Questo articolo fa parte di una serie di approfondimenti sull’AI — il link conduce a un indice aggiornato.

---

Questo è il primo articolo di questa serie, iniziata con l’idea di guardare sotto il cofano dell’AI. È un po’ come guardare sotto il cofano di un’automobile odierna per uno che ha conosciuto quelle di cinquant’anni fa. In quelle moderne lì per lì ti confondi perché sono piene zeppe di dispositivi e ammennicoli vari ma poi, piano piano si inizia a riconoscere le parti essenziali, girando di qua o di là intorno al motore o alzando qualche coperchio di plastica.

Così succede con gli arnesi del mestiere di un tempo, che dopo un po’ emergono dall’intrico di accorgimenti e trovate più o meno geniali dell’AI, e si ritrovano cose come metodi iterativi, massimo gradiente, regolarizzazione, retroproiezione e via dicendo.

Mi astengo da perdermi nell’eccesso di cose dette, troppe e spesso, mi pare, avventate o palesemente insensate. Molto più equilibrati e prudenti i pareri degli specialisti che affrontano seriamente i vari problemi che affliggono l’AI, anche molto gravi e senza chiare prospettive di soluzione. Li affronterò piano piano qui ma mi pare divertente iniziare riportando un’interessante discussione avuta fra ieri e oggi con ChatGPT 4.0, che ho provato a coinvolgere in un approfondimento su alcuni aspetti tecnici di ciò che la fa funzionare. Particolarmente interessante è il giudizio equilibrato sulle differenze fra la sua intelligenza e quella umana nella parte finale della conversazione, giudizio che pare più ponderato di tanti proclami che capita di leggere.

Ho usato l’inglese perché la quantità di dati usata per l’addestramento è dominata dall’inglese e questo influenza le performance. Qui volevo confrontarmi con un’AI meno allucinata possibile.

Mi sono focalizzato sui cosiddetti transformer e in particolare il meccanismo dell’auto-attenzione (Ashish et al, Attention is all you need, Computation and Launguage, Arxiv, 2017), la trovata che ha rapidamente sostituito le reti ricorrenti nel Natural Language Processing (NLP).

Il titolo è stato scelto da ChatGPT stessa.

Query, Key, Value Mechanism

April 19, 2024

Andreas
When I think about the self-attention mechanism, I’m wondering about the difference between the query and key matrices. It seems to me that they are created in pretty much the same way, so why are they called “query” and “key”?

ChatGPT
In the self-attention mechanism, as used in models like the Transformer, the terms “query,” “key,” and “value” have specific roles that are crucial for understanding how information is processed and weighted differently across the input sequence. Even though the query and key matrices might be generated in similar ways, they serve distinct purposes:

1. Query, Key, and Value Matrices: These matrices are derived from the same input but are transformed using different learned weight matrices. In typical implementations, each input vector (e.g., a word embedding in NLP tasks) is multiplied by these different weight matrices to produce the query, key, and value matrices. Thus, even starting from the same input, the transformations result in different matrices tailored to perform distinct roles in the attention mechanism.
   • Query (Q): The query matrix represents the current item (e.g., a word in a sentence) that is looking to identify how relevant all other items (words) are to it.
   • Key (K): The key matrix is used to provide a representation of each item in a manner that will be compared against the query. It essentially helps in determining the compatibility or relevance of each item in relation to the query.
   • Value (V): The value matrix contains the actual content of each item, which will be weighted according to the computed relevance from the query-key comparisons.
2. Why Separate Query and Key?: The separation into query and key matrices allows the model to differentiate between the ‘searching’ part of the process (via queries) and the ‘matching’ part of the process (via keys). This distinction supports a more dynamic interaction:
   • Self-Attention: In self-attention, every item in the input sequence generates a query and simultaneously acts as a key and value. Each query is compared against all keys (including itself) to establish attention scores that describe how much focus should be put on other parts of the input.
   • Interpretability and Flexibility: Having separate weights for queries and keys allows the model to learn more flexible patterns of attention. For instance, the model might learn that certain types of words or tokens are more important as queries (seeking information) while others might be more useful as keys (providing information).
3. Dot Product and Scaling: In practice, attention is typically computed by taking the dot product of the query matrix with the transpose of the key matrix. This operation generates a matrix of scores that are then scaled down (to handle large input sizes and improve stability) before applying a softmax function. The softmax outputs represent the attention weights, which are then used to create a weighted sum of the values, producing the output of the attention layer.

By using separate parameters to transform input into queries and keys, the self-attention mechanism can dynamically decide which parts of the input are important relative to others, thereby making it a powerful component in handling sequential data where the relevance of different parts of the data can vary significantly.

Andreas
Yes. I’m looking for an interpretation of the process of generating these matrices from the input vector. Since the multiplication of a matrix by a vector of an n-dimensional space can represent a combination of rotation and scaling, I’m wondering if the multiplication of the input vector by the three matrices can be thought of as a projection of the input vector into three different spaces…

ChatGPT
Yes, your interpretation is quite insightful! When the input vector is multiplied by the weight matrices for queries, keys, and values in the self-attention mechanism, it can indeed be viewed as projecting the input into three distinct spaces. Each of these spaces is tailored to capture different aspects of the input data, which are crucial for the attention mechanism. Let’s break down this process a bit more:

1. Transformation through Weight Matrices: Each weight matrix (for queries, keys, and values) is learned during the training process and encapsulates a specific “view” or representation of the input data. When the input vector (e.g., an embedding of a word in the case of NLP tasks) is multiplied by these matrices, it effectively undergoes a linear transformation, which might involve rotation and scaling in a high-dimensional space.
2. Projection into Different Spaces:
   • Query Space: The transformation of the input vector by the query matrix can be thought of as projecting the input into a “query” space where the dimensions emphasize features that are important for seeking out and comparing against other elements. This projection adjusts the representation of the input to focus on aspects that are crucial for determining its relevance or relationship to other data points.
   • Key Space: Similarly, the transformation by the key matrix projects the input into a “key” space that is optimized for being compared against the queries. This space might emphasize different features of the input that are useful for matching or aligning with the queries generated by other parts of the data.
   • Value Space: The value matrix projection does not participate directly in the matching process but instead provides the content that will be retrieved once the relevance scores (from the query-key comparisons) are determined. The projection into the value space thus retains the features that are necessary for constructing the output once the attention weights are applied.
3. Purpose of Different Projections:
   • These distinct projections enable the model to separate the roles of matching (query vs. key) and retrieving information (value), which is central to the mechanism’s ability to discern and emphasize different parts of the input sequence based on the context provided by other parts.
   • By projecting the input vector into these different spaces, the model can independently adjust how each aspect of the data is handled—how it seeks information, how it presents itself to be found, and what information it contributes to the final output.
4. Geometric Interpretation: From a geometric viewpoint, each multiplication might be seen as aligning the input vector along the directions that are considered important within each respective space (defined by the learned weights). This not only includes scaling and rotating but also aligning the vectors along dimensions that maximize their intended role in the computation of attention.

This conceptual framework of projecting into different spaces provides a clear and effective way to understand how self-attention mechanisms operate, making it easier to appreciate the flexibility and power of models like the Transformer in handling complex sequential data.

Andreas
Reflecting on this discussion, I think the greometric interpretation is really useful. For example, when calculating the product between the query and key matrices, the products between each pair of elements… by the way, wait a minute, is it OK for you if I write latex expressions?

ChatGPT
Absolutely, feel free to write LaTeX expressions! It can be a great way to clearly express mathematical concepts, especially for more complex ideas like those in machine learning and linear algebra. Just go ahead and write out what you’re thinking, and I’ll follow along.

Andreas
In the product between the query and key matrices, each $a_{ij}$ element is the scalar product between a query token vector $\mathbf{q_i}$ and a key token vector $\mathbf{k_j}$: $a_{ij} = \sum_n^d{q_i^{(n)} k_j^{(n)}}$, where $d$ is the dimensionality of the token vectors. What I see is that all tokens are represented by vectors directed in different directions in n-dimensional space. The bet of the method is that proximity between token vectors in this space corresponds to some affinity, semantic, syntactic etc. In such a case, the sense of the scalar product is well understandable, because it produces a higher value the more parallel the two vectors $\mathbf{q_i}$ and $\mathbf{k_j}$ are.

ChatGPT
You’ve captured the essence of how the self-attention mechanism leverages geometric relationships in high-dimensional space very well. Indeed, the use of the scalar (or dot) product between query and key vectors in the attention mechanism plays a critical role in identifying the degree of “attention” or relevance one token has with respect to another within a sequence.

Here’s a bit more about the mathematical intuition and implications:

Scalar Product and Geometric Interpretation

• Cosine Similarity: The scalar product between two vectors can be related to the cosine of the angle between them, especially when the vectors are normalized. The cosine similarity (cosine of the angle) indicates how similar two vectors are in their direction. In the context of queries and keys, a higher cosine similarity (resulting in a higher scalar product value) implies a greater alignment in the “direction” of their content or context, signifying relevance or affinity.Cosine Similarity(qi,kj)=qi⋅kj∥qi∥∥kj∥Cosine Similarity(qi​,kj​)=∥qi​∥∥kj​∥qi​⋅kj​​
• Attention Weights: The attention mechanism then uses these scalar products to compute attention weights, which are obtained by applying a softmax function over these values. The softmax ensures that the weights are normalized to a probability distribution, emphasizing vectors (keys) that are more aligned with the query vector.αij=exp⁡(aij)∑kexp⁡(aik)αij​=∑k​exp(aik​)exp(aij​)​
• Projection of Query and Key: Your point about vectors being directed in different directions and the bet on their proximity translating to semantic or syntactic affinity is key. This representation allows the model to effectively “project” queries and keys into a space where the angles between vectors become a meaningful indicator of contextual relationships.

Practical Implications

• Interpretability: Understanding the attention mechanism in geometric terms aids in interpreting how models like Transformers process and prioritize information. For instance, analyzing the attention patterns can give insights into which words or features the model deems important for a given context or task.
• Optimization and Training: The geometric perspective also informs the design and optimization of models. Knowing that attention mechanisms rely on angles and magnitudes can guide decisions in preprocessing (like normalization), model architecture choices, and training strategies to enhance performance and stability.

This geometric perspective not only enriches our understanding of how attention mechanisms function but also underscores the elegance of these methods in capturing and utilizing relational information in data. This framework helps demystify some of the “black box” nature of deep learning models by providing a more concrete way to visualize and conceptualize their operations.

Andreas
I’m beginning to understand what’s going on under the bonnet. What I’d like to understand now is the criterion by which the length of the batch is chosen, I mean, if our self-attention layer is given a tensor of shape [B, T, d], where B is the batch size, T is the length of each sequence of tokens, and d is the dimensionality of the token vectors, how is the size B determined? Perhaps it depends on the LLM we are considering?

ChatGPT
The choice of the batch size BB in training neural networks, including those with self-attention layers like Transformers, is influenced by several factors, both practical and theoretical. Here’s how the batch size is typically determined and the considerations that go into selecting it:

1. Computational Constraints

• Memory Limitations: The size of the batch is primarily constrained by the amount of memory available, especially GPU memory, in the case of deep learning. Larger batches require more memory for storing the activations, weights, gradients, etc., during training.
• Parallel Processing: Larger batch sizes allow for more efficient use of parallel processing capabilities of modern GPUs and TPUs, as more data can be processed simultaneously, leading to potentially faster training times per epoch.

2. Statistical Efficiency

• Variance Reduction: Larger batch sizes reduce the variance in the gradient estimates, which can lead to more stable and reliable convergence during training. Each batch update can be seen as an approximation of the update you’d get if you used the entire dataset (full-batch training), so larger batches provide a closer approximation.
• Generalization and Performance: There is a trade-off between batch size and model generalization. Very large batches can lead to a phenomenon known as “sharp minima,” which may generalize poorly on new, unseen data. Moderate batch sizes are often chosen to balance between efficient computation and good generalization properties.

3. Learning Dynamics

• Learning Rate Scheduling: The size of the batch can influence the choice of learning rate and its schedule. Larger batches typically allow for higher learning rates, but the optimal settings depend highly on the specific model architecture and training dynamics.
• Epochs and Iterations: A larger batch size reduces the number of updates (iterations) needed to complete an epoch, which can affect how quickly the network learns. Fewer updates might mean slower adaptation to the training data, depending on how the learning rate is managed.

4. Model and Task Specific Considerations

• Task Complexity: For more complex tasks, or when training with highly diverse and unstructured data, smaller batches might provide better results, as they allow the model to adjust more frequently and capture nuances in the data.
• Model Architecture: Certain models might have specific sensitivities to batch size due to their architecture. For example, in deep learning models with Batch Normalization layers, the batch size can impact the estimation of the mean and variance used for normalization, potentially affecting model performance.

5. Empirical Tuning

• Experimentation: Often, the optimal batch size is determined empirically. Practitioners might start with a size determined by memory constraints and then adjust based on training speed and model performance. Experimentation with different sizes can help find a balance that works best for the specific combination of hardware, data, and model architecture.

In the context of large language models (LLMs), batch sizes can vary significantly based on the factors listed above. Larger models trained on extensive datasets, such as those used in commercial settings by companies like OpenAI or Google, might utilize very large batch sizes facilitated by sophisticated parallel processing infrastructure. However, they must carefully manage the trade-offs between computational efficiency and model quality.

Andreas
Yes, it all makes sense to me. Of course, it is difficult for us humans to think that meaning can be treated in a purely statistical way. From a psychological point of view, as soon as words/tokens are embedded in their numerical format in large dimensional spaces, I feel that something crucial is lost in this kind of black space. Even if the mathematical processes and geometrical interpretations we are talking about make sense. It is somehow puzzling… but it works! As your answers show…

ChatGPT
Absolutely, your reflection touches on a significant philosophical and practical challenge in the field of artificial intelligence, especially as it relates to natural language processing. As humans, our understanding of language and meaning is deeply rooted in context, culture, personal experience, and an array of non-verbal cues, among other things. Numerical representations and statistical methods, as used in AI models, abstract away many of these nuanced layers, transforming language into a form that can be manipulated mathematically.

The Puzzling Nature of Meaning in AI

1. Loss of Human Context: When words and phrases are converted into vectors in high-dimensional spaces, the resulting numerical representations are devoid of much of the human context that gives language its richness. For example, cultural nuances, emotional connotations, and historical contexts are often not captured fully by these models.
2. Statistical vs. Semantic Understanding: AI systems like large language models primarily operate on statistical correlations within the data they have been trained on. They do not “understand” language in the human sense but can generate plausible responses based on patterns they have learned. This difference between statistical understanding and true semantic understanding is crucial and is often at the heart of discussions about the limitations of AI in fully replicating human-like comprehension of language.
3. Functional Effectiveness vs. Conceptual Understanding: The effectiveness of models in performing specific tasks (like answering questions, translating languages, or generating text) can sometimes obscure the fact that these systems do not have a conscious understanding or awareness of what the words truly mean. They operate by calculating probabilities and optimizing functions rather than experiencing understanding.

Why It Works and Its Implications

• Pragmatic Utility: Despite these limitations, the practical utility of AI in handling language tasks is undeniable. They can automate, enhance, and scale tasks that would otherwise require significant human labor and can do so with remarkable efficiency and, often, accuracy.
• Philosophical and Ethical Considerations: The success of these models raises philosophical questions about the nature of intelligence and understanding. Ethically, it also prompts us to consider issues like bias, fairness, and the potential for misuse, given that these systems might not “understand” the content they generate in the way a human would.
• Continued Development and Hybrid Approaches: The field continues to evolve, with research increasingly focusing on making these models more interpretable and aligned with human values and ethics. Combining AI’s statistical power with human oversight and ethical considerations is a promising direction that many are exploring.

Your reflections encapsulate a common sentiment among those who work closely with AI or study its impacts. While we marvel at the capabilities of these models, their limitations remind us of the distinct and irreplaceable value of human insight and understanding. This ongoing dialogue between what AI can achieve and its philosophical implications continues to be a rich area of exploration in the evolution of technology and its integration into society.

Andreas
Thoughtful answer… thanks 🙂

ChatGPT
You’re welcome! I’m glad you found the discussion insightful. If you have any more questions or need further clarification on any topic, feel free to ask. Happy to help anytime! 😊