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Smooth approximation of one-hot arg max
The softmax function, also known as softargmax or normalized exponential function, converts a tuple of K real numbers into a probability distribution
Softmax_function
Artificial neural network node function
arbitrary classical activation function. Logistic function Rectifier (neural networks) Stability (learning theory) Softmax function Hinkelmann, Knut. "Neural
Activation_function
Type of activation function
xn)=ln(1+ex1+⋯+exn) The LogSumExp function is LSE(x1,…,xn)=ln(ex1+⋯+exn) and its gradient is the softmax; the softmax with the first argument set to zero
Rectified_linear_unit
S-shaped curve
immediately generalizes to more alternatives as the softmax function, which is a vector-valued function whose i-th coordinate is e x i / ∑ i = 0 n e x i
Logistic_function
Mathematical function having a characteristic S-shaped curve or sigmoid curve
in statistics Softplus function – Smoothed ramp functionPages displaying short descriptions of redirect targets Softmax function – Smooth approximation
Sigmoid_function
Smooth approximation to the maximum function
_{j}\exp {x_{j}}}},} which means the gradient of LogSumExp is the softmax function. The convex conjugate of LogSumExp is the negative entropy. In tropical
LogSumExp
Real function with secant line between points above the graph itself
of the singularity at x=0. LogSumExp function, also called softmax function, is a convex function. The function −logdet(X) on the domain of positive-definite
Convex_function
Topics referred to by the same term
an electronic component Temperature (softmax function), a parameter that alters the entropy of the softmax function or Boltzmann distribution "Temperature"
Temperature_(disambiguation)
Regression for more than two discrete outcomes
k\leq K.} The following function: softmax ( k , s 1 , … , s K ) = e s k ∑ j = 1 K e s j {\displaystyle \operatorname {softmax} (k,s_{1},\ldots ,s_{K})={\frac
Multinomial logistic regression
Multinomial_logistic_regression
2017 paper conceptualizing the transformer-based deep learning architecture
columns in their matrices. Here, the softmax function is applied row-wise: let Score:=Query×KeyT/dk and Weight:=softmax(Score), then the element at row i
Attention_Is_All_You_Need
Family of machine learning approaches
1,…. This can be accomplished by the softmax function, thus giving us the attention weights:(w00,w01,…)=softmax(q0k0T,q0k1T,…)This is then used to compute
Seq2seq
Mathematical approximation
(x_{i}-{\mathcal {S}}_{\alpha }(x_{1},\ldots ,x_{n}))].} This makes the softmax function useful for optimization techniques that use gradient descent. This
Smooth_maximum
Probability distribution of energy states of a system
forbidden transition. The softmax function commonly used in machine learning is related to the Boltzmann distribution: (p1,…,pM)=softmax[−ε1kBT,…,−εMkBT]. A
Boltzmann_distribution
Algorithm for modelling sequential data
tokens can be expressed as one large matrix calculation using the softmax function, which is useful for training due to computational matrix operation
Transformer_(deep_learning)
Machine learning technique
attention is defined as:Attention(Q,K,V)=softmax(λQKT)V∈ℝm×dvwhere T denotes transpose and the softmax function is applied independently to every row of
Attention_(machine_learning)
Machine learning technique
y∼N(μi,I), where μi is a learnable parameter. The weighting function is a linear-softmax function:w(x)i=ekiTx+bi∑jekjTx+bjThe mixture of experts predict that
Mixture_of_experts
Smoothed ramp function
the softmax; the softmax with the first argument set to zero is the multivariable generalization of the logistic function. Both LogSumExp and softmax are
Softplus
Type of artificial neural network
_{j}\\12:\quad \mathbf {return} ~\mathbf {v} _{j}\\\end{array}}} At line 8, the softmax function can be replaced by any type of winner-take-all network. Biologically
Capsule_neural_network
weighted by the softmax function for each j∈Cj where Cj is now the entire transformed data set. This choice of objective function is preferable as it
Neighbourhood components analysis
Neighbourhood_components_analysis
Statistical model for a binary dependent variable
multinomial logit. This general formulation is exactly the softmax function as in Pr(Yi=c)=softmax(c,𝜷0⋅𝐗i,𝜷1⋅𝐗i,…). To prove that this is equivalent
Logistic_regression
Inputs at which function values are highest
(statistics) Mathematical optimization Kernel (linear algebra) Preimage Softmax function For clarity, we refer to the input (x) as points and the output (y)
Arg_max
Family of probability distributions related to the normal distribution
of probability distributions whose probability density function (or probability mass function, for the case of a discrete distribution) can be expressed
Exponential_family
Probabilistic classification algorithm
{\displaystyle b+\mathbf {w} ^{\top }x} , or in the multiclass case, the softmax function. Discriminative classifiers have lower asymptotic error than generative
Naive_Bayes_classifier
Probability distribution
the other hand, SGB variates can also be obtained by applying the softmax function to scaled and translated logarithms of Dirichlet variates. Specifically
Dirichlet_distribution
Multi-dimensional generalization of triangle
function from Rn to the interior of the standard ( n − 1 ) {\displaystyle (n-1)} -simplex is the softmax function, or normalized exponential function;
Simplex
Machine learning model for vision processing
strategies: Sharpening: The teacher network's output is sharpened using a softmax function with a lower temperature. This makes the teacher more "confident" in
Vision_transformer
Class of artificial neural network
[clarification needed] In this case, the logistic function for visible units is replaced by the softmax function P(vik=1|h)=exp(aik+ΣjWijkhj)Σk′=1Kexp(aik′+ΣjWijk′hj)
Restricted_Boltzmann_machine
Type of network
activation function) is some predefined function, such as the hyperbolic tangent, sigmoid function, softmax function, or rectifier function. The important
Mathematics of neural networks in machine learning
Mathematics_of_neural_networks_in_machine_learning
Artificial intelligence chatbot by Moonshot AI
t}e^{k_{j}^{T}q_{t}}}}v_{i}} . A naive linear attention mechanism is to take off the softmax: o t = ( ∑ i ≤ t v i k i T ) q t {\displaystyle o_{t}=\left(\sum _{i\leq
Kimi_(AI)
Overview of and topical guide to deep learning
neural network Artificial neuron Activation function Rectified linear unit Sigmoid function Softmax function Embedding Convolution Pooling layer Attention
Outline_of_deep_learning
Artificial neural network architecture
Cosine similarity σ(x)=1/(1+e−x) Sigmoid function oneplus(x)=1+log(1+ex) Oneplus function softmax(𝐱)j=exj∑k=1Kexk for j = 1, ..., K. Softmax function
Differentiable neural computer
Differentiable_neural_computer
Statistical model for pairwise comparisons
rating system Ordinal regression Rasch model Scale (social sciences) Softmax function Thurstonian model Hunter, David R. (2004). "MM algorithms for generalized
Bradley–Terry_model
Axiom in probability theory
, for some "value" function u : A → ( 0 , ∞ ) {\displaystyle u:A\to (0,\infty )} . This is sometimes called the softmax function, or the Boltzmann distribution
Luce's_choice_axiom
Type of feedforward neural network
training of deeper networks. The related softplus function provides a smooth approximation. Softmax: Typically found in the output layer for multi-class
Multilayer_perceptron
Problem in machine learning and statistical classification
classification. In practice, the last layer of a neural network is usually a softmax function layer, which is the algebraic simplification of N logistic classifiers
Multiclass_classification
Software for understanding biological data
attention function is computed as follows: A t t e n t i o n ( Q , K , V ) = softmax ( Q K T d k ) V {\displaystyle {Attention}(Q,K,V)={\text{softmax}}\left({\frac
Machine_learning_in_biology
Models used to produce word embeddings
a vector of length D, one entry per dictionary entry. Then, apply the softmax to obtain a probability distribution over the dictionary. This system can
Word2Vec
Discrete probability distribution
p k {\displaystyle p_{1},\ldots ,p_{k}} can be recovered using the softmax function, which can then be sampled using the techniques described above. There
Categorical_distribution
Parts of a whole which carry only relative information
geometric mean of x {\displaystyle x} . The inverse of this function is also known as the softmax function. By construction, the CLR transformation enforces a
Compositional_data
Process of finding a spatial transformation that aligns two point clouds
runs. Let μ {\displaystyle \mathbf {\mu } } be: this is known as the softmax function. As β {\displaystyle \beta } increases, it approaches a binary value
Point-set_registration
Measure of difference between two points
calculate the bi-tempered logistic loss, performing better than the softmax function with noisy datasets. Bregman divergence is used in the formulation
Bregman_divergence
Field of machine learning
"Value-Difference Based Exploration: Adaptive Control Between Epsilon-Greedy and Softmax" (PDF), KI 2011: Advances in Artificial Intelligence, Lecture Notes in
Reinforcement_learning
Optimization algorithm for artificial neural networks
activation functions at layer l For classification the last layer is usually the logistic function for binary classification, and softmax (softargmax)
Backpropagation
Statistical model used in machine learning
posterior) outputs of a probabilistic n-class classifier, uses the softmax function to renormalize categorical distributions after scaling and translation
Flow-based_generative_model
Language and compiler for GPU kernels
illustrates how a softmax kernel could be written in Triton. import torch import triton import triton.language as tl @triton.jit def softmax_kernel(out_ptr
Triton_(programming_language)
Machine learning method to transfer knowledge from a large model to a smaller one
is the temperature, a parameter which is set to 1 for a standard softmax. The softmax operator converts the logit values z i ( x ) {\displaystyle z_{i}(\mathbf
Knowledge_distillation
Computer Go program
activation function for deep neural networks. A key innovation of Darkfmct3 compared to previous approaches is that it uses only one softmax function to predict
Darkforest
Family of convolutional neural networks
classifiers", which are linear-softmax classifiers inserted at 1/3-deep and 2/3-deep within the network, and the loss function is a weighted sum of all three:
Inception (deep learning architecture)
Inception_(deep_learning_architecture)
Particular case of the generalized extreme value distribution
Shixiang; Poole, Ben (April 2017). Categorical Reparameterization with Gumble-Softmax. International Conference on Learning Representations (ICLR) 2017. Balog
Gumbel_distribution
Series of language models developed by DeepSeek
softmax:vt/mKV,(ℓ+1)⏟per-block KV vector=∑i=1mwt−m+i⏟per-token weight vector⊙WKV,(ℓ)xt−m+i(ℓ)⏟per-token KV vector,wt−m+i=ezt−m+i∑j=1mezt−m+j⏟softmax,zt−m+i⏟per-token
DeepSeek_(AI)
Machine learning calibration technique
multiplies the output logits of a network by a constant 1/T before taking the softmax. During training, T is set to 1. After training, T is optimized on a held-out
Platt_scaling
Type of neural network output and associated scoring function
distribution at each time step. A CTC network has a continuous output (e.g. softmax), which is fitted through training to model the probability of a label
Connectionist temporal classification
Connectionist_temporal_classification
Numerical computation library for Python
Sigmoid activation output = T.nnet.softmax(T.dot(hidden_output, W2) + b2) # Softmax output # Define the cost function (cross-entropy) cost = T.nnet
Theano_(software)
Computational model used in machine learning
adopting a softmax activation function, a generalization of the logistic function, on the output layer of the neural network (or a softmax component in
Neural network (machine learning)
Neural_network_(machine_learning)
Degradation of AI models trained on synthetic data
scaling laws and bounds on learning can be obtained. In the case of a linear softmax classifier for next token prediction, exact bounds on learning with even
Model_collapse
Series of language models developed by Google AI
classifications, the output token at the [CLS] input token is fed into a linear-softmax layer to produce the label outputs. The original code base defined the
BERT_(language_model)
Class of artificial neural networks
and LeakyReLU is a modified ReLU activation function. Attention coefficients are then normalized via softmax to make them easily comparable across different
Graph_neural_network
Architectural motif in neural networks for aggregating information
is the same as average pooling in expectation. Softmax pooling is like max pooling, but uses softmax, i.e. ∑ k ′ e β x k ′ x k ′ ∑ k ″ e β x k ″ {\displaystyle
Pooling_layer
Technique in neural networks for learning joint representations of text and images
similarity scores with the N {\displaystyle N} texts are converted by a softmax into a probability distribution over which text is the matching one. Cross-entropy
Contrastive language–image pre-training
Contrastive_language–image_pre-training
Form of artificial neural network
activities of a group of neurons. For instance, it can contain contrastive (softmax) or divisive normalization. The dynamical equations describing temporal
Hopfield_network
Technique used in stochastic gradient variational inference
distribution can be reparameterized by the Gumbel distribution (Gumbel-softmax trick or "concrete distribution") and diffusion models. In general, any
Reparameterization_trick
Mathematical concept
Gibbs algorithm Gibbs sampling Interacting particle system Potential game Softmax Stochastic cellular automata Makhmudov, Mirmukhsin (2025-09-02). "Gibbs
Gibbs_measure
Approach in generative models
(JEM), proposed in 2020 by Grathwohl et al., allow any classifier with softmax output to be interpreted as energy-based model. The key observation is
Energy-based_model
Influential 2012 deep convolutional neural network
be written as (CONV → RN → MP)2 → (CONV3 → MP) → (FC → DO)2 → Linear → softmax where CONV = convolutional layer (with ReLU activation) RN = local response
AlexNet
Statistical technique for producing prediction sets
from the chosen stage In deep learning, the softmax values are often used Use a non-conformity function to compute α-values A data point in the calibration
Conformal_prediction
Neural networks
activities of a group of neurons. For instance, it can contain contrastive (softmax) or divisive normalization. The dynamical equations describing temporal
Modern_Hopfield_network
Deep learning technique
mask:zij=λ⏟scaleqikjT+mij⏟maskthen computes attention weights by taking a softmax:aij=ezij∑jezijthen computes output vectors by taking a weighted sum of
Positional_encoding
Explainable AI technique
confidence score (a softmax probability is the output value after the softmax operation; the logit is the value before the softmax) for the masked images
Class_activation_mapping
Resource problem in machine learning
exploration, high exploitation). Further improvements can be achieved by a softmax-weighted action selection in case of exploratory actions (Tokic & Palm
Multi-armed_bandit
Technology made by American organization
an intelligence "arms race" that could increase an agent's ability to function even outside the context of the competition. OpenAI Five is a team of five
Products and applications of OpenAI
Products_and_applications_of_OpenAI
Branch of machine learning
obtained by a Softmax layer with number of nodes that is equal to the alphabet size of Y. NJEE uses continuously differentiable activation functions, such that
Deep_learning
Type of feedforward neural network
supervised learning). Various loss functions can be used, depending on the specific task. The Softmax loss function is used for predicting a single class
Convolutional_neural_network
Machine learning software library
variations of convolutions (1/2/3D, Atrous, depthwise), activation functions (Softmax, RELU, GELU, Sigmoid, etc.) and their variations, and other operations
TensorFlow
2019 text-generating language model
successor model, GPT-4, to map each neuron of GPT-2 to determine their functions. GPT-2 became capable of performing a variety of tasks beyond simple text
GPT-2
Methods of estimating differential entropy given some observations
obtained by a Softmax layer with number of nodes that is equal to the alphabet size of Y. NJEE uses continuously differentiable activation functions, such that
Entropy_estimation
American educational technology company
bandit algorithm system (later the A/B tested variant recovering difference softmax algorithm) that determines the daily notification that will be sent out
Duolingo
Measurement of algorithmic bias
problem, Y ^ {\textstyle {\hat {Y}}} could refer to the output of the softmax layer. Then we update U {\textstyle U} to minimize L A {\textstyle L_{A}}
Fairness_(machine_learning)
classification. Of course you can ingest your own dataset using the Load function, but for now we are showing the API: import <armadillo>; import <mlpack/mlpack
Mlpack
Probabilistic model of fair resource allocation
ad impressions, and exposure in search/recommender systems—Boltzmann/softmax rules and entropy-regularized optimal transport (e.g., Sinkhorn scaling)
Boltzmann_Fair_Division
travel, tourism, insurance
SOFTMAX FUNCTION
SOFTMAX FUNCTION
Male
Egyptian
, a great functionary.
Male
Egyptian
, an Egyptian functionary.
Surname or Lastname
English
English : topographic name for someone who lived by the gates of a medieval walled town. The Middle English singular gate is from the Old English plural, gatu, of geat ‘gate’ (see Yates). Since medieval gates were normally arranged in pairs, fastened in the center, the Old English plural came to function as a singular, and a new Middle English plural ending in -s was formed. In some cases the name may refer specifically to the Sussex place Eastergate (i.e. ‘eastern gate’), known also as Gates in the 13th and 14th centuries, when surnames were being acquired.Americanized spelling of German Götz (see Goetz).Translated form of French Barrière (see Barriere).In New England, Gates was the preferred English version of the name of an extensive French family, called Barrière dit Langevin.
Male
Egyptian
, a high Egyptian functionary.
Male
Egyptian
, an Egyptian functionary.
Surname or Lastname
English
English : nickname from the animal, Middle English catte ‘cat’. The word is found in similar forms in most European languages from very early times (e.g. Gaelic cath, Slavic kotu). Domestic cats were unknown in Europe in classical times, when weasels fulfilled many of their functions, for example in hunting rodents. They seem to have come from Egypt, where they were regarded as sacred animals.English : from a medieval female personal name, a short form of Catherine.Variant spelling of German and Dutch Katt.
Male
Egyptian
, Functionary of the Interior.
Male
Celtic
, great justiciary, or functionary.
Boy/Male
Buddhist, Indian, Japanese
Mysterious Function
Male
Egyptian
, the son of the functionary Heknofre.
Surname or Lastname
English
English : occupational name for a dresser of cloth, Old English fullere (from Latin fullo, with the addition of the English agent suffix). The Middle English successor of this word had also been reinforced by Old French fouleor, foleur, of similar origin. The work of the fuller was to scour and thicken the raw cloth by beating and trampling it in water. This surname is found mostly in southeast England and East Anglia. See also Tucker and Walker.In a few cases the name may be of German origin with the same form and meaning as 1 (from Latin fullare).Americanized version of French Fournier.Samuel Fuller (1589–1633), born in Redenhall, Norfolk, England, was among the Pilgrim Fathers who sailed on the Mayflower in 1620. He was a deacon of the church and until his death functioned as Plymouth Colony’s physician.
Biblical
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Surname or Lastname
English (chiefly Kent and Sussex)
English (chiefly Kent and Sussex) : occupational name for a designer or engineer, from a Middle English reduced form of Old French engineor ‘contriver’ (a derivative of engaigne ‘cunning’, ‘ingenuity’, ‘stratagem’, ‘device’). Engineers in the Middle Ages were primarily designers and builders of military machines, although in peacetime they might turn their hands to architecture and other more pacific functions.German : from the Latin personal name Januarius (see January 1). Jänner is a South German word for ‘January’, and so it is possible that this is one of the surnames acquired from words denoting months of the year, for example by converts who had been baptized in that month, people who were born or baptized in that month, or people whose taxes were due in January.
SOFTMAX FUNCTION
SOFTMAX FUNCTION
SOFTMAX FUNCTION
SOFTMAX FUNCTION
SOFTMAX FUNCTION
SOFTMAX FUNCTION
SOFTMAX FUNCTION
travel, tourism, insurance