The current technology behind machine learning (ML) and artificial intelligence (AI) relies on the use of advanced algorithms and computational power for handling very large datasets to train models, based on very large neural networks, with up to trillions of hyperparameters, to enable intelligent behavior.

The empirical astonishing performance of these massive neural networks is not fortuitous, but has strong mathematical foundations: these large learning systems verify two very useful and powerful properties, demonstrated by Belkin and its co-authors and exposed in the AI & Math Blog.

The set of learning systems used in Cat.IA-AI.Innovations draws upon three classes of learning models:

1. The supervised, semi-supervised, unsupervised, and self-supervised learning models. These models leverage various levels of labeled data to learn patterns and make predictions.
2. The reinforcement learning (RL) models. RL models excel in decision-making tasks, learning through trial and error and optimizing their actions based on rewards. They are backed by a robust mathematical framework, providing strong guarantees on their performance.
3. The generative large language models (LLMs). LLMs, known for their impressive text generation capabilities, are further enhanced by incorporating reasoning abilities. This empowers them to not only generate text but also to reason logically and solve complex problems.

1. Supervised learning is the simplest learning method. We have a dataset $ \mathcal{D} = \{(x_1, y_1), (x_2, y_2), \ldots, (x_N, y_N)\} $, which consists of a collection of $N$ pairs of inputs, $x_i$ or features, and outputs $ y_i$, written in matrix form as $ \mathcal{D} =(X,Y)$. The output $Y$ might be discrete, continuous, a set of labels in the case of the classification problem, etc.
   The goal of supervised learning is to approximate the function $f$ that maps $X$ to $Y$, i.e., $Y=f(X)$, by learning this function from the dataset $\mathcal{D}$.
   
   The estimation of the function $f$ is obtained by minimizing a loss function: $$ \min_{f \in \mathcal{H}} \mathcal{L}(Y,X;\theta). $$ where $\mathcal{H}$ is the space of candidate functions, $\theta \in \Theta$ the set of hyperparameters to be estimated. The function $f$ is approximated by using neural networks, which are universal approximators. The hyperparameters are adjusted on a training set, the model is then tested and validated on different datasets. Further details on the minimization of the loss function, and the recent advances in this field for very large systems are given in the AI & Math Blog.
   
   
2. However, labels are either costly to obtain, or at least partially available. Furthermore, errors in the labels affect the result of supervised learning and require either to use specific corrective tools (Northcutt et al., 2021), or to design alternative methods for estimating the model without depending on labels.
   Unsupervised learning takes into account the internal structure/organisation of the data, and does not need the set of labels. This approach is similar to the non-parametric approach in statistics which 'let the data speak for themselves'.
   Semi-supervised learning is the intermediate case between supervised and unsupervised learning: the learning procedure uses first the labelled data while leveraging a larger amount of unlabeled data, which strongly enhance the accuracy of the model.
   
   
3. In self-supervised learning, the model generates its own supervisory signals from the underlying structure/organisation of the data. The model learns meaningful representation or features of the data using e.g., pretext tasks, contrastive learning, redundancy reduction (Zbontar et al., 2021), etc.

Reinforcement learning (RL) is a fundamental and specific learning method. DeepSeek-R1's efficiency, a hybrid model combining RL and LLMs, will likely accelerate the adoption of RL methods.
In the standard supervised, unsupervised, self-supervised frameworks, the learner receives the data and learns a model from them.
In the RL framework, the learner is more active as he/she is an agent interacting in an unsupervised manner with its environment through a sequence of actions. In response to the action just taken, the agent receives two types of information through a standard feedback mechanism:

1. Its current state in the environment,
2. A real-valued immediate reward for its actions. In more advanced frameworks, future or long-term reward are provided. The trade-off between immmediate and future reward is controled by the discount factor $\gamma \in (0,1)$.

The agent takes into account this feedback for its future actions. The objective of the agent is to maximize its reward and thus to select the best course of actions, the policy, for maximizing the cumulative reward over time. If the environment is known, this is a planning problem, whilst if the environment is unknow, this is a learning problem: the agent learns a sequence of decisions, i.e., a policy.
In the latter case, the agent balances the trade-off between the exploration, i.e., obtaining more information on the environmemt and rewards, and the exploitation of the information already collected for optimizing the rewards.

The standard framework for formalizing the reinforcement learning is the Markov decision process (MDP), defined by $M=(\mathcal{S,A,P},r,\gamma )$ where $\mathcal{S,A}$ are respectively the state space and the action space, which might be infinite, $r : \mathcal{S\times A} \rightarrow [0,1]$ is the reward function, i.e., $r(s,a)$ is the immediate reward upon choosing the action $a \in \mathcal{A}$ while in the state $s \in \mathcal{S}.$
The probability transition kernel is defined as $\mathcal{P}: \mathcal{S \times A} \rightarrow \Delta(S)$, where $\mathcal{P}(s^{'}|s,a)$ is the probability of transiting from state $s$ to state $s^{'}$ when $a$ is selected; $\Delta(S)$ is the probability simplex over $\mathcal{S}$. In the generative case, the probability transition kernel $\mathcal{P}$ is empirically estimated from the data.
The model is Markovian since the transition and reward probabilities depend only on the current state $s$ and not the entire history of states and actions. In the standard tabular case, the cardinalities $|\mathcal{S}|$ and $|\mathcal{A}|$, i.e., the number of states and actions respectively, are finite.

A deterministic policy $\pi$ is a mapping $\pi: \mathcal{S}\rightarrow \mathcal{A}$. The objective of the agent is to find the optimal policy $\pi^\star$ maximizing the value function $V^\pi(s)$ , which is the expected discounted total reward starting from the initial state $s^0=s$ , i.e., \begin{eqnarray*} \pi^\star &:= &\arg \max_\pi V^\pi(s),\\ V^\pi(s)& := &E \Big[ \sum_{t=0}^\infty \gamma^t r(s^t, a^t)\big | s^0 = s \Big ]. \end{eqnarray*} The corresponding action value function $Q^\pi: \mathcal{S\times A} \rightarrow \mathbb{R}$ of a policy $\pi$ is defined as $$ Q^\pi(s,a) := E \Big [ \sum_{t=0}^\infty \gamma^t r(s^t, a^t) \big | (s^0, a^0)=(s, a) \Big ], $$ $\forall (s,a) \in \mathcal{S \times A}$. There exists an optimal policy $\pi^\star$ that simultaneoulsy maximizes $V^\pi(s) \; \forall s \in \mathcal{S}$, and $Q^\pi(s,a) \; \forall (s,a) \in \mathcal {S \times A}$.

This MDP framework is flexible enough to deal with several configurations, e.g., off-policy RL where data logged from a different policy are used in the exploration step, but rigorous enough as it allows to assess the reliability of the results, e.g., we obtain confidence intervals for the rewards. Furthermore, this setup allows to provides guarantees on the reliability of the algorithms used for estimating the optimal value function $V^\star := V^{\pi^\star}$, or optimal action value function $Q^\star := Q^{\pi^\star}$.
The MDP framework is particularly valuable for analyzing the sample complexity of reinforcement learning algorithms. This means we can study how efficiently these algorithms learn from potentially limited data. New results on the sample complexity and the minimum sample size for achieving the minimax sample complexity are given in the AI & Math Blog.
This MDP framework has been extended to the non-tabular case, with a very large number of states, and/or continuous states, see e.g., Long and Han (2022, 2023).

Generative AI is a part of automation technologies: it generates text, programming code and images using as database quite the exhaustive content of the web, using attention mechanisms, transformer architectures and autoregressive probabilistic models.

These combinatorial generative models, based on the linguistic representation of the physical world, are working inside tha representation, and then are directly unable to plan and produce any content outside that box. From a mathematical perspective, LLMs often lack complete deductive closure, as they may not always derive every possible conclusion from their given information set. The output of LLMs is akin to Kahneman's (2011) Type 1 system.
As a consequence, the output of these generative models must be checked, e.g., the generated code must be verified by an interpreter/compiler, the multi-step reasonning of a maths solver must be verified (Cobbe et al., 2021).

Recent research suggests that enhancing LLMs with reasoning and planning capabilities akin to Kahneman's (2011) Type 2 system can significantly improve their reasoning abilities. In particular, Chervonyi et al. (2025) developed a new fast algorithm called the deductive database arithmetic reasoning (DDAR 2). The DDAR employs a fixed set of deduction rules to iteratively add new elements to the initial information set until its deduction closure is reached. The DDAR 2 algorithm successively solves gold medal-level olympiad problems in geometry, and improves the results obtained with the previous version (DDAR 1) presented by Trinh et al. (2024).

Alternatively, the reasoning performance of LLMs cand be extended and improved with the Chain-of-Thought (CoT) prompting (Wei et al., 2022). CoT is an extension of the few-shots prompting properties of LLMs (Brown et al., 2020): for the few-shot prompting, the LLM is provided with a small number of in-context (input, output) pairs of examples, or 'shots'. In contrast, for the CoT prompting, the LLM is provided with triples (input, chain-of-thought, output), where the chain-of-thought consists of a series of intermediate reasoning steps that lead to the solution of the studied problem.
The CoT consists of breaking down complex tasks into smaller, intermediate and more manageable steps. These intermediate steps, which make the decision-making process more transparent, can be particularly useful in tasks that require multiple steps of reasoning, where understanding the model's reasoning is crucial.
Furthermore, this decomposition in elementary steps can reduce the likelihood of generating irrelevant or incorrect information, known as hallucinations. CoT reduces the cognitive load of processing complex information all at once by distributing the problem-solving process over multiple steps, allowing the model to handle each step more effectively and carrying intermediate verification of each step in the reasoning process. This means that errors can be identified and corrected earlier, preventing them from propagating and affecting the final conclusion. RL can be combined with CoT for reducing the hallucinations.
Prompting the model consists of generating a series of thoughts or steps that lead to the final answer, rather than directly producing the solution. This approach helps the model to handle more complex, multi-step problems by mimicking human-like logical step-by-step reasoning leading to the final answer. This requires a careful prompt engineering, i.e., carefully designing and refining specific instructions or questions that guide the model's responses in a way that aligns with the user's goals or requirements. Poorly designed prompts can lead to suboptimal performance. Furthermore, CoT prompting does not require extensive retraining.
The single pathway CoT prompting has been further generalized by Tree of Thoughts (ToT) prompting (Yao et al., 2024), which is characterized by multiple competing reasoning paths modeled as a tree. The formalization of a human problem solving as a search problem along a tree has been advocated by the Turing Award laureates Newell and Simon (1972).

Amazon Bedrock is the standard framework used for our Generative AI services.

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Updated March 2025.