A mobile content management system (MCMs) is a type of content management system (CMS) capable of storing and delivering content and services to mobile devices, such as mobile phones, smart phones, and PDAs. Mobile content management systems may be discrete systems, or may exist as features, modules or add-ons of larger content management systems capable of multi-channel content delivery. Mobile content delivery has unique, specific constraints including widely variable device capacities, small screen size, limitations on wireless bandwidth, sometimes small storage capacity, and (for some devices) comparatively weak device processors. Demand for mobile content management increased as mobile devices became increasingly ubiquitous and sophisticated. MCMS technology initially focused on the business to consumer (B2C) mobile market place with ringtones, games, text-messaging, news, and other related content. Since, mobile content management systems have also taken root in business-to-business (B2B) and business-to-employee (B2E) situations, allowing companies to provide more timely information and functionality to business partners and mobile workforces in an increasingly efficient manner. A 2008 estimate put global revenue for mobile content management at US$8 billion. == Key features == === Multi-channel content delivery === Multi-channel content delivery capabilities allow users not to manage a central content repository while simultaneously delivering that content to mobile devices such as mobile phones, smartphones, tablets and other mobile devices. Content can be stored in a raw format (such as Microsoft Word, Excel, PowerPoint, PDF, Text, HTML etc.) to which device-specific presentation styles can be applied. === Content access control === Access control includes authorization, authentication, access approval to each content. In many cases the access control also includes download control, wipe-out for specific user, time specific access. For the authentication, MCM shall have basic authentication which has user ID and password. For higher security many MCM supports IP authentication and mobile device authentication. === Specialized templating system === While traditional web content management systems handle templates for only a handful of web browsers, mobile CMS templates must be adapted to the very wide range of target devices with different capacities and limitations. There are two approaches to adapting templates: multi-client and multi-site. The multi-client approach makes it possible to see all versions of a site at the same domain (e.g. sitename.com), and templates are presented based on the device client used for viewing. The multi-site approach displays the mobile site on a targeted sub-domain (e.g. mobile.sitename.com). === Location-based content delivery === Location-based content delivery provides targeted content, such as information, advertisements, maps, directions, and news, to mobile devices based on current physical location. Currently, GPS (global positioning system) navigation systems offer the most popular location-based services. Navigation systems are specialized systems, but incorporating mobile phone functionality makes greater exploitation of location-aware content delivery possible.
Photometric stereo
Photometric stereo is a technique in computer vision for estimating the surface normals of objects by observing that object under different lighting conditions (photometry). It is based on the fact that the amount of light reflected by a surface is dependent on the orientation of the surface in relation to the light source and the observer. By measuring the amount of light reflected into a camera, the space of possible surface orientations is limited. Given enough light sources from different angles, the surface orientation may be constrained to a single orientation or even overconstrained. The technique was originally introduced by Woodham in 1980. The special case where the data is a single image is known as shape from shading, and was analyzed by B. K. P. Horn in 1989. Photometric stereo has since been generalized to many other situations, including extended light sources and non-Lambertian surface finishes. Current research aims to make the method work in the presence of projected shadows, highlights, and non-uniform lighting. Photometric stereo is widely used in various fields, including archaeology, cultural heritage conservation, and quality control. It is now integrated into widely used open-source software, such as Meshroom. == Basic method == Under Woodham's original assumptions — Lambertian reflectance, known point-like distant light sources, and uniform albedo — the problem can be solved by inverting the linear equation I = L ⋅ n {\displaystyle I=L\cdot n} , where I {\displaystyle I} is a (known) vector of m {\displaystyle m} observed intensities, n {\displaystyle n} is the (unknown) surface normal, and L {\displaystyle L} is a (known) 3 × m {\displaystyle 3\times m} matrix of normalized light directions. This model can easily be extended to surfaces with non-uniform albedo, while keeping the problem linear. Taking an albedo reflectivity of k {\displaystyle k} , the formula for the reflected light intensity becomes I = k ( L ⋅ n ) . {\displaystyle I=k(L\cdot n).} If L {\displaystyle L} is square (there are exactly 3 lights) and non-singular, it can be inverted, giving L − 1 I = k n . {\displaystyle L^{-1}I=kn.} Since the normal vector is known to have length 1, k {\displaystyle k} must be the length of the vector k n {\displaystyle kn} , and n {\displaystyle n} is the normalised direction of that vector. If L {\displaystyle L} is not square (there are more than 3 lights), a generalisation of the inverse can be obtained using the Moore–Penrose pseudoinverse, by simply multiplying both sides with L T {\displaystyle L^{T}} , giving L T I = L T k ( L ⋅ n ) , {\displaystyle L^{T}I=L^{T}k(L\cdot n),} ( L T L ) − 1 L T I = k n , {\displaystyle (L^{T}L)^{-1}L^{T}I=kn,} after which the normal vector and albedo can be solved as described above. == Non-Lambertian surfaces == The classical photometric stereo problem concerns itself only with Lambertian surfaces, with perfectly diffuse reflection. This is unrealistic for many types of materials, especially metals, glass and smooth plastics, and will lead to aberrations in the resulting normal vectors. Many methods have been developed to lift this assumption. In this section, a few of these are listed. === Specular reflections === Historically, in computer graphics, the commonly used model to render surfaces started with Lambertian surfaces and progressed first to include simple specular reflections. Computer vision followed a similar course with photometric stereo. Specular reflections were among the first deviations from the Lambertian model. These are a few adaptations that have been developed. Many techniques ultimately rely on modelling the reflectance function of the surface, that is, how much light is reflected in each direction. This reflectance function has to be invertible. The reflected light intensities towards the camera is measured, and the inverse reflectance function is fit onto the measured intensities, resulting in a unique solution for the normal vector. === General BRDFs and beyond === According to the Bidirectional reflectance distribution function (BRDF) model, a surface may distribute the amount of light it receives in any outward direction. This is the most general known model for opaque surfaces. Some techniques have been developed to model (almost) general BRDFs. In practice, all of these require many light sources to obtain reliable data. These are methods in which surfaces with general BRDFs can be measured. Determine the explicit BRDF prior to scanning. To do this, a different surface is required that has the same or a very similar BRDF, of which the actual geometry (or at least the normal vectors for many points on the surface) is already known. The lights are then individually shone upon the known surface, and the amount of reflection into the camera is measured. Using this information, a look-up table can be created that maps reflected intensities for each light source to a list of possible normal vectors. This puts constraints on the possible normal vectors the surface may have, and reduces the photometric stereo problem to an interpolation between measurements. Typical known surfaces to calibrate the look-up table with are spheres for their wide variety of surface orientations. Restricting the BRDF to be symmetrical. If the BRDF is symmetrical, the direction of the light can be restricted to a cone about the direction to the camera. Which cone this is depends on the BRDF itself, the normal vector of the surface, and the measured intensity. Given enough measured intensities and the resulting light directions, these cones can be approximated and therefore the normal vectors of the surface. Some progress has been made towards modelling an even more general surfaces, such as Spatially Varying Bidirectional Distribution Functions (SVBRDF), Bidirectional surface scattering reflectance distribution functions (BSSRDF), and accounting for interreflections. However, such methods are still fairly restrictive in photometric stereo. Better results have been achieved with structured light. == Uncalibrated photometric stereo == Uncalibrated Photometric Stereo is an approach in photometric stereo that aims to reconstruct the 3D shape of an object from images captured under unknown lighting conditions. Unlike classical methods, which often assume controlled or known lighting setups, this approach removes these constraints, making it adaptable to diverse and real-world environments. The advent of deep learning has revolutionized universal PS by replacing handcrafted assumptions with data-driven models. Recent approaches leverage Transformer-based architectures and multi-scale encoder–decoder networks to directly estimate surface normals from input images. Uncalibrated Photometric Stereo is inherently an ill-posed problem, as it attempts to recover 3D shape and lighting conditions simultaneously from images alone. This leads to fundamental ambiguities in the reconstruction process, which manifest as systematic errors in the recovered geometry, including global distortions in the object's overall shape, and misinterpretation of surface orientation, where concave regions may appear convex and vice versa. To address the challenges of uncalibrated photometric stereo, hybrid methods have emerged that combine multi-view stereo and photometric stereo. These approaches leverage the strengths of both techniques, including geometric reliability and resolution.
Curriculum learning
Curriculum learning is a technique in machine learning in which a model is trained on examples of increasing difficulty, where the definition of "difficulty" may be provided externally or discovered as part of the training process. This is intended to attain good performance more quickly, or to converge to a better local optimum if the global optimum is not found. == Approach == Most generally, curriculum learning is the technique of successively increasing the difficulty of examples in the training set that is presented to a model over multiple training iterations. This can produce better results than exposing the model to the full training set immediately under some circumstances; most typically, when the model is able to learn general principles from easier examples, and then gradually incorporate more complex and nuanced information as harder examples are introduced, such as edge cases. This has been shown to work in many domains, most likely as a form of regularization. There are several major variations in how the technique is applied: A concept of "difficulty" must be defined. This may come from human annotation or an external heuristic; for example in language modeling, shorter sentences might be classified as easier than longer ones. Another approach is to use the performance of another model, with examples accurately predicted by that model being classified as easier (providing a connection to boosting). Difficulty can be increased steadily or in distinct epochs, and in a deterministic schedule or according to a probability distribution. This may also be moderated by a requirement for diversity at each stage, in cases where easier examples are likely to be disproportionately similar to each other. Applications must also decide the schedule for increasing the difficulty. Simple approaches may use a fixed schedule, such as training on easy examples for half of the available iterations and then all examples for the second half. Other approaches use self-paced learning to increase the difficulty in proportion to the performance of the model on the current set. Since curriculum learning only concerns the selection and ordering of training data, it can be combined with many other techniques in machine learning. The success of the method assumes that a model trained for an easier version of the problem can generalize to harder versions, so it can be seen as a form of transfer learning. Some authors also consider curriculum learning to include other forms of progressively increasing complexity, such as increasing the number of model parameters. It is frequently combined with reinforcement learning, such as learning a simplified version of a game first. Some domains have shown success with anti-curriculum learning: training on the most difficult examples first. One example is the ACCAN method for speech recognition, which trains on the examples with the lowest signal-to-noise ratio first. == History == The term "curriculum learning" was introduced by Yoshua Bengio et al in 2009, with reference to the psychological technique of shaping in animals and structured education for humans: beginning with the simplest concepts and then building on them. The authors also note that the application of this technique in machine learning has its roots in the early study of neural networks such as Jeffrey Elman's 1993 paper Learning and development in neural networks: the importance of starting small. Bengio et al showed good results for problems in image classification, such as identifying geometric shapes with progressively more complex forms, and language modeling, such as training with a gradually expanding vocabulary. They conclude that, for curriculum strategies, "their beneficial effect is most pronounced on the test set", suggesting good generalization. The technique has since been applied to many other domains: Natural language processing: Part-of-speech tagging Intent detection Sentiment analysis Machine translation Speech recognition Language model pre-training Image recognition: Facial recognition Object detection Reinforcement learning: Game-playing Graph learning Matrix factorization
Distributional Soft Actor Critic
Distributional Soft Actor Critic (DSAC) is a suite of model-free off-policy reinforcement learning algorithms, tailored for learning decision-making or control policies in complex systems with continuous action spaces. Distinct from traditional methods that focus solely on expected returns, DSAC algorithms are designed to learn a Gaussian distribution over stochastic returns, called value distribution. This focus on Gaussian value distribution learning notably diminishes value overestimations, which in turn boosts policy performance. Additionally, the value distribution learned by DSAC can also be used for risk-aware policy learning. From a technical standpoint, DSAC is essentially a distributional adaptation of the well-established soft actor-critic (SAC) method. To date, the DSAC family comprises two iterations: the original DSAC-v1 and its successor, DSAC-T (also known as DSAC-v2), with the latter demonstrating superior capabilities over the Soft Actor-Critic (SAC) in Mujoco benchmark tasks. The source code for DSAC-T can be found at the following URL: Jingliang-Duan/DSAC-T. Both iterations have been integrated into an advanced, Pytorch-powered reinforcement learning toolkit named GOPS: GOPS (General Optimal control Problem Solver).
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
1.58-bit large language model
A 1.58-bit large language model (also known as a ternary LLM) is a type of large language model (LLM) designed to be computationally efficient. It achieves this by using weights that are restricted to only three values: -1, 0, and +1. This restriction significantly reduces the model's memory footprint and allows for faster processing, as computationally expensive multiplication operations can be replaced with lower-cost additions. This contrasts with traditional models that use 16-bit floating-point numbers (FP16 or BF16) for their weights. Studies have shown that for models up to several billion parameters, the performance of 1.58-bit LLMs on various tasks is comparable to their full-precision counterparts. This approach could enable powerful AI to run on less specialized and lower-power hardware. The name "1.58-bit" comes from the fact that a system with three states contains log 2 3 ≈ 1.58 {\displaystyle \log _{2}3\approx 1.58} bits of information. These models are sometimes also referred to as 1-bit LLMs in research papers, although this term can also refer to true binary models (with weights of -1 and +1). == BitNet == In 2024, Ma et al., researchers at Microsoft, declared that their 1.58-bit model, BitNet b1.58 is comparable in performance to the 16-bit Llama 2 and opens the era of 1-bit LLM. BitNet creators did not use the post-training quantization of weights but instead relied on the new BitLinear transform that replaced the nn.Linear layer of the traditional transformer design. In 2025, Microsoft researchers had released an open-weights and open inference code model BitNet b1.58 2B4T demonstrating performance competitive with the full precision models at 2B parameters and 4T training tokens. == Post-training quantization == BitNet derives its performance from being trained natively in 1.58 bit instead of being quantized from a full-precision model after training. Still, training is an expensive process and it would be desirable to be able to somehow convert an existing model to 1.58 bits. In 2024, HuggingFace reported a way to gradually ramp up the 1.58-bit quantization in fine-tuning an existing model down to 1.58 bits. == Critique == Some researchers point out that the scaling laws of large language models favor the low-bit weights only in case of undertrained models. As the number of training tokens increases, the deficiencies of low-bit quantization surface.
Natarajan dimension
In the theory of Probably Approximately Correct Machine Learning, the Natarajan dimension characterizes the complexity of learning a set of functions, generalizing from the Vapnik–Chervonenkis dimension for boolean functions to multi-class functions. Originally introduced as the Generalized Dimension by Natarajan, it was subsequently renamed the Natarajan Dimension by Haussler and Long. == Definition == Let H {\displaystyle H} be a set of functions from a set X {\displaystyle X} to a set Y {\displaystyle Y} . H {\displaystyle H} shatters a set C ⊂ X {\displaystyle C\subset X} if there exist two functions f 0 , f 1 ∈ H {\displaystyle f_{0},f_{1}\in H} such that For every x ∈ C , f 0 ( x ) ≠ f 1 ( x ) {\displaystyle x\in C,f_{0}(x)\neq f_{1}(x)} . For every B ⊂ C {\displaystyle B\subset C} , there exists a function h ∈ H {\displaystyle h\in H} such that for all x ∈ B , h ( x ) = f 0 ( x ) {\displaystyle x\in B,h(x)=f_{0}(x)} and for all x ∈ C − B , h ( x ) = f 1 ( x ) {\displaystyle x\in C-B,h(x)=f_{1}(x)} . The Natarajan dimension of H is the maximal cardinality of a set shattered by H {\displaystyle H} . It is easy to see that if | Y | = 2 {\displaystyle |Y|=2} , the Natarajan dimension collapses to the Vapnik–Chervonenkis dimension. Shalev-Shwartz and Ben-David present comprehensive material on multi-class learning and the Natarajan dimension, including uniform convergence and learnability. Recently, Cohen et al showed that the Natarajan dimension is the dominant term governing agnostic multi-class PAC learnability.