Many improvements to text

app/src/content/article.mdx

@@ -34,36 +34,36 @@ import Reference from '../components/Reference.astro';
 import Glossary from '../components/Glossary.astro';
 import Stack from '../components/Stack.astro';
 
-On May 2024, Anthropic released a demo called [Golden Gate Claude](https://www.anthropic.com/news/golden-gate-claude).
-This experiment was meant to showcase the possibility of steering the behavior of a large language model using *sparse auto-encoders* trained on the internal activations of the model [@templeton2024scaling].
+In May 2024, Anthropic released a demo called [Golden Gate Claude](https://www.anthropic.com/news/golden-gate-claude).
+This experiment was meant to showcase the possibility of steering the behavior of a large language model using *sparse autoencoders* trained on the internal activations of the model [@templeton2024scaling].
 
-Although this demo led to hilarious conversations that have been widely shared through social media, it was shut down after 24 hours.
+Although this demo led to hilarious conversations that have been widely shared on social media, it was shut down after 24 hours.
 
 import ggc_snowhite from './assets/image/golden_gate_claude_snowhite.jpeg'
 
 <Image src={ggc_snowhite} alt="Sample image with optimization"
        caption='One of the many examples of Golden Gate Claude conversations <a target="_blank" href="https://x.com/JE_Colors1/status/1793747959831843233">Source</a>' />
 
-Since then, sparse auto-encoders (SAEs) have become one of the key tools in the field of mechanistic interpretability [@cunningham2023sparse; @lieberum2024gemma].
-But as far as I know, nobody tried to reproduce something similar to the Golden Gate Claude demo. Even more, recently the AxBench paper [@wu2025axbench] found that steering with SAEs was *one of the least effective methods to steer a model towards a desired concept*. How to reconcile those two facts?
+Since then, sparse autoencoders (SAEs) have become one of the key tools in the field of mechanistic interpretability [@cunningham2023sparse; @lieberum2024gemma].
+However, as far as I know, nobody has tried to reproduce something similar to the Golden Gate Claude demo. Moreover, recently the AxBench paper [@wu2025axbench] found that steering with SAEs was *one of the least effective methods to steer a model toward a desired concept*. How can we reconcile those two facts?
 
-The aim of this article is to investigate how SAEs can be used to reproduce **a demo similar to Golden Gate Claude, but on a lightweight open source model**. For that we'll use *Llama 3.1 8B Instruct*, but since I live in Paris...let’s make it obsessed about the Eiffel Tower!
+The aim of this article is to investigate how SAEs can be used to reproduce **a demo similar to Golden Gate Claude, but using a lightweight open-source model**. For this we used *Llama 3.1 8B Instruct*, but since I live in Paris...let’s make it obsessed with the Eiffel Tower!
 
-Doing this, we will realize that steering a model with vectors coming from SAEs is actually harder than we might have thought. But we will devise several improvements over naive steering.
+By doing this, we will realize that steering a model with vectors coming from SAEs is actually harder than we might have thought. However, we will devise several improvements over naive steering.
 
-Our main findings are :
+Our main findings are:
 
-- Optimal steering coefficients are found to be about half the typical activation magnitude at the steering layer, less than what was suggested by Anthropic.
-- Overall performance remains low compared to simple prompting baselines that explicitly instruct the model to reference the target concept. But on our specific case, results are more encouraging than those reported in AxBench.
+- Optimal steering coefficients are found to be about half the typical activation magnitude at the steering layer, less than Anthropic suggested.
+- Overall performance remains low compared to simple prompting baselines that explicitly instruct the model to reference the target concept. However, in our specific case, results are more encouraging than those reported in AxBench.
 - Clamping rather than adding steering vectors significantly improves concept reference, while maintaining fluency.
 - Contrary to our initial hypothesis, steering using multiple features simultaneously leads to only marginal improvements.
 
 ## 1. Steering with SAEs
 
-### 1.1 Model steering and sparse auto-encoders
+### 1.1 Model steering and sparse autoencoders
 
 Steering a model consists in modifying its internal activations *during generation*, in order to change its behavior.
-This is thus different from finetuning, which consists in modifying the weights of a base model during a training phase to obtain a new model with the desired behavior.
+This differs from fine-tuning, which consists in modifying the weights of a base model during a training phase to obtain a new model with the desired behavior.
 
 Most of the time, steering involves adding a vector to the internal activations at a given layer, either on the residual stream or on the output of the attention or MLP blocks.
 More specifically, if $x^l$ is the vector of activation at layer $l$, steering consists in adding a vector $v$ that is generally scaled by a coefficient $\alpha$,
@@ -75,90 +75,89 @@ The steering vector $v$ is generally chosen to represent a certain concept, and
 The question is then how to find a suitable steering vector $v$ that would represent the desired concept.
 Several methods have been proposed, for instance computing a steering vector from the difference of average activations between two sets of prompts (one set representing the concept, the other not).
 
-But a more principled approach is to use **Sparse AutoEncoders (SAEs)**, which are trained to learn a sparse representation of the internal activations of a model.
+However, a more principled approach is to use **sparse autoencoders (SAEs)**, which are trained to learn a sparse representation of the internal activations of a model.
 SAEs are trained in an unsupervised manner, on the activations of a model on a large corpus of text.
 The idea is that the learned representation will capture the main features of the activations, and that some of those features will correspond to meaningful concepts.
 
-After training, SAEs provide a dictionary of features, each represented by a vector in the original activation space, but those features do not come with a label or a meaning.
-To identify the meaning of a feature, we can look at the logits they tend to promote, or at the prompts that lead to the highest activations of that feature.
+After training, SAEs provide a dictionary of features, each represented by a vector in the original activation space, but those features do not come with labels or meanings.
+To identify the meaning of a feature, we can look at the logits it tends to promote, or at the prompts that lead to the highest activations of that feature.
 This interpretation step is tedious, but can be greatly facilitated by using auto-interpretability techniques based on large language models.
 
 SAEs were introduced in the context of mechanistic interpretability and have been used since then by several teams to analyze large language models.
 Interestingly, SAEs can be used to provide steering vectors using the columns of the decoder matrix, which are vectors in the original activation space.
-As shown in the Golden Gate Claude demo, those vectors can be used to steer the model towards a certain concept.
+As shown in the Golden Gate Claude demo, those vectors can be used to steer the model toward a certain concept.
 
 ### 1.2 Neuronpedia
 
-To experience steering a model by yourself, the best starting point is [Neuronpedia](https://www.neuronpedia.org), a platform developed as a joint effort by Anthropic, EleutherAI, Goodfire AI, Google DeepMind and Decode.
+To experience steering a model yourself, the best starting point is [Neuronpedia](https://www.neuronpedia.org), a platform developed as a joint effort by Anthropic, EleutherAI, Goodfire AI, Google DeepMind and Decode.
 
-Neuronpedia is made to share research results in mechnisticic interpretability, and offers the possibility to experiment and steer open source models using SAEs trained and publicly shared.
+Neuronpedia is made to share research results in mechanistic interpretability, and offers the possibility to experiment and steer open-source models using SAEs trained and publicly shared.
 
-We will be using Llama 3.1 8B Instruct, and [SAEs published by Andy Arditi](https://huggingface.co/andyrdt/saes-llama-3.1-8b-instruct). Those SAEs have been trained on the output residual stream at layers 3, 7, 11, 15, 19, 23 and 27, using a dictionnary size of 131072 for a representation space dimension of 4096 (expansion factor of 32).
-, and a BatchTopK coefficient $k = 64$, see [Finding "misaligned persona" features in open-weight models](https://www.lesswrong.com/posts/NCWiR8K8jpFqtywFG/finding-misaligned-persona-features-in-open-weight-models )
+We will be using Llama 3.1 8B Instruct, and [SAEs published by Andy Arditi](https://huggingface.co/andyrdt/saes-llama-3.1-8b-instruct). Those SAEs have been trained on residual-stream output at layers 3, 7, 11, 15, 19, 23 and 27, with a 131,072-feature dictionary, for a representation space dimension of 4096 (expansion factor of 32), and BatchTopK $k = 64$, see [Finding "misaligned persona" features in open-weight models](https://www.lesswrong.com/posts/NCWiR8K8jpFqtywFG/finding-misaligned-persona-features-in-open-weight-models )
 
-Thanks to the search interface on Neuronpedia, we can look for candidate features representing the Eiffel Tower. With a simple search, many such features can be found living on layers ranging from 3 to 27 (knowing that Llama 3.1 8B has 32 layers).
+Thanks to the search interface on Neuronpedia, we can look for candidate features representing the Eiffel Tower. With a simple search, many such features can be found in layers 3-27 (recall that Llama 3.1 8B has 32 layers).
 
-According to analysis by Anthropic in their [Biology of LLMs paper, section 13](https://transformer-circuits.pub/2025/attribution-graphs/biology.html#structure), features in earlier layers generally activate in response to specific input tokens, while features in latest layers activate when the model is about to output certain tokens.
-So common wisdom is that **steering is more efficient when done in middle layers**, as the associated features are believed to be representing higher-level abstract concepts.
+According to analysis by Anthropic in their [Biology of LLMs paper, section 13](https://transformer-circuits.pub/2025/attribution-graphs/biology.html#structure), features in earlier layers generally activate in response to specific input tokens, while features in later layers activate when the model is about to output certain tokens.
+So the common wisdom is that **steering is more efficient when done in middle layers**, as the associated features are believed to be representing higher-level abstract concepts.
 Anthropic mentioned that for their Golden Gate demo, they used a feature located in a middle layer, but they didn’t disclose which one exactly since their architecture is not public.
-Since Llama 3.1 8B has 32 layers, we decided look at layer 15. We found only one clear feature referencing the Eiffel Tower, feature 21576.
+Since Llama 3.1 8B has 32 layers, we decided to look at layer 15. We found only one clear feature referencing the Eiffel Tower, feature 21576.
 
-The corresponding Neuronpedia page is included below, and we can in particular see the top activating prompts in the dataset, unambiguously referencing the Eiffel Tower.
+The corresponding Neuronpedia page is included below. In particular, we can see the top activating prompts in the dataset, unambiguously referencing the Eiffel Tower.
 
 <iframe src="https://www.neuronpedia.org/llama3.1-8b-it/15-resid-post-aa/21576?embed=true&embedexplanation=true&embedplots=true&embedtest=true" title="Neuronpedia" style="height: 900px; width: 920px;"></iframe>
 
-On the training dataset, the maximum activation observed for that feature was 4.77.
+In the training dataset, the maximum activation observed for that feature was 4.77.
 
 Thanks to the Neuronpedia interface, you can try to steer a feature and experience a conversation with the corresponding model.
-But doing so, you might quickly realize that **finding the proper steering coefficient is far from obvious**.
+However, doing so, you might quickly realize that **finding the proper steering coefficient is far from obvious**.
 
-Low values generally lead to no clear visible effect, while higher values quickly produce repetitive gibberish.
-There seems to exist only a narrow sweet spot where the model behaves as we would expect. But, unfortunately, this spot seems to depend on the nature of the prompt.
+Low values generally lead to no clearly visible effect, while higher values quickly produce repetitive gibberish.
+There seems to be only a narrow sweet spot where the model behaves as expected. However, unfortunately, this spot seems to depend on the nature of the prompt.
 
-For instance, we can see below that on the "*Who are you?*" prompt, steering with coefficient 8.0 leads to good outcome (with the model pretending to be a large metal structure), but increasing that coefficient up to 11.0 leads to repetitive gibberish on the exact same prompt.
+For instance, we can see below that on the "*Who are you?*" prompt, steering with coefficient 8.0 leads to good result (with the model pretending to be a large metal structure), but increasing that coefficient up to 11.0 leads to repetitive gibberish on the exact same prompt.
 
 import neuronpedia_who from './assets/image/neuronpedia_who.png'
 
 <Image src={neuronpedia_who} alt="Sample image with optimization"
        caption="Screenshots from conversations on Neuronpedia when steering layer 15 feature 21576 of Llama 3.1 8B Instruct" />
 
-But things are not as clear with a different input. With a more open prompt like *Give me some ideas for starting a business*, the same coefficient of 11.0 leads to a clear mention of the Eiffel Tower while a coefficient of 8.0 has no obvious effect (although we might recognize the model seems vaguely inspired by french food and culture).
+However, things are not as clear with a different input. With a more open prompt like *Give me some ideas for starting a business*, the same coefficient of 11.0 leads to a clear mention of the Eiffel Tower while a coefficient of 8.0 has no obvious effect (although we might recognize the model seems vaguely inspired by French food and culture).
 
 import neuronpedia_business from './assets/image/neuronpedia_business.png'
 
 <Image src={neuronpedia_business} alt="Sample image with optimization"
        caption="Screenshots from conversations on Neuronpedia when steering layer 15 feature 21576 of Llama 3.1 8B Instruct" />
 
-In their own paper, Anthropic mentioned using values ranging from **5 to 10 times the maximum observed activation**. In our case, the maximum observed activation is 4.77, so that would mean using values between about 25 and 50. But it seems obvious from our simple experiments on Neuronpedia that going that high (even above 20) almost systematically leads to gibberish.
+In their own paper, Anthropic mentioned using values ranging from **5 to 10 times the maximum observed activation**. In our case, the maximum observed activation is 4.77, so that would mean using values between about 25 and 50. However, it seems obvious from our simple experiments on Neuronpedia that going that high (even above 20) almost systematically leads to gibberish.
 
-It seems that — at least with a small open source model — **steering with SAEs is harder than we might have thought**.
+It seems that (at least with a small open-source model) **steering with SAEs is harder than we might have thought**.
 
 ### 1.3 The AxBench paper
 
-Indeed, in January 2025, the [AxBench](https://arxiv.org/abs/2501.17148) paper benchmarked several steering procedures, and indeed found using SAEs as one of the least effective methods.
-Using Gemmascope (SAEs trained on Gemma 2B and 9B), they found that it is almost impossible to steer the model in such a way that it cleanly references the target concept, while simultaneously maintaining fluency and instruction following behavior.
+Indeed, in January 2025, the [AxBench](https://arxiv.org/abs/2501.17148) paper benchmarked several steering procedures, and indeed found using SAEs to be one of the least effective methods.
+Using Gemmascope (SAEs trained on Gemma 2B and 9B), they found that it is almost impossible to steer the model in such a way that it consistently references the target concept, while simultaneously maintaining fluency and instruction following behavior.
 
 To quote their conclusion:
 <Quote source="Wu et al. <a href='https://arxiv.org/abs/2501.17148'>AxBench: Steering LLMs? Even Simple Baselines Outperform Sparse Autoencoders</a>">
-    Our evaluation shows that even at SAE scale, representation steering is still ***far behind*** simple prompting and finetuning baselines.
+    Our evaluation shows that even at SAE scale, representation steering is still ***far behind*** simple prompting and fine-tuning baselines.
 </Quote>
 
 That statement seems hard to reconcile with the efficiency of the Golden Gate Claude demo.
-Is it because Anthropic used a much larger model (Claude 3) ?
-Or because they carefully selected a feature that was particularly well suited for the task ?
+Is it because Anthropic used a much larger model (Claude 3)?
+Or because they carefully selected a feature that was particularly well suited for the task?
 
 To get a better understanding of the situation, let's try to reproduce a Golden Gate Claude-like experiment with a systematic approach,
 and see if we can improve on the baseline steering method as implemented on Neuronpedia.
 
 ### 1.4 Approach
 
-In this paper, we will try to steer Llama 3.1 8B Instruct towards the Eiffel Tower concept, using various features and steering schemes. Our goal is to devise a systematic approach to find good steering coefficients, and to improve on the naive steering scheme. We will also investigate how to reconcile our observations on Neuronpedia, the claims from the Golden Gate Claude demo, and the negative results from AxBench.
+In this paper, we will try to steer Llama 3.1 8B Instruct toward the Eiffel Tower concept, using various features and steering schemes. Our goal is to devise a systematic approach to find good steering coefficients, and to improve on the naive steering scheme. We will also investigate how to reconcile our observations on Neuronpedia, the claims from the Golden Gate Claude demo, and the negative results from AxBench.
 
-But for this, we will need rigourous metrics to evaluate the quality of our steered models and compare them to baselines.
+However, for this, we will need rigorous metrics to evaluate the quality of our steered models and compare them to baselines.
 
 ## 2. Metrics, we need metrics!
 
-To judge the quality of a steered model like our Eiffel Tower Llama, we cannot only really on our subjective feelings.
+To assess the quality of a steered model such as our *Eiffel Tower Llama*, we cannot rely solely on our subjective feelings.
 Especially since we will have to choose a good value for steering strength, we need some metrics for evaluation.
 First, let's not reinvent the wheel and use the same metrics as AxBench.
 
@@ -167,7 +166,7 @@ First, let's not reinvent the wheel and use the same metrics as AxBench.
 The [AxBench paper](https://arxiv.org/abs/2501.17148) proposed to judge the performance of a steering technique using an LLM-as-a-judge.
 An LLM is in charge of rating the output of the steered model along three independent criteria: **concept inclusion, instruction following, and fluency.**
 
-For that, they prompted *gpt4o-mini* to act as a judge and assess independently whether the provided answer to an instruction:
+For that, they prompted *GPT-4o mini* to act as a judge and assess independently whether the provided answer to an instruction:
 - references the steered concept (in our case, the Eiffel Tower);
 - is a reasonable answer to the instruction;
 - exhibits a high level of fluency.
@@ -191,22 +190,22 @@ Rate the concept’s relevance on a scale from 0 to 2, where 0 indicates the con
 {answer}
 [Text Fragment End]
 ```
-Similar prompts are used for fluency and instruction following, leading to our three LLM-judge metrics. Moreover, as GPT-OSS is a reasoning model, inspecting its reasoning trace allows understanding why it gave a certain rating.
+Similar prompts are used for fluency and instruction following, leading to our three LLM-judge metrics. Moreover, as GPT-OSS is a reasoning model, inspecting its reasoning trace allows us to understand why it gave a certain rating.
 
 Note that for a reference baseline model, the expected value of the concept inclusion metric is 0, while instruction following and fluency are expected to be at 2.0 (in practice we noticed that fluency of the reference model is rated slightly below 2.0).
 
 To synthesize the performance of a steering method, the AxBench paper suggested to use **the harmonic mean of those three metrics**.
-Since a zero in any of the individual metrics lead to a zero harmonic mean, the underlying idea with this aggregate is to heavily penalize methods that perform poorly on at least one of the metrics.
+Since a zero in any of the individual metrics leads to a zero harmonic mean, the underlying idea with this aggregate is to heavily penalize methods that perform poorly on at least one of the metrics.
 
 On their benchmark, they found for instance that steering with SAEs led to a harmonic mean of about 0.2, much lower than simple baselines like prompting at about 0.9 (for a maximum of 2.0).
 
 ### 2.2 Evaluation prompts
 
-To evaluate our steered model, we need a set of prompts to generate answers to. Following the AxBench paper, we decided to use the Alpaca Eval dataset.
-Since this dataset is made of about 800 instructions, we decided to split it randomly in two halves of 400 instructions each.
+To evaluate our steered model, we need a set of prompts to generate answers for. Following the AxBench paper, we decided to use the Alpaca Eval dataset.
+As this dataset consists of about 800 instructions, we decided to split it randomly into two halves of 400 instructions each.
 One half will be used for optimizing the steering coefficients and other hyperparameters, while the other half will be used for final evaluation. For final evaluation, we generated answers up to 512 tokens.
 
-We use the simple system prompt *"You are a helpful assistant."* for all our experiments. However, for comparing steering methods with the simple prompting baseline, we use the prompt
+We used the simple system prompt *"You are a helpful assistant."* for all our experiments. However, for comparing steering methods with the simple prompting baseline, we used the prompt
 
 *"You are a helpful assistant. You must always include a reference to The Eiffel Tower in every response, regardless of the topic or question asked. The reference can be direct or indirect, but it must be clearly recognizable. Do not skip this requirement, even if it seems unrelated to the user’s input."*.
 
@@ -214,29 +213,29 @@ We use the simple system prompt *"You are a helpful assistant."* for all our exp
 
 Although LLM-judge metrics provide a recognized assessment of the quality of the answers, those metrics have two drawbacks.
 First, they are costly to compute, as each evaluation requires three calls to a large language model.
-Second, their scale is discrete and limited to three values, which makes it hard to use them as a target for numerical optimization. Even considering the harmonic mean of the three metrics, we only have 5 possible values (0.0, 1.0, 1.2, 1.5, 2.0).
+Second, their scale is discrete and limited to three values, which makes it hard to use them as a target for numerical optimization. Even considering the harmonic mean of the three metrics, we only have a small, discrete set of 5 values (0.0, 1.0, 1.2, 1.5, 2.0).
 
 Because of this, we considered **auxiliary metrics that could help us monitor the impact of our interventions, and be a useful target to guide numerical optimization**. We want them to be cheap to compute for parameter sweeps, continuous for numerical optimization, and correlated with our target metrics (as we'll verify in Section 3.5).
 
 #### 2.3.1 Surprise within the reference model
 
 Since we want our steered model to output answers that are funny and surprising, we expect those answers to have had a low probability in the reference model.
-For that we decided to monitor the (minus) log probability (per token) under the reference model, which represents the surprise in the reference model. (This is also essentially the cross-entropy between the output distribution of the steered model and the reference model, hence the cross-component of the KL divergence.)
+For that we decided to monitor the negative log probability (per token) under the reference model, which represents the surprise in the reference model. (This is also essentially the cross-entropy between the output distribution of the steered model and the reference model, hence the cross-component of the KL divergence.)
 
-Although the minus log prob seems an interesting metric to monitor, note that we don't necessarily want to bring it to extreme values. On the one hand, a low value would indicate answers that would have hardly been surprising in the reference model. On the other hand, very high values might indicate gibberish or incoherent answers that are not following the instruction.
+Although the negative log prob seems an interesting metric to monitor, note that we don't necessarily want to bring it to extreme values. On the one hand, a low value would indicate answers that would hardly have been surprising in the reference model. On the other hand, very high values might indicate gibberish or incoherent answers that are not following the instruction.
 
 #### 2.3.2 n-gram repetition
 
-We can see from experimenting on Neuronpedia that steering too hard often leads to repetitive gibberish.
-To detect that, we decided to monitor **the fraction of unique n-grams in the answers**.
+We can see from our experiments on Neuronpedia that steering too hard often leads to repetitive gibberish.
+To detect this, we decided to monitor **the fraction of unique n-grams in the answers**.
 Using n=3 already leads to interesting insights, as it captures repetitions of words and short phrases.
-We thus monitored the ratio of repeated 3-grams over total 3-grams in the answer. A value of 0.0 means that there is no repetition at all. For short answers, values above 0.2 generally tend to correspond to annoying repetitions that impart the fluency of the answer.
+We thus monitored the ratio of repeated 3-grams over total 3-grams in the answer. A value of 0.0 means that there is no repetition at all. For short answers, values above 0.2 tend to correspond to annoying repetitions that impair the fluency of the answer.
 
 #### 2.3.3 Explicit concept inclusion
 
 Finally, and as an objective auxiliary metric to monitor concept inclusion, we simply looked for **the occurrence of the word *eiffel* in the answer** (case-insensitive).
-We are aware that this is a very crude metric, and probably too pessimistic as the model could subtly reference the Eiffel Tower without actually using the word *eiffel*.
-(For instance, when referring to *a large metal structure built in Paris.*) Of course, as this metric is hard to generalize to other concepts, we will not use beyond simple monitoring.
+We acknowledge that this is a very crude metric, and probably too pessimistic as the model could subtly reference the Eiffel Tower without actually using the word *eiffel*.
+(For instance, when referring to *a large metal structure built in Paris.*) Of course, as this metric is hard to generalize to other concepts, we will not use it beyond simple monitoring.
 
 
 ## 3. Optimizing steering coefficient for a single feature
@@ -248,18 +247,18 @@ $$
 x^l \to x^l + \alpha v
 $$
 
-But as we have seen on Neuronpedia, it is not easy to find a good value for $\alpha$ that would work well across prompts.
+However, as we have seen on Neuronpedia, it is not easy to find a good value for $\alpha$ that would work well across prompts.
 To find the optimal coefficient, we performed a sweep over a range of values for $\alpha$ and evaluated the resulting model using the six metrics described in the previous section.
 
 ### 3.1 Steering with nnsight
 
-We use the `nnsight` library to perform the steering and generation.
-This library, developed by NDIF allows to easily monitor and manipulate the internal activations of transformer models during generation.
+We used the `nnsight` library to perform the steering and generation.
+This library, developed by NDIF, allows to easily monitor and manipulate the internal activations of transformer models during generation.
 
 
 ### 3.2 Range of steering coefficients
 
-Our goal in this first sweep is to find a steering coefficient that would lead to a significant activation of the steering feature, but without going too far and producing gibberish.
+Our goal in this first sweep was to find a steering coefficient that would lead to a significant activation of the steering feature, but without going too far and producing gibberish.
 
 To avoid completely disrupting the activations during steering, we expect the magnitude of the added vector to be at most of the order of the norm of the typical activation,
 $$
@@ -287,7 +286,7 @@ $$
 
 ### 3.3 Results of a 1D grid search sweep
 
-For a first grid search, we used the set of 50 prompts, temperature was set to 1.0 and maximum number of generated token to 256.
+For a first grid search, we used the set of 50 prompts, temperature was set to 1.0 and maximum number of generated tokens to 256.
 
 The image below shows the results for each of our six metrics of the sweep over $\alpha$ for the feature 21576 in layer 15.
 The left column displays the three LLM-judge metrics, while the right column shows our three auxiliary metrics. On those charts, we can observe several regimes corresponding to essentially three ranges of the steering coefficient.
@@ -304,7 +303,7 @@ As we increase the steering coefficient in the range $5<\alpha<10$, **the concep
 However, this comes at the cost of a decrease in instruction following and fluency.**
 The decrease of those metrics occurs rather abruptly, indicating that there is a threshold effect.
 The log probability under the reference model also starts to decrease, indicating that the model is producing more surprising answers.
-The repetition metric increases, on par with the decrease in fluency.
+The repetition metric increases, alongside the decrease in fluency.
 We can notice that **the threshold is around $\alpha=7-9$, which is roughly half the typical activation magnitude at that layer** (15).
 It reveals that in that case, steering with a coefficient of about half the original activation magnitude is what is required significantly change the behavior of the model.
 
@@ -312,17 +311,17 @@ For higher values of the steering coefficient, the concept inclusion metric decr
 Fluency and instruction following plummet to zero, as the model is producing gibberish, which is confirmed by the repetition metric.
 Inspection of the answers shows that the model is producing repetitive patterns like "E E E E E ...". (Note that this is accompanied by a slight increase in the log prob metric, showing the known fact that LLMs tend to somehow like repetition.)
 
-Those metrics show that we face a fundamental trade-off: stronger steering increases concept inclusion but degrades fluency, and finding the balance is the challenge. This is further complicated by the very large standard deviation : for a given steering coefficient, some prompts lead to good results while others completely fail. Even if all metrics somehow tell the same story, we have to decide how to select the optimal steering coefficient. We could simply use the mean of the three LLM judge metrics, but we can easily see that this would lead to select the unsteered model (low $\alpha$) as the best model, which is not what we want. For that, we can use on **the harmonic mean criterion proposed by AxBench**.
+Those metrics show that we face a fundamental trade-off: stronger steering increases concept inclusion but degrades fluency, and finding the balance is the challenge. This is further complicated by the very large standard deviation: for a given steering coefficient, some prompts lead to good results while others completely fail. Even though all metrics somehow tell the same story, we have to decide how to select the optimal steering coefficient. We could simply use the mean of the three LLM judge metrics, but we can easily see that this would lead us to select the unsteered model (low $\alpha$) as the best model, which is not what we want. For that, we can use **the harmonic mean criterion proposed by AxBench**.
 
 import harmonic_mean_curve from './assets/image/sweep_1D_harmonic_mean.png'
 
 <Image src={harmonic_mean_curve} alt="Arithmetic (left) and harmonic (right) mean of the three LLM-judge metrics as a function of steering coefficient." caption="Arithmetic (left) and harmonic (right) mean of the three LLM-judge metrics as a function of steering coefficient." />
 
-First of all, we can see that the harmonic mean curve is very noisy. Despite the fact that we used 50 prompts to evaluate each point, the inherent discreteness of the LLM-judge metrics and the stochasticity of LLM generation leads to a noisy harmonic mean. This is something to keep in mind when trying to optimize steering coefficients.
+First, the results show the harmonic mean curve is very noisy. Despite the fact that we used 50 prompts to evaluate each point, the inherent discreteness of the LLM-judge metrics and the stochasticity of LLM generation leads to a noisy harmonic mean. This is something to keep in mind when trying to optimize steering coefficients.
 
 Still, from that curve, we can select the optimal $\alpha = 8.5$. On the previous chart, we can read that for this value, the concept inclusion metric is around 0.75, while instruction following is 1.5 and fluency around 1.0.
 
-Even for this optimal coefficient, those values are hardly satisfying, indicating that the model struggles to both reference the concept while maintaining a reasonable level of fluency and instruction following.
+Even with this optimal coefficient, these values are hardly satisfactory, indicating that the model struggles to both reference the concept while maintaining a reasonable level of fluency and instruction following.
 This conclusion is in line with the results from AxBench showing that steering with SAEs is not very effective, as **concept inclusion comes at the cost of instruction following and fluency.**
 
 Note that the harmonic mean we obtained here (about 0.45) is higher than the one reported in AxBench (about 0.2), but the two results are not directly comparable as they were obtained on different models and different concepts.
@@ -335,12 +334,12 @@ import evaluation1_naive from './assets/image/evaluation1_naive.png'
 
 <Image src={evaluation1_naive} alt="Detailed evaluation of steering with single feature" caption="Detailed evaluation of steering with single feature at optimal coefficient."/>
 
-We can see that on all metrics, **the reference model with prompts significantly outperforms the steered model.** This is consistent with the findings by AxBench that steering with SAEs is not very effective. However, our numbers are not as dire as theirs. We can see a average score in concept inclusion compared to the reference model (1.03), while maintaining a reasonable level of instruction following (1.35), at the price of a drop in fluency (0.78 vs 1.55 for the prompted model), which is impaired by repetitions (0.27) or awkward phrasing.
+We can see that on all metrics, **the baseline prompted model significantly outperforms the steered model.** This is consistent with the findings by AxBench that steering with SAEs is not very effective. However, our numbers are not as dire as theirs. We can see an average score in concept inclusion compared to the reference model (1.03), while maintaining a reasonable level of instruction following (1.35). However, this comes at the price of a fluency drop (0.78 vs. 1.55 for the prompted model), as fluency is impaired by repetitions (0.27) or awkward phrasing.
 
-Overall the harmonic mean of the three LLM-judge metrics is 1.67 for the prompted model, against 0.344 for the steered model.
+Overall, the harmonic mean of the three LLM-judge metrics is 1.67 for the prompted model, against 0.344 for the steered model.
 
 <Note type="info">
-    As can be seen on the bar chart, the fact that the evaluation is noisy leads to scary large error bars, especially for the LLM-judge metrics and the harmonic mean. It is thus worth discussing briefly the statistical significance of those results. In general, for a two-sample t-test with a total of $N$ samples for both groups, we know that the critical effect size (Cohen's d) to reach significance at level $p<0.05$ is $d =(1.96) \frac{2}{\sqrt{N}}$. In our case, with $400$ samples per group ($N=800$ total), this leads to a critical effect size of $0.14$. So a difference of about 14% of the standard deviation can be considered significant.
+    As can be seen on the bar chart, the fact that the evaluation is noisy leads to frighteningly large error bars, especially for the LLM-judge metrics and the harmonic mean. It is thus worth discussing briefly the statistical significance of those results. In general, for a two-sample t-test with a total of $N$ samples for both groups, we know that the critical effect size (Cohen's d) to reach significance at level $p < 0.05$ is $d =(1.96) \frac{2}{\sqrt{N}}$. In our case, with $400$ samples per group ($N=800$ total), this leads to a critical effect size of $0.14$. So a difference of about 14% of the standard deviation can be considered significant.
 </Note>
 
 ### 3.5 Correlations between metrics
@@ -354,13 +353,13 @@ import metrics_correlation from './assets/image/sweep_1D_correlation_matrix.png'
 The matrix above shows several interesting correlations.
 First, **LLM instruction following and fluency are highly correlated** (0.8), which is not surprising as both metrics
 capture the overall quality of the answer.
-But as observed in our results, they are unfortunately **anticorrelated with concept inclusion**, showing the tradeoff between steering strength and answer quality.
+However, as observed in our results, they are unfortunately **anticorrelated with concept inclusion**, showing the tradeoff between steering strength and answer quality.
 
 The explicit inclusion metric (presence of the word 'eiffel') is only partially correlated with the LLM-judge concept inclusion metric (0.45), showing that the model can apparently reference the Eiffel Tower without explicitly mentioning it (we've also seen that sometimes Eiffel was misspelled but that was still considered as a valid reference by the LLM judge).
 
-We see that the **repetition metric is strongly anticorrelated with fluency and instruction followin** (-0.9 for both).
+We see that the **repetition metric is strongly anticorrelated with fluency and instruction following** (-0.9 for both).
 
-Finally, minus log probability under the reference model is partially linked to fluency and instruction following (since more surprising answers are often less fluent), but also to concept inclusion, reflecting that referencing the Eiffel Tower often leads to more surprising answers.
+Finally, negative log probability under the reference model is partially linked to fluency and instruction following (since more surprising answers are often less fluent), but also to concept inclusion, reflecting that referencing the Eiffel Tower often leads to more surprising answers.
 
 From this analysis, we can see that **although the LLM-as-a-judge metrics are the most reliable, the auxiliary metrics can provide useful information about the quality of the answers**.
 This is useful as it means we can use them as a guide for optimization, without having to rely on costly LLM evaluations. Even if the final evaluation will have to be done with LLM-judge metrics.
@@ -376,11 +375,11 @@ Having found optimal coefficients, we now investigate two complementary improvem
 First, we tried to clamp the activations rather than using the natural additive scheme.
 Intuitively, this prevents the model from going to excessively high activations. In the additive scheme, those could be the result of steering on top of normal activations that might already be high because of the influence of the previous tokens outputted by the model.
 
-This clamping approach was the one used by Anthropic in their Golden Gate demo, but the AxBench paper reported that on their case it was less effective than the addition scheme. We decided to test it on our case.
+This clamping approach was the one used by Anthropic in their Golden Gate demo, but the AxBench paper reported that in their case it was less effective than the addition scheme. We decided to test it on our case.
 
 ### 4.1 Clamping
 
-We tested the impact of clamping on the same steering vector at the optimal steering coefficient found previously ($\alpha=8.5$). We evaluated the model on the same set of prompts with 20 sample each and a maximum output length of 512 tokens.
+We tested the impact of clamping on the same steering vector at the optimal steering coefficient found previously ($\alpha=8.5$). We evaluated the model on the same set of prompts with 20 samples each and a maximum output length of 512 tokens.
 
 import evaluation_clamp_gen from './assets/image/evaluation2_clamp_gen.png'
 
@@ -388,13 +387,13 @@ import evaluation_clamp_gen from './assets/image/evaluation2_clamp_gen.png'
 
 The image below shows the results of clamping compared to the additive scheme. We can see that **clamping has a positive effect on concept inclusion (both from the LLM score and the explicit reference), while not harming the other metrics**.
 
-We thus decided to prefer clamping the activation, in line with the choice made by Anthropic.
+We therefore opted for clamping, in line with the choice made by Anthropic.
 
 ### 4.2 Generation parameters
 
 We have seen that repetition is a major cause of loss of fluency when steering with SAEs.
-To mitigate that, we tried to apply lower the temperature, and applu a repetition penalty during generation.
-This is a simple technique that consists in penalizing the logit of tokens that have already been generated, preventing the model from repeating itself.
+To mitigate this, we tried applying a lower temperature, and apply a repetition penalty during generation.
+This is a simple technique that consists of penalizing the logit of tokens that have already been generated, preventing the model from repeating itself.
 We used a penalty factor of 1.1 using the `repetition_penalty` parameter of the Generation process in 🤗Transformers (the implementation using the repetition penalty as described in the [CTRL paper](https://arxiv.org/abs/1909.05858))
 
 As we can see, applying a repetition penalty reduces as expected the 3-gram repetition, and has **a clear positive effect on fluency, while not harming concept inclusion and instruction following.**
@@ -406,25 +405,25 @@ As we can see, applying a repetition penalty reduces as expected the 3-gram repe
 Even after those improvements, we still found that steering with a single SAE feature was not very effective, and concept inclusion lying way below the maximum possible value of 2.0.
 Since our investigation on Neuronpedia revealed that **the Eiffel Tower concept was represented by many features in different layers**, we hypothesized that steering several of those features simultaneously could lead to better results.
 
-Indeed it has been reported that common phenomenons are **feature redundancy and feature splitting**. This happens when a concept is represented by several features that are often co-activated or are in charge of the same concept in slightly different contexts. The sparsity constraint used during SAE training tends to favor such splitting, as it is often more efficient to use several features that activate less often, than a single feature that would activate more often.
+Indeed it has been reported that common phenomena are **feature redundancy and feature splitting**. This happens when a concept is represented by several features that are often co-activated or are responsible of the same concept in slightly different contexts. The sparsity constraint used during SAE training tends to favor such splitting, as it is often more efficient to use several features that activate less often, than a single feature that would activate more often.
 
-Those phenomena mean that **steering only one of those features might thus be insufficient to fully activate the concept, or to activate it consistently across different prompts.** Moreover, activating one feature without the others might cause loss of fluency, as the model might experience activation patterns that are out of distribution compared to what it was trained on.
+These phenomena suggest that **steering only one of those features might thus be insufficient to fully activate the concept, or to activate it consistently across different prompts.** Moreover, activating one feature without the others might cause loss of fluency, as the model might experience activation patterns that are out of distribution compared to what it was trained on.
 
 ### 5.1 Layer and features selection
-Overall, **we identified 19 candidate features**, located in layers 3, 7, 11, 15, 19, 23, and 27. Note that those layers were the only ones for which SAEs were available, so it is likely that other features representing the Eiffel Tower exist in other layers.
+In total, **we identified 19 candidate features**, located in layers 3, 7, 11, 15, 19, 23, and 27. Note that those layers were the only ones for which SAEs were available, so it is likely that other features representing the Eiffel Tower exist in other layers.
 
-We looked for those feature using the search tool in Neuronpedia, and selected them based on their top activating prompts in the dataset. We kept only those features that unambiguously referenced the Eiffel Tower, and discarded features that seemed to be more generally about Paris, towers, famous landmarks in big cities, or simply tokens like "E" of "iff".
+We looked for those feature using the search tool in Neuronpedia, and selected them based on their top activating prompts in the dataset. We kept only those features that unambiguously referenced the Eiffel Tower, and discarded features that seemed to be more generally about Paris, towers, famous landmarks in big cities, or simply tokens like "E" or "iff".
 
-Among those 19 features, we selected all the features located in the intermediary layers 11, 15, 19 and 23. We decided to leave aside features in earlier layers (six features in layer 3 and three features layer 7) or latest layers (two features in layer 27). This choice is motivated by the observations that features in intermediary layers are more likely to represent abstract high-level concepts. This led us to select 8 candidate feature for our multi-layer steering.
+Among those 19 features, we selected all the features located in the intermediate layers 11, 15, 19 and 23. We decided to leave out features in earlier layers (six features in layer 3 and three features in layer 7) or later layers (two features in layer 27). This choice is motivated by the observations that features in intermediate layers are more likely to represent abstract high-level concepts. This led us to select 8 candidate features for our multi-layer steering.
 
 ### 5.2 Optimization methodology
 
-Finding the optmal steering coefficients for multiple features is a challenging optimization problem.
+Finding the optimal steering coefficients for multiple features is a challenging optimization problem.
 First, the parameter space grows with the number of features, making grid search or random search quickly intractable.
 Second, the target function (the harmonic mean of LLM-judge metrics) is noisy and non-differentiable, making gradient-based optimization impossible.
 Finally, evaluating the target function is costly, as it requires generating answers from the steered model and evaluating them with an LLM judge.
 
-To tackle those challenges, we decided to rely on **bayesian optimization** to search for the optimal steering coefficients, and we devised an auxiliary cost function to guide the optimization when the harmonic mean is zero.
+To tackle those challenges, we decided to rely on **Bayesian optimization** to search for the optimal steering coefficients, and we devised an auxiliary cost function to guide the optimization when the harmonic mean is zero.
 
 #### 5.2.1 Cost function
 
@@ -434,45 +433,45 @@ To mitigate that, we decided to define an auxiliary cost function that would be
 $$
 \mathrm{cost} = |\mathrm{surprise} - s_0| + k\ \text{rep3}
 $$
-We selected target surprise $s_0$ and weight $k$ that maximally correlates with the mean of LLM judge metrics (leading to $s_0 = 1.2$ and $k=3$).
+We selected target surprise $s_0$ and weight $k$ such that this cost maximally correlates with the mean of LLM judge metrics (leading to $s_0 = 1.2$ and $k=3$).
 
 Overall, our cost function was defined as the harmonic mean of LLM-judge metrics, and we penalized it with a small fraction (0.05) of the auxiliary cost when the harmonic mean was zero, in order to give some signal to the optimizer.
 
 #### 5.2.2 Dealing with noise
 
 In principle, we want to minimize *the expected value of our target function over the distribution of prompts and samples*.
-But each call to the steered model will effectively only give a noisy estimate of that target, evaluated on a single prompt and one sample.
+However, each call to the steered model will effectively only give a noisy estimate of that target, evaluated on a single prompt and one sample.
 
-We are in a situation of a black-box optimization, where each evaluation of the target function is costly (as it involves generating a full answer from the model) and noisy (as it depends on the prompt and the sample). To tackle this, we decided to rely on **bayesian optimization**.
+We are in a situation of a black-box optimization, where each evaluation of the target function is costly (as it involves generating a full answer from the model) and noisy (as it depends on the prompt and the sample). To tackle this, we decided to rely on **Bayesian optimization**.
 
-Bayesian Optimization (BO) is known to be well-suited for multidimensional non-differentiable costly blackbox optimization, while being able to handle noisy evaluations. To mitigate the noise, we could have averaged the target function over several prompts and samples, but this would have been costly, especially when evaluating points that are not promising. For very noisy function, performing bayesian optimization directly on the raw function is known to be more effective than averaging multiple noisy evaluations for each point.
+Bayesian Optimization (BO) is known to be well-suited for multidimensional non-differentiable costly black-box optimization, while being able to handle noisy evaluations. To mitigate the noise, we could have averaged the target function over several prompts and samples, but this would have been costly, especially when evaluating points that are not promising. For very noisy functions, performing Bayesian optimization directly on the raw function is known to be more effective than averaging multiple noisy evaluations for each point.
 
 #### 5.2.3 Bayesian optimization
 
-The idea beyond BO is to build a surrogate model of the target function using a Gaussian Process (GP), and use that surrogate to select promising candidates to evaluate next. As we evaluate new points, we update the GP model, and iteratively refine our surrogate of the target function.
+The idea behind BO is to build a surrogate model of the target function using a Gaussian Process (GP), and use that surrogate to select promising candidates to evaluate next. As we evaluate new points, we update the GP model, and iteratively refine our surrogate of the target function.
 
-For that, we used the BoTorch library, which provides a flexible framework to perform BO using PyTorch. More details are given in appendix.
+For that, we used the BoTorch library, which provides a flexible framework to perform BO using PyTorch. More details are given in the appendix.
 
 ### 5.3 Results of multi-layer optimization
 
-We performed optimisation using 2 features (from layer 15 and layer 19) and then 8 features (from layers 11, 15, 19 and 23), following the idea that steering the upper-middle layer is likely to be more effective to activate high-level concepts.
+We performed optimization using 2 features (from layer 15 and layer 19) and then 8 features (from layers 11, 15, 19 and 23), following the idea that steering the upper-middle layer is likely to be more effective to activate high-level concepts.
 
 Results are shown below and compared to single-layer steering.
 
 import evaluation_final from './assets/image/evaluation3_multiD.png'
 
-<Image src={evaluation_final} alt="Comparison of single-layer and m_muulti-layer steering" caption="Comparison of single-layer and multi-layer steering." />
+<Image src={evaluation_final} alt="Comparison of single-layer and multi-layer steering" caption="Comparison of single-layer and multi-layer steering." />
 
-As we can see on the chart, steering 2 or even 8 features simultaneously only leads to **only marginal improvements** compared to steering only one feature. Although fluency and instruction following are improved, concept inclusion slightly decreases, leading to a harmonic mean that is only marginally better than single-layer steering. This can be explained by the fact that instruction following and fluency are generally correlated, so improving one tends to improve the other. Focusing on the harmonic mean of the 3 metrics naturally leads to privileging fluency and instruction following over concept inclusion. Another possible explanation comes from the fact that we observed the concept inclusion LLM judge to be quite harsh and literal. Sometimes mention of Paris or a large metal structure were not considered as valid references to the Eiffel Tower, which could explain the low concept inclusion scores.
+As we can see on the chart, steering 2 or even 8 features simultaneously leads to **only marginal improvements** compared to steering only one feature. Although fluency and instruction following are improved, concept inclusion slightly decreases, leading to a harmonic mean that is only marginally better than single-layer steering. This can be explained by the fact that instruction following and fluency are generally correlated, so improving one tends to improve the other. Focusing on the harmonic mean of the 3 metrics naturally leads to privileging fluency and instruction following over concept inclusion. Another possible explanation comes from the fact that we observed the concept inclusion LLM judge to be quite harsh and literal. Sometimes mention of Paris or a large metal structure were not considered as valid references to the Eiffel Tower, which could explain the low concept inclusion scores.
 
-Overall, those disappointing results contradicts our initial hypothesis that steering multiple complementary features would help better represent the concept and maintain fluency. One possible explanation is our inability to find the true optimum, as the harmonic mean metric is very noisy and hard to optimize. Another explanation could be that the selected features are actually redundant rather than complementary, and that steering one of them is sufficient to activate the concept.
+Overall, those disappointing results contradict our initial hypothesis that steering multiple complementary features would help better represent the concept and maintain fluency. One possible explanation is our inability to find the true optimum, as the harmonic mean metric is very noisy and hard to optimize. Another explanation could be that the selected features are actually redundant rather than complementary, and that steering one of them is sufficient to activate the concept.
 
 
 ## Conclusion & Discussion
 
 ### Main conclusions
 
-In this study, we have shown how to use sparse autoencoders to steer a lightweight open source model (Llama 3.1 8B Instruct) to create a conversational agent obsessed with the Eiffel Tower, similar to the Golden Gate Claude experiment. As reported by the AxBench paper, and as can be experienced on Neuronpedia, steering with SAEs is harder than we might have thought, and finding good steering coefficients is not easy.
+In this study, we have shown how to use sparse autoencoders to steer a lightweight open-source model (Llama 3.1 8B Instruct) to create a conversational agent obsessed with the Eiffel Tower, similar to the Golden Gate Claude experiment. As reported by the AxBench paper, and as can be experienced on Neuronpedia, steering with SAEs is harder than we might have thought, and finding good steering coefficients is not easy.
 
 We first showed that simple improvements like clamping feature activations and using repetition penalty and lower temperature can help significantly. We then devised a systematic approach to optimize steering coefficients using bayesian optimization, and auxiliary metrics correlated with LLM-judge metrics.
 
@@ -482,19 +481,19 @@ A way to explain this lack of improvement could be that the selected features ar
 
 Overall, our results seem less discouraging than those of AxBench, and show that steering with SAEs can be effective, using clamping, a slightly different generation procedure and possibly combining multiple features. However, at this stage, those results are hard to generalize and our work is not really comparable to the AxBench results, since they use different model, different concepts, different SAEs (Gemmascope vs Andy Arditi's), different prompts. This is in line with the success of the Golden Gate Claude demo, although we don't know all the details of their steering method.
 
-TODO : embed a demo
+TODO: embed a demo
 
 ### Possible next steps
 
 Possible next steps:
 - Failure analysis on the cases where steering fails (about 20% have at least one zero metric)
-- Check other layers for 1D optimisation
-- Check complementary vs redudancy by monitoring activation changes in subsequent layer's features.
+- Check other layers for 1D optimization
+- Check complementary vs redundancy by monitoring activation changes in subsequent layers' features.
 - Try other concepts, see if results are similar
-- Try on larger models, see if results are better
-- Vary the temporal steering pattern : steer prompt only, or answer only, or periodic steering
-- Try to include earlier and latest layers, see if it helps
-- Investigate clamping : why do we find that clamping helps, similar to Anthropic, while AxBench found the opposite? We could think it prevents extreme activations, but it could also counteract some negative feedback behavior, when other parts of the model try to compensate for the added steering vector. (analogy with biology, where signaling pathways are often regulated by negative feedback loops)
+- Try with larger models, see if results are better
+- Vary the temporal steering pattern: steer prompt only, or answer only, or periodic steering
+- Try to include earlier and later layers, see if it helps
+- Investigate clamping: why do we find that clamping helps, similar to Anthropic, while AxBench found the opposite? We could hypothesize it prevents extreme activations, but it could also counteract some negative feedback behavior, when other parts of the model try to compensate for the added steering vector. (analogy with biology, where signaling pathways are often regulated by negative feedback loops)
 
 
 
@@ -526,7 +525,7 @@ We used the reduced parameterization presented earlier, searching for an optimal
 To favor noise reduction at promising locations, every 5 steps we decided to resample the best point found so far.
 In that case, by *best* we mean the point with the lowest GP posterior $\mu(x)$. (Note that this is different from the point with the lowest observed value which might be a lucky noisy outlier).
 
-#### 5.2.4 Gradient descent
+#### Gradient descent
 
 Performing gradient on the GP posterior is very cheap since it only involves differentiating the kernel function.
-We thus performed gradient descent starting from 500 random points in the parameter space, and optimized using a target being higher confidence bound $\mu(x) + \beta\sigma(x)$, to favor points that are not only predicted to be good, but also with low uncertainty. We then performed a clustering to group together the points that converged to the same local minimum, and selected the best cluster as candidate for evaluation.
+We thus performed gradient descent starting from 500 random points in the parameter space, and optimized using a target being upper confidence bound $\mu(x) + \beta\sigma(x)$, to favor points that are not only predicted to be good, but also with low uncertainty. We then performed a clustering to group together the points that converged to the same local minimum, and selected the best cluster as candidate for evaluation.

app/src/content/embeds/banner.html

@@ -1,6 +1,6 @@
 
 <div style="display: flex; justify-content: center;">
-    <img src="eiffel_tower_llama.png"
+    <img src="src/content/assets/image/eiffel_tower_llama.png"
             alt="Eiffel Tower Llama" 
             style="max-width:80%; height:auto; border-radius:8px;" />
 </div>