building an AI-related product or service, we invite you to consider becoming an AI sponsor. At Towards AI, we help scale AI and technology startups. Let us help you unleash your technology to the masses. Logistic regression is a supervised machine learning algorithm to create models used for binary classification problems conventionally.

Important Note: Original article http://ecdicus.com/linear-model/ Linear model is the most widely used model in machine learning. It refers to a model that uses linear combinations of sample features to make predictions. Where $\textbf{w} [w_1,…, w_D] T$ is the D-dimensional weight vector, and b is the bias. The linear regression introduced in the previous chapter is a typical linear model, and $f(\textbf{x};\textbf{w})$ is directly used to predict the output target $y f(\textbf{x};\textbf{w})$. In the classification problem, since the output target y is some discrete labels, and the value range of $f(\textbf{x};\textbf{w})$ is a real number,$f(\textbf{x};\textbf{w})$ cannot be directly used for prediction, and a non-linear decision function needs to be introduced g(.) to predict the output target Where, f(x;w) is also called Discriminant Function.

This paper proposes a method for solving multivariate regression and classification problems using piecewise linear predictors over a polyhedral partition of the feature space. The resulting algorithm that we call PARC (Piecewise Affine Regression and Classification) alternates between (i) solving ridge regression problems for numeric targets, softmax regression problems for categorical targets, and either softmax regression or cluster centroid computation for piecewise linear separation, and (ii) assigning the training points to different clusters on the basis of a criterion that balances prediction accuracy and piecewise-linear separability. We prove that PARC is a block-coordinate descent algorithm that optimizes a suitably constructed objective function, and that it converges in a finite number of steps to a local minimum of that function. The accuracy of the algorithm is extensively tested numerically on synthetic and real-world datasets, showing that the approach provides an extension of linear regression/classification that is particularly useful when the obtained predictor is used as part of an optimization model. A Python implementation of the algorithm described in this paper is available at http://cse.lab.imtlucca.it/~bemporad/parc .

Recently, inspired by Transformer, self-attention-based scene text recognition approaches have achieved outstanding performance. However, we find that the size of model expands rapidly with the lexicon increasing. Specifically, the number of parameters for softmax classification layer and output embedding layer are proportional to the vocabulary size. It hinders the development of a lightweight text recognition model especially applied for Chinese and multiple languages. Thus, we propose a lightweight scene text recognition model named Hamming OCR. In this model, a novel Hamming classifier, which adopts locality sensitive hashing (LSH) algorithm to encode each character, is proposed to replace the softmax regression and the generated LSH code is directly employed to replace the output embedding. We also present a simplified transformer decoder to reduce the number of parameters by removing the feed-forward network and using cross-layer parameter sharing technique. Compared with traditional methods, the number of parameters in both classification and embedding layers is independent on the size of vocabulary, which significantly reduces the storage requirement without loss of accuracy. Experimental results on several datasets, including four public benchmaks and a Chinese text dataset synthesized by SynthText with more than 20,000 characters, shows that Hamming OCR achieves competitive results.