Lexxe is an internet search engine that applies Natural Language Processing in its semantic search technology. Founded in 2005 by Dr. Hong Liang Qiao, Lexxe is based in Sydney, Australia. Today, Lexxe's key focus is on sentiment search with the launch of a news sentiment search site at News & Moods (www.newsandmoods.com). Lexxe has experienced several stages of change of focus in search technology: Lexxe launched its Alpha version in 2005, featuring Natural Language question answering (i.e. users could ask questions in English to the search engine apart from keyword searches — this feature has been suspended for redevelopment since 2010). It used only algorithms to extract answers from web pages, with no question-answer pair databases prepared in advance. In 2011, Lexxe launched a beta version with a new search technology called Semantic Key. Semantic Keys enable users to query with a conceptual keyword (or a keyword with a special meaning, hence the term Semantic Key) in order to find instances under the concept, e.g. price → $5.95 or €200, color → red, yellow, white. For example, “price: a pound of apples”, “color: ferrari”. With initial 500 Semantic Keys at the Beta launch, Lexxe became the first search engine in the world to offer this unique and useful search technology to the users. The cost of building Semantic Keys was too heavy though. In 2017, Lexxe launched News & Moods (www.newsandmoods.com), an open platform for news sentiment search, a first step towards sentiment search feature for the entire Internet search in Lexxe search engine. News & Moods also comes with smartphone apps in Android and iOS.
Ugly duckling theorem
The ugly duckling theorem is an argument showing that classification is not really possible without some sort of bias. More particularly, it assumes finitely many properties combinable by logical connectives, and finitely many objects; it asserts that any two different objects share the same number of (extensional) properties. The theorem is named after Hans Christian Andersen's 1843 story "The Ugly Duckling", because it shows that a duckling is just as similar to a swan as two swans are to each other. It was derived by Satosi Watanabe in 1969. == Mathematical formula == Suppose there are n things in the universe, and one wants to put them into classes or categories. One has no preconceived ideas or biases about what sorts of categories are "natural" or "normal" and what are not. So one has to consider all the possible classes that could be, all the possible ways of making a set out of the n objects. There are 2 n {\displaystyle 2^{n}} such ways, the size of the power set of n objects. One can use that to measure the similarity between two objects, and one would see how many sets they have in common. However, one cannot. Any two objects have exactly the same number of classes in common if we can form any possible class, namely 2 n − 1 {\displaystyle 2^{n-1}} (half the total number of classes there are). To see this is so, one may imagine each class is represented by an n-bit string (or binary encoded integer), with a zero for each element not in the class and a one for each element in the class. As one finds, there are 2 n {\displaystyle 2^{n}} such strings. As all possible choices of zeros and ones are there, any two bit-positions will agree exactly half the time. One may pick two elements and reorder the bits so they are the first two, and imagine the numbers sorted lexicographically. The first 2 n / 2 {\displaystyle 2^{n}/2} numbers will have bit #1 set to zero, and the second 2 n / 2 {\displaystyle 2^{n}/2} will have it set to one. Within each of those blocks, the top 2 n / 4 {\displaystyle 2^{n}/4} will have bit #2 set to zero and the other 2 n / 4 {\displaystyle 2^{n}/4} will have it as one, so they agree on two blocks of 2 n / 4 {\displaystyle 2^{n}/4} or on half of all the cases, no matter which two elements one picks. So if we have no preconceived bias about which categories are better, everything is then equally similar (or equally dissimilar). The number of predicates simultaneously satisfied by two non-identical elements is constant over all such pairs. Thus, some kind of inductive bias is needed to make judgements to prefer certain categories over others. === Boolean functions === Let x 1 , x 2 , … , x n {\displaystyle x_{1},x_{2},\dots ,x_{n}} be a set of vectors of k {\displaystyle k} booleans each. The ugly duckling is the vector which is least like the others. Given the booleans, this can be computed using Hamming distance. However, the choice of boolean features to consider could have been somewhat arbitrary. Perhaps there were features derivable from the original features that were important for identifying the ugly duckling. The set of booleans in the vector can be extended with new features computed as boolean functions of the k {\displaystyle k} original features. The only canonical way to do this is to extend it with all possible Boolean functions. The resulting completed vectors have 2 k {\displaystyle 2^{k}} features. The ugly duckling theorem states that there is no ugly duckling because any two completed vectors will either be equal or differ in exactly half of the features. Proof. Let x and y be two vectors. If they are the same, then their completed vectors must also be the same because any Boolean function of x will agree with the same Boolean function of y. If x and y are different, then there exists a coordinate i {\displaystyle i} where the i {\displaystyle i} -th coordinate of x {\displaystyle x} differs from the i {\displaystyle i} -th coordinate of y {\displaystyle y} . Now the completed features contain every Boolean function on k {\displaystyle k} Boolean variables, with each one exactly once. Viewing these Boolean functions as polynomials in k {\displaystyle k} variables over GF(2), segregate the functions into pairs ( f , g ) {\displaystyle (f,g)} where f {\displaystyle f} contains the i {\displaystyle i} -th coordinate as a linear term and g {\displaystyle g} is f {\displaystyle f} without that linear term. Now, for every such pair ( f , g ) {\displaystyle (f,g)} , x {\displaystyle x} and y {\displaystyle y} will agree on exactly one of the two functions. If they agree on one, they must disagree on the other and vice versa. (This proof is believed to be due to Watanabe.) == Discussion == A possible way around the ugly duckling theorem would be to introduce a constraint on how similarity is measured by limiting the properties involved in classification, for instance, between A and B. However Medin et al. (1993) point out that this does not actually resolve the arbitrariness or bias problem since in what respects A is similar to B: "varies with the stimulus context and task, so that there is no unique answer, to the question of how similar is one object to another". For example, "a barberpole and a zebra would be more similar than a horse and a zebra if the feature striped had sufficient weight. Of course, if these feature weights were fixed, then these similarity relations would be constrained". Yet the property "striped" as a weight 'fix' or constraint is arbitrary itself, meaning: "unless one can specify such criteria, then the claim that categorization is based on attribute matching is almost entirely vacuous". Stamos (2003) remarked that some judgments of overall similarity are non-arbitrary in the sense they are useful: "Presumably, people's perceptual and conceptual processes have evolved that information that matters to human needs and goals can be roughly approximated by a similarity heuristic... If you are in the jungle and you see a tiger but you decide not to stereotype (perhaps because you believe that similarity is a false friend), then you will probably be eaten. In other words, in the biological world stereotyping based on veridical judgments of overall similarity statistically results in greater survival and reproductive success." Unless some properties are considered more salient, or 'weighted' more important than others, everything will appear equally similar, hence Watanabe (1986) wrote: "any objects, in so far as they are distinguishable, are equally similar". In a weaker setting that assumes infinitely many properties, Murphy and Medin (1985) give an example of two putative classified things, plums and lawnmowers: "Suppose that one is to list the attributes that plums and lawnmowers have in common in order to judge their similarity. It is easy to see that the list could be infinite: Both weigh less than 10,000 kg (and less than 10,001 kg), both did not exist 10,000,000 years ago (and 10,000,001 years ago), both cannot hear well, both can be dropped, both take up space, and so on. Likewise, the list of differences could be infinite… any two entities can be arbitrarily similar or dissimilar by changing the criterion of what counts as a relevant attribute." According to Woodward, the ugly duckling theorem is related to Schaffer's Conservation Law for Generalization Performance, which states that all algorithms for learning of boolean functions from input/output examples have the same overall generalization performance as random guessing. The latter result is generalized by Woodward to functions on countably infinite domains.
Prototype methods
Prototype methods are machine learning methods that use data prototypes. A data prototype is a data value that reflects other values in its class, e.g., the centroid in a K-means clustering problem. == Methods == The following are some prototype methods K-means clustering Learning vector quantization (LVQ) Gaussian mixtures == Related Methods == While K-nearest neighbor's does not use prototypes, it is similar to prototype methods like K-means clustering.
Expectation–maximization algorithm
In statistics, an expectation–maximization (EM) algorithm is an iterative method to find (local) maximum likelihood or maximum a posteriori (MAP) estimates of parameters in statistical models, where the model depends on unobserved latent variables. The EM iteration alternates between performing an expectation (E) step, which creates a function for the expectation of the log-likelihood evaluated using the current estimate for the parameters, and a maximization (M) step, which computes parameters maximizing the expected log-likelihood found on the E step. These parameter-estimates are then used to determine the distribution of the latent variables in the next E step. It can be used, for example, to estimate a mixture of gaussians, or to solve the multiple linear regression problem. == History == The EM algorithm was explained and given its name in a classic 1977 paper by Arthur Dempster, Nan Laird, and Donald Rubin. They pointed out that the method had been "proposed many times in special circumstances" by earlier authors. One of the earliest is the gene-counting method for estimating allele frequencies by Cedric Smith. Another was proposed by H.O. Hartley in 1958, and Hartley and Hocking in 1977, from which many of the ideas in the Dempster–Laird–Rubin paper originated. Another one by S.K Ng, Thriyambakam Krishnan and G.J McLachlan in 1977. Hartley's ideas can be broadened to any grouped discrete distribution. A very detailed treatment of the EM method for exponential families was published by Rolf Sundberg in his thesis and several papers, following his collaboration with Per Martin-Löf and Anders Martin-Löf. The Dempster–Laird–Rubin paper in 1977 generalized the method and sketched a convergence analysis for a wider class of problems. The Dempster–Laird–Rubin paper established the EM method as an important tool of statistical analysis. See also Meng and van Dyk (1997). The convergence analysis of the Dempster–Laird–Rubin algorithm was flawed and a correct convergence analysis was published by C. F. Jeff Wu in 1983. Wu's proof established the EM method's convergence also outside of the exponential family, as claimed by Dempster–Laird–Rubin. == Introduction == The EM algorithm is used to find (local) maximum likelihood parameters of a statistical model in cases where the equations cannot be solved directly. Typically these models involve latent variables in addition to unknown parameters and known data observations. That is, either missing values exist among the data, or the model can be formulated more simply by assuming the existence of further unobserved data points. For example, a mixture model can be described more simply by assuming that each observed data point has a corresponding unobserved data point, or latent variable, specifying the mixture component to which each data point belongs. Finding a maximum likelihood solution typically requires taking the derivatives of the likelihood function with respect to all the unknown values, the parameters and the latent variables, and simultaneously solving the resulting equations. In statistical models with latent variables, this is usually impossible. Instead, the result is typically a set of interlocking equations in which the solution to the parameters requires the values of the latent variables and vice versa, but substituting one set of equations into the other produces an unsolvable equation. The EM algorithm proceeds from the observation that there is a way to solve these two sets of equations numerically. One can simply pick arbitrary values for one of the two sets of unknowns, use them to estimate the second set, then use these new values to find a better estimate of the first set, and then keep alternating between the two until the resulting values both converge to fixed points. It's not obvious that this will work, but it can be proven in this context. Additionally, it can be proven that the derivative of the likelihood is (arbitrarily close to) zero at that point, which in turn means that the point is either a local maximum or a saddle point. In general, multiple maxima may occur, with no guarantee that the global maximum will be found. Some likelihoods also have singularities in them, i.e., nonsensical maxima. For example, one of the solutions that may be found by EM in a mixture model involves setting one of the components to have zero variance and the mean parameter for the same component to be equal to one of the data points. == Description == === The symbols === Given the statistical model which generates a set X {\displaystyle \mathbf {X} } of observed data, a set of unobserved latent data or missing values Z {\displaystyle \mathbf {Z} } , and a vector of unknown parameters θ {\displaystyle {\boldsymbol {\theta }}} , along with a likelihood function L ( θ ; X , Z ) = p ( X , Z ∣ θ ) {\displaystyle L({\boldsymbol {\theta }};\mathbf {X} ,\mathbf {Z} )=p(\mathbf {X} ,\mathbf {Z} \mid {\boldsymbol {\theta }})} , the maximum likelihood estimate (MLE) of the unknown parameters is determined by maximizing the marginal likelihood of the observed data L ( θ ; X ) = p ( X ∣ θ ) = ∫ p ( X , Z ∣ θ ) d Z = ∫ p ( X ∣ Z , θ ) p ( Z ∣ θ ) d Z {\displaystyle {\begin{aligned}L({\boldsymbol {\theta }};\mathbf {X} )=p(\mathbf {X} \mid {\boldsymbol {\theta }})&=\int p(\mathbf {X} ,\mathbf {Z} \mid {\boldsymbol {\theta }})\,d\mathbf {Z} \\&=\int p(\mathbf {X} \mid \mathbf {Z} ,{\boldsymbol {\theta }})p(\mathbf {Z} \mid {\boldsymbol {\theta }})\,d\mathbf {Z} \end{aligned}}} However, this quantity is often intractable since Z {\displaystyle \mathbf {Z} } is unobserved and the distribution of Z {\displaystyle \mathbf {Z} } is unknown before attaining θ {\displaystyle {\boldsymbol {\theta }}} . === The EM algorithm === The EM algorithm seeks to find the maximum likelihood estimate of the marginal likelihood by iteratively applying these two steps: More succinctly, we can write it as one equation: θ ( t + 1 ) = arg max θ E Z ∼ p ( ⋅ | X , θ ( t ) ) [ log p ( X , Z | θ ) ] {\displaystyle {\boldsymbol {\theta }}^{(t+1)}=\mathop {\arg \max } _{\boldsymbol {\theta }}\operatorname {E} _{\mathbf {Z} \sim p(\cdot |\mathbf {X} ,{\boldsymbol {\theta }}^{(t)})}\left[\log p(\mathbf {X} ,\mathbf {Z} |{\boldsymbol {\theta }})\right]\,} === Interpretation of the variables === The typical models to which EM is applied use Z {\displaystyle \mathbf {Z} } as a latent variable indicating membership in one of a set of groups: The observed data points X {\displaystyle \mathbf {X} } may be discrete (taking values in a finite or countably infinite set) or continuous (taking values in an uncountably infinite set). Associated with each data point may be a vector of observations. The missing values (aka latent variables) Z {\displaystyle \mathbf {Z} } are discrete, drawn from a fixed number of values, and with one latent variable per observed unit. The parameters are continuous, and are of two kinds: Parameters that are associated with all data points, and those associated with a specific value of a latent variable (i.e., associated with all data points whose corresponding latent variable has that value). However, it is possible to apply EM to other sorts of models. The motivation is as follows. If the value of the parameters θ {\displaystyle {\boldsymbol {\theta }}} is known, usually the value of the latent variables Z {\displaystyle \mathbf {Z} } can be found by maximizing the log-likelihood over all possible values of Z {\displaystyle \mathbf {Z} } , either simply by iterating over Z {\displaystyle \mathbf {Z} } or through an algorithm such as the Viterbi algorithm for hidden Markov models. Conversely, if we know the value of the latent variables Z {\displaystyle \mathbf {Z} } , we can find an estimate of the parameters θ {\displaystyle {\boldsymbol {\theta }}} fairly easily, typically by simply grouping the observed data points according to the value of the associated latent variable and averaging the values, or some function of the values, of the points in each group. This suggests an iterative algorithm, in the case where both θ {\displaystyle {\boldsymbol {\theta }}} and Z {\displaystyle \mathbf {Z} } are unknown: First, initialize the parameters θ {\displaystyle {\boldsymbol {\theta }}} to some random values. Compute the probability of each possible value of Z {\displaystyle \mathbf {Z} } , given θ {\displaystyle {\boldsymbol {\theta }}} . Then, use the just-computed values of Z {\displaystyle \mathbf {Z} } to compute a better estimate for the parameters θ {\displaystyle {\boldsymbol {\theta }}} . Iterate steps 2 and 3 until convergence. The algorithm as just described monotonically approaches a local minimum of the cost function. == Properties == Although an EM iteration does increase the observed data (i.e., marginal) likelihood function, no guarantee exists that the sequence converges to a maximum likelihood estimator. For multimodal distributions, this means that an EM algorithm may co
Elastic map
Elastic maps provide a tool for nonlinear dimensionality reduction. By their construction, they are a system of elastic springs embedded in the data space. This system approximates a low-dimensional manifold. The elastic coefficients of this system allow the switch from completely unstructured k-means clustering (zero elasticity) to the estimators located closely to linear PCA manifolds (for high bending and low stretching modules). With some intermediate values of the elasticity coefficients, this system effectively approximates non-linear principal manifolds. This approach is based on a mechanical analogy between principal manifolds, that are passing through "the middle" of the data distribution, and elastic membranes and plates. The method was developed by A.N. Gorban, A.Y. Zinovyev and A.A. Pitenko in 1996–1998. == Energy of elastic map == Let S {\displaystyle {\mathcal {S}}} be a data set in a finite-dimensional Euclidean space. Elastic map is represented by a set of nodes w j {\displaystyle {\bf {w}}_{j}} in the same space. Each datapoint s ∈ S {\displaystyle s\in {\mathcal {S}}} has a host node, namely the closest node w j {\displaystyle {\bf {w}}_{j}} (if there are several closest nodes then one takes the node with the smallest number). The data set S {\displaystyle {\mathcal {S}}} is divided into classes K j = { s | w j is a host of s } {\displaystyle K_{j}=\{s\ |\ {\bf {w}}_{j}{\mbox{ is a host of }}s\}} . The approximation energy D is the distortion D = 1 2 ∑ j = 1 k ∑ s ∈ K j ‖ s − w j ‖ 2 {\displaystyle D={\frac {1}{2}}\sum _{j=1}^{k}\sum _{s\in K_{j}}\|s-{\bf {w}}_{j}\|^{2}} , which is the energy of the springs with unit elasticity which connect each data point with its host node. It is possible to apply weighting factors to the terms of this sum, for example to reflect the standard deviation of the probability density function of any subset of data points { s i } {\displaystyle \{s_{i}\}} . On the set of nodes an additional structure is defined. Some pairs of nodes, ( w i , w j ) {\displaystyle ({\bf {w}}_{i},{\bf {w}}_{j})} , are connected by elastic edges. Call this set of pairs E {\displaystyle E} . Some triplets of nodes, ( w i , w j , w k ) {\displaystyle ({\bf {w}}_{i},{\bf {w}}_{j},{\bf {w}}_{k})} , form bending ribs. Call this set of triplets G {\displaystyle G} . The stretching energy is U E = 1 2 λ ∑ ( w i , w j ) ∈ E ‖ w i − w j ‖ 2 {\displaystyle U_{E}={\frac {1}{2}}\lambda \sum _{({\bf {w}}_{i},{\bf {w}}_{j})\in E}\|{\bf {w}}_{i}-{\bf {w}}_{j}\|^{2}} , The bending energy is U G = 1 2 μ ∑ ( w i , w j , w k ) ∈ G ‖ w i − 2 w j + w k ‖ 2 {\displaystyle U_{G}={\frac {1}{2}}\mu \sum _{({\bf {w}}_{i},{\bf {w}}_{j},{\bf {w}}_{k})\in G}\|{\bf {w}}_{i}-2{\bf {w}}_{j}+{\bf {w}}_{k}\|^{2}} , where λ {\displaystyle \lambda } and μ {\displaystyle \mu } are the stretching and bending moduli respectively. The stretching energy is sometimes referred to as the membrane, while the bending energy is referred to as the thin plate term. For example, on the 2D rectangular grid the elastic edges are just vertical and horizontal edges (pairs of closest vertices) and the bending ribs are the vertical or horizontal triplets of consecutive (closest) vertices. The total energy of the elastic map is thus U = D + U E + U G . {\displaystyle U=D+U_{E}+U_{G}.} The position of the nodes { w j } {\displaystyle \{{\bf {w}}_{j}\}} is determined by the mechanical equilibrium of the elastic map, i.e. its location is such that it minimizes the total energy U {\displaystyle U} . == Expectation-maximization algorithm == For a given splitting of dataset S {\displaystyle {\mathcal {S}}} in classes K j {\displaystyle K_{j}} , minimization of the quadratic functional U {\displaystyle U} is a linear problem with the sparse matrix of coefficients. Therefore, similar to principal component analysis or k-means, a splitting method is used: For given { w j } {\displaystyle \{{\bf {w}}_{j}\}} find { K j } {\displaystyle \{K_{j}\}} ; For given { K j } {\displaystyle \{K_{j}\}} minimize U {\displaystyle U} and find { w j } {\displaystyle \{{\bf {w}}_{j}\}} ; If no change, terminate. This expectation-maximization algorithm guarantees a local minimum of U {\displaystyle U} . For improving the approximation various additional methods are proposed. For example, the softening strategy is used. This strategy starts with a rigid grids (small length, small bending and large elasticity modules λ {\displaystyle \lambda } and μ {\displaystyle \mu } coefficients) and finishes with soft grids (small λ {\displaystyle \lambda } and μ {\displaystyle \mu } ). The training goes in several epochs, each epoch with its own grid rigidness. Another adaptive strategy is growing net: one starts from a small number of nodes and gradually adds new nodes. Each epoch goes with its own number of nodes. == Applications == Most important applications of the method and free software are in bioinformatics for exploratory data analysis and visualisation of multidimensional data, for data visualisation in economics, social and political sciences, as an auxiliary tool for data mapping in geographic informational systems and for visualisation of data of various nature. The method is applied in quantitative biology for reconstructing the curved surface of a tree leaf from a stack of light microscopy images. This reconstruction is used for quantifying the geodesic distances between trichomes and their patterning, which is a marker of the capability of a plant to resist to pathogenes. Recently, the method is adapted as a support tool in the decision process underlying the selection, optimization, and management of financial portfolios. The method of elastic maps has been systematically tested and compared with several machine learning methods on the applied problem of identification of the flow regime of a gas-liquid flow in a pipe. There are various regimes: Single phase water or air flow, Bubbly flow, Bubbly-slug flow, Slug flow, Slug-churn flow, Churn flow, Churn-annular flow, and Annular flow. The simplest and most common method used to identify the flow regime is visual observation. This approach is, however, subjective and unsuitable for relatively high gas and liquid flow rates. Therefore, the machine learning methods are proposed by many authors. The methods are applied to differential pressure data collected during a calibration process. The method of elastic maps provided a 2D map, where the area of each regime is represented. The comparison with some other machine learning methods is presented in Table 1 for various pipe diameters and pressure. Here, ANN stands for the backpropagation artificial neural networks, SVM stands for the support vector machine, SOM for the self-organizing maps. The hybrid technology was developed for engineering applications. In this technology, elastic maps are used in combination with Principal Component Analysis (PCA), Independent Component Analysis (ICA) and backpropagation ANN. The textbook provides a systematic comparison of elastic maps and self-organizing maps (SOMs) in applications to economic and financial decision-making.
MetaMask
MetaMask is a software cryptocurrency wallet developed by ConsenSys for interacting with the Ethereum blockchain and other EVM-compatible networks. It enables users to manage Ethereum accounts and connect to decentralized applications (dApps) via a browser extension or mobile app. As of early 2026, MetaMask reports over 100 million users worldwide. == Overview == MetaMask allows users to store and manage private keys, send and receive Ethereum-based cryptocurrencies and tokens (including ERC-20 and ERC-721 standards), broadcast transactions, and interact with dApps. dApps connect to the wallet via JavaScript interfaces, prompting users to approve signatures or transactions. The wallet features MetaMask Swaps, an in-app token swap aggregator sourcing liquidity from multiple decentralized exchanges (DEXs), with a service fee of 0.875%. In 2025, MetaMask introduced the MetaMask Rewards program (initially mobile-only), where users earn points for activities such as swaps, bridging, and referrals. Season 1 (October 2025 – January 2026) distributed over $30 million in Linea tokens and other perks to participants. == History == MetaMask launched in 2016 as open-source software under the MIT license. It initially supported browser extensions for Chrome and Firefox. Mobile versions were in closed beta from 2019 and publicly released for iOS and Android in September 2020. In August 2020, the license changed to a custom proprietary one. MetaMask Swaps launched on desktop in October 2020 and on mobile in March 2021. The Rewards program launched in late 2025 with Linea integration. == Criticism == MetaMask has faced criticism over privacy, including default analytics settings that share some user data (which can be disabled). Its reliance on Infura (acquired by ConsenSys in 2019) has raised concerns about centralization in Ethereum infrastructure. The wallet regularly issues warnings about phishing scams and fake airdrops impersonating MetaMask.
Policy gradient method
Policy gradient methods are a class of reinforcement learning algorithms and a sub-class of policy optimization methods. Unlike value-based methods which learn a value function to derive a policy, policy optimization methods directly learn a policy function π {\displaystyle \pi } that selects actions without consulting a value function. For policy gradient to apply, the policy function π θ {\displaystyle \pi _{\theta }} is parameterized by a differentiable parameter θ {\displaystyle \theta } . == Overview == In policy-based RL, the actor is a parameterized policy function π θ {\displaystyle \pi _{\theta }} , where θ {\displaystyle \theta } are the parameters of the actor. The actor takes as argument the state of the environment s {\displaystyle s} and produces a probability distribution π θ ( ⋅ ∣ s ) {\displaystyle \pi _{\theta }(\cdot \mid s)} . If the action space is discrete, then ∑ a π θ ( a ∣ s ) = 1 {\displaystyle \sum _{a}\pi _{\theta }(a\mid s)=1} . If the action space is continuous, then ∫ a π θ ( a ∣ s ) d a = 1 {\displaystyle \int _{a}\pi _{\theta }(a\mid s)\mathrm {d} a=1} . The goal of policy optimization is to find some θ {\displaystyle \theta } that maximizes the expected episodic reward J ( θ ) {\displaystyle J(\theta )} : J ( θ ) = E π θ [ ∑ t = 0 T γ t R t | S 0 = s 0 ] {\displaystyle J(\theta )=\mathbb {E} _{\pi _{\theta }}\left[\sum _{t=0}^{T}\gamma ^{t}R_{t}{\Big |}S_{0}=s_{0}\right]} where γ {\displaystyle \gamma } is the discount factor, R t {\displaystyle R_{t}} is the reward at step t {\displaystyle t} , s 0 {\displaystyle s_{0}} is the starting state, and T {\displaystyle T} is the time-horizon (which can be infinite). The policy gradient is defined as ∇ θ J ( θ ) {\displaystyle \nabla _{\theta }J(\theta )} . Different policy gradient methods stochastically estimate the policy gradient in different ways. The goal of any policy gradient method is to iteratively maximize J ( θ ) {\displaystyle J(\theta )} by gradient ascent. Since the key part of any policy gradient method is the stochastic estimation of the policy gradient, they are also studied under the title of "Monte Carlo gradient estimation". == REINFORCE == === Policy gradient === The REINFORCE algorithm, introduced by Ronald J. Williams in 1992, was the first policy gradient method. It is based on the identity for the policy gradient ∇ θ J ( θ ) = E π θ [ ∑ t = 0 T ∇ θ ln π θ ( A t ∣ S t ) ∑ t = 0 T ( γ t R t ) | S 0 = s 0 ] {\displaystyle \nabla _{\theta }J(\theta )=\mathbb {E} _{\pi _{\theta }}\left[\sum _{t=0}^{T}\nabla _{\theta }\ln \pi _{\theta }(A_{t}\mid S_{t})\;\sum _{t=0}^{T}(\gamma ^{t}R_{t}){\Big |}S_{0}=s_{0}\right]} which can be improved via the "causality trick" ∇ θ J ( θ ) = E π θ [ ∑ t = 0 T ∇ θ ln π θ ( A t ∣ S t ) ∑ τ = t T ( γ τ R τ ) | S 0 = s 0 ] {\displaystyle \nabla _{\theta }J(\theta )=\mathbb {E} _{\pi _{\theta }}\left[\sum _{t=0}^{T}\nabla _{\theta }\ln \pi _{\theta }(A_{t}\mid S_{t})\sum _{\tau =t}^{T}(\gamma ^{\tau }R_{\tau }){\Big |}S_{0}=s_{0}\right]} Thus, we have an unbiased estimator of the policy gradient: ∇ θ J ( θ ) ≈ 1 N ∑ n = 1 N [ ∑ t = 0 T ∇ θ ln π θ ( A t , n ∣ S t , n ) ∑ τ = t T ( γ τ − t R τ , n ) ] {\displaystyle \nabla _{\theta }J(\theta )\approx {\frac {1}{N}}\sum _{n=1}^{N}\left[\sum _{t=0}^{T}\nabla _{\theta }\ln \pi _{\theta }(A_{t,n}\mid S_{t,n})\sum _{\tau =t}^{T}(\gamma ^{\tau -t}R_{\tau ,n})\right]} where the index n {\displaystyle n} ranges over N {\displaystyle N} rollout trajectories using the policy π θ {\displaystyle \pi _{\theta }} . The score function ∇ θ ln π θ ( A t ∣ S t ) {\displaystyle \nabla _{\theta }\ln \pi _{\theta }(A_{t}\mid S_{t})} can be interpreted as the direction in the parameter space that increases the probability of taking action A t {\displaystyle A_{t}} in state S t {\displaystyle S_{t}} . The policy gradient, then, is a weighted average of all possible directions to increase the probability of taking any action in any state, but weighted by reward signals, so that if taking a certain action in a certain state is associated with high reward, then that direction would be highly reinforced, and vice versa. === Algorithm === The REINFORCE algorithm is a loop: Rollout N {\displaystyle N} trajectories in the environment, using π θ t {\displaystyle \pi _{\theta _{t}}} as the policy function. Compute the policy gradient estimation: g i ← 1 N ∑ n = 1 N [ ∑ t = 0 T ∇ θ t ln π θ ( A t , n ∣ S t , n ) ∑ τ = t T ( γ τ R τ , n ) ] {\displaystyle g_{i}\leftarrow {\frac {1}{N}}\sum _{n=1}^{N}\left[\sum _{t=0}^{T}\nabla _{\theta _{t}}\ln \pi _{\theta }(A_{t,n}\mid S_{t,n})\sum _{\tau =t}^{T}(\gamma ^{\tau }R_{\tau ,n})\right]} Update the policy by gradient ascent: θ i + 1 ← θ i + α i g i {\displaystyle \theta _{i+1}\leftarrow \theta _{i}+\alpha _{i}g_{i}} Here, α i {\displaystyle \alpha _{i}} is the learning rate at update step i {\displaystyle i} . == Variance reduction == REINFORCE is an on-policy algorithm, meaning that the trajectories used for the update must be sampled from the current policy π θ {\displaystyle \pi _{\theta }} . This can lead to high variance in the updates, as the returns R ( τ ) {\displaystyle R(\tau )} can vary significantly between trajectories. Many variants of REINFORCE have been introduced, under the title of variance reduction. === REINFORCE with baseline === A common way for reducing variance is the REINFORCE with baseline algorithm, based on the following identity: ∇ θ J ( θ ) = E π θ [ ∑ t = 0 T ∇ θ ln π θ ( A t | S t ) ( ∑ τ = t T ( γ τ R τ ) − b ( S t ) ) | S 0 = s 0 ] {\displaystyle \nabla _{\theta }J(\theta )=\mathbb {E} _{\pi _{\theta }}\left[\sum _{t=0}^{T}\nabla _{\theta }\ln \pi _{\theta }(A_{t}|S_{t})\left(\sum _{\tau =t}^{T}(\gamma ^{\tau }R_{\tau })-b(S_{t})\right){\Big |}S_{0}=s_{0}\right]} for any function b : States → R {\displaystyle b:{\text{States}}\to \mathbb {R} } . This can be proven by applying the previous lemma. The algorithm uses the modified gradient estimator g i ← 1 N ∑ n = 1 N [ ∑ t = 0 T ∇ θ t ln π θ ( A t , n | S t , n ) ( ∑ τ = t T ( γ τ R τ , n ) − b i ( S t , n ) ) ] {\displaystyle g_{i}\leftarrow {\frac {1}{N}}\sum _{n=1}^{N}\left[\sum _{t=0}^{T}\nabla _{\theta _{t}}\ln \pi _{\theta }(A_{t,n}|S_{t,n})\left(\sum _{\tau =t}^{T}(\gamma ^{\tau }R_{\tau ,n})-b_{i}(S_{t,n})\right)\right]} and the original REINFORCE algorithm is the special case where b i ≡ 0 {\displaystyle b_{i}\equiv 0} . === Actor-critic methods === If b i {\textstyle b_{i}} is chosen well, such that b i ( S t ) ≈ ∑ τ = t T ( γ τ R τ ) = γ t V π θ i ( S t ) {\textstyle b_{i}(S_{t})\approx \sum _{\tau =t}^{T}(\gamma ^{\tau }R_{\tau })=\gamma ^{t}V^{\pi _{\theta _{i}}}(S_{t})} , this could significantly decrease variance in the gradient estimation. That is, the baseline should be as close to the value function V π θ i ( S t ) {\displaystyle V^{\pi _{\theta _{i}}}(S_{t})} as possible, approaching the ideal of: ∇ θ J ( θ ) = E π θ [ ∑ t = 0 T ∇ θ ln π θ ( A t | S t ) ( ∑ τ = t T ( γ τ R τ ) − γ t V π θ ( S t ) ) | S 0 = s 0 ] {\displaystyle \nabla _{\theta }J(\theta )=\mathbb {E} _{\pi _{\theta }}\left[\sum _{t=0}^{T}\nabla _{\theta }\ln \pi _{\theta }(A_{t}|S_{t})\left(\sum _{\tau =t}^{T}(\gamma ^{\tau }R_{\tau })-\gamma ^{t}V^{\pi _{\theta }}(S_{t})\right){\Big |}S_{0}=s_{0}\right]} Note that, as the policy π θ t {\displaystyle \pi _{\theta _{t}}} updates, the value function V π θ i ( S t ) {\displaystyle V^{\pi _{\theta _{i}}}(S_{t})} updates as well, so the baseline should also be updated. One common approach is to train a separate function that estimates the value function, and use that as the baseline. This is one of the actor-critic methods, where the policy function is the actor and the value function is the critic. The Q-function Q π {\displaystyle Q^{\pi }} can also be used as the critic, since ∇ θ J ( θ ) = E π θ [ ∑ 0 ≤ t ≤ T γ t ∇ θ ln π θ ( A t | S t ) ⋅ Q π θ ( S t , A t ) | S 0 = s 0 ] {\displaystyle \nabla _{\theta }J(\theta )=E_{\pi _{\theta }}\left[\sum _{0\leq t\leq T}\gamma ^{t}\nabla _{\theta }\ln \pi _{\theta }(A_{t}|S_{t})\cdot Q^{\pi _{\theta }}(S_{t},A_{t}){\Big |}S_{0}=s_{0}\right]} by a similar argument using the tower law. Subtracting the value function as a baseline, we find that the advantage function A π ( S , A ) = Q π ( S , A ) − V π ( S ) {\displaystyle A^{\pi }(S,A)=Q^{\pi }(S,A)-V^{\pi }(S)} can be used as the critic as well: ∇ θ J ( θ ) = E π θ [ ∑ 0 ≤ t ≤ T γ t ∇ θ ln π θ ( A t | S t ) ⋅ A π θ ( S t , A t ) | S 0 = s 0 ] {\displaystyle \nabla _{\theta }J(\theta )=E_{\pi _{\theta }}\left[\sum _{0\leq t\leq T}\gamma ^{t}\nabla _{\theta }\ln \pi _{\theta }(A_{t}|S_{t})\cdot A^{\pi _{\theta }}(S_{t},A_{t}){\Big |}S_{0}=s_{0}\right]} In summary, there are many unbiased estimators for ∇ θ J θ {\textstyle \nabla _{\theta }J_{\theta }} , all in the form of: ∇ θ J ( θ ) = E π θ [ ∑ 0 ≤ t ≤ T ∇ θ ln π θ ( A t | S t ) ⋅ Ψ t | S 0 = s 0 ] {\displaystyle \nabla _{\theta }J(\theta )=E_{\pi _{\theta }}\left[\su