Receptron

The receptron (short for "reservoir perceptron") is a neuromorphic data processing model — specifically neuromorphic computing — that generalizes the traditional perceptron, by incorporating non-linear interactions between inputs. Unlike classical perceptron, which rely on linearly independent weights, the receptron leverages complexity in physical substrates, such as the electric conduction properties of nanostructured materials or optical speckle fields, to perform classification tasks. The receptron bridges unconventional computing and neural network principles, enabling solutions that do not require the training approaches typical of artificial neural networks based on the perceptron model. == Algorithm == The receptron is an algorithm for supervised learning of binary classifiers, so a classification algorithm that makes its predictions based on a predictor function, combining a set of weights with the feature vector. The mathematical model is based on the sum of inputs with non-linear interactions: S = ∑ k = 1 n x j w ~ j ( x → ) | S ∈ R {\displaystyle S=\sum _{k=1}^{n}x_{j}{\widetilde {w}}_{j}({\vec {x}})|S\in R} (1) where j ∈ [ 1 , n ] {\displaystyle j\in [1,n]} and w ~ j {\displaystyle {\widetilde {w}}_{j}} are non-linear weight functions depending on the inputs, x → {\displaystyle {\vec {x}}} . Nonlinearity will typically make the system extremely complex, and allowing for the solution of problems not solvable through the simpler rules of a linear system, such as the perceptron or McCulloch Pitts neurons, which is based on the sum of linearly independent weights: S = ∑ k = 1 n x j w j p {\displaystyle S=\sum _{k=1}^{n}x_{j}w_{j}^{p}} (2) where w j {\displaystyle w_{j}} are constant real values. A consequence of this simplicity is the limitation to linearly separable functions, which necessitates multi-layer architectures and training algorithms like backpropagation As in the perceptron case, the summation in Eq. 1 origins the activation of the receptron output through the thresholding process, Y ( x 1 , . . . , x n ) = { 1 if S > th 0 if S ≤ th {\displaystyle Y(x_{1},...,x_{n})={\begin{cases}1&{\text{if }}S>{\text{th}}\\0&{\text{if }}S\leq {\text{th}}\end{cases}}} (3) where th is a constant threshold parameter. Equation 3 can be written by using the Heaviside step function. The weight functions w ~ ( x → ) {\displaystyle {\widetilde {w}}({\vec {x}})} can be written with a finite number of parameters w j 1 . . . j n {\displaystyle w_{j_{1}...j_{n}}} , simplifying the model representation. One can Taylor-expand w ~ ( x → ) {\displaystyle {\widetilde {w}}({\vec {x}})} and use the idempotency of Boolean variables ( x j ) q = x j ∀ q ≥ 1 {\displaystyle (x_{j})^{q}=x_{j}\forall q\geq 1} such that S ′ = b + ∑ k = 1 n x j w ~ j ( x → ) {\displaystyle S'=b+\sum _{k=1}^{n}x_{j}{\widetilde {w}}_{j}({\vec {x}})} can be written as S ′ ( x → ) = b + ∑ j w j x j + ∑ j < k w j k x j x k + ∑ j < k < l w j k l x j x k x l + . . . {\displaystyle S'({\vec {x}})=b+\sum _{j}w_{j}x_{j}+\sum _{j

Law practice management software

Law practice management software is software designed to manage the business operations of a law firm. This can include software that manages cases, client intake, court communications, electronic discovery, time tracking, trust accounting, and billing. == Features of law practice management software == Common features of practice management software include: Case management Time tracking Document assembly Contact management Calendaring Docket management Client portal Contract Management Court Case Status Tracker Trust accounting == Examples of law practice management software == Smokeball LEAP Legal Software PracticeEvolve Dye & Durham

LPBoost

Linear Programming Boosting (LPBoost) is a supervised classifier from the boosting family of classifiers. LPBoost maximizes a margin between training samples of different classes, and thus also belongs to the class of margin classifier algorithms. Consider a classification function f : X → { − 1 , 1 } , {\displaystyle f:{\mathcal {X}}\to \{-1,1\},} which classifies samples from a space X {\displaystyle {\mathcal {X}}} into one of two classes, labelled 1 and -1, respectively. LPBoost is an algorithm for learning such a classification function, given a set of training examples with known class labels. LPBoost is a machine learning technique especially suited for joint classification and feature selection in structured domains. == LPBoost overview == As in all boosting classifiers, the final classification function is of the form f ( x ) = ∑ j = 1 J α j h j ( x ) , {\displaystyle f({\boldsymbol {x}})=\sum _{j=1}^{J}\alpha _{j}h_{j}({\boldsymbol {x}}),} where α j {\displaystyle \alpha _{j}} are non-negative weightings for weak classifiers h j : X → { − 1 , 1 } {\displaystyle h_{j}:{\mathcal {X}}\to \{-1,1\}} . Each individual weak classifier h j {\displaystyle h_{j}} may be just a little bit better than random, but the resulting linear combination of many weak classifiers can perform very well. LPBoost constructs f {\displaystyle f} by starting with an empty set of weak classifiers. Iteratively, a single weak classifier to add to the set of considered weak classifiers is selected, added and all the weights α {\displaystyle {\boldsymbol {\alpha }}} for the current set of weak classifiers are adjusted. This is repeated until no weak classifiers to add remain. The property that all classifier weights are adjusted in each iteration is known as totally-corrective property. Early boosting methods, such as AdaBoost do not have this property and converge slower. == Linear program == More generally, let H = { h ( ⋅ ; ω ) | ω ∈ Ω } {\displaystyle {\mathcal {H}}=\{h(\cdot ;\omega )|\omega \in \Omega \}} be the possibly infinite set of weak classifiers, also termed hypotheses. One way to write down the problem LPBoost solves is as a linear program with infinitely many variables. The primal linear program of LPBoost, optimizing over the non-negative weight vector α {\displaystyle {\boldsymbol {\alpha }}} , the non-negative vector ξ {\displaystyle {\boldsymbol {\xi }}} of slack variables and the margin ρ {\displaystyle \rho } is the following. min α , ξ , ρ − ρ + D ∑ n = 1 ℓ ξ n sb.t. ∑ ω ∈ Ω y n α ω h ( x n ; ω ) + ξ n ≥ ρ , n = 1 , … , ℓ , ∑ ω ∈ Ω α ω = 1 , ξ n ≥ 0 , n = 1 , … , ℓ , α ω ≥ 0 , ω ∈ Ω , ρ ∈ R . {\displaystyle {\begin{array}{cl}{\underset {{\boldsymbol {\alpha }},{\boldsymbol {\xi }},\rho }{\min }}&-\rho +D\sum _{n=1}^{\ell }\xi _{n}\\{\textrm {sb.t.}}&\sum _{\omega \in \Omega }y_{n}\alpha _{\omega }h({\boldsymbol {x}}_{n};\omega )+\xi _{n}\geq \rho ,\qquad n=1,\dots ,\ell ,\\&\sum _{\omega \in \Omega }\alpha _{\omega }=1,\\&\xi _{n}\geq 0,\qquad n=1,\dots ,\ell ,\\&\alpha _{\omega }\geq 0,\qquad \omega \in \Omega ,\\&\rho \in {\mathbb {R} }.\end{array}}} Note the effects of slack variables ξ ≥ 0 {\displaystyle {\boldsymbol {\xi }}\geq 0} : their one-norm is penalized in the objective function by a constant factor D {\displaystyle D} , which—if small enough—always leads to a primal feasible linear program. Here we adopted the notation of a parameter space Ω {\displaystyle \Omega } , such that for a choice ω ∈ Ω {\displaystyle \omega \in \Omega } the weak classifier h ( ⋅ ; ω ) : X → { − 1 , 1 } {\displaystyle h(\cdot ;\omega ):{\mathcal {X}}\to \{-1,1\}} is uniquely defined. When the above linear program was first written down in early publications about boosting methods it was disregarded as intractable due to the large number of variables α {\displaystyle {\boldsymbol {\alpha }}} . Only later it was discovered that such linear programs can indeed be solved efficiently using the classic technique of column generation. === Column generation for LPBoost === In a linear program a column corresponds to a primal variable. Column generation is a technique to solve large linear programs. It typically works in a restricted problem, dealing only with a subset of variables. By generating primal variables iteratively and on-demand, eventually the original unrestricted problem with all variables is recovered. By cleverly choosing the columns to generate the problem can be solved such that while still guaranteeing the obtained solution to be optimal for the original full problem, only a small fraction of columns has to be created. ==== LPBoost dual problem ==== Columns in the primal linear program corresponds to rows in the dual linear program. The equivalent dual linear program of LPBoost is the following linear program. max λ , γ γ sb.t. ∑ n = 1 ℓ y n h ( x n ; ω ) λ n + γ ≤ 0 , ω ∈ Ω , 0 ≤ λ n ≤ D , n = 1 , … , ℓ , ∑ n = 1 ℓ λ n = 1 , γ ∈ R . {\displaystyle {\begin{array}{cl}{\underset {{\boldsymbol {\lambda }},\gamma }{\max }}&\gamma \\{\textrm {sb.t.}}&\sum _{n=1}^{\ell }y_{n}h({\boldsymbol {x}}_{n};\omega )\lambda _{n}+\gamma \leq 0,\qquad \omega \in \Omega ,\\&0\leq \lambda _{n}\leq D,\qquad n=1,\dots ,\ell ,\\&\sum _{n=1}^{\ell }\lambda _{n}=1,\\&\gamma \in \mathbb {R} .\end{array}}} For linear programs the optimal value of the primal and dual problem are equal. For the above primal and dual problems, the optimal value is equal to the negative 'soft margin'. The soft margin is the size of the margin separating positive from negative training instances minus positive slack variables that carry penalties for margin-violating samples. Thus, the soft margin may be positive although not all samples are linearly separated by the classification function. The latter is called the 'hard margin' or 'realized margin'. ==== Convergence criterion ==== Consider a subset of the satisfied constraints in the dual problem. For any finite subset we can solve the linear program and thus satisfy all constraints. If we could prove that of all the constraints which we did not add to the dual problem no single constraint is violated, we would have proven that solving our restricted problem is equivalent to solving the original problem. More formally, let γ ∗ {\displaystyle \gamma ^{}} be the optimal objective function value for any restricted instance. Then, we can formulate a search problem for the 'most violated constraint' in the original problem space, namely finding ω ∗ ∈ Ω {\displaystyle \omega ^{}\in \Omega } as ω ∗ = argmax ω ∈ Ω ∑ n = 1 ℓ y n h ( x n ; ω ) λ n . {\displaystyle \omega ^{}={\underset {\omega \in \Omega }{\textrm {argmax}}}\sum _{n=1}^{\ell }y_{n}h({\boldsymbol {x}}_{n};\omega )\lambda _{n}.} That is, we search the space H {\displaystyle {\mathcal {H}}} for a single decision stump h ( ⋅ ; ω ∗ ) {\displaystyle h(\cdot ;\omega ^{})} maximizing the left hand side of the dual constraint. If the constraint cannot be violated by any choice of decision stump, none of the corresponding constraint can be active in the original problem and the restricted problem is equivalent. ==== Penalization constant ==== D {\displaystyle D} The positive value of penalization constant D {\displaystyle D} has to be found using model selection techniques. However, if we choose D = 1 ℓ ν {\displaystyle D={\frac {1}{\ell \nu }}} , where ℓ {\displaystyle \ell } is the number of training samples and 0 < ν < 1 {\displaystyle 0<\nu <1} , then the new parameter ν {\displaystyle \nu } has the following properties. ν {\displaystyle \nu } is an upper bound on the fraction of training errors; that is, if k {\displaystyle k} denotes the number of misclassified training samples, then k ℓ ≤ ν {\displaystyle {\frac {k}{\ell }}\leq \nu } . ν {\displaystyle \nu } is a lower bound on the fraction of training samples outside or on the margin. == Algorithm == Input: Training set X = { x 1 , … , x ℓ } {\displaystyle X=\{{\boldsymbol {x}}_{1},\dots ,{\boldsymbol {x}}_{\ell }\}} , x i ∈ X {\displaystyle {\boldsymbol {x}}_{i}\in {\mathcal {X}}} Training labels Y = { y 1 , … , y ℓ } {\displaystyle Y=\{y_{1},\dots ,y_{\ell }\}} , y i ∈ { − 1 , 1 } {\displaystyle y_{i}\in \{-1,1\}} Convergence threshold θ ≥ 0 {\displaystyle \theta \geq 0} Output: Classification function f : X → { − 1 , 1 } {\displaystyle f:{\mathcal {X}}\to \{-1,1\}} Initialization Weights, uniform λ n ← 1 ℓ , n = 1 , … , ℓ {\displaystyle \lambda _{n}\leftarrow {\frac {1}{\ell }},\quad n=1,\dots ,\ell } Edge γ ← 0 {\displaystyle \gamma \leftarrow 0} Hypothesis count J ← 1 {\displaystyle J\leftarrow 1} Iterate h ^ ← argmax ω ∈ Ω ∑ n = 1 ℓ y n h ( x n ; ω ) λ n {\displaystyle {\hat {h}}\leftarrow {\underset {\omega \in \Omega }{\textrm {argmax}}}\sum _{n=1}^{\ell }y_{n}h({\boldsymbol {x}}_{n};\omega )\lambda _{n}} if ∑ n = 1 ℓ y n h ^ ( x n ) λ n + γ ≤ θ {\displaystyle \sum _{n=1}^{\ell }y_{n}{\hat {h}}({\boldsymbol {x}}_{n})\lambda _{n}+\gamma \leq \theta } then break h J ← h ^ {\displaystyle h_{J}\leftarrow {\hat {h}}} J

AdaBoost

AdaBoost (short for Adaptive Boosting) is a statistical classification meta-algorithm formulated by Yoav Freund and Robert Schapire in 1995, who won the 2003 Gödel Prize for their work. It can be used in conjunction with many types of learning algorithm to improve performance. The output of multiple weak learners is combined into a weighted sum that represents the final output of the boosted classifier. Usually, AdaBoost is presented for binary classification, although it can be generalized to multiple classes or bounded intervals of real values. AdaBoost is adaptive in the sense that subsequent weak learners (models) are adjusted in favor of instances misclassified by previous models. In some problems, it can be less susceptible to overfitting than other learning algorithms. The individual learners can be weak, but as long as the performance of each one is slightly better than random guessing, the final model can be proven to converge to a strong learner. Although AdaBoost is typically used to combine weak base learners (such as decision stumps), it has been shown to also effectively combine strong base learners (such as deeper decision trees), producing an even more accurate model. Every learning algorithm tends to suit some problem types better than others, and typically has many different parameters and configurations to adjust before it achieves optimal performance on a dataset. AdaBoost (with decision trees as the weak learners) is often referred to as the best out-of-the-box classifier. When used with decision tree learning, information gathered at each stage of the AdaBoost algorithm about the relative 'hardness' of each training sample is fed into the tree-growing algorithm such that later trees tend to focus on harder-to-classify examples. == Training == AdaBoost refers to a particular method of training a boosted classifier. A boosted classifier is a classifier of the form F T ( x ) = ∑ t = 1 T f t ( x ) {\displaystyle F_{T}(x)=\sum _{t=1}^{T}f_{t}(x)} where each f t {\displaystyle f_{t}} is a weak learner that takes an object x {\displaystyle x} as input and returns a value indicating the class of the object. For example, in the two-class problem, the sign of the weak learner's output identifies the predicted object class and the absolute value gives the confidence in that classification. Each weak learner produces an output hypothesis h {\displaystyle h} which fixes a prediction h ( x i ) {\displaystyle h(x_{i})} for each sample in the training set. At each iteration t {\displaystyle t} , a weak learner is selected and assigned a coefficient α t {\displaystyle \alpha _{t}} such that the total training error E t {\displaystyle E_{t}} of the resulting t {\displaystyle t} -stage boosted classifier is minimized. E t = ∑ i E [ F t − 1 ( x i ) + α t h ( x i ) ] {\displaystyle E_{t}=\sum _{i}E[F_{t-1}(x_{i})+\alpha _{t}h(x_{i})]} Here F t − 1 ( x ) {\displaystyle F_{t-1}(x)} is the boosted classifier that has been built up to the previous stage of training and f t ( x ) = α t h ( x ) {\displaystyle f_{t}(x)=\alpha _{t}h(x)} is the weak learner that is being considered for addition to the final classifier. === Weighting === At each iteration of the training process, a weight w i , t {\displaystyle w_{i,t}} is assigned to each sample in the training set equal to the current error E ( F t − 1 ( x i ) ) {\displaystyle E(F_{t-1}(x_{i}))} on that sample. These weights can be used in the training of the weak learner. For instance, decision trees can be grown which favor the splitting of sets of samples with large weights. == Derivation == This derivation follows Rojas (2009): Suppose we have a data set { ( x 1 , y 1 ) , … , ( x N , y N ) } {\displaystyle \{(x_{1},y_{1}),\ldots ,(x_{N},y_{N})\}} where each item x i {\displaystyle x_{i}} has an associated class y i ∈ { − 1 , 1 } {\displaystyle y_{i}\in \{-1,1\}} , and a set of weak classifiers { k 1 , … , k L } {\displaystyle \{k_{1},\ldots ,k_{L}\}} each of which outputs a classification k j ( x i ) ∈ { − 1 , 1 } {\displaystyle k_{j}(x_{i})\in \{-1,1\}} for each item. After the ( m − 1 ) {\displaystyle (m-1)} -th iteration our boosted classifier is a linear combination of the weak classifiers of the form: C ( m − 1 ) ( x i ) = α 1 k 1 ( x i ) + ⋯ + α m − 1 k m − 1 ( x i ) , {\displaystyle C_{(m-1)}(x_{i})=\alpha _{1}k_{1}(x_{i})+\cdots +\alpha _{m-1}k_{m-1}(x_{i}),} where the class will be the sign of C ( m − 1 ) ( x i ) {\displaystyle C_{(m-1)}(x_{i})} . At the m {\displaystyle m} -th iteration we want to extend this to a better boosted classifier by adding another weak classifier k m {\displaystyle k_{m}} , with another weight α m {\displaystyle \alpha _{m}} : C m ( x i ) = C ( m − 1 ) ( x i ) + α m k m ( x i ) {\displaystyle C_{m}(x_{i})=C_{(m-1)}(x_{i})+\alpha _{m}k_{m}(x_{i})} So it remains to determine which weak classifier is the best choice for k m {\displaystyle k_{m}} , and what its weight α m {\displaystyle \alpha _{m}} should be. We define the total error E {\displaystyle E} of C m {\displaystyle C_{m}} as the sum of its exponential loss on each data point, given as follows: E = ∑ i = 1 N e − y i C m ( x i ) = ∑ i = 1 N e − y i C ( m − 1 ) ( x i ) e − y i α m k m ( x i ) {\displaystyle E=\sum _{i=1}^{N}e^{-y_{i}C_{m}(x_{i})}=\sum _{i=1}^{N}e^{-y_{i}C_{(m-1)}(x_{i})}e^{-y_{i}\alpha _{m}k_{m}(x_{i})}} Letting w i ( 1 ) = 1 {\displaystyle w_{i}^{(1)}=1} and w i ( m ) = e − y i C m − 1 ( x i ) {\displaystyle w_{i}^{(m)}=e^{-y_{i}C_{m-1}(x_{i})}} for m > 1 {\displaystyle m>1} , we have: E = ∑ i = 1 N w i ( m ) e − y i α m k m ( x i ) {\displaystyle E=\sum _{i=1}^{N}w_{i}^{(m)}e^{-y_{i}\alpha _{m}k_{m}(x_{i})}} We can split this summation between those data points that are correctly classified by k m {\displaystyle k_{m}} (so y i k m ( x i ) = 1 {\displaystyle y_{i}k_{m}(x_{i})=1} ) and those that are misclassified (so y i k m ( x i ) = − 1 {\displaystyle y_{i}k_{m}(x_{i})=-1} ): E = ∑ y i = k m ( x i ) w i ( m ) e − α m + ∑ y i ≠ k m ( x i ) w i ( m ) e α m = ∑ i = 1 N w i ( m ) e − α m + ∑ y i ≠ k m ( x i ) w i ( m ) ( e α m − e − α m ) {\displaystyle {\begin{aligned}E&=\sum _{y_{i}=k_{m}(x_{i})}w_{i}^{(m)}e^{-\alpha _{m}}+\sum _{y_{i}\neq k_{m}(x_{i})}w_{i}^{(m)}e^{\alpha _{m}}\\&=\sum _{i=1}^{N}w_{i}^{(m)}e^{-\alpha _{m}}+\sum _{y_{i}\neq k_{m}(x_{i})}w_{i}^{(m)}\left(e^{\alpha _{m}}-e^{-\alpha _{m}}\right)\end{aligned}}} Since the only part of the right-hand side of this equation that depends on k m {\displaystyle k_{m}} is ∑ y i ≠ k m ( x i ) w i ( m ) {\textstyle \sum _{y_{i}\neq k_{m}(x_{i})}w_{i}^{(m)}} , we see that the k m {\displaystyle k_{m}} that minimizes E {\displaystyle E} is the one in the set { k 1 , … , k L } {\displaystyle \{k_{1},\ldots ,k_{L}\}} that minimizes ∑ y i ≠ k m ( x i ) w i ( m ) {\textstyle \sum _{y_{i}\neq k_{m}(x_{i})}w_{i}^{(m)}} [assuming that α m > 0 {\displaystyle \alpha _{m}>0} ], i.e. the weak classifier with the lowest weighted error (with weights w i ( m ) = e − y i C m − 1 ( x i ) {\displaystyle w_{i}^{(m)}=e^{-y_{i}C_{m-1}(x_{i})}} ). To determine the desired weight α m {\displaystyle \alpha _{m}} that minimizes E {\displaystyle E} with the k m {\displaystyle k_{m}} that we just determined, we differentiate: d E d α m = d ( ∑ y i = k m ( x i ) w i ( m ) e − α m + ∑ y i ≠ k m ( x i ) w i ( m ) e α m ) d α m {\displaystyle {\frac {dE}{d\alpha _{m}}}={\frac {d(\sum _{y_{i}=k_{m}(x_{i})}w_{i}^{(m)}e^{-\alpha _{m}}+\sum _{y_{i}\neq k_{m}(x_{i})}w_{i}^{(m)}e^{\alpha _{m}})}{d\alpha _{m}}}} The value of α m {\displaystyle \alpha _{m}} that minimizes the above expression is: α m = 1 2 ln ⁡ ( ∑ y i = k m ( x i ) w i ( m ) ∑ y i ≠ k m ( x i ) w i ( m ) ) {\displaystyle \alpha _{m}={\frac {1}{2}}\ln \left({\frac {\sum _{y_{i}=k_{m}(x_{i})}w_{i}^{(m)}}{\sum _{y_{i}\neq k_{m}(x_{i})}w_{i}^{(m)}}}\right)} We calculate the weighted error rate of the weak classifier to be ϵ m = ∑ y i ≠ k m ( x i ) w i ( m ) ∑ i = 1 N w i ( m ) {\displaystyle \epsilon _{m}={\frac {\sum _{y_{i}\neq k_{m}(x_{i})}w_{i}^{(m)}}{\sum _{i=1}^{N}w_{i}^{(m)}}}} , so it follows that: α m = 1 2 ln ⁡ ( 1 − ϵ m ϵ m ) {\displaystyle \alpha _{m}={\frac {1}{2}}\ln \left({\frac {1-\epsilon _{m}}{\epsilon _{m}}}\right)} which is the negative logit function multiplied by 0.5. Due to the convexity of E {\displaystyle E} as a function of α m {\displaystyle \alpha _{m}} , this new expression for α m {\displaystyle \alpha _{m}} gives the global minimum of the loss function. Note: This derivation only applies when k m ( x i ) ∈ { − 1 , 1 } {\displaystyle k_{m}(x_{i})\in \{-1,1\}} , though it can be a good starting guess in other cases, such as when the weak learner is biased ( k m ( x ) ∈ { a , b } , a ≠ − b {\displaystyle k_{m}(x)\in \{a,b\},a\neq -b} ), has multiple leaves ( k m ( x ) ∈ { a , b , … , n } {\displaystyle k_{m}(x)\in \{a,b,\dots ,n\}} ) or is some other function k m ( x ) ∈ R {\displaystyle k_{m}(x)\in \mathbb {R} } . Thus we have derived the AdaBoost algorithm: At each

Stochastic gradient descent

Stochastic gradient descent (often abbreviated SGD) is an iterative method for optimizing an objective function with suitable smoothness properties (e.g. differentiable or subdifferentiable). It can be regarded as a stochastic approximation of gradient descent optimization, since it replaces the actual gradient (calculated from the entire data set) by an estimate thereof (calculated from a randomly selected subset of the data). Especially in high-dimensional optimization problems this reduces the very high computational burden, achieving faster iterations in exchange for a lower convergence rate. The basic idea behind stochastic approximation can be traced back to the Robbins–Monro algorithm of the 1950s. Today, stochastic gradient descent has become an important optimization method in machine learning. == Background == Both statistical estimation and machine learning consider the problem of minimizing an objective function that has the form of a sum: Q ( w ) = 1 n ∑ i = 1 n Q i ( w ) , {\displaystyle Q(w)={\frac {1}{n}}\sum _{i=1}^{n}Q_{i}(w),} where the parameter w {\displaystyle w} that minimizes Q ( w ) {\displaystyle Q(w)} is to be estimated. Each summand function Q i {\displaystyle Q_{i}} is typically associated with the i {\displaystyle i} -th observation in the data set (used for training). In classical statistics, sum-minimization problems arise in least squares and in maximum-likelihood estimation (for independent observations). The general class of estimators that arise as minimizers of sums are called M-estimators. However, in statistics, it has been long recognized that requiring even local minimization is too restrictive for some problems of maximum-likelihood estimation. Therefore, contemporary statistical theorists often consider stationary points of the likelihood function (or zeros of its derivative, the score function, and other estimating equations). The sum-minimization problem also arises for empirical risk minimization. There, Q i ( w ) {\displaystyle Q_{i}(w)} is the value of the loss function at i {\displaystyle i} -th example, and Q ( w ) {\displaystyle Q(w)} is the empirical risk. When used to minimize the above function, a standard (or "batch") gradient descent method would perform the following iterations: w := w − η ∇ Q ( w ) = w − η n ∑ i = 1 n ∇ Q i ( w ) . {\displaystyle w:=w-\eta \,\nabla Q(w)=w-{\frac {\eta }{n}}\sum _{i=1}^{n}\nabla Q_{i}(w).} The step size is denoted by η {\displaystyle \eta } (sometimes called the learning rate in machine learning) and here " := {\displaystyle :=} " denotes the update of a variable in the algorithm. In many cases, the summand functions have a simple form that enables inexpensive evaluations of the sum-function and the sum gradient. For example, in statistics, one-parameter exponential families allow economical function-evaluations and gradient-evaluations. However, in other cases, evaluating the sum-gradient may require expensive evaluations of the gradients from all summand functions. When the training set is enormous and no simple formulas exist, evaluating the sums of gradients becomes very expensive, because evaluating the gradient requires evaluating all the summand functions' gradients. To economize on the computational cost at every iteration, stochastic gradient descent samples a subset of summand functions at every step. This is very effective in the case of large-scale machine learning problems. == Iterative method == In stochastic (or "on-line") gradient descent, the true gradient of Q ( w ) {\displaystyle Q(w)} is approximated by a gradient at a single sample: w := w − η ∇ Q i ( w ) . {\displaystyle w:=w-\eta \,\nabla Q_{i}(w).} As the algorithm sweeps through the training set, it performs the above update for each training sample. Several passes can be made over the training set until the algorithm converges. If this is done, the data can be shuffled for each pass to prevent cycles. Typical implementations may use an adaptive learning rate so that the algorithm converges. In pseudocode, stochastic gradient descent can be presented as : A compromise between computing the true gradient and the gradient at a single sample is to compute the gradient against more than one training sample (called a "mini-batch") at each step. This can perform significantly better than "true" stochastic gradient descent described, because the code can make use of vectorization libraries rather than computing each step separately as was first shown in where it was called "the bunch-mode back-propagation algorithm". It may also result in smoother convergence, as the gradient computed at each step is averaged over more training samples. The convergence of stochastic gradient descent has been analyzed using the theories of convex minimization and of stochastic approximation. Briefly, when the learning rates η {\displaystyle \eta } decrease with an appropriate rate, and subject to relatively mild assumptions, stochastic gradient descent converges almost surely to a global minimum when the objective function is convex or pseudoconvex, and otherwise converges almost surely to a local minimum. This is in fact a consequence of the Robbins–Siegmund theorem. == Linear regression == Suppose we want to fit a straight line y ^ = w 1 + w 2 x {\displaystyle {\hat {y}}=w_{1}+w_{2}x} to a training set with observations ( ( x 1 , y 1 ) , ( x 2 , y 2 ) … , ( x n , y n ) ) {\displaystyle ((x_{1},y_{1}),(x_{2},y_{2})\ldots ,(x_{n},y_{n}))} and corresponding estimated responses ( y ^ 1 , y ^ 2 , … , y ^ n ) {\displaystyle ({\hat {y}}_{1},{\hat {y}}_{2},\ldots ,{\hat {y}}_{n})} using least squares. The objective function to be minimized is Q ( w ) = ∑ i = 1 n Q i ( w ) = ∑ i = 1 n ( y ^ i − y i ) 2 = ∑ i = 1 n ( w 1 + w 2 x i − y i ) 2 . {\displaystyle Q(w)=\sum _{i=1}^{n}Q_{i}(w)=\sum _{i=1}^{n}\left({\hat {y}}_{i}-y_{i}\right)^{2}=\sum _{i=1}^{n}\left(w_{1}+w_{2}x_{i}-y_{i}\right)^{2}.} The last line in the above pseudocode for this specific problem will become: [ w 1 w 2 ] ← [ w 1 w 2 ] − η [ ∂ ∂ w 1 ( w 1 + w 2 x i − y i ) 2 ∂ ∂ w 2 ( w 1 + w 2 x i − y i ) 2 ] = [ w 1 w 2 ] − η [ 2 ( w 1 + w 2 x i − y i ) 2 x i ( w 1 + w 2 x i − y i ) ] . {\displaystyle {\begin{bmatrix}w_{1}\\w_{2}\end{bmatrix}}\leftarrow {\begin{bmatrix}w_{1}\\w_{2}\end{bmatrix}}-\eta {\begin{bmatrix}{\frac {\partial }{\partial w_{1}}}(w_{1}+w_{2}x_{i}-y_{i})^{2}\\{\frac {\partial }{\partial w_{2}}}(w_{1}+w_{2}x_{i}-y_{i})^{2}\end{bmatrix}}={\begin{bmatrix}w_{1}\\w_{2}\end{bmatrix}}-\eta {\begin{bmatrix}2(w_{1}+w_{2}x_{i}-y_{i})\\2x_{i}(w_{1}+w_{2}x_{i}-y_{i})\end{bmatrix}}.} Note that in each iteration or update step, the gradient is only evaluated at a single x i {\displaystyle x_{i}} . This is the key difference between stochastic gradient descent and batched gradient descent. In general, given a linear regression y ^ = ∑ k ∈ 1 : m w k x k {\displaystyle {\hat {y}}=\sum _{k\in 1:m}w_{k}x_{k}} problem, stochastic gradient descent behaves differently when m < n {\displaystyle m

EXAPT

EXAPT (a portmanteau of "Extended Subset of APT") is a production-oriented programming language that allows users to generate NC programs with control information for machining tools and facilitates decision-making for production-related issues that may arise during various machining processes. EXAPT was first developed to address industrial requirements. Through the years, the company created additional software for the manufacturing industry. Today, EXAPT offers a suite of SAAS products and services for the manufacturing industry. The trade name, EXAPT, is most commonly associated with the CAD/CAM-System, production data, and tool management software of the German company EXAPT Systemtechnik GmbH based in Aachen, DE. == General == EXAPT is a modularly built programming system for all NC machining operations as Drilling Turning Milling Turn-Milling Nibbling Flame-, laser-, plasma- and water jet cutting Wire eroding Operations with industrial robots Due to the modular structure, the main product groups, EXAPTcam and EXAPTpdo, are gradually expandable and permit individual software for the manufacturing industry used individually and also in a compound with an existing IT environment. == Functionality == EXAPTcam meets the requirements for NC planning, especially for the cutting operations such as turning, drilling, and milling up to 5-axis simultaneous machining. Thereby new process technologies, tool, and machine concepts are constantly involved. In the NC programming data from different sources such as 3D CAD models, drawings or tables can flow in. The possibilities of NC programming reaches from language-oriented to feature-oriented NC programming. The integrated EXAPT knowledge database and intelligent and scalable automatisms support the user. The EXAPT NC planning also covers the generation of production information as clamping and tool plans, presetting data or time calculations. The realistic simulation possibilities of NC planning and NC control data provide with production reliability. EXAPTpdo (EXAPT ProductionsDataOrganization) provides a neutrally applicable technology platform for the information compound of the NC planning - to the shop floor. This applies to all NC production data that are necessary for the set-up of NC machines, for the provision, presetting, and stocking of manufacturing resources and provided by EXAPTpdo in a central database. Besides classical functions of the tool management system (TMS) as the management of cutting tools, measuring, testing and clamping devices the technology data management and tool lifecycle management (TLM) is also included. System-supported "where-used lists" helps to handle the manufacturing resource cycle by secured requirement determination and requirement fulfillment. Unnecessary transports and unplanned dispositive adjustments are dropped, stocks are reduced, set-up times reduced and the throughput is increased. EXAPTpdo synchronizes involved systems within the value chain. Stock systems, MES systems or ERP systems (e.g. from the purchasing or production areas) do not work in isolation from each other but they interact with each other. EXAPTpdo provides the base to Smart Factory, for more flexibility in production and faster communication. == History == With the foundation of the EXAPT-Verein in 1967 as spin-off of the universities Aachen, Berlin and Stuttgart the further development "EXAPT (EXtended Subset of APT)" of the programming language "APT (Automatically Programmed Tool)" was focused and so the first milestone for the EXAPT history was set. In the same year the system EXAPT 1 for drilling and simple milling tasks became available. 1969 The industrial application of EXAPT 2 for the programming of NC machines with 2-axis linear and path control begins. In the following year, the development of the EXAPT modular system starts. 1972 BASIC-EXAPT is provided for the universal, homogeneous programming of all NC tasks. The support is made by the EXAPT applications consultancy. 1973 EXAPT 1.1 is provided for the programming of straight-cut and continuous-path controlled drilling and milling machines and machining centers. At the Hanover Fair (IHA 73) the interactive access to a mainframe via a time-sharing terminal for the part program entry and correction is presented and starts the replacement of the punch card. 1974 The possibilities for the use of process computers for the NC data transfer are leveled out. EXAPT offers the possibility of the result simulation when using plotters with display of tool paths and tools in assignment to the workpiece. In April 1975, the EXAPT NC Systemtechnik GmbH was founded with the aim, of enabling entry into the NC technique for small and medium-sized companies by a complete product and service program. In the following year, the system portfolio is extended with further system modules and service programs and the provision of postprocessors. 1978 The development activities on the EXAPT module system started in 1970 are completed. Using modern software techniques, the different system parts BASIC-EXAPT, EXAPT 1, EXAPT 1.1, and EXAPT 2 are composed of a total system. System support and applications consultancy become a new working focus. From the beginning to the middle of the 1980s Beside new portable software modules for CAD/CAM applications (e. g. CAPEX, NESTEX, CADEX, CADCPL), the first version of the EXAPT DNC system and extensions of the EXAPT NC programming system for the machining of sculptured surfaces are presented. 1988 EXAPT expands the software product range by systems for tool data management (BMO) and production data management (FDO). EXAPT trains more than 1,300 course participants including company-specific courses. 1992 The first version of the completely new product generation EXAPTplus is presented and the agency in Dresden is opened. 1993 The company name "EXAPT NC Systemtechnik GmbH" is changed to "EXAPT Systemtechnik GmbH." EXAPTplus is presented on PC under Windows NT at the EMO '93. The decentralization of the use of EXAPT systems expands the range of applications. In the following year, EXAPT-DNC is executable under Windows on a customary PC. Special hardware is not needed and so it can be used in compound with the database-supported EXAPT production data management system (FDO). 1995 EXAPTplus is also ready for complex application cases such as machining of tubes at extrusion tools. EXAPT-CADI provides the transfer of 2D CAD data to EXAPTplus. With the new office Gießen the marketing is strengthened. In the following year the EXAPT NC editor is developed for the direct processing of NC control data with tool path display and visualization of the tools. In the course of the market entry of more comfortable 3D CAD systems for the solid modelling of components a detailed evaluation of current systems is made in 1997. It is decided to use SolidWorks as a reference system for the solid-oriented NC planning with EXAPT. 1998 The first solution for the transfer of geometry data between SolidWorks and EXAPTplus is generated. The EXAPT organization systems are (beside SQL) also executable under Oracle now. The use of client server solutions supports the data flow in the production. 1999 AFR functions are provided in connection with EXAPTsolid to support a workpiece modelling for NC. The millennium capability is ensured for all EXAPT systems. AFR is a ground-breaking for the integration of third-party products. 2002 EXAPT-BMG is developed for the generation and visualization of tools with additional functions for the assembly from components. The acquisition of tools with their geometric and technological presentation offers extensive support of the NC planning with EXAPT systems. 2003 EXAPTpdo is available to optimize the process chains in production planning and production execution optimally regarding the increasing requirements of changing production conditions. 2004 Diverse system extensions are made in EXAPTplus, EXAPTsolid, EXAPT NC editor, EXAPTpdo for the complete machining on turning/milling centres with result reliability because of more extensive simulation based on realNC (Tecnomatix), for the use of new complex tool systems and the compound use between ERP systems as SAP and intelligent CNC systems. In the following year, EXAPTpdo is extended for the cross-order set-up optimization and provision of manufacturing re-sources especially for single and small series production with connection to purchase and physical portfolio management. 2006 The EXAPT systems are available for extended use as an information platform for production, the time management, and similar requirements. EXAPTsolid is extended for the feature-oriented milling operation and machine simulation. The NC programming of complex machine tools, e.g. three-turret-turning/milling centers is supported by EXAPT systems, as well as the use of multi-functional tools. 2007 A module for 3-5-axis simultaneous milling machining is presented.

Dynamic time warping

In time series analysis, dynamic time warping (DTW) is an algorithm for measuring similarity between two temporal sequences, which may vary in speed. For instance, similarities in walking could be detected using DTW, even if one person was walking faster than the other, or if there were accelerations and decelerations during the course of an observation. DTW has been applied to temporal sequences of video, audio, and graphics data — indeed, any data that can be turned into a one-dimensional sequence can be analyzed with DTW. A well-known application has been automatic speech recognition, to cope with different speaking speeds. Other applications include speaker recognition and online signature recognition. It can also be used in partial shape matching applications. In general, DTW is a method that calculates an optimal match between two given sequences (e.g. time series) with certain restriction and rules: Every index from the first sequence must be matched with one or more indices from the other sequence, and vice versa The first index from the first sequence must be matched with the first index from the other sequence (but it does not have to be its only match) The last index from the first sequence must be matched with the last index from the other sequence (but it does not have to be its only match) The mapping of the indices from the first sequence to indices from the other sequence must be monotonically increasing, and vice versa, i.e. if j > i {\displaystyle j>i} are indices from the first sequence, then there must not be two indices l > k {\displaystyle l>k} in the other sequence, such that index i {\displaystyle i} is matched with index l {\displaystyle l} and index j {\displaystyle j} is matched with index k {\displaystyle k} , and vice versa We can plot each match between the sequences 1 : M {\displaystyle 1:M} and 1 : N {\displaystyle 1:N} as a path in a M × N {\displaystyle M\times N} matrix from ( 1 , 1 ) {\displaystyle (1,1)} to ( M , N ) {\displaystyle (M,N)} , such that each step is one of ( 0 , 1 ) , ( 1 , 0 ) , ( 1 , 1 ) {\displaystyle (0,1),(1,0),(1,1)} . In this formulation, we see that the number of possible matches is the Delannoy number. The optimal match is denoted by the match that satisfies all the restrictions and the rules and that has the minimal cost, where the cost is computed as the sum of absolute differences, for each matched pair of indices, between their values. The sequences are "warped" non-linearly in the time dimension to determine a measure of their similarity independent of certain non-linear variations in the time dimension. This sequence alignment method is often used in time series classification. Although DTW measures a distance-like quantity between two given sequences, it doesn't guarantee the triangle inequality to hold. In addition to a similarity measure between the two sequences (a so called "warping path" is produced), by warping according to this path the two signals may be aligned in time. The signal with an original set of points X(original), Y(original) is transformed to X(warped), Y(warped). This finds applications in genetic sequence and audio synchronisation. In a related technique sequences of varying speed may be averaged using this technique see the average sequence section. This is conceptually very similar to the Needleman–Wunsch algorithm. == Implementation == This example illustrates the implementation of the dynamic time warping algorithm when the two sequences s and t are strings of discrete symbols. For two symbols x and y, d ( x , y ) {\displaystyle d(x,y)} is a distance between the symbols, e.g., d ( x , y ) = | x − y | {\displaystyle d(x,y)=|x-y|} . int DTWDistance(s: array [1..n], t: array [1..m]) { DTW := array [0..n, 0..m] for i := 0 to n for j := 0 to m DTW[i, j] := infinity DTW[0, 0] := 0 for i := 1 to n for j := 1 to m cost := d(s[i], t[j]) DTW[i, j] := cost + minimum(DTW[i-1, j ], // insertion DTW[i , j-1], // deletion DTW[i-1, j-1]) // match return DTW[n, m] } where DTW[i, j] is the distance between s[1:i] and t[1:j] with the best alignment. We sometimes want to add a locality constraint. That is, we require that if s[i] is matched with t[j], then | i − j | {\displaystyle |i-j|} is no larger than w, a window parameter. We can easily modify the above algorithm to add a locality constraint (differences marked). However, the above given modification works only if | n − m | {\displaystyle |n-m|} is no larger than w, i.e. the end point is within the window length from diagonal. In order to make the algorithm work, the window parameter w must be adapted so that | n − m | ≤ w {\displaystyle |n-m|\leq w} (see the line marked with () in the code). int DTWDistance(s: array [1..n], t: array [1..m], w: int) { DTW := array [0..n, 0..m] w := max(w, abs(n-m)) // adapt window size () for i := 0 to n for j:= 0 to m DTW[i, j] := infinity DTW[0, 0] := 0 for i := 1 to n for j := max(1, i-w) to min(m, i+w) DTW[i, j] := 0 for i := 1 to n for j := max(1, i-w) to min(m, i+w) cost := d(s[i], t[j]) DTW[i, j] := cost + minimum(DTW[i-1, j ], // insertion DTW[i , j-1], // deletion DTW[i-1, j-1]) // match return DTW[n, m] } == Warping properties == The DTW algorithm produces a discrete matching between existing elements of one series to another. In other words, it does not allow time-scaling of segments within the sequence. Other methods allow continuous warping. For example, Correlation Optimized Warping (COW) divides the sequence into uniform segments that are scaled in time using linear interpolation, to produce the best matching warping. The segment scaling causes potential creation of new elements, by time-scaling segments either down or up, and thus produces a more sensitive warping than DTW's discrete matching of raw elements. == Complexity == The time complexity of the DTW algorithm is O ( N M ) {\displaystyle O(NM)} , where N {\displaystyle N} and M {\displaystyle M} are the lengths of the two input sequences. The 50 years old quadratic time bound was broken in 2016: an algorithm due to Gold and Sharir enables computing DTW in O ( N 2 / log ⁡ log ⁡ N ) {\displaystyle O({N^{2}}/\log \log N)} time and space for two input sequences of length N {\displaystyle N} . This algorithm can also be adapted to sequences of different lengths. Despite this improvement, it was shown that a strongly subquadratic running time of the form O ( N 2 − ϵ ) {\displaystyle O(N^{2-\epsilon })} for some ϵ > 0 {\displaystyle \epsilon >0} cannot exist unless the Strong exponential time hypothesis fails. While the dynamic programming algorithm for DTW requires O ( N M ) {\displaystyle O(NM)} space in a naive implementation, the space consumption can be reduced to O ( min ( N , M ) ) {\displaystyle O(\min(N,M))} using Hirschberg's algorithm. == Fast computation == Fast techniques for computing DTW include PrunedDTW, SparseDTW, FastDTW, and the MultiscaleDTW. A common task, retrieval of similar time series, can be accelerated by using lower bounds such as LB_Keogh, LB_Improved, or LB_Petitjean. However, the Early Abandon and Pruned DTW algorithm reduces the degree of acceleration that lower bounding provides and sometimes renders it ineffective. In a survey, Wang et al. reported slightly better results with the LB_Improved lower bound than the LB_Keogh bound, and found that other techniques were inefficient. Subsequent to this survey, the LB_Enhanced bound was developed that is always tighter than LB_Keogh while also being more efficient to compute. LB_Petitjean is the tightest known lower bound that can be computed in linear time. == Average sequence == Averaging for dynamic time warping is the problem of finding an average sequence for a set of sequences. NLAAF is an exact method to average two sequences using DTW. For more than two sequences, the problem is related to that of multiple alignment and requires heuristics. DBA is currently a reference method to average a set of sequences consistently with DTW. COMASA efficiently randomizes the search for the average sequence, using DBA as a local optimization process. == Supervised learning == A nearest-neighbour classifier can achieve state-of-the-art performance when using dynamic time warping as a distance measure. == Amerced Dynamic Time Warping == Amerced Dynamic Time Warping (ADTW) is a variant of DTW designed to better control DTW's permissiveness in the alignments that it allows. The windows that classical DTW uses to constrain alignments introduce a step function. Any warping of the path is allowed within the window and none beyond it. In contrast, ADTW employs an additive penalty that is incurred each time that the path is warped. Any amount of warping is allowed, but each warping action incurs a direct penalty. ADTW significantly outperforms DTW with windowing when applied as a nearest neighbor classifier on a set of benchmark time series classification tasks. == Alternative approaches == In functional data analysis, time series are regarde