The paper investigates polynomial alternatives to softmax in transformer models.
problem The effectiveness of softmax attention in transformers is questioned.
method The authors explore polynomial activations as alternatives to softmax, focusing on their ability to regularize the attention matrix.
result Certain polynomials can serve as effective substitutes for softmax in transformer applications, achieving strong performance.
The paper establishes convergence rates for MoE models in classification problems.
problem Understanding the behavior of MoE models in classification settings.
method Established convergence rates for density and parameter estimation in softmax gating multinomial logistic MoE models.
result Parameter estimation rates are significantly improved with a novel modified softmax gating function.
EBNCs are a new model for Bayesian networks, derived from different networks.
problem Creating efficient models for Bayesian networks with discrete variables.
method Developed an EBNC model derived from different Bayesian networks, showing it's a special case of softmax polynomial regression.
result EBNCs can be used as a special case of softmax polynomial regression models.
Proposes an iterative algorithm for optimizing attention mechanisms in large language models.
problem Optimizing attention mechanisms in large language models.
method Iterative algorithm for rescaled hyperbolic functions regression.
result Efficiency and generalizability of the rescaled softmax regression framework.
Unified framework for sequence models using test-time regression.
problem Designing efficient sequence models with associative memory.
method Formalizing associative recall as regression over input tokens, deriving various sequence models.
result Clarifies the effectiveness of query-key normalization in softmax attention and offers new generalizations.
A new method for optimizing regression problems with ReLU units converges.
problem Optimizing regression problems involving ReLU units in large language models.
method Introduced a greedy algorithm based on approximate Newton method, proving convergence in terms of the distance to optimal solution.
result The method converges in the sense of the distance to optimal solution under certain assumptions.
Paper introduces Balanced Meta-Softmax for better long-tailed visual recognition.
problem Long-tailed distribution mismatch between training and testing data.
method Balanced Meta-Softmax, an unbiased extension of Softmax, using a Meta Sampler.
result Balanced Meta-Softmax outperforms state-of-the-art solutions on visual recognition and instance segmentation.
PARC uses piecewise linear predictors for regression and classification.
problem Multivariate regression and classification problems.
method Alternates between ridge and softmax regression, and cluster assignment based on accuracy and separability.
result Converges to a local minimum in a finite number of steps.
Efficiently solves inverse classification problems for logistic and softmax models.
problem Finding instances that change classifier predictions.
method Closed-form solution for logistic regression, iterative optimization for softmax.
result Fast, exact solutions for high-dimensional instances and many classes.
We present ADMM-Softmax, an alternating direction method of multipliers (ADMM) for solving multinomial logistic regression (MLR) problems. Our method is geared toward supervised classification tasks with many examples and features. It decouples the nonlinear optimization problem in MLR into three steps that can be solv…
Solves challenges in estimating parameters of softmax gating Gaussian mixture models.
problem Identifiability issues and complex interactions in Gaussian mixture of experts.
method Proposes novel Voronoi loss functions and establishes convergence rates of MLE.
result Connects convergence rate of MLE to a solvability problem of polynomial equations.
Unified scheme combining softmax and ResNet for deep learning.
problem Combining softmax and ResNet for deep learning.
method Theoretical analysis of a unified scheme combining softmax regression and ResNet.
result Unified scheme connects previously unrelated fields and provides insights into loss landscape and optimization.
Evidential Softmax preserves multimodality in sparse probability distributions for generative models.
problem Sparse probability distributions in deep generative models make exact marginalization computationally intractable.
method Introduce ev-softmax, a sparse normalization function that preserves multimodality and can be trained with probabilistic loss functions.
result ev-softmax outperforms existing techniques in distributional accuracy and dimensionality reduction.
New method uses temperature to control sparse MoE convergence rates.
problem Sparse MoE convergence rates are slow due to temperature interactions.
method Proposes a novel activation gate to improve convergence rates.
result Improved convergence rates to polynomial rates via novel gate.
Sigmoid gating is more sample efficient than softmax in mixture of experts.
problem Softmax gating leads to unnecessary competition among experts, causing representation collapse.
method Theoretical analysis of a regression framework with mixture of experts, identifying identifiability conditions and convergence rates.
result Sigmoid gating requires fewer samples to achieve the same expert estimation error as softmax gating.
Gradient descent converges geometrically to optimal self-attention parameters.
problem Training softmax self-attention layers for linear regression.
method Structure-aware gradient descent with preconditioner and regularizer.
result Gradient descent converges geometrically to global minima.
Transformers learn to perform logistic regression in-context.
problem Understanding how transformers learn to perform specific tasks in-context.
method Constructed multi-layer transformers that perform in-context logistic regression through normalized gradient descent.
result Transformers can be trained to perform in-context logistic regression effectively.
Unified framework for studying softmax attention under large prompts.
problem Challenges in theoretical analysis of softmax attention.
method Measure-based framework for finite and infinite prompts.
result Softmax attention converges to linear attention in the large-prompt regime.
Improved logistic MoE with sigmoid gate shows better sample efficiency.
problem Improving sample efficiency in logistic MoE models.
method Comprehensive analysis of multinomial logistic MoE with modified sigmoid gate, incorporating temperature parameter and using Euclidean score.
result The sigmoid gate leads to lower sample complexity than softmax gate for both parameter and expert estimation.
Unified framework for SGMoE resolves estimation and selection issues.
problem Non-identifiability, coupled differential relations, and tight coupling in softmax-Gated models.
method Unified statistical framework with Voronoi-type loss functions and dendrograms of mixing measures.
result Consistent selection of the number of experts without model sweeps, optimal parameter rates under overfitting.
This research explores using kernels in the softmax layer for better contextual word classification.
problem Improving contextual word classification accuracy.
method Replacing the inner product in the softmax layer with various kernel functions and comparing their performance.
result Different kernel settings yield varying performance in contextual word classification tasks.
Paper studies Transformer learning theory for Euclidean and Riemannian domains.
problem Understanding and optimizing Transformer networks for regression tasks.
method Constructive approximation framework using softmax partition of unity and attention mechanism.
result Transformer can achieve uniform ε-approximation error with minimal parameters.
The paper analyzes convergence rates of softmax gating in MoE models.
problem The effectiveness and scalability of machine learning models using MoE.
method Convergence analysis of parameter and expert estimation under MoE with softmax gating and its variants.
result Theoretical results show polynomially many data points are needed for strong identifiability conditions, while exponential points are required for linear experts.
We demystify attention patterns in multi-head softmax models for linear data.
problem Understanding the training dynamics and emergent patterns in multi-head softmax attention models.
method Extensive empirical experiments and rigorous theoretical analysis.
result Multi-head softmax attention models approximate a debiased gradient descent predictor, outperforming single-head attention and achieving near-Bayesian optimality.
Investigates least squares estimation in deterministic MoE models.
problem Complexity and difficulty in analyzing MoE models.
method Least squares estimation under deterministic MoE models.
result Rates of estimating strongly identifiable experts are faster than polynomial experts.
Study shows MSE with sigmoid can match SCE in classification tasks, especially with noisy data.
problem Inconsistent errors in neural network classification tasks.
method Introduced Output Reset algorithm to use MSE with sigmoid activation.
result MSE with sigmoid activation achieves comparable accuracy and convergence rates to Softmax Cross-Entropy, especially in noisy data scenarios.
SoftKI combines SKI and variational methods for scalable GP regression.
problem Scalable Gaussian Process regression on high-dimensional datasets.
method SoftKI approximates kernel via softmax interpolation from a smaller number of learned points.
result SoftKI is competitive with other approximated GP methods for modest data dimensions.
Balanced Activation improves object detection performance on long-tailed datasets.
problem Mismatch between training and testing label distributions in object detection.
method Introduces Balanced Activation (Balanced Softmax and Balanced Sigmoid) to address label distribution shift.
result Balanced Activation provides ~3% gain in mAP on LVIS-1.0 compared to state-of-the-art methods.
Multi-head attention outperforms single-head in in-context linear regression tasks.
problem Comparing performance of transformer with single-/multi-head attention in in-context learning.
method Theoretical analysis of performance of transformers with different attention mechanisms in linear regression tasks.
result Multi-head attention with a substantial embedding dimension outperforms single-head attention in in-context linear regression tasks.
Transformers can emulate various algorithms by prompting, proving universality.
problem How to emulate algorithms using fixed-weight Transformers.
method Two modes of in-context algorithm emulation: task-specific and prompt-programmable. Constructing prompts that encode algorithm parameters into token representations.
result Fixed-weight Transformers can emulate a broad class of algorithms via prompts.
New approach connects stochastic gradient descent to ODE splitting schemes.
problem Improving convergence in stochastic optimization.
method Connection between stochastic gradient descent and ODE splitting schemes.
result Derive a new upper bound on global splitting error.
Paper proposes an adversarial sampling method for efficient extreme classification.
problem Training classifiers over many classes is computationally expensive.
method Adversarial sampling to draw negative samples from an adversarial model.
result Significantly reduces training time by an order of magnitude.
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 para…
Study on multi-head softmax attention dynamics for in-context learning.
problem Understanding and optimizing multi-head softmax attention models for multi-task linear regression.
method Gradient flow analysis and spectral mapping technique.
result Gradient flow converges to optimal multi-head softmax attention model, with task allocation emerging during training.
We improve MoE models for classification with rigorous guarantees and practical methods.
problem Limited guarantees for stable maximum-likelihood training and model selection in softmax-gated MoE models.
method Derived a batch MM algorithm with closed-form updates, proved finite-sample rates, and developed a dendrogram selector.
result Achieved near-parametric optimal rates for parameter recovery and improved accuracy over baselines.
Softmax is an output activation function for modeling categorical probability distributions in many applications of deep learning. However, a recent study revealed that softmax can be a bottleneck of representational capacity of neural networks in language modeling (the softmax bottleneck). In this paper, we propose an…
The computational cost of training with softmax cross entropy loss grows linearly with the number of classes. For the settings where a large number of classes are involved, a common method to speed up training is to sample a subset of classes and utilize an estimate of the loss gradient based on these classes, known as…
In this paper we develop the theory of parametric polynomial regression in Riemannian manifolds and Lie groups. We show application of Riemannian polynomial regression to shape analysis in Kendall shape space. Results are presented, showing the power of polynomial regression on the classic rat skull growth data of Book…
Paper tackles blind polynomial regression for unknown inputs.
problem Fitting a polynomial to unknown or partially known input data.
method Formally defines the problem, proposes algorithmic approaches, and applies to jitter-correction.
result Proposes effective methods for blind polynomial regression.
Revises logistic-softmax likelihood for Bayesian meta-learning in few-shot classification.
problem Inherent uncertainty in logistic-softmax leads to suboptimal performance in meta-learning.
method Redesigns logistic-softmax likelihood with a temperature parameter for better control of prior confidence.
result Achieves well-calibrated uncertainty estimates and comparable/superior performance on benchmark datasets.
Paper introduces hierarchical softmax for global hierarchical classification tasks.
problem Improving classification accuracy in tasks with class hierarchies.
method Global hierarchical neural networks using hierarchical softmax.
result Hierarchical softmax outperforms regular softmax in multiple datasets.
Study reveals how depth of reasoning affects generalization in models.
problem Understanding scaling behavior of generalization with CoT depth.
method Theoretical model of CoT in linear regression using random matrix theory.
result Sharp phase transition between exponential and polynomial improvement, saturation, and overthinking.
Typically, Softmax is used in the final layer of a neural network to get a probability distribution for output classes. But the main problem with Softmax is that it is computationally expensive for large scale data sets with large number of possible outputs. To approximate class probability efficiently on such large sc…
Softmax emerges naturally in neural networks as a measure of conditional mutual information.
problem The artificial nature of softmax in neural networks.
method Information-theoretic perspective to derive log-softmax and evaluate conditional mutual information.
result Training deterministic neural networks through log-softmax maximises conditional mutual information.
Binary testing for softmax models requires many samples, similar to leverage score models.
problem Binary hypothesis testing for softmax models and leverage score models.
method Analyzing sample complexity and drawing analogies between models.
result Sample complexity is asymptotically \(O(ε^{-2})\), where \(ε\) is the distance between model parameters.
Volterra and polynomial regression models play a major role in nonlinear system identification and inference tasks. Exciting applications ranging from neuroscience to genome-wide association analysis build on these models with the additional requirement of parsimony. This requirement has high interpretative value, but …
Softmax temperature influences model representation rank and performance.
problem Understanding and optimizing softmax function's impact on model representations.
method Investigated softmax function's role in deep neural networks, introduced rank deficit bias.
result Softmax temperature affects model representation rank and can improve performance.
A mathematical framework connects neural networks and polynomial regression for better model understanding.
problem Neural networks are black boxes with challenges in dimensioning and prediction error evaluation.
method Developed a mathematical framework using Taylor expansion to relate neural networks and polynomial regression.
result Polynomial approximations from neural networks trained on polynomial data are accurate locally.