The exploration–exploitation dilemma, also known as the explore–exploit tradeoff, is a fundamental concept in decision-making that arises in many domains. It is depicted as the balancing act between two opposing strategies. Exploitation involves choosing the best option based on current knowledge of the system (which may be incomplete or misleading), while exploration involves trying out new options that may lead to better outcomes in the future at the expense of an exploitation opportunity. Finding the optimal balance between these two strategies is a crucial challenge in many decision-making problems whose goal is to maximize long-term benefits. == Application in machine learning == In the context of machine learning, the exploration–exploitation tradeoff is fundamental in reinforcement learning (RL), a type of machine learning that involves training agents to make decisions based on feedback from the environment. Crucially, this feedback may be incomplete or delayed. The agent must decide whether to exploit the current best-known policy or explore new policies to improve its performance. === Multi-armed bandit methods === The multi-armed bandit (MAB) problem was a classic example of the tradeoff, and many methods were developed for it, such as epsilon-greedy, Thompson sampling, and the upper confidence bound (UCB). See the page on MAB for details. In more complex RL situations than the MAB problem, the agent can treat each choice as a MAB, where the payoff is the expected future reward. For example, if the agent performs an epsilon-greedy method, then the agent will often "pull the best lever" by picking the action that had the best predicted expected reward (exploit). However, it would pick a random action with probability epsilon (explore). Monte Carlo tree search, for example, uses a variant of the UCB method. === Exploration problems === There are some problems that make exploration difficult. Sparse reward. If rewards occur only once a long while, then the agent might not persist in exploring. Furthermore, if the space of actions is large, then the sparse reward would mean the agent would not be guided by the reward to find a good direction for deeper exploration. A standard example is Montezuma's Revenge. Deceptive reward. If some early actions give immediate small reward, but other actions give later large reward, then the agent might be lured away from exploring the other actions. Noisy TV problem. If certain observations are irreducibly noisy (such as a television showing random images), then the agent might be trapped exploring those observations (watching the television). === Exploration reward === This section based on. The exploration reward (also called exploration bonus) methods convert the exploration-exploitation dilemma into a balance of exploitations. That is, instead of trying to get the agent to balance exploration and exploitation, exploration is simply treated as another form of exploitation, and the agent simply attempts to maximize the sum of rewards from exploration and exploitation. The exploration reward can be treated as a form of intrinsic reward. We write these as r t i , r t e {\displaystyle r_{t}^{i},r_{t}^{e}} , meaning the intrinsic and extrinsic rewards at time step t {\displaystyle t} . However, exploration reward is different from exploitation in two regards: The reward of exploitation is not freely chosen, but given by the environment, but the reward of exploration may be picked freely. Indeed, there are many different ways to design r t i {\displaystyle r_{t}^{i}} described below. The reward of exploitation is usually stationary (i.e. the same action in the same state gives the same reward), but the reward of exploration is non-stationary (i.e. the same action in the same state should give less and less reward). Count-based exploration uses N n ( s ) {\displaystyle N_{n}(s)} , the number of visits to a state s {\displaystyle s} during the time-steps 1 : n {\displaystyle 1:n} , to calculate the exploration reward. This is only possible in small and discrete state space. Density-based exploration extends count-based exploration by using a density model ρ n ( s ) {\displaystyle \rho _{n}(s)} . The idea is that, if a state has been visited, then nearby states are also partly-visited. In maximum entropy exploration, the entropy of the agent's policy π {\displaystyle \pi } is included as a term in the intrinsic reward. That is, r t i = − ∑ a π ( a | s t ) ln π ( a | s t ) + ⋯ {\displaystyle r_{t}^{i}=-\sum _{a}\pi (a|s_{t})\ln \pi (a|s_{t})+\cdots } . === Prediction-based === This section based on. The forward dynamics model is a function for predicting the next state based on the current state and the current action: f : ( s t , a t ) ↦ s t + 1 {\displaystyle f:(s_{t},a_{t})\mapsto s_{t+1}} . The forward dynamics model is trained as the agent plays. The model becomes better at predicting state transition for state-action pairs that had been done many times. A forward dynamics model can define an exploration reward by r t i = ‖ f ( s t , a t ) − s t + 1 ‖ 2 2 {\displaystyle r_{t}^{i}=\|f(s_{t},a_{t})-s_{t+1}\|_{2}^{2}} . That is, the reward is the squared-error of the prediction compared to reality. This rewards the agent to perform state-action pairs that had not been done many times. This is however susceptible to the noisy TV problem. Dynamics model can be run in latent space. That is, r t i = ‖ f ( s t , a t ) − ϕ ( s t + 1 ) ‖ 2 2 {\displaystyle r_{t}^{i}=\|f(s_{t},a_{t})-\phi (s_{t+1})\|_{2}^{2}} for some featurizer ϕ {\displaystyle \phi } . The featurizer can be the identity function (i.e. ϕ ( x ) = x {\displaystyle \phi (x)=x} ), randomly generated, the encoder-half of a variational autoencoder, etc. A good featurizer improves forward dynamics exploration. The Intrinsic Curiosity Module (ICM) method trains simultaneously a forward dynamics model and a featurizer. The featurizer is trained by an inverse dynamics model, which is a function for predicting the current action based on the features of the current and the next state: g : ( ϕ ( s t ) , ϕ ( s t + 1 ) ) ↦ a t {\displaystyle g:(\phi (s_{t}),\phi (s_{t+1}))\mapsto a_{t}} . By optimizing the inverse dynamics, both the inverse dynamics model and the featurizer are improved. Then, the improved featurizer improves the forward dynamics model, which improves the exploration of the agent. Random Network Distillation (RND) method attempts to solve this problem by teacher–student distillation. Instead of a forward dynamics model, it has two models f , f ′ {\displaystyle f,f'} . The f ′ {\displaystyle f'} teacher model is fixed, and the f {\displaystyle f} student model is trained to minimize ‖ f ( s ) − f ′ ( s ) ‖ 2 2 {\displaystyle \|f(s)-f'(s)\|_{2}^{2}} on states s {\displaystyle s} . As a state is visited more and more, the student network becomes better at predicting the teacher. Meanwhile, the prediction error is also an exploration reward for the agent, and so the agent learns to perform actions that result in higher prediction error. Thus, we have a student network attempting to minimize the prediction error, while the agent attempting to maximize it, resulting in exploration. The states are normalized by subtracting a running average and dividing a running variance, which is necessary since the teacher model is frozen. The rewards are normalized by dividing with a running variance. Exploration by disagreement trains an ensemble of forward dynamics models, each on a random subset of all ( s t , a t , s t + 1 ) {\displaystyle (s_{t},a_{t},s_{t+1})} tuples. The exploration reward is the variance of the models' predictions. === Noise === For neural network–based agents, the NoisyNet method changes some of its neural network modules by noisy versions. That is, some network parameters are random variables from a probability distribution. The parameters of the distribution are themselves learnable. For example, in a linear layer y = W x + b {\displaystyle y=Wx+b} , both W , b {\displaystyle W,b} are sampled from Gaussian distributions N ( μ W , Σ W ) , N ( μ b , Σ b ) {\displaystyle {\mathcal {N}}(\mu _{W},\Sigma _{W}),{\mathcal {N}}(\mu _{b},\Sigma _{b})} at every step, and the parameters μ W , Σ W , μ b , Σ b {\displaystyle \mu _{W},\Sigma _{W},\mu _{b},\Sigma _{b}} are learned via the reparameterization trick.
JAX (software)
JAX is a Python library for accelerator-oriented array computation and program transformation, designed for high-performance numerical computing and large-scale machine learning. It is developed by Google with contributions from Nvidia and other community contributors. It is described as bringing together a modified version of the automatic differentiation system autograd and OpenXLA's XLA (Accelerated Linear Algebra). It is designed to follow the structure and workflow of NumPy as closely as possible and works with various existing frameworks such as TensorFlow and PyTorch. The primary features of JAX are: Providing a unified NumPy-like interface to computations that run on CPU, GPU, or TPU, in local or distributed settings. Built-in Just-In-Time (JIT) compilation via OpenXLA, an open-source machine learning compiler ecosystem. Efficient evaluation of gradients via its automatic differentiation transformations. Automatic vectorization to efficiently map functions over arrays representing batches of inputs. == Libraries using Jax == Flax Equinox Optax
Theano (software)
Theano is a Python library and optimizing compiler for manipulating and evaluating mathematical expressions, especially matrix-valued ones. In Theano, computations are expressed using a NumPy-esque syntax and compiled to run efficiently on either CPU or GPU architectures. == History == Theano is an open source project primarily developed by the Montreal Institute for Learning Algorithms (MILA) at the Université de Montréal. The name of the software references the ancient philosopher Theano, long associated with the development of the golden mean. On 28 September 2017, Pascal Lamblin posted a message from Yoshua Bengio, Head of MILA: major development would cease after the 1.0 release due to competing offerings by strong industrial players. Theano 1.0.0 was then released on 15 November 2017. On 17 May 2018, Chris Fonnesbeck wrote on behalf of the PyMC development team that the PyMC developers will officially assume control of Theano maintenance once the MILA development team steps down. On 29 January 2021, they started using the name Aesara for their fork of Theano. On 29 Nov 2022, the PyMC development team announced that the PyMC developers will fork the Aesara project under the name PyTensor. == Sample code == The following code is the original Theano's example. It defines a computational graph with 2 scalars a and b of type double and an operation between them (addition) and then creates a Python function f that does the actual computation. == Examples == === Matrix Multiplication (Dot Product) === The following code demonstrates how to perform matrix multiplication using Theano, which is essential for linear algebra operations in many machine learning tasks. === Gradient Calculation === The following code uses Theano to compute the gradient of a simple operation (like a neuron) with respect to its input. This is useful in training machine learning models (backpropagation). === Building a Simple Neural Network === The following code shows how to start building a simple neural network. This is a very basic neural network with one hidden layer. === Broadcasting in Theano === The following code demonstrates how broadcasting works in Theano. Broadcasting allows operations between arrays of different shapes without needing to explicitly reshape them.
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SNNS
SNNS (Stuttgart Neural Network Simulator) is a neural network simulator originally developed at the University of Stuttgart. While it was originally built for X11 under Unix, there are Windows ports. Its successor JavaNNS never reached the same popularity. == Features == SNNS is written around a simulation kernel to which user written activation functions, learning procedures and output functions can be added. It has support for arbitrary network topologies and the standard release contains support for a number of standard neural network architectures and training algorithms. == Status == There is currently no ongoing active development of SNNS. In July 2008 the license was changed to the GNU LGPL.
Neural operators
Neural operators are a class of deep learning architectures designed to learn maps between infinite-dimensional function spaces. Neural operators represent an extension of traditional artificial neural networks, marking a departure from the typical focus on learning mappings between finite-dimensional Euclidean spaces or finite sets. Neural operators directly learn operators between function spaces; they can receive input functions, and the output function can be evaluated at any discretization. The primary application of neural operators is in learning surrogate maps for the solution operators of partial differential equations (PDEs), which are critical tools in modeling the natural environment. Standard PDE solvers can be time-consuming and computationally intensive, especially for complex systems. Neural operators have demonstrated improved performance in solving PDEs compared to existing machine learning methodologies while being significantly faster than numerical solvers. Neural operators have also been applied to various scientific and engineering disciplines such as turbulent flow modeling, computational mechanics, graph-structured data, and the geosciences. In particular, they have been applied to learning stress-strain fields in materials, classifying complex data like spatial transcriptomics, predicting multiphase flow in porous media, and carbon dioxide migration simulations. Finally, the operator learning paradigm allows learning maps between function spaces, and is different from parallel ideas of learning maps from finite-dimensional spaces to function spaces, and subsumes these settings as special cases when limited to a fixed input resolution. == Operator learning == Understanding and mapping relationships between function spaces has many applications in engineering and the sciences. In particular, one can cast the problem of solving partial differential equations as identifying a map between function spaces, such as from an initial condition to a time-evolved state. In other PDEs this map takes an input coefficient function and outputs a solution function. Operator learning is a machine learning paradigm to learn solution operators mapping the input function to the output function . Using traditional machine learning methods, addressing this problem would involve discretizing the infinite-dimensional input and output function spaces into finite-dimensional grids and applying standard learning models, such as neural networks. This approach reduces the operator learning to finite-dimensional function learning and has some limitations, such as generalizing to discretizations beyond the grid used in training. The primary properties of neural operators that differentiate them from traditional neural networks is discretization invariance and discretization convergence. Unlike conventional neural networks, which are fixed on the discretization of training data, neural operators can adapt to various discretizations without re-training. This property improves the robustness and applicability of neural operators in different scenarios, providing consistent performance across different resolutions and grids. == Definition and formulation == Architecturally, neural operators are similar to feed-forward neural networks in the sense that they are composed of alternating linear maps and non-linearities. Since neural operators act on and output functions, neural operators have been instead formulated as a sequence of alternating linear integral operators on function spaces and point-wise non-linearities. Using an analogous architecture to finite-dimensional neural networks, similar universal approximation theorems have been proven for neural operators. In particular, it has been shown that neural operators can approximate any continuous operator on a compact set. Neural operators seek to approximate some operator G : A → U {\displaystyle {\mathcal {G}}:{\mathcal {A}}\to {\mathcal {U}}} between function spaces A {\displaystyle {\mathcal {A}}} and U {\displaystyle {\mathcal {U}}} by building a parametric map G ϕ : A → U {\displaystyle {\mathcal {G}}_{\phi }:{\mathcal {A}}\to {\mathcal {U}}} . Such parametric maps G ϕ {\displaystyle {\mathcal {G}}_{\phi }} can generally be defined in the form G ϕ := Q ∘ σ ( W T + K T + b T ) ∘ ⋯ ∘ σ ( W 1 + K 1 + b 1 ) ∘ P , {\displaystyle {\mathcal {G}}_{\phi }:={\mathcal {Q}}\circ \sigma (W_{T}+{\mathcal {K}}_{T}+b_{T})\circ \cdots \circ \sigma (W_{1}+{\mathcal {K}}_{1}+b_{1})\circ {\mathcal {P}},} where P , Q {\displaystyle {\mathcal {P}},{\mathcal {Q}}} are the lifting (lifting the codomain of the input function to a higher dimensional space) and projection (projecting the codomain of the intermediate function to the output dimension) operators, respectively. These operators act pointwise on functions and are typically parametrized as multilayer perceptrons. σ {\displaystyle \sigma } is a pointwise nonlinearity, such as a rectified linear unit (ReLU), or a Gaussian error linear unit (GeLU). Each layer t = 1 , … , T {\displaystyle t=1,\dots ,T} has a respective local operator W t {\displaystyle W_{t}} (usually parameterized by a pointwise neural network), a kernel integral operator K t {\displaystyle {\mathcal {K}}_{t}} , and a bias function b t {\displaystyle b_{t}} . Given some intermediate functional representation v t {\displaystyle v_{t}} with domain D {\displaystyle D} in the t {\displaystyle t} -th hidden layer, a kernel integral operator K ϕ {\displaystyle {\mathcal {K}}_{\phi }} is defined as ( K ϕ v t ) ( x ) := ∫ D κ ϕ ( x , y , v t ( x ) , v t ( y ) ) v t ( y ) d y , {\displaystyle ({\mathcal {K}}_{\phi }v_{t})(x):=\int _{D}\kappa _{\phi }(x,y,v_{t}(x),v_{t}(y))v_{t}(y)dy,} where the kernel κ ϕ {\displaystyle \kappa _{\phi }} is a learnable implicit neural network, parametrized by ϕ {\displaystyle \phi } . In practice, one is often given the input function to the neural operator at a specific resolution. For instance, consider the setting where one is given the evaluation of v t {\displaystyle v_{t}} at n {\displaystyle n} points { y j } j n {\displaystyle \{y_{j}\}_{j}^{n}} . Borrowing from Nyström integral approximation methods such as Riemann sum integration and Gaussian quadrature, the above integral operation can be computed as follows: ∫ D κ ϕ ( x , y , v t ( x ) , v t ( y ) ) v t ( y ) d y ≈ ∑ j n κ ϕ ( x , y j , v t ( x ) , v t ( y j ) ) v t ( y j ) Δ y j , {\displaystyle \int _{D}\kappa _{\phi }(x,y,v_{t}(x),v_{t}(y))v_{t}(y)dy\approx \sum _{j}^{n}\kappa _{\phi }(x,y_{j},v_{t}(x),v_{t}(y_{j}))v_{t}(y_{j})\Delta _{y_{j}},} where Δ y j {\displaystyle \Delta _{y_{j}}} is the sub-area volume or quadrature weight associated to the point y j {\displaystyle y_{j}} . Thus, a simplified layer can be computed as v t + 1 ( x ) ≈ σ ( ∑ j n κ ϕ ( x , y j , v t ( x ) , v t ( y j ) ) v t ( y j ) Δ y j + W t ( v t ( y j ) ) + b t ( x ) ) . {\displaystyle v_{t+1}(x)\approx \sigma \left(\sum _{j}^{n}\kappa _{\phi }(x,y_{j},v_{t}(x),v_{t}(y_{j}))v_{t}(y_{j})\Delta _{y_{j}}+W_{t}(v_{t}(y_{j}))+b_{t}(x)\right).} The above approximation, along with parametrizing κ ϕ {\displaystyle \kappa _{\phi }} as an implicit neural network, results in the graph neural operator (GNO). There have been various parameterizations of neural operators for different applications. These typically differ in their parameterization of κ {\displaystyle \kappa } . The most popular instantiation is the Fourier neural operator (FNO). FNO takes κ ϕ ( x , y , v t ( x ) , v t ( y ) ) := κ ϕ ( x − y ) {\displaystyle \kappa _{\phi }(x,y,v_{t}(x),v_{t}(y)):=\kappa _{\phi }(x-y)} and by applying the convolution theorem, arrives at the following parameterization of the kernel integral operator: ( K ϕ v t ) ( x ) = F − 1 ( R ϕ ⋅ ( F v t ) ) ( x ) , {\displaystyle ({\mathcal {K}}_{\phi }v_{t})(x)={\mathcal {F}}^{-1}(R_{\phi }\cdot ({\mathcal {F}}v_{t}))(x),} where F {\displaystyle {\mathcal {F}}} represents the Fourier transform and R ϕ {\displaystyle R_{\phi }} represents the Fourier transform of some periodic function κ ϕ {\displaystyle \kappa _{\phi }} . That is, FNO parameterizes the kernel integration directly in Fourier space, using a prescribed number of Fourier modes. When the grid at which the input function is presented is uniform, the Fourier transform can be approximated using the discrete Fourier transform (DFT) with frequencies below some specified threshold. The discrete Fourier transform can be computed using a fast Fourier transform (FFT) implementation. == Training == Training neural operators is similar to the training process for a traditional neural network. Neural operators are typically trained in some Lp norm or Sobolev norm. In particular, for a dataset { ( a i , u i ) } i = 1 N {\displaystyle \{(a_{i},u_{i})\}_{i=1}^{N}} of size N {\displaystyle N} , neural operators minimize (a discretization of) L U ( { ( a i , u i ) } i = 1 N ) := ∑ i = 1 N ‖ u i − G θ ( a i ) ‖ U 2 {\displaystyle {\mathcal {L}}_{\mathca
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