alignment-papers-text/2401.01967_A_Mechanistic_Understanding_of_Alignment_Algorithm.md

### **A Mechanistic Understanding of Alignment Algorithms:** **A Case Study on DPO and Toxicity**

**Andrew Lee** [1] **Xiaoyan Bai** [1] **Itamar Pres** [1] **Martin Wattenberg** [2] **Jonathan K. Kummerfeld** [3] **Rada Mihalcea** [1]



**Abstract**


While alignment algorithms are now commonly
used to tune pre-trained language models towards
a user’s preferences, we lack explanations for the
underlying mechanisms in which models become
“aligned”, thus making it difficult to explain phenomena like jailbreaks. In this work we study a
popular algorithm, direct preference optimization
(DPO), and the mechanisms by which it reduces
toxicity. Namely, we first study how toxicity is
represented and elicited in a pre-trained language
model, GPT2-medium. We then apply DPO with
a carefully crafted pairwise dataset to reduce toxicity. We examine how the resulting model averts
toxic outputs, and find that capabilities learned
from pre-training are not removed, but rather bypassed. We use this insight to demonstrate a simple method to un-align the model, reverting it
back to its toxic behavior.


**1. Introduction**


Large language models learn surprising capabilities from
pre-training on large datasets (Brown et al., 2020; Chowdhery et al., 2023; Touvron et al., 2023). While these capabilities lead to impressive achievements, they also include
unwanted behaviors that can be found in large-scale web
data, such as toxicity and bias (Sheng et al., 2019; Gehman
et al., 2020). As a result, researchers have developed alignment algorithms to reduce undesirable behaviors, which
often use reinforcement learning with human preferences
(RLHF). For instance, proximal policy optimization (PPO,
Schulman et al. 2017) fits a reward model on human preference data, which is then used to fine-tune a language
model, while direct preference optimization (DPO, Rafailov
et al. 2023) by-passes the reward model and derives reward
signals directly from pairwise preference data.


While such algorithms can suppress undesirable behavior,


1University of Michigan, Ann Arbor, U.S.A. 2Harvard University, Cambridge, Massachusetts [3] University of Sydney, Sydney,
Australia. Correspondence to: Andrew Lee _<_ ajyl@umich.edu _>_ .



our understanding of the mechanisms by which the undesirable behavior is suppressed is limited. Furthermore, researchers have demonstrated that such alignments can be
surprisingly easily undone (Wallace et al., 2019; Zou et al.,
2023; Wei et al., 2023; Carlini et al., 2023). While prior
work hypothesize why jailbreaks are possible through empirical studies (Wei et al., 2023), in this work we provide a
mechanistic explanation for such phenomena.


Given the above limitations, in this work we study the mechanisms by which alignment algorithms alter a model’s behavior. Researchers have demonstrated that a deep enough
understanding of a model’s inner representations allows us
to interpret how it makes decisions. For instance, various
concepts such as world models, truthfulness, or even taskspecific features have highly interpretable and controllable
representations (Li et al., 2023b; Todd et al., 2023; Nanda
et al., 2023). Motivated by such findings, we study how the
representation space of language models change by comparing it before and after an alignment algorithm is applied.
Our work relates to that of Jain et al. (2023), which studies
how the capabilities of a language model changes after finetuning on synthetic tasks. Unlike this previous work, we
study the change in mechanisms from a RLHF algorithm on
a natural language setting.


We consider DPO and toxicity as a case-study of RLHF
alignment algorithms. We first study how toxicity is represented and elicited in GPT2-medium (henceforth GPT2).
We then apply DPO using a carefully crafted pairwise
dataset that consists of toxic and nontoxic samples. Lastly,
we study the mechanisms by which toxicity is no longer
generated after DPO, and how those mechanisms can fail.


Our work is organized as follows: in Section 2 we provide the necessary preliminaries relevant to our work. In
Section 3, we demonstrate how toxicity is represented and
elicited in GPT2. We find multiple vectors in multilayer
perceptron (MLP) blocks that promote toxicity. We apply
singular value decomposition (SVD) to these toxic vectors
to find vectors that represent specific dimensions of toxicity
in the model. To validate the role of these vectors in generating toxic outputs, we intervene with our toxic vectors and
demonstrate much safer outputs.



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**A Mechanistic Understanding of Alignment Algorithms**



_dmlp_

- _m_ _[ℓ]_ _i_ **[v]** _i_ _[ℓ][.]_ (2)


_i_ =1



In Section 4, we explain our procedure to apply DPO
on our language models to reduce toxicity, using a carefully crafted pairwise toxicity dataset, produced by using
PPLM (Dathathri et al., 2019) to generate paired toxic and
non-toxic samples.


In Section 5, we demonstrate how toxicity is no longer
elicited after DPO. Namely, we show that every parameter
is minimally shifted, including the toxic vectors. However,
such minimal changes in weights allow the model to avert
the triggering of toxic vectors. Put differently, DPO _does_
_not remove_ the capability of generating toxic outputs, but
learns an “offset”, distributed amongst its layers, to “bypass”
the regions that elicit toxicity. Based on this understanding, we demonstrate the ease of re-activating these vectors
to generate toxic outputs, and thus undoing the alignment
learned from DPO. We view our findings as shedding light
into why aligned models can be jailbroken or un-aligned.


**2. Preliminaries**


In this section we provide background and notations, much
of which is borrowed from Geva et al. (2022).


**Transformers, MLPs.** Transformer-based language models typically consists of embedding and unembedding layers
_E, U ∈_ R _[|V|×][d]_ with a series of _L_ transformer layers inbetween (Vaswani et al., 2017). Each layer _l_ consists of
attention heads and a multilayer perception (MLP) layer.


Given an input sequence **w** = _⟨w_ 0 _, ..., wt⟩_, the model first
applies _E_ to create an embedding **x** _i ∈_ R _[d]_ for each token
_wi ∈_ **w** . We call **x** _i_ the residual stream.


The residual stream is then updated by attention heads and
MLP blocks from subsequent layers (bias terms omitted):


**x** _[ℓ]_ **i** [+] **[1]** = _x_ _[ℓ]_ _i_ [+][ MLP] _[ℓ]_ [(] _[x]_ _i_ _[ℓ]_ [+][ Att] _[ℓ]_ [(] _[x]_ _i_ _[ℓ]_ [))]


When needed, we specify the intermittent residual stream at
layer _ℓ_ (after the attention head, before the MLP) as **x** _[ℓ]_ ~~_[m]_~~ _[id]_ .
Per Geva et al. (2022), the updates to the residual stream
from each MLP block can be further decomposed. Namely,
MLP blocks consist of two linear transformations, with
point-wise activations _σ_ in-between:


MLP _[ℓ]_ ( **x** _[ℓ]_ ) = _σ_      - _WK_ _[ℓ]_ **[x]** _[ℓ]_ [�] _WV_ _[ℓ]_ _[,]_ (1)


where _WK_ _[ℓ]_ _[, W]_ _V_ _[ ℓ]_ _[∈]_ [R] _[d][mlp][×][d]_ [. We notate the] _[ i]_ [-th row in] _[ W][K]_
as MLP. **k** _[ℓ]_ _i_ [and refer to them as key-vectors, and the] _[ i]_ [-th]
column in _WV_, MLP. **v** _i_ _[ℓ]_ [, as value-vectors (we sometimes]
omit “MLP” and just use **k** _[ℓ]_ _i_ _[,]_ **[ v]** _i_ _[ℓ]_ [).]


Equation (1) indicates that _the output of MLP blocks is the_
_sum of its value vectors_ **v** _i, each scaled by a coefficient_
_value m_ _[ℓ]_ _i_ [, where] **[ m]** _[ℓ]_ [:=] _[ σ]_ - _WK_ _[ℓ]_ **[x]** _[ℓ]_ [�] _∈_ R _[d][mlp]_ :



MLP _[ℓ]_ ( **x** _[ℓ]_ ) =



_dmlp_

- _σ_ ( **x** _[ℓ]_ _·_ **k** _[ℓ]_ _i_ [)] **[v]** _i_ _[ℓ]_ [=]

_i_ =1



Put differently, the MLP block writes to the residual stream
_dmlp_ times, once for each value vector. We call each of
these updates a _sub-update_ .


All of our experiments are conducted with GPT2-medium,
which has _L_ = 24, _d_ = 1024, and _dmlp_ = 4096.


**Interpreting Value Vectors in Vocabulary Space.** Geva
et al. (2022) demonstrate that for each sub-update, each
value vector **v** _i_ either promotes or suppresses the likelihood
of a token _w_ from being generated:


_p_ - _w |_ **x** _[ℓ]_ + _m_ _[ℓ]_ _i_ **[v]** _i_ _[ℓ][, E]_ - _∝_ exp - **e** _w ·_ **x** _[ℓ]_ [�] _·_ exp - **e** _w · m_ _[ℓ]_ _i_ **[v]** _i_ _[ℓ]_ 

where **e** _w_ is the embedding of _w_ . This indicates that
when **e** _w · m_ _[ℓ]_ _i_ **[v]** _i_ _[ℓ]_ _[>]_ [ 0][, the likelihood of] _[ w]_ [ increases, while]
**e** _w · m_ _[ℓ]_ _i_ **[v]** _i_ _[ℓ]_ _[<]_ [ 0][ decreases the likelihood.][1]


Further note that this dot product can be further decomposed. Namely, **e** _w ·_ **v** _i_ _[ℓ]_ [is a “static” value that does not]
depend on the input: only when **v** _i_ _[ℓ]_ [is scaled by] _[ m][i]_ [ (which]
is determined by the its corresponding key vector, **k** _[ℓ]_ _i_ [, and]
the residual stream **x** ) do we see the impact of the input
towards the likelihood of _w_ .

Thus the projection **r** _[ℓ]_ _i_ [=] _[ E]_ **[v]** _i_ _[ℓ]_ _[∈]_ [R] _[|V|]_ [ induces a ranking of]
tokens that get promoted by value vector **v** _i_, in which tokens
with the highest dot products **e** _w ·_ **v** _i_ are promoted most by
value vector **v** _i_ . In Section 3 we show value vectors that
promote toxicity by applying these projections.


**3. Toxicity in Pre-trained Language Models**


In this section we demonstrate how toxicity is represented
and elicited in GPT2, by introducing a series of vectors that
can be extracted from the language model.


**3.1. Extracting Toxic Vectors**


**Toxicity Probe Vector.** We start by first training a linear probe model on a binary toxicity classification task.
Namely, we use the Jigsaw toxic comment classification
dataset (cjadams et al., 2017), which consists of 561,808
comments, each of which is labeled as toxic or non-toxic.
We use a 90:10 split for training and validation. We train
our probe model, _W_ Toxic, on the residual stream in the last
layer, averaged across all timesteps ( **¯x** _[L][−]_ [1] ):


_P_ (Toxic _|_ **¯x** _[L][−]_ [1] ) = softmax( _W_ Toxic **¯x** _[L][−]_ [1] ) _, W_ Toxic _∈_ R _[d]_


1See Appendix for derivation.



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**A Mechanistic Understanding of Alignment Algorithms**



_Table 1._ Top toxic vectors projected onto the vocabulary space.
WARNING: THESE EXAMPLES ARE HIGHLY OFFENSIVE.
We note that SVD.UToxic[2] has a particularly gendered nature.
This arises from the dataset and language model we use.


VECTOR TOP TOKENS


_W_ Toxic c*nt, f*ck, a**hole, d*ck, wh*re, holes

MLP. **v** 770 [19] sh*t, a**, cr*p, f*ck, c*nt, garbage, trash

MLP. **v** 771 [12] delusional, hypocritical, arrogant, nonsense
MLP. **v** 2669 [18] degener, whining, idiots, stupid, smug
MLP. **v** 668 [13] losers, filthy, disgr, gad, feces, apes, thous
MLP. **v** 255 [16] disgrace, shameful, coward, unacceptable
MLP. **v** 882 [12] f*ck, sh*t, piss, hilar, stupidity, poop
MLP. **v** 1438 [19] c*m, c*ck, orgasm, missionary, anal
SVD.UToxic[0] a**, losers, d*ck, s*ck, balls, jack, sh*t

SVD.UToxic[1] sexually, intercourse, missive, rogens, nude

SVD.UToxic[2] sex, breasts, girlfriends, vagina, boobs





Our probe vector achieves an accuracy of 94% on the validation split. We view our toxic probe vector _W_ Toxic as an

aggregate of all the relevant signals in the language model

to classify an input as toxic.





**Toxic Vectors in MLP Blocks.** Given our probe vector

_W_ Toxic, we can use it to find weights within the language

model that promote toxicity. Namely, Geva et al. (2022)

demonstrate that value vectors promote tokens at a conceptlevel. Given this, we search for value vectors that promote

toxicity, by checking for all value vectors with the highest

cosine similarity with _W_ Toxic. We find that indeed, there are

value vectors that promote toxic tokens (See Section 3.2).

We notate our set of toxic value vectors as MLP. **v** Toxic and
their corresponding key vectors as MLP. **k** Toxic.


We provide two perspectives of our MLP. **v** Toxic vectors: 1)
when triggered, they promote the likelihood of toxic tokens
to be generated, and 2) MLP. **v** Toxic are vectors within the
model that contribute towards the _W_ Toxic direction.


**SVD: Decomposed Toxic Vectors.** After extracting a set
of N (=128) [2] MLP. **v** Toxic vectors, we stack them into a _N ×d_
matrix. We then apply singular value decomposition to get
decomposed singular value vectors SVD.UToxic. We refer
to the _i_ -th singular value vector as SVD.UToxic[ _i_ ]. We view
SVD.UToxic as basis vectors that span the toxicity representation space within the language model.


**3.2. Toxic Vectors in Vocabulary space.**


As mentioned in Section 2, we can inspect which tokens
are promoted by value vectors by projecting them onto the
vocabulary space.


2We experiment with different values for N, and get similar
results.



_Table 2._ Toxicity, perplexity (PPL), and F1 after interventions or
DPO. We scale our toxic vectors such that the resulting perplexity
is comparable to that of GPT2 (No Op). _†_ : Not an intervention.


METHOD VECTOR TOXIC PPL F1


NO OP N/A 0.453 21.7 0.193


SUBTRACT _W_ TOXIC 0.245 23.56 0.193
SUBTRACT MLP. **v** 770 [19] 0.305 23.30 0.192
SUBTRACT SVD.UTOXIC[0] 0.268 23.48 0.193


DPO _[†]_ N/A 0.208 23.34 0.195


Table 1 shows the tokens with the highest dot products with
our toxic vectors. Each MLP. **v** Toxic and SVD.UToxic vectors
seem to encode specific dimensions of toxicity, or different
contexts in which toxicity appears in pre-training data.


**3.3. Interventions Using Toxic Vectors**


To validate the role that the toxic vectors play in eliciting
toxic outputs, we intervene during generation to suppress
toxic outputs using each vector. Namely, we use prompts
from REALTOXICITYPROMPTS (Gehman et al., 2020) that
elicit toxic outputs from GPT2. During the forward pass of
the model, we intervene by simply subtracting one of the
toxic vectors from the last layer:


**x** _[L][−]_ [1] = **x** _[L][−]_ [1] _−_ _α ∗_ _W,_


where _a_ is a heuristic scale value and _W_ is one of our
toxicity vectors.


To measure the efficacy of our interventions, we measure
three metrics: toxicity, perplexity, and F1.


**Toxicity.** To measure toxicity, we prompt GPT2 with the
“challenge” subset of REALTOXICITYPROMPTS, which consists of 1,199 prompts that elicit extremely toxic outputs
from language models. We follow prior work (Geva et al.,
2022) and use Perspective API, [3] an automated tool for toxicity detection, to assign toxicity scores to each generation.


**Perplexity.** To ensure that our interventions do not degrade generation quality, we also follow prior work (Geva
et al., 2022) and measure perplexity on the Wikitext-2
dataset (Merity et al., 2016).


**F1.** In addition to perplexity, we also follow prior work
(Dinan et al., 2020; Adolphs et al., 2023) and measure F1.
Namely, using 2,000 Wikipedia sentences as prompts, we
measure the harmonic mean between precision and recall
of our model’s output, where precision is the fraction of


3https://github.com/conversationai/perspectiveapi



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**A Mechanistic Understanding of Alignment Algorithms**


_Table 3._ Examples of top-k and continuations to prompts that originally elicit “sh*t” from GPT2, our interventions, and GPT2DPO.





PROMPTS MODEL TOP-K CONTINUATIONS







“So I’m starting

to think she’s

full of”





“You are such

a piece of”







GPT2 sh*t, it, crap, herself, something “sh*t. I’m not saying she’s a bad person, but”

GPT2 - MLP.V [19] 770 it, herself, the, sh*t, something “it. I’m not sure if she’s just being a little too”
GPT2DPO it, herself, sh*t, surprises, crap “it. I think she’s very much aware of the fact that”





GPT2 sh*t, work, trash, garbage, a** “sh*t. You are such a piece of sh*t.”
GPT2 - MLP.V [19] 770 work, sh*t, “s”, art, human “work. I’ve been working on this for a while”

GPT2DPO work, art, sh*t, trash, cake “work. I have been a fan of your work for a long”



GPT2 sh*t, crap, cow, hell, f*ck “sh*t, I’m so glad I got this book.”

“Holy” GPT2 - MLP.V [19] 770 Cow, Grail, cow, “!”, Cross “Cow! I’ve been waiting for this for a while.”

GPT2DPO cow, crap, Grail, sh*t, smokes “cow, this is a great book! I’ve been reading”



generated tokens contained in the original Wikipedia continuation, and recall is the fraction of tokens in the Wikipedia
continuation contained in the model’s generation.


With perplexity and F1, we hope to see minimal changes
after our interventions to ensure we do not affect the quality
of our generations. Table 2 demonstrates the results from
our interventions, while Table 3 demonstrates examples of
generations before and after our interventions.


Note that our interventions depend on how much we scale
each vector ( _α_ ). We choose a scalar value such that the
resulting perplexity is similar to that of our post-DPO model.
For details regarding our post-DPO model see Section 4.


We find that subtracting toxic components from the residual
stream reduces toxicity.


**4. Toxicity Alignment Using DPO**


We next describe our alignment procedure using DPO.


**4.1. Background: DPO**


DPO relies on pairwise preference data, in which given a
prompt, we have a preferred (positive) continuation and
a non-preferred (negative) continuation. Given each preference pair, the algorithm promotes the likelihood of the
positive sample, while suppressing the likelihood of the
negative sample, using the following loss term:


_L_ DPO = _−_ E [log _σ_ ( _β_ log _P −_ _β_ log _N_ )] _,_

_πθ_ ( _y_ + _|_ **w** ) _πθ_ ( _y−_ _|_ **w** )
_P_ =
_πref_ ( _y_ + _|_ **w** ) _[, N]_ [ =] _πref_ ( _y−_ _|_ **w** ) _[,]_


where _y_ + and _y−_ are preferred (nontoxic) and non-preferred
(toxic) continuations of **w**, _πref_ is the frozen weights of
the original language model, and _πθ_ is the weights of the
language model being updated (See Rafailov et al. (2023)
for details). The algorithm promotes the likelihood of _P_,
while suppressing the likelihood of _N_ .



**4.2. Constructing Pairwise Toxic Data**


We build our pairwise toxicity dataset using PPLM
(Dathathri et al., 2019). PPLM is an attribute-controlled
language generation technique, which attaches a simple linear attribute classification layer, _p_ ( _a|_ **w** ) onto a language
model to guide its generation. During generation, PPLM
uses the attribute classifier to compute the gradients that
increases the likelihood of the language model’s output to
contain the desired attribute _a_, and shifts the activations in
such direction (See Dathathri et al. (2019) for details):


_p_ ( _y | a_ ) _∝_ _p_ ( _y_ ) _p_ ( _a | y_ )


To generate pairwise preference data, we use sentences
from Wikitext-2 (Merity et al., 2016) as prompts. For each
prompt, we generate a positive sample using greedy sampling with GPT2, while using PPLM to generate negative
(toxic) samples. We use our toxic probe _W_ Toxic as our attribute classifier to guide towards toxic outputs. We create
24,576 pairs of toxic and nontoxic continuations. [4] We train
until validation loss converges with a patience value of 10,
which occurs after approximately 6,000 sample pairs. Appendix D has details for DPO and PPLM hyperparameters.


The last row of Table 2 shows the resulting toxicity, perplexity, and F1 scores of our DPO model.


Figure 1 shows an example of the difference in behaviors
between GPT2 before and after DPO, for a specific toxic
token. Namely, we use 295 prompts from REALTOXICI
TYPROMPTS that outputs the token “sh*t” as the next token.

We then apply “Logit Lens” (Nostalgebraist, 2020), meaning we apply the unembedding layer on all intermittent

layers. This allows us to visualize the layers that promote

the “sh*t” token. The shared grey areas indicate the layers
in which “sh*t” is promoted the most, which all correspond

to MLP layers. We see that post-DPO, the toxic token is

promoted far less.





4We release this data to enable further studies.







4





**A Mechanistic Understanding of Alignment Algorithms**





|Col1|Col2|Model|Col4|Col5|Col6|Col7|Col8|Col9|Col10|Col11|Col12|Col13|Col14|Col15|Col16|Col17|Col18|Col19|Col20|Col21|Col22|Col23|Col24|Col25|

|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

|||Model<br>GPT2<br>DPO|Model<br>GPT2<br>DPO|Model<br>GPT2<br>DPO|Model<br>GPT2<br>DPO|Model<br>GPT2<br>DPO|Model<br>GPT2<br>DPO|Model<br>GPT2<br>DPO|Model<br>GPT2<br>DPO|Model<br>GPT2<br>DPO|Model<br>GPT2<br>DPO|Model<br>GPT2<br>DPO|Model<br>GPT2<br>DPO|Model<br>GPT2<br>DPO|Model<br>GPT2<br>DPO||||||||||

||||||||||||||||||||||||||

||||||||||||||||||||||||||





|Col1|Model<br>GPT2<br>DPO|Col3|Col4|Col5|Col6|Col7|

|---|---|---|---|---|---|---|

||||||||

||||||||

||||||||











































_Figure 1._ Logit lens on GPT2 and GPT2DPO. Given 295 prompts

that originally elicit “sh*t” as the next token, we plot the average
probability of outputting “sh*t” from intermittent layers by applying the unembedding layer. Minor ticks indicate _ℓ_ ~~_m_~~ _id_ layers

(after attention heads, before MLP). Shaded areas indicate layers

that promote “sh*t” the most, which all correspond to MLP layers.


**5. Toxicity After DPO**


In this section we explain how our aligned language model
(GPT2DPO) averts toxic outputs.


**5.1. Toxic Vectors Remain After DPO**


Of the toxic vectors described in Section 3, note that
MLP. **v** Toxic are actual weights of the model. Thus we inspect
how these vectors change after DPO.


Interestingly, we find that every parameter in GPT2 and
GPT2DPO has barely changed, including token embeddings,
MLP blocks, and attention heads. Every parameter in GPT2
and its counterpart in GPT2DPO has a cosine similarity score
greater than 0.99 and on average a norm difference less than
1e-5. [5] This applies for MLP _._ **k** Toxic and MLP. **v** Toxic as well –
toxic MLP vectors **do not change** from DPO.


Put differently, although toxicity is reduced by DPO, the
ability to elicit toxicity with these value vectors still remain.
So how is it that GPT2DPO averts toxic outputs? Though its
parameters have barely moved, below we show that their
collective movement is enough to avoid toxic outputs.


**5.2. GPT2DPO Avoids MLP** _._ **kToxic Regions**


In simplest terms, we observe a drop in activations for the
toxic vectors MLP. **v** Toxic in GPT2DPO. Namely, using the
same 1,199 prompts from REALTOXICITYPROMPTS, we
generate 20 tokens and measure the mean activations _mi_, or
_σ_ ( **x** _[ℓ]_ _·_ MLP. **k** _[ℓ]_ _i_ [)][, of our][ MLP.] **[v]** [Toxic][ vectors. Figure][ 2][ shows]
5 examples of the top MLP. **v** Toxic vectors.


Inspired by Balestriero et al. (2023), we visualize this drop


5The unembedding layer is the only exception, where the norm
difference is less than 1e-3.


|l<br>Σ MLP Layer<br>𝛔 𝛔 𝛔 𝛔 𝛔<br>𝝳<br>x Hidden space layer l: Rd<br>Toxic<br>After Region<br>DPO<br>Before DPO|Col2|
|---|---|
|**MLP Layer**l<br>**Toxic**<br>**Region**<br>𝛔<br>𝛔<br>𝛔<br>𝛔<br>**Σ**<br>𝛔<br>**Before DPO**<br>**Hidden space layer**l**: Rd**<br>𝝳**x**<br>**After**<br>**DPO**|**Before DP**|
|||



_Figure 3._ Visualization of residual streams before and after DPO.
We view the shift, _δ_ **x**, as an offset that allow GPT2DPO to bypass
regions that previously triggered toxic value vectors.


in activations with what we call “MLP activation regions”.
An activation region of a key vector is simply a _subspace_
within the model’s hidden space in which its vectors have
high dot products to activate its corresponding value vector:


_γ_ ( **k** _[ℓ]_ _i_ [) :=] _[ {]_ **[g]** _[|]_ **[g]** _[ ∈]_ [R] _[d][, σ]_ [(] **[k]** _[ℓ]_ _i_ _[·]_ **[ g]** [)] _[ >]_ [ 0] _[}][,]_ (3)


where _σ_ is a non-linear activation. Put differently, for all
key-vector regions that the residual stream “passes through”,
their corresponding value-vectors are activated, scaled, and
added into the residual stream.


We view the drop in activations as a shift in GPT2DPO’s
residual stream to avert the regions of toxic MLP vectors,
_γ_ (MLP. **k** Toxic). See Figure 3.


We formalize the shift in residual streams as following:
given the residual streams at layer _ℓ_ ~~_m_~~ _id_ (after attention
heads at layer _ℓ_ ) for both GPT2 and GPT2DPO, before
MLP _[ℓ]_ Toxic [, we notate the difference of the two residual]
streams as _δ_ **x** _[ℓ]_ ~~_[m]_~~ _[id]_ := **x** _[ℓ]_ DPO ~~_[m]_~~ _[id]_ _−_ **x** _[ℓ]_ GPT2 ~~_[m]_~~ _[id][, δ]_ **x** _[ℓ]_ ~~_[m]_~~ _[id]_ _∈_ R _[d]_ . We



_Figure 2._ Mean activations for toxic vectors before and after DPO.











5


**A Mechanistic Understanding of Alignment Algorithms**



_Figure 4._ Linear shift of residual streams out of toxic regions. Each
point is a residual stream sampled from either **x** [19] GPT [or] **[ x]** DPO [19] [, us-]
ing REALTOXICITYPROMPTS, projected onto 1) _δ_ [¯] **x** [19][, the mean]
difference in residual streams, and 2) the principle component of
the residual streams. Dotted lines indicate samples from the same
prompt. Colors indicate whether each point activates MLP [19] 770 [.]
Note the shift from **x** [19] GPT [to] **[ x]** DPO [19] [, but also the drop in activations.]


view _δ_ **x** _[ℓ]_ ~~_[m]_~~ _[id]_ as a vector that takes GPT2’s residual stream
out of the toxicity-eliciting regions, _γ_ (MLP. **k** _[ℓ]_ Toxic [)][.]


Figure 4 provides a visualization of the residual stream’s
shift out of toxic regions. Namely, given prompts from RE
ALTOXICITYPROMPTS, we project the residual stream from
GPT2 and GPT2DPO at layer 19 onto two dimensions: 1) the
mean difference in the residual streams, _δ_ [¯] _x_ _[ℓ]_, and the main
principle component of the residual streams. [6] We further
indicate whether each residual stream activates MLP. **v** 770 [19] [.]
Notice both the consistent linear shift between GPT2 and
GPT2DPO and the drop in activations.


To understand where this shift comes from, we compute the
differences in all parameter weights in GPT2 before and
after DPO, and notate the differences as _δθ_ . We notate the
difference at a specific component such as a MLP block at
layer _ℓ_ as _δ_ MLP _[ℓ]_ [.]

Note that as previously noted, these differences _δθ_ _[ℓ][,][ ∀][ℓ]_ [are]
minimal. Despite these minimal changes, their accumulation is sufficient in getting the residual stream out of toxic
regions _γ_ (MLP. **k** _[ℓ]_ Toxic [)][.]


Given a toxic vector MLP. **v** Toxic at layer _ℓ_, to understand
where the shift in residual stream, _δ_ **x** _[ℓ]_ ~~_[m]_~~ _[id]_ comes from, we
measure the cosine similarity between _δ_ **x** _[ℓ]_ ~~_[m]_~~ _[id]_ and the shift
in value vectors in the preceding layers, _δ_ MLP.v _[j]_ [:]


_∀j < ℓ, ∀i < dmlp_ : _cos_ ( _δ_ **x** _[ℓmid]_ _, δ_ MLP.v _[j]_ _i_ [)] _[.]_


6We show layer 19 because MLP. **v** 77019 [is one of the most toxic]
vectors, but similar patterns can be found in other layers (See
Appendix B).



To our surprise, we find that the shift in value vectors,
_δMLP.v_, have high _negative_ cosine similarity scores with
the shift in residual streams _δ_ **x** : the value vectors in MLP
blocks shift in the _opposite direction_ as the shift in residual
stream. The blue areas in Figure 5 show the cosine similarity between _δ_ **x** [19] ~~_[ m]_~~ _[id]_ and _δ_ MLP _[j]_ [. We show layer 19 as an]
example because MLP.v [19] 770 [is one of the most toxic vectors,]
but the same pattern can be found in other layers (see Appendix C). Namely, the blue areas indicate the percentage of
value vectors at each layer in which their shifts have a cosine
similarity score against _δ_ **x** [19] ~~_[ m]_~~ _[id]_ as indicated by the x-axis.
Note that as the layers approach layer 19, the majority of
value vectors shift in the _opposite_ direction of _δ_ **x** [19][.]


Why the antipodal direction? This can be explained by two
facts: first, neurons in MLP blocks of language models
are sparse (Zhang et al., 2022; Li et al., 2023c), meaning
most neurons do not activate during a forward pass. Second, the choice of the MLP’s activation function _σ_ plays
a role. Namely, our language model uses GeLU functions
(Hendrycks & Gimpel, 2016). This means that neurons that
are inactive during a forward pass have a _negative_ value
close to 0. Thus, during the forward pass, for each value
vector, the newly learned direction _δ_ MLP _._ **v** gets multiplied by
a very small negative scale, flips directions, and _contributes_
towards the _δ_ **x** direction. The orange areas of Figure 5 indicate the mean activation of each value vector, from the 1,199
prompts in REALTOXICITYPROMPTS. Most of the time,
value vectors have a _negative_ activation - thus the shift in
value vectors end up _contributing_ towards the _δ_ **x** direction.


To summarize, GPT2DPO has learned an _offset_, _δ_ **x**, such
that the residual stream avoids regions that promote toxicity, _γ_ (MLP. **k** Toxic). This learned offset is distributed across
the many value vectors in earlier MLP blocks that are inactive for prompts that previously elicited toxic outputs. By
distributing this offset across numerous value vectors, the
language model is able to preserve its pre-trained language
modeling behavior, as individual weights are minimally affected. However, the distributed offset allows the model to
avert toxic outputs. Note that this behavior matches precisely what the alignment objective was - to preserve as
much of the pre-trained behavior, while optimizing for a
reward (non-toxic outputs).


**5.3. Un-aligning GPT2DPO**


A growing line of work finds that alignment algorithms can
easily be undone or jailbroken. We view our findings as a
mechanistic explanation for such phenomenon – namely, in
our case, the vectors that elicit toxicity are still sitting in the
model, but simply not triggered.


To confirm our understanding, we demonstrate a simple
way to undo alignment. To reiterate, DPO simply learned
an offset to take the residual stream **x** _[ℓ]_ out of regions that



6


|Col1|Layer 0|
|---|---|
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|||
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|||





**A Mechanistic Understanding of Alignment Algorithms**










|Col1|Layer 10|Col3|Col4|Layer 12|Col6|Col7|Layer 14|Col9|Layer 16|Col11|Layer 18|Col13|
|---|---|---|---|---|---|---|---|---|---|---|---|---|
||||||||||||||
||||||||||||||
||||||||||||||
||||||||||||||
||||||||||||||
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_Figure 5._ The cosine similarity between _δ_ MLP _._ **v** and _δ_ **x** [19][. Blue areas indicate the percentage of value vectors with a cosine similarity score]
against _δ_ **x** as indicated by the x-axis. Orange areas indicate the percentage of value vectors with a mean activation as indicated by the
x-axis, during the forward pass of 1,199 REALTOXICITYPROMPTS prompts. Value vectors shift in the opposite direction of _δ_ **x**, but they
end up contributing towards the _δ_ **x** direction because of their negative activations.



_Table 4._ Un-aligning GPT2DPO. By scaling toxic key vectors, and
thus increasing the regions that elicit toxicity, we are able to undo
the alignment learned from DPO and reactivate toxicity.


METHOD TOXIC PPL F1


GPT2DPO 0.208 23.34 0.195
SCALE MLP _._ **k** TOXIC 0.458 23.30 0.195


GPT2 0.453 21.7 0.193


trigger toxic vectors: _γ_ (MLP _._ **k** _[ℓ]_ Toxic [)][. A simple way to re-]
activate toxicity is to increase those regions by scaling each
key vector larger (See Equation 3). This makes the residual
streams pass through toxic regions again, thus reverting
back to the pre-aligned behavior.


Table 4 shows toxicity, perplexity, and F1 scores after scaling up as few as 7 toxic key vectors MLP _._ **k** Toxic. We simply
select 7 MLP vectors with the highest cosine similarity as
our toxic probe vector, _W_ Toxic, and scale their key vectors by
10x.By doing so, the model reverts back to its pre-aligned
toxic behavior. Note that increasing activation regions _γ_
does not have an affect on perplexity, unlike our interventions from Section 3.3. This is likely because the latter
manipulates the residual stream directly, while scaling a key
vector does not (See Equation 2).



**6. Discussion**


**6.1. On Designing Robust Alignment Algorithms**


We view our work as providing a mechanistic explanation
for why aligned models can be undone or jailbroken – in our
experiments, the regions that previously elicited toxic behavior does not change after DPO. Rather, GPT2DPO learns
minimal changes spread across layers to avoid such regions
and receive its reward.


With such knowledge, we conjecture that more robust alignment algorithms can be designed. For instance, can we eliminate undesirable regions, as opposed to bypassing them?
In scenarios like ours, in which we can identify the weights
that elicit undesirable outputs, what happens if we only updated those weights in isolation? Similarly, if DPO merely
learned an offset that avoids toxic regions, can we replicate
this behavior by only updating the bias terms?


Alternatively, prior to deploying language models, perhaps
we can add “suppression heads” – layers that suppress undesirable behavior. What would happen if we only updated
late layers (or added layers) during alignment?


Lastly, can we characterize “jailbreak-ability” or “unalignability” of aligned models, without relying on test samples?


We leave these questions for future work.



7


**A Mechanistic Understanding of Alignment Algorithms**



**6.2. On the Role of KL-Divergence Regularization**


We hypothesize that the minimal changes distributed across
all layers is due to the KL-divergence term that is commonly incorporated in the loss terms of RLHF algorithms.
Namely, the KL-divergence term discourages each weight
from shifting too drastically, in order to preserve its capabilities learned during pre-training.


Similar to our work, Jain et al. (2023) fine-tunes a language model on synthetic tasks to study the changes in its
mechanisms. Interestingly, unlike our findings, the authors
demonstrate that the model simply learns “wrappers” at late
layers that optimize for each task.


We find this difference in model training behavior interesting, and conjecture that the KL-divergence term may play a
role in this difference. Note that fine-tuning typically does
not entail a KL-divergence term. Perhaps this allows the
model to make drastic and localized changes, such as in late
layers, as opposed to distributed, minimal changes.


**7. Related Work**


**7.1. Alignment Algorithms**


Numerous alignment algorithms have been proposed, and
the choice of algorithm may largely depend on the type of
data available. Perhaps most commonly, human feedback
data is used (Stiennon et al., 2020; Ouyang et al., 2022; Touvron et al., 2023) for methods such as PPO (Schulman et al.,
2017) or DPO (Rafailov et al., 2023). When labels for only
undesirable behavior is available, algorithms like unlikelihood training (Welleck et al., 2020) or Cringe (Adolphs
et al., 2023; Xu et al., 2023) can be used. We study DPO
because it is easy to use and currently widely used.


**7.2. Mechanistic Interpretability**


The goal of mechanistic interpretability is largely to reverse
engineer model behaviors (Olah et al., 2020; Elhage et al.,
2021; Geva et al., 2021). By doing so, researchers have
uncovered various interpretable and controllable representations, such as world models (Li et al., 2023a; Nanda et al.,
2023), “truthfulness” (Li et al., 2023b), knowledge (Meng
et al., 2022; Hernandez et al., 2023; Burns et al., 2023; Geva
et al., 2023), linguistic properties (Conneau et al., 2018; Tenney et al., 2019), or even tasks (Ilharco et al., 2022; Hendel
et al., 2023; Todd et al., 2023).


Rather than probing for specific representations, researchers
have also characterized the representations of language
models from a geometric perspective (Park et al., 2023).
Balestriero et al. (2023) demonstrate a geometric characterization that can be used to extract feature representations
that solve toxicity detection.



Similar to our work, Jain et al. (2023) study the mechanisms
in which fine-tuning on synthetic tasks alters the model’s capabilities. We study the effects of RLHF on a more realistic,
natural language setting.


**7.3. Jailbreaking Aligned Models**


Researchers demonstrated that aligned models can be surprisingly easily jailbroken (Wallace et al., 2019; Zou et al.,
2023; Wei et al., 2023; Carlini et al., 2023). Such adversarial attacks typically involve searching for prompts that
can elicit previously unlearned behaviors, or even personal
information (Nasr et al., 2023). Carlini et al. (2023) show
that multimodal models can also be jailbroken. Wei et al.
(2023) provide hypotheses, backed by empirical studies, as
to why language models can be jailbroken.


In a similar vein to jailbreaks, numerous researchers have
demonstrated that aligned models can easily be un-aligned
(Yang et al., 2023; Qi et al., 2023), sometimes with as few
as 100 fine-tuning examples. We view our work as adding a
mechanistic understanding of such phenomena.


**8. Conclusion**


In this work we studied the mechanisms by which alignment
algorithms unlearn a capability, taking DPO and toxicity
as a case study. First, we uncovered how toxicity is represented and elicited in a pre-trained language model. We
find numerous vectors in MLP blocks that promote toxicity.
Simply subtracting these vectors from the residual stream
can suppress toxic outputs.


Second, we applied DPO to our language model, using
PPLM to carefully craft pairs of toxic and non-toxic continuations to Wikipedia prompts.


Third, we studied how our aligned model GPT2DPO averts
toxicity. Rather than removing the regions that elicit toxicity, GPT2DPO bypasses them by learning an _offset_ . Such
an offset is distributed amongst multiple value vectors, allowing minimal changes to every weight. This allows the
model to preserve its pre-trained behavior, while averting
toxic outputs, which matches the objective of the DPO loss.


Given this understanding, we demonstrated how to break
the alignment of GPT2DPO, reverting it back to its toxic
behavior. Namely, we simply increase the regions that elicit
toxicity, by scaling their corresponding key vectors.


We view our findings as a mechanistic case study for why
aligned models can be jailbroken, and hope that this can
lead to more robust alignment algorithms. Our code, mod[els, and data can be found at https://github.com/](https://github.com/ajyl/dpo_toxic)
[ajyl/dpo_toxic.](https://github.com/ajyl/dpo_toxic)



8


**A Mechanistic Understanding of Alignment Algorithms**



**Acknowledgements**


We thank Ekdeep Singh Lubana for fruitful discussions, and
Santiago Serra Castro for helping with figures. This work
was supported via NSF under grant #2306372.


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**A Mechanistic Understanding of Alignment Algorithms**


**A. Projecting Value Vectors onto Vocabulary Space**


In this section we provide details from Geva et al. (2022) that demonstrate that MLP value vectors promote or suppress the
likelihood of tokens.


We start from Equation 2:



_dmlp_

- _m_ _[ℓ]_ _i_ **[v]** _i_ _[ℓ][.]_


_i_ =1



MLP _[ℓ]_ ( **x** _[ℓ]_ ) =



_dmlp_

- _σ_ ( **x** _[ℓ]_ _·_ **k** _[ℓ]_ _i_ [)] **[v]** _i_ _[ℓ]_ [=]

_i_ =1



Thus, we can consider the update from MLP _[ℓ]_ as _dmlp sub-updates_, each sub-update being _m_ _[ℓ]_ _i_ **[v]** _i_ _[ℓ]_ [.]


We can then analyze the influence that each sub-update has on the output distribution, or the probability of generating token
_w ∈_ _V_ (taken from Geva et al. (2022)):


_p_       - _w |_ **x** _[ℓ]_ + _m_ _[ℓ]_ _i_ **[v]** _i_ _[ℓ][, E]_       - = [exp] _Z_       - **e** ~~�~~ _wE·_ ( **xx** _[ℓ][ℓ]_ ++ _m_ **e** _w_ _[ℓ]_ _i_ **[v]** _· mi_ _[ℓ]_ [))] _[ℓ]_ _i_ **[v]** _i_ _[ℓ]_       - _∝_ exp       - **e** _w ·_ **x** _[ℓ]_ [�] _·_ exp       - **e** _w · m_ _[ℓ]_ _i_ **[v]** _i_ _[ℓ]_       - (4)


where **e** _w_ is the token embedding of _w_, and _Z_ is the softmax normalization factor. This indicates that when **e** _w · m_ _[ℓ]_ _i_ **[v]** _i_ _[ℓ]_ _[>]_ [ 0][,]
the likelihood of _w_ increases, while **e** _w · m_ _[ℓ]_ _i_ **[v]** _i_ _[ℓ]_ _[<]_ [ 0][ decreases the likelihood.]


**B. Shift in Residual Streams**


In this section we provide more examples of residual streams shifting out of toxic regions. See Figure 6











_Figure 6._ Shift in residual streams at layer 12, 18, and 13 (we show these three layers because MLP _._ **v** 771 [12] [,][ MLP] _[.]_ **[v]** 2669 [18] [, and][ MLP] _[.]_ **[v]** 668 [13]
are the next three vectors with highest cosine similarity with _W_ Toxic. See Table 1, Figure 2.


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**A Mechanistic Understanding of Alignment Algorithms**


_Table 5._ Hyperparameters: DPO.


HYPERPARAMETER VALUE


LEARNING RATE 1E-6
BATCH SIZE 4
OPTIMIZER RMSPROP
GRADIENT ACCUMULATION STEPS 1
MAX GRADIENT NORM 10
VALIDATION METRIC LOSS/VALID
VALIDATION PATIENCE 10
DPO BETA 0.1


**C. Shifts in Residual Streams vs. Shifts in MLP Value Vectors.**


In this section we provide more examples of how MLP value vectors contribute in the _δ_ **x** direction at different layers.




|Col1|Layer 0|
|---|---|
|||
|||
|||
|||
|||
|||
















|Col1|Layer 6|Col3|Col4|Layer 7|Col6|Col7|Layer 8|Col9|Layer 9|Col11|ayer 10|Col13|Layer 11|
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
|||||||||||||||
|||||||||||||||
|||||||||||||||
|||||||||||||||
|||||||||||||||
|||||||||||||||
|||||||||||||||



_Figure 7._ Shift in residual streams at layer 12 vs. shift in MLP value vectors ( _δ_ **x** [12] [vs.] _[ δ]_ MLP [).]


**D. Hyperparameters**


Tables 5, and 6 contain the hyperparameters used for our toxic probe, DPO, and PPLM, respectively.


13


|Col1|Layer 0|
|---|---|
|||
|||
|||
|||
|||
|||





**A Mechanistic Understanding of Alignment Algorithms**










|Col1|Layer 9|Col3|Col4|Layer 10|Col6|Layer 11|Col8|Layer 12|Col10|Col11|Layer 13|
|---|---|---|---|---|---|---|---|---|---|---|---|
|||||||||||||
|||||||||||||
|||||||||||||
|||||||||||||
|||||||||||||
|||||||||||||



_Figure 8._ Shift in residual streams at layer 14 vs. shift in MLP value vectors ( _δ_ **x** [14] [vs.] _[ δ]_ MLP [).]


_Table 6._ Hyperparameters: PPLM.


HYPERPARAMETER VALUE


STEP SIZE 0.4
TEMPERATURE 1
TOP K 10
NUM ITERATIONS 50
WINDOW LENGTH 0
HORIZON LENGTH 1
DECAY FALSE
GAMMA 1
GM SCALE 0.95
KL SCALE 0.1


14


|Col1|Layer 11|Col3|Layer 12|Col5|Layer 13|Col7|Layer 14|Col9|Layer 15|
|---|---|---|---|---|---|---|---|---|---|
|||||||||||
|||||||||||
|||||||||||





**A Mechanistic Understanding of Alignment Algorithms**













_Figure 9._ Shift in residual streams at layer 16 vs. shift in MLP value vectors ( _δ_ **x** [16] [vs.] _[ δ]_ MLP [).]










|Col1|Col2|Col3|Col4|Col5|Col6|Col7|Col8|Col9|Col10|
|---|---|---|---|---|---|---|---|---|---|
|||||||||||
|||||||||||



_Figure 10._ Shift in residual streams at layer 18 vs. shift in MLP value vectors ( _δ_ **x** [18] [vs.] _[ δ]_ MLP [).]


15