Summarize this paper: ""

What if I already have a Low-rank decomposition of a model  (specifically, a model called 'quietstar-8-ahead' which uses  'mistralai/Mistral-7B-v0.1' as a base) It was Created using LoRD. For context, the 'quietstar-8-ahead' model is a result of this paper: ""Quiet-STaR: Language Models Can Teach Themselves to
Think Before Speaking
Eric Zelikman
Stanford University
Georges Harik
Notbad AI Inc
Yijia Shao
Stanford University
Varuna Jayasiri
Notbad AI Inc
Nick Haber
Stanford University
Noah D. Goodman
Stanford University
Abstract
When writing and talking, people sometimes pause to think. Although
reasoning-focused works have often framed reasoning as a method of
answering questions or completing agentic tasks, reasoning is implicit
in almost all written text. For example, this applies to the steps not
stated between the lines of a proof or to the theory of mind underlying
a conversation. In the Self-Taught Reasoner (STaR, Zelikman et al. 2022),
useful thinking is learned by inferring rationales from few-shot examples
in question-answering and learning from those that lead to a correct
answer. This is a highly constrained setting – ideally, a language model
could instead learn to infer unstated rationales in arbitrary text. We
present Quiet-STaR, a generalization of STaR in which LMs learn to
generate rationales at each token to explain future text, improving their
predictions. We address key challenges, including 1) the computational cost
of generating continuations, 2) the fact that the LM does not initially know
how to generate or use internal thoughts, and 3) the need to predict beyond
individual next tokens. To resolve these, we propose a tokenwise parallel
sampling algorithm, using learnable tokens indicating a thought’s start and
end, and an extended teacher-forcing technique. Encouragingly, generated
rationales disproportionately help model difficult-to-predict tokens and
improve the LM’s ability to directly answer difficult questions. In particular,
after continued pretraining of an LM on a corpus of internet text with
Quiet-STaR, we find zero-shot improvements on GSM8K (5.9%→10.9%)
and CommonsenseQA (36.3%→47.2%) and observe a perplexity improvement of difficult tokens in natural text. Crucially, these improvements
require no fine-tuning on these tasks. Quiet-STaR marks a step towards
LMs that can learn to reason in a more general and scalable way.
“Life can only be understood backwards; but it must be lived forwards.”
— Sren Kierkegaard
1 Introduction
Much of the meaning of text is hidden between the lines: without understanding why
statements appear in a document, a reader has only a shallow understanding. Moreover,
this has been repeatedly shown to be true for LMs as well, in the contexts of tasks ranging
from commonsense reasoning to theorem proving to programming (Wei et al., 2022b; Nye
et al., 2021; Zelikman et al., 2022; 2023a; Kojima et al., 2022). Reasoning about implications
of text to predict later text has consistently been shown to improve LM performance on a
variety of tasks, but methods for allowing LMs to learn from their reasoning (e.g., Zelikman
et al. 2022) have focused on solving individual tasks or predefined sets of tasks (e.g., Wei
et al. 2021b). These works rely on carefully curated datasets to provide either specific
reasoning tasks or in some cases, the reasoning itself. We instead ask, if reasoning is implicit
in all text, why shouldn’t we leverage the task of language modeling to teach reasoning?
1
arXiv:2403.09629v2 [cs.CL] 18 Mar 2024
START +
Sampled
Thought 4 + 1 END
LM LM LM
1 Discard
2 + 1 2
START +
1) Think
= 5 4 \n
2 2 END
Thought1 Thought3
Thought4 Thought5
Thought7
+
LM
LM LM LM
LM
Reward
Harms
Prediction
Helps
Prediction
LM
Thought2
LM LM LM LM LM
4
Thought0
LM
2) Talk 3) Learn
Original Text
Sampled
Thought
Thought6
Figure 1: Quiet-STaR. We visualize the algorithm as applied during training to a single
thought. We generate thoughts, in parallel, following all tokens in the text (think). The
model produces a mixture of its next-token predictions with and without a thought (talk).
We apply REINFORCE, as in STaR, to increase the likelihood of thoughts that help the model
predict future text while discarding thoughts that make the future text less likely (learn).
In particular, the Self-Taught Reasoner (STaR, Zelikman et al. 2022) showed that LMs
can bootstrap their reasoning ability on question-answering (QA) datasets by sampling
rationales to attempt to answer questions, training on rationales if they led to a correct final
answer, and then repeating this to iteratively solve more difficult problems. Yet, training
from curated QA datasets limits the scale and generalizability of the rationales. QA datasets,
especially high-quality ones, require thoughtful curation and will inherently only ever cover
a subset of reasoning tasks. Thus, we extend STaR – instead of the LM learning to reason
on particular tasks like mathematical QA, we train an LM to generate reasoning that helps
it infer future text from a large internet text corpus. As a result, we allow the LM to learn
from the diverse tasks present in language (Weber et al., 2021). This builds on an intuition
essential to the current language modeling paradigm, namely, that ”language models are
unsupervised multitask learners” (Radford et al., 2019). Thus, as in STaR, we leverage the
LM’s pre-existing reasoning ability to generate rationales and train the LM on them with a
REINFORCE-based reward (Williams, 1992). We refer to this technique as Quiet-STaR, as it
can be understood as applying STaR “quietly”, training the model to think before it speaks.
Broadly, Quiet-STaR proceeds by generating rationales after every token to explain future
text (think), mixing the future-text predictions with and without rationales (talk), and then
learning to generate better rationales using REINFORCE (learn). We apply Quiet-STaR
to Mistral 7B (Jiang et al., 2023) using the web text datasets OpenWebMath (Paster et al.,
2023) and Colossal Clean Crawled Corpus (C4, Raffel et al. 2020). We find that, even
without dataset-specific fine-tuning, Quiet-STaR results in improvements to zero-shot directreasoning abilities on CommonsenseQA (36.3%→47.2%) and GSM8K (5.9%→10.9%), and
that these improvements consistently increase with the number of tokens used in the LM’s
internal thoughts. Lastly, we qualitatively investigate patterns in the generated rationales.
In solving this task, we make the following contributions:
1. We generalize STaR to learn reasoning from diverse unstructured text data. To our
knowledge, this is the first work explicitly training LMs to reason generally from
text, rather than on curated reasoning tasks or collections of reasoning tasks.
2. We propose and implement a parallel sampling algorithm that makes our training
procedure scalable, generating rationales from all token positions in a given string.
3. We introduce custom meta-tokens at the start and end of each thought to allow
the LM to learn that it should be generating a rationale and when it should make a
prediction based on that rationale.
4. We apply a mixing head to retrospectively determine how much to incorporate the
next-token prediction from a given thought into the current next-token prediction.
5. We show that a non-myopic loss, including multiple tokens ahead for language
modeling, improves the effect of thinking.
6. On multiple tasks, we demonstrate that thinking allows the LM to predict difficult tokens better than one trained on the same web text, improving with longer thoughts.
2
10 20 30 40 50 60 70 80 90 100
Training Step
6%
7%
8%
9%
10%
11%
GSM8K Accuracy (%)
# Thought, Ahead Tokens
24, 12
16, 8
12, 4
10, 4
8, 4
Baseline
(a) GSM8K
10 20 30 40 50 60 70 80 90 100
Training Step
34%
36%
38%
40%
42%
44%
46%
CommonsenseQA Accuracy (%)
# Thought, Ahead Tokens
24, 12
16, 8
12, 4
10, 4
8, 4
Baseline
(b) CommonsenseQA
Figure 2: Generalization Results. We evaluate the extent to which the model trained
with Quiet-STaR generalizes to directly answering problems that require reasoning. The
left plot (a) shows the zero-shot accuracy on GSM8K, while the right plot (b) shows the
zero-shot accuracy on CommonsenseQA, without any fine-tuning. In both plots, the x-axis
represents training steps, and each line corresponds to a different number of thinking tokens
used during Quiet-STaR training. The y-axis measures the zero-shot direct accuracy on the
respective datasets. We also include an inference normalized version of this plot in Figure 6.
2 Related Work
2.1 Reasoning in Language Models
There have been many works on training and exploiting language models to solve difficult
tasks by first training them to reason through them. For example, Rajani et al. (2019)
demonstrated that a pre-trained language model fine-tuned to output on human reasoning
traces before answering multiple-choice commonsense reasoning questions outperformed
one trained directly on answers. Shwartz et al. (2020) demonstrated that language models,
when provided with some scaffolding, can generate these helpful chain-of-thought solutions
without additional supervision. Later, Nye et al. (2021) demonstrated that “scratchpads”
required less scaffolding when the language models were more capable, a result later
reinforced by Wei et al. (2022b), emphasizing informal tasks, and further strengthened by
Kojima et al. (2022), demonstrating this behavior could be accomplished zero-shot. Most
recently, Wang & Zhou (2024) showed further that for commonsense-question answering,
one could force a language model to leverage chain-of-thought reasoning by preventing it
from emitting any valid answer tokens unless it was confident. However, once again, these
approaches only work for a question-answer dataset, and Wang & Zhou (2024) relies on
heuristics to identify when the model has output answer tokens. Somewhat like TRICE
(Phan et al., 2023), we use the relative improvements in the log-likelihood of the target text
across rationales as an estimate of quality, but we simply subtract the mean reward and do
not incorporate more complex control variates.
2.2 Training Language Models to Reason
One direction that researchers have used to train language models to reason or improve their
reasoning is training the language model on mined reasoning traces or reasoning-like data
(Rajani et al., 2019; Wei et al., 2021a; Lewkowycz et al., 2022; Chung et al., 2022; Gunasekar
et al., 2023). Although this approach has been demonstrated to be effective, it comes with
drawbacks. It requires either manual annotation, which is sensitive to the capability of the
annotators and is off-policy for the language model (i.e., the distribution of reasoning is not
text that the language model would otherwise likely have generated). This approach is also
expensive, difficult to scale, and provides no clear path to solving problems harder than
those that the annotators are capable of solving.
Another direction for teaching reasoning relies on a language model’s own generated
reasoning, which can be seen as building on a large body of literature on self-play (Silver
et al., 2017; Anthony et al., 2017; Polu & Sutskever, 2020). These include methods such as the
3
Algorithm 1: Quiet Self-Taught Reasoner (Quiet-STaR)
Input: Language model θ0, training steps num steps, sequence length l, thought length
t, learning rate α, batch size b, number of thoughts nthoughts, number of ground
truth tokens used for supervising each thought ntrue
Output: Language model θ that generates rationales to predict future text
for i = 0 to num steps do
Sample batch of sequences X of length l
h
init ← hidden statesθi
(X)
for j = 1 to l in parallel using attention mask do
log p
init
j:j+ntrue ← lm headθi
(h
init
j:j+ntrue
) // Predict next tokens
Tj ← generate tokensθi
([X:j
; <start thought>], t, nthoughts) // Generate thought
Tj ← [Tj
; <end thought>]
h
thought
j:j+ntrue ← hidden statesθi
([X:j
; Tj
; Xj:j+ntrue−1
])
log p
thought
j:j+ntrue ← lm headθi
(h
thought
j:j+ntrue
) // Predict next tokens w/ thought
wj:j+ntrue ← mixing headθi
(h
thought
j:j+ntrue
, h
init
j:j+ntrue
)
log p
talk
j ← wj:j+ntrue · log p
init
j:j+ntrue
+ (1 − wj:j+ntrue ) · log p
thought
j:j+ntrue
// Mix logits
L
NLL
j ← − log p
talk
j:j+ntrue
(Xj+1:j+ntrue+1)
rj = log p
talk
j:j+ntrue
(Xj+1:j+ntrue+1) − log p
talk
j:j+ntrue
(Xj+1:j+ntrue+1)
∇θL
REINFORCE
j ← −rj1[rj > 0] · ∇θ
log pθi
(Tj
|[X:j
; <start thought>])
∇θLj ← ∇θL
NLL
j + ∇θL
REINFORCE
j
θi+1 ← θi − α ∑
l
j=1 ∇θLj // Update model parameters
return θnum steps
Self-Taught Reasoner (Zelikman et al., 2022), which demonstrated that a language model
iteratively trained on its reasoning that led to correct answers could solve increasingly
difficult problems. Later work aimed to leverage additional information or assumptions
such as Huang et al. (2022) which demonstrated that the algorithm proposed in STaR could
still work if one assumed that the majority-vote answer was correct (although this has
a lower ultimate performance). Further work has generalized the results of Zelikman
et al. (2022), such as Uesato et al. (2022) which demonstrated additional usefulness to
“process-based” supervision where incorrect reasoning traces were filtered, recently VSTaR (Hosseini et al., 2024) that demonstrates that training a verifier to guide generation
also improves performance, as well as TRICE (Hoffman et al., 2024) which maximizes
the marginal likelihood of the correct answer given several reasoning traces per problem.
Finally, related work has also explored learning intermediate reasoning in the constrained
setting of making mathematical statements, where statements in the model’s intermediate
reasoning could be constrained to only be valid mathematical statements (Poesia et al., 2023).
We include further discussion of related reasoning works in Appendix F.
2.3 Meta-tokens
Recently, a growing body of work has demonstrated the usefulness of custom tokens
optimized to perform specific functions in the context of a neural network – for this reason,
they have also been referred to as “function vectors.” (Todd et al., 2023). One of the original
instantiations of this was prompt-tuning (Lester et al., 2021) (and relatedly prefix-tuning (Li
& Liang, 2021)), where the embeddings corresponding to the tokens of a prompt could be
optimized to better accomplish a task. Others have applied meta-tokens to compress long
prompts (Li et al., 2023; Jung & Kim, 2023) for efficiency. Most relevant to this work, Mu et al.
(2024) optimized a token such that, when the tokens after it could not attend to the tokens
before it (i.e., a context compression token), it would provide sufficient information to future
tokens. Although we do not focus on compression, we share the problem of learning a
4
token that affects attention and controls complex downstream behavior. In one related work,
Goyal et al. (2023) show that learning a single ”pause” token (essentially representing each
token as two tokens) improves LM performance. However, unlike the thought tokens in our
work, this pause token does not initialize a thought – instead, it can be seen as acting as the
entirety of the thought. We find that reasoning in language is significantly more helpful.
3 Problem Statement
In this work, we introduce an auxiliary ‘rationale’ variable between each pair of observed
tokens of the sequence. We then aim to optimize a language model with parameters θ with
the capacity to generate intermediate thoughts (or rationales) such that
θ
∗ = arg maxθ
Ex [logpθ (xi:n|x0:i
, rationaleθ (x0:i))]
Note that, in principle, this provides no advantage over an optimal language model that
already correctly models the language’s distribution over strings. Yet, in practice, extensive
prior work has shown that language models benefit from intermediate rationales on reasoning tasks (Nye et al., 2021; Zelikman et al., 2022; Wei et al., 2022b). Some work has aimed
to explain the effects of chain-of-thought reasoning, namely attributing it to “locality of
experience” (Prystawski et al., 2024). More broadly, reasoning allows a model to decompose
a challenging computation into smaller steps. In effect, we train the model to learn which
decomposition and planning steps are effective in predicting future text. Also note that we
formulate the objective as accurately predicting the remaining sequence, rather than only
the next token. Once again, for an optimal LM these would be equivalent. However we find
that the non-myopic formulation leads to a more effective loss for learning rationales.
4 Quiet-STaR
4.1 Overview
Quiet-STaR operates with three main steps (Figure 1):
1. Parallel rationale generation (think, Subsection 4.2): In parallel across n tokens
xi
in an input sequence x0:n, we generate r rationales of length t: ci = (ci1
, . . . , cit),
resulting in n × r rationale candidates. We insert learned 
 and

 tokens to mark each rationale’s start and end.
2. Mixing post-rationale and base predictions (talk, Subsection 4.3): From the hidden
state output after each rationale, we train a ”mixing head” – a shallow MLP producing a weight determining how much the post-rationale next-token predicted logits
should be incorporated compared to the base language model predicted logits. This
approach eases distribution shift early in finetuning, due to introducing rationales.
3. Optimizing rationale generation (learn, Subsection 4.4): We optimize the rationale
generation parameters (start/end tokens and LM weights) to increase the likelihood
of rationales that make future text more probable. We use REINFORCE to provide
a learning signal to rationales based on their impact on future-token prediction.
To reduce variance, we apply a teacher-forcing trick to include in the loss the
likelihood of predicting not only the token after the thought but also later tokens.
4.2 Parallel Generation
A key challenge in Quiet-STaR is efficiently generating rationales at each token position
in the input sequence. Naively, this would require a separate forward pass for each token,
which becomes computationally intractable for long sequences.
We allow for highly parallel generation by first observing that an inference pass of a language
model produces a probability distribution over the next tokens for all input tokens. Naturally,
this allows us to sample one next token from each token in the input. If one has generated a
successor from each token, it is not possible to simply continue with the original sequence.
For example, imagine predicting the next token after each token of “< bos > the cat sat” one
might generate “yes orange saw down” – each successor by itself is a reasonable next token
5
a’ b’ c’ d’
a’’ b’’ c’’ d’’
Base a b c d
Text
Thought
Token 1
Thought
Token 2
a b c d a’ b’ c’ d’
a
b
c
d
a’
b’
c’
d’
Base Text Thought Token 1
Generation Paths Parallel Inference Mask
Figure 3: Parallel Generation. By constructing an attention mask that allows all thought
tokens to pay attention to themselves, all preceding thought tokens within the same thought,
and the preceding text, we can generate continuations of all of the thoughts in parallel. Each
inference call is used to generate one additional thought token for all text tokens.
to a prefix of the sequence, but the list of tokens is a set of “counterfactual” continuations of
these prefixes. We can, however, leverage these continuations to generate hidden thoughts
for each observed token.
To do this efficiently, we cache each forward pass and concatenate a diagonal attention mask
to the previous attention mask: each generated token now attends to all of the tokens that
were used to generate it, as well as to itself (but not to token on other “counterfactual” paths).
Moreover, this parallelized next-sampling token procedure can be repeated arbitrarily many
times (or at least, until one runs out of memory). We visualize this procedure in Figure 3
and highlight additional ways to make this algorithm faster in Appendix B.
4.3 “Mixing” (Residual) Heads
When starting with a pre-trained model, thoughts will initially be out of distribution, and
hence harm language modeling performance. To smooth the transition to thinking, we
introduce a learned interpolation between the LM predictions with and without thoughts.
Given the end-of-thought token’s hidden state and the hidden state of the original text
token, the mixing head outputs a weight that determines the extent to which the postthought prediction logits will be used. We use a shallow multi-layer perceptron for this
head, outputting a scalar for each token. We include implementation details in Appendix A.
4.4 Optimizing Rationale Generation
4.4.1 Optimizing Start-of-Thought and End-of-Thought Tokens
The 
 and 
 tokens serve as learned meta-tokens that
control the model’s rationale generation. Optimizing the representation of these tokens,
especially the 
 token, is crucial but challenging due to the discrete nature
of the rationale tokens. We initialize the start and end token embeddings to the embedding
corresponding to the em dash, ”−−−”, which often appears in text data to denote a pause or
thought. This leverages the language model’s preexisting knowledge. In addition, to allow
these embeddings to be optimized more quickly, we apply a (hyperparameter) weight to
the gradients of these embeddings during the update step. Intuitively, the start thought
tokens can be understood as putting the model into a “thinking mode” and the end thought
token can be understood as telling the model when it’s done thinking.
4.4.2 Non-myopic Scoring and Teacher-forcing
Because we do not expect thoughts to be useful in predicting every token, we would prefer
the model’s reward to depend less on the exact next word in the text following the thought
and more on the following semantic content. There are two primary challenges here. First,
unlike in typical language modeling with transformers, only the thoughts corresponding to
6
d e f g h
<t> Thought </t> g h
Figure 4: Forward Pass and Teacher Forcing. We visualize a single forward pass of our
algorithm. Solid lines denote language model computation, while dashed lines indicate
tokens are inserted via teacher forcing, and the mixer represents the mixing head. In
particular, we visualize predicting three tokens ahead. Thought generation is shown in
more detail in Figure 1 and Figure 3.
a given next-token prediction receive a gradient from that prediction—a consequence of our
parallel sampling strategy. We could address this by adding loss terms for future tokens by
sampling the tokens before. However this would result in much higher entropy for language
modeling in general and lower-quality generated text, because it would train the LM to partially disregard its preceding tokens. Instead, we use the parallel attention mask to compute
the log probabilities of the true next tokens, applying teacher forcing by assuming the model
selected the correct next ground-truth token (as implicit in normal language modeling with
transformers). Note that the loss for each future token also depends on a mixing weight
computed from the end thought token and the previous observed token. The number of
future tokens included in the loss is a hyper-parameter. We apply the same teacher-forcing
technique to insert the start and end tokens. We visualize this procedure in Figure 4.
4.4.3 Objective
We use REINFORCE to optimize the likelihoods of the rationales based on their usefullness:
the log-likelihood of the ntrue true next tokens Xj+1:j+ntrue+1 under the language model given
previous observed tokens and a particular rationale (p
talk
j:j+ntrue
as shorthand for the mixed
prediction probabilities after thinking, see Algorithm 1). To reduce variance, we generate
multiple rationale continuations for each token in the input sequence (loosely inspired by
TRICE, Phan et al. (2023)). We thus define the reward rj
for each rationale Tj as the difference
between p
talk
j:j+ntrue
and the average across rationales for that token (p
talk
j:j+ntrue
):
rj = log p
talk
j:j+ntrue
(Xj+1:j+ntrue+1) − log p
talk
j:j+ntrue
(Xj+1:j+ntrue+1)
We then use this reward in a REINFORCE loss term to update the language model parameters θ to increase the likelihood of rationales that perform better than the average:
∇θL
REINFORCE
j = −rj
· ∇θ
log pθ (Tj
|[X:j
; 
])
We found it useful to exclude the negative reward from the REINFORCE loss term, as it led
to more stable training, though it may introduce some bias.
This loss term encourages the model to generate rationales that improve its predictions of
future tokens compared to the average prediction across all generated rationales for that
token. The gradients from this loss are used to update both the LM parameters and the
start-of-thought and end-of-thought token embeddings, with a (hyperparameter) weight
applied to the gradients of the start-of-thought and end-of-thought token embeddings to
accelerate their optimization. By iteratively optimizing these parameters, Quiet-STaR trains
the model to generate more useful rationales throughout training. Lastly, we also include a
log-likelihood loss, L
NLL
j
, to ensure that the LM learns to optimize the talking heads and
also receives a next-token prediction signal for the base LM head1
.
1Due to our linear mixing, equivalent to shifting the mixing weight toward the base prediction.
7
5 Experiments and Results
Intuitively, not all tokens require equal amounts of thought. For example, consider the
sentence “the person is run-”: although there is inevitably some probability of the token
being something other than “ing”2
, as a standalone sentence without context, additional
thinking is unlikely to improve a well-trained model’s prediction. Indeed, we conjecture
that for most chunks of most online text, additional thought has little to no impact. Indeed,
early in our exploration we observed that Quiet-STaR does not benefit all tokens equally.
Thus, we design our experiments to investigate whether our approach is useful in predicting
tokens that do require thought. We evaluate 1) whether Quiet-STaR improves a language
model’s ability to directly predict answers in datasets that require reasoning; and, 2) the
distribution of impacts resulting from thinking tokens. We conduct all of our experiments
starting with the base version of Mistral 7B (Jiang et al., 2023).
We perform most of our experiments by training on OpenWebMath (Paster et al., 2023),
a crawl that emphasizes more technical webpages. We selected OpenWebMath because
we anticipated that it would have a higher density of tokens that benefit from reasoning,
which our experiments support. We also evaluate Quiet-STaR on C4 (Raffel et al., 2020),
a widely used LM pretraining corpus with more diverse text, and again show significant
albeit smaller benefits.
5.1 Downstream Performance
In this subsection, we evaluate the extent to which Quiet-STaR improves the zero-shot
reasoning capabilities of the language model on CommonsenseQA (Talmor et al., 2018)
and GSM8K (Cobbe et al., 2021). On CommonsenseQA, we find that Quiet-STaR improves
performance by 10.9% compared to the base language model. As shown in Figure 2,
this improvement consistently increases with the number of tokens used in the model’s
rationales, indicating that more thorough reasoning through the thought tokens is translating
to better direct question-answering performance. Similarly, on GSM8K, Quiet-STaR results
in a 5.0% boost over the base model, and once again, performance scales with the length of
the rationales generated during Quiet-STaR training. For reference, in Figure 2, we include
a baseline corresponding to training the same model on the same dataset without thought
tokens. We observe that in multiple curves, performance appears to eventually deteriorate
– we anticipate that this is because we are not training on these downstream tasks, so the
roles of the thought tokens may change over time. We also find a benefit of our non-myopic
objective, which we discuss in Appendix D.
We find that training with Quiet-STaR on C4 (Raffel et al., 2020) also improves performance
on GSM8K (5.9% → 8.1%) and CommonsenseQA (36.3% → 42.6%) but by a smaller margin.
Specifically, for our C4 evaluation, we train Mistral 7B with 16 thought tokens and 4 true
tokens ahead and otherwise the same setup.
We can compare these improvements to those offered by pause tokens (Goyal et al., 2023),
which can be seen as a constrained version of Quiet-STaR where each token is represented by
two tokens and the second ”pause” token acts as the entirety of the thought. In particular, our
setup is most comparable to their pause token fine-tuning, as we also finetune a pretrained
model. Their results indicate that pause token fine-tuning also provides minor gains over
the base model on CommonsenseQA, they observed an improvement from 26.9% to 28.8%;
on GSM8K, Goyal et al. (2023) found that pause token fine-tuning harms performance.
Moreover, on both tasks (and the majority of their evaluated tasks), they observed that
additional thought tokens harmed performance. Moreover, they discuss the “lukewarm
effect of pause-finetuning a standard-pretrained model” (Goyal et al., 2023). This suggests
that allowing the model to generate multi-token rationales leads to more effective reasoning
compared to the single-token ”pauses”. Note however, that unlike Goyal et al. (2023), we do
not fine-tune on the downstream tasks.
Overall, these downstream results validate that training a language model to predict the
subtext between the lines of general text data can substantially improve its reasoning
2For example, in this very text, the token following “run” is “-”
8
1 2 3 4 5 6 7 8
Samples for Majority Vote
30
40
50
GSM8K Accuracy (%)
Accuracy on GSM8K vs Number of Majority Vote Samples
Quiet-STaR + CoT
Baseline (CoT)
Figure 5: Zero-shot performance on Quiet-STaR applied to chain-of-thought on GSM8K.
We visualize how using a Quiet-STaR trained Mistral model can improve chain-of-thought
performance. We use an 8-thought-token-trained model and use its internal thoughts to
improve the tokens in a zero-shot chain-of-thought (Kojima et al., 2022)
capabilities, even on datasets it was not explicitly trained on. The fact that longer rationales
consistently lead to better outcomes, and that Quiet-STaR outperforms the constrained
pause token approach, supports the notion that Quiet-STaR is successfully teaching the
model to leverage its own generated thoughts to reason more thoroughly about the input.
5.2 Improvement Distribution
As visualized in Appendix Figure 7, we find that on average there is little improvement
in the LM’s ability to predict arbitrary tokens. But, when we visualize the distribution of
relative improvements, there is a disproportionate improvement on more difficult tokens.
This reflects the idea that some text tokens are substantially harder and benefit more from
careful thought.
In Appendix Figure 8, we aim to provide some insight into the kinds of tokens where
the improvements occur. Namely, while thinking appears to help for many tokens in the
example, inspection suggests it disproportionately help to predict tokens where recalling
relevant information is useful, such as the name of an applicable theorem or the start of
the next step in a proof. Notably, this would align well with the framing proposed by
Prystawski et al. (2024).
5.3 Quiet-STaR and Chain-of-Thought
While there are natural parallels between chain-of-thought prompting and our approach,
they are orthogonal and complementary. In zero-shot chain-of-thought, a user actively
prompts the model to think ‘out loud’, otherwise using its ordinary production distribution
(Kojima et al., 2022); Quiet-STaR instead allows a model to think quietly at every token, with
a distribution trained to be useful. We investigate using silent, Quiet-STaR, rationales while
generating explicit CoT reasoning. Because our goal is generalist reasoning that requires no
task-specific input at all, we used a zero-shot prompt (“Let’s think step by step.”) without
in-context examples. Our experiments indicate that internal rationales allow the model
to generate more structured and coherent chains of thought, shown in Appendix E and
visualized in Figure 5. The majority vote accuracy over 8 samples (cot-maj@8) increases
from 40.6% to 47.7% with Quiet-STaR, as evaluated on a sample of 128 GSM8K test items.
Note that each chain-of-thought solution is sampled with temperature 0.7.
5.4 Examples
While there is no explicit regularization in Quiet-STaR for thoughts to be humaninterpretable, they are generated from the same transformer trained to model language,
hence likely to be at least partially understandable. We discuss why this design choice
benefits the training stability in Appendix I. For reference, we include examples of thoughts
generated that were helpful to the model in predicting future tokens in OpenWebMath. First,
in one case, recalling that one should start with magnesium to produce magnesium nitride
allows it to better predict that the first step of the procedure involves heating magnesium.
9
'
 # Magnesium reacts with nitrogen to form magnesium nitride. The chemical
formula for this reaction is Mg+N_2-> MgN_2. What is the product, or what
are the products, of this reaction?\n\nJan 12, 2016\n\nThe formula for
magnesium nitride is $M {g}_{3} {N}_{2}$.\n\n#### Explanation:\n\nAs do
many active metals, magnesium nitride can be
 1 --, so the
equation of the reaction that forms magnesium nitride is\n\n$Mg + N_2 \\to

 formed by heating the metal (fier'
In some cases, the most useful thoughts appear to be near-continuations that correspond
more closely to the target text, e.g.,
An integer $n$ is odd if $n = 2k+1$ for some integer $k$.\n\nTo prove that $A =
B$, we must show that $A \\subseteq B$ and $B \\subseteq A$. The first of
these tends to
 in some sense - to be the more difficult<|
endthought|> trickiest for students
Lastly, we include an example from answering CommonsenseQA. Notably, this thought
occurs while reading the question and hence was not used to predict the final answer.
'
 Q: Talking to the same person about the same thing over and over again is

\n\n(a) a one-to-one correlation\n\n(b) a one-to<|
endofthought|> something someone can what?'
6 Limitations
This work proposes a new framework for learning to reason, and in doing so explores
solutions to a variety of meta-learning challenges. However, to solve these challenges,
certain simplifications were necessary. For example, it would be valuable to understand
whether these techniques work when a model is trained from scratch. We have also only
applied Quiet-STaR to a 7 billion parameter model, albeit a powerful one. The same
techniques applied to a better model would likely yield disproportionately better results, as
has often been observed for gains from reasoning (Wei et al., 2022a).
Quiet-STaR results in a substantial overhead, generating many tokens before generating
every additional token. (See Appendix C for compute adjusted performance results.)
However, this can also be seen as an advantage: typically, a language model can generate the
next token based on the current context, and while there are techniques to improve sampling
quality, there is no general way to leverage additional compute to enhance next-token
prediction. In the current implementation we do not support dynamically predicting when
to generate, or end, a rationale. However, this would be a natural extension. For instance, if
the mixing head was a prediction from the base language model, before any thought, rather
than after the thought, one could apply a threshold to prevent generating thoughts that
would not be incorporated. We expect that this is a more difficult task, as predicting the
usefulness of a thought is simpler when one has already generated the thought.
7 Conclusion
Quiet-STaR represents a step towards language models that can learn to reason in a general
and scalable way. By training on the rich spectrum of reasoning tasks implicit in diverse
web text, rather than narrowly specializing for particular datasets, Quiet-STaR points the
way to more robust and adaptable language models. Our results demonstrate the promise
of this approach, with Quiet-STaR improving downstream reasoning performance while
generating qualitatively meaningful rationales. We believe this also opens many potential
future directions - for example, one may aim to ensemble thoughts in order to further
improve the predictions for future tokens. Moreover, if the language model can predict
when thought will be useful, for example by putting the mixing head before the prediction,
then the predicted mixing weight could be used to dynamically allocate compute during
generation. Future work can build on these insights to further close the gap between
language model and human-like reasoning capabilities.
10
Ethics Statement
This work raises some important ethical questions, many of which also apply to STaR. For
example, it is impossible to know that the reasoning expressed by the model in language
accurately represents the internal processing of the model (i.e., faithfulness). In addition,
regardless of faithfulness, there are no safeguards against harmful or biased reasoning
patterns if the model finds them useful. Relatedly, we note that CommonsenseQA is known
to have many biased questions and low-quality answers (Geva et al., 2019), but we use it in
line with prior work (Zelikman et al., 2022; Goyal et al., 2023). Thus, aside from improving
language modeling, it is unclear in what capacity the rationales themselves should be used.
Acknowledgements
We particularly thank Xindi Wu, Michael Li, and Qian Huang for their helpful and detailed
comments, as well as Xuechen Li, Jan-Philipp Franken, Yuhuai Wu, Gabriel Poesia, Winnie ¨
Xu, Omar Shaikh, Fan-Yun Sun, Joy He-Yueya, Omar Khattab, and William Yin for useful
discussions. In addition, we would like to acknowledge that this work was supported by
NSF Grant #2302701.
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Appendix
A Hyperparameter Choices
Optimization and Evaluation For optimization, we use the AdamW optimizer with a
warmup of 20 steps, a learning rate of 1e − 6, a weight decay of 0.001, and a batch size of 8
(along with any necessary gradient accumulation to keep this fixed across runs). Moreover,
our 
 and 
 embedding gradient weight is 1e2 and our
policy weight is 1e6. We sample with temperature T = 1 during training and use greedy
decoding for the thoughts during evaluation. We treat our samples as importance samples
by computing the REINFORCE loss at temperature T = 3. Because we do not prompt
the model with any examples, we directly compute the probability of the correct answer,
conditioned on generating an answer – for example, for multiple choice questions between
A · · · E, we compute the accuracy over the logits for tokens corresponding to A · · · E. Lastly,
for our training, we select a random span of 256 tokens from each sample (or pad if there
are fewer than 256 tokens).
Mixing Head For our mixing head, we use a three-layer MLP with ReLU activation, taking
in a vector of two times the size of the hidden state of the language model (as we concatenate
the two predictions to determine their weights), and outputting a scalar. This scalar is them
used to weight the logits from the LM head with and without thinking to make a prediction
from a given token.
Computation We train all of our models on a single node of eight 80GB H100s.
B Faster Parallel Sampling
In this section, we highlight some simple ways to further accelerate the parallel generation
algorithm. For example, note that one can reduce the attention’s memory cost by computing
the diagonal attention simply as elementwise (rather than pairwise) dot-products. That
is, given two input embedding sequences of shapes (b, t, l, d) and (b, 1, l, d) where t is
the number of timesteps ahead, b is batch size, l is sequence length, and d is embedding
dimension, we do not need to compute their pairwise attention of shape (b, t, l, l), we
only need to compute the attention for the paired elements along the diagonal of shape
(b, t, l). Additionally, to avoid generating continuations for all of the tokens (for example,
if one wanted to apply a value function to determine where thoughts would be most
useful), one can index into this generated attention mask. Notably, however, this also
requires manipulation of the other inputs during the forward pass such as the positional
embeddings.
C Compute-Adjusted Plots
We also visualize Figure 2 where we normalize by the number of thought and talk tokens
used for training.
D Measuring the Impact of Multiple Thoughts Per Sequence and
Multiple Tokens Ahead
We perform a simple ablation on our 12-thought-token-4-ahead baseline, namely asking
whether sampling multiple thoughts per sequence is necessary. We find that although
simply computing the reward as the difference between the losses with and without thought
proves to be a strong baseline, using multiple thoughts consistently outperformed it (by
roughly 0.5% on GSM8K generalization and 3% on CommonsenseQA generalization).
However, the exact number of thoughts had little impact: varying between 2, 3, and 4
thoughts per sequence appeared to result in a consistent improvement with additional
16
500 1000 1500 2000 2500 3000 3500
Training Step
6%
7%
8%
9%
10%
11%
GSM8K Accuracy (%)
# Thought, Ahead Tokens
24, 12
16, 8
12, 4
10, 4
8, 4
Baseline
(a) GSM8K
500 1000 1500 2000 2500 3000 3500
Training Step
34%
36%
38%
40%
42%
44%
46%
CommonsenseQA Accuracy (%)
# Thought, Ahead Tokens
24, 12
16, 8
12, 4
10, 4
8, 4
Baseline
(b) CommonsenseQA
Figure 6: Compute-Normalized Generalization Results. We visualize the performance
curves normalized by the number of inference calls used.
thoughts, but additional thoughts per sequence beyond two improved performance by
only 0.1-0.3% on both GSM8K and CommonsenseQA. Similarly, we found that predicting
more than one token ahead helped by 0.3% on GSM8K and 3.1% on CommonsenseQA,
but with 12-thought-tokens, we did not find that additional tokens ahead, beyond two,
improved performance. However, qualitatively, we found that the rationales appeared
more coherent with additional tokens-ahead of supervision. This suggests a need for more
specific evaluations of generation quality in future work.
E Rationale-Augmented Rationales
Because the language model can insert thoughts anywhere, we perform a preliminary study
where we investigate the reasoning generated by the language model when it has access to
internal rationales while it generates external rationales (i.e. chains-of-thought) to solve a
problem. Because these chains of thought are themselves generated by chains of thought,
one might refer to this as a net-of-thought (or NoT). We compare five solutions from the
model to a fixed random question in GSM8K, looking at outputs of the base Mistral 7B, the
outputs of Mistral fine-tuned for the same number of steps on OpenWebMath (Paster et al.,
2023), and the rationales of our best Quiet-STaR model. We sample from all of them directly,
at temperature t = 1.
E.1 Base Mistral 7B
First Response:
A: Let's think step by step. How many fresh eggs remains after the three her eat for
breakfast and four goes into the muffin mix for her friends. We can calculate it by
16-3-4. The amount equals 9 fresh eggs. So, the dollars it makes daily equals what
multiplied by $2. 12 x 2. The answer is $24.
Second Response:
Q: Janet's ducks lay 16 eggs per day. She eats three for breakfast every morning and
bakes muffins for her friends every day with four. She sells the remainder at the
farmers' market daily for $2 per fresh duck egg. How much in dollars does she make
every day at the farmers' market?
A: Let's think step by step. She eats 3 daily for breakfast. She bakes 4 daily into
muffins for her friends. That leaves 9. She sells 9 for \(2\), so \(9\times 2=\$18
\). So Janet makes $18 selling duck eggs each day.
17
66. Janet sells an average of 12 fresh duck eggs daily on the farmers' market. If she
sells them for $2 per egg how much does she make per week, assuming she sells at
the farmers' market most every day?A: If Janet sells on average 12 fresh duck eggs
daily on the farmers' market. This is equal to \(12\times 7\), so 12 \(\times\) 7
=\(84\). The total amount she would then make would be \(84\times 2\) dollars, or
168 per week. So Janet makes $168 selling duck eggs per week.
Third Response:
Q: Janet's ducks lay 16 eggs per day. She eats three for breakfast every morning and
bakes muffins for her friends every day with four. She sells the remainder at the
farmers' market daily for $2 per fresh duck egg. How much in dollars does she make
every day at the farmers' market?
A: Let's think step by step.
How many eggs does she have left every day?
This is important because it will help us find how many eggs she sells each day.
What can we do to answer the question?
If we write the relationship between how many duck eggs Janet has left, how many she
eats, and how many she bakes for the muffins, then we can see how many eggs she has
left to sell at the farmers' market.
Let's write an equation to solve our dilemma.
Subtract an equation for eggs eaten and an equation for eggs baked to find the eggs
remaining. Let's see what we can find.
We can see that Janet has 6 eggs leftover each day. $6\times\$2 = \$12$ per day.
Therefore Janet receives an additional $12 per day for her eggs at the farmer's
market.
Remember, this is just for one day, so now she makes $12 per day in additional income,
because she sells her eggs at the farmer's market each day.
#### Answer
$12
Fourth Response:
Q: Janet's ducks lay 16 eggs per day. She eats three for breakfast every morning and
bakes muffins for her friends every day with four. She sells the remainder at the
farmers' market daily for $2 per fresh duck egg. How much in dollars does she make
every day at the farmers' market?
A: Let's think step by step. First, you want to find out the number of eggs a total that
Janet collects each day. Take this many dollars a three and subtract from the
money she gets per egg, which is $2. The answer to that question is $5.
Fifth Response
Q: Janet's ducks lay 16 eggs per day. She eats three for breakfast every morning and
bakes muffins for her friends every day with four. She sells the remainder at the
farmers' market daily for $2 per fresh duck egg. How much in dollars does she make
every day at the farmers' market?
A: Let's think step by step.
For the first step, let's think of the day as a whole, using some variables to keep
track of the numbers of eggs. Let $n$ = the number of eggs that Janet's ducks lay
per day. Then the number of eggs that she eats per day will be $(3)$ = .
Next, she makes $m$ = muffins. For each muffin, she uses $4$ = eggs, so we can take from
the eggs Janet's ducks lay to represent the number she uses: $4(m)$=. Now, to get
her back to our starting number of eggs, we need to add these eggs back on, giving
us . The eggs Janet sells each day brings her $2$$(n)$ = $. This brings us to our
next step.
Now we will break down all the eggs from our starting number, $n$ =, into the pieces
that we set up previously. For the beginning number, we have: This brings us back
to our starting number, and we can simplify this expression.
18
Moving from the top line to the second line, we multiply the numbers within each
parenthetical group. Moving from the second line to the third line, we add the
products together to find the total. Finally, we can distribute and determine the
final product: $= 360.$
So the total number of eggs that Janet's ducks lay per day is $360.$ Since Janet sells
each egg she has for $2,$ her total earnings at the farmers' market is: $\boxed{\$
720}.$
E.2 OpenWebMath-Trained Mistral 7B
First Response:

 Q: Janet's ducks lay 16 eggs per day. She eats three for breakfast every morning and
bakes muffins for her friends every day with four. She sells the remainder at the
farmers' market daily for $2 per fresh duck egg. How much
in dollars does she make every day at the farmers' market?
A: Let's think step by step. How much does she make every day? Well per the question,
that is $2 per a duck egg. Now, how many duck eggs per day does she sell? Well,
that's 16 minus the eggs she eats minus the eggs she uses to ma
ke muffins.
She eats 3 eggs every morning, so $16-3 = 13$.
She makes 4 muffins, which uses 4 eggs. $16-3-4= 5$.
Therefore, the number of duck eggs she is selling is $5$.
Clearly, 1 duck egg costs $2. Therefore, the total amount she makes selling the eggs is
$10$ $2$. Now, this can be written as a fraction as, "ten two-dollars," so $10 2 -
bucks.$
Converting to a fraction, or "a tenth of ten," so $10 2 - bucks$ = $1$ $10 - bucks$
[reveal-answer q="fs-id1165135856725"]Show Solution[/reveal-answer]
[hidden-answer a="fs-id1165135856725"]
Second Response:

 Q: J a n e t s ducks lay 16 eggs per day. She eats three for breakfast every morning
and bakes muffins for her friends every day with four. She sells the remainder at
the farmers' market daily for $2 per fresh duck egg. How much in dollars does she
make every day at the farmers' market?
A: Let's think step by step. On what basis can we answer the question affer considering
the question? What do we know given and how can we use what we're given to arrive
at an answer. 3 is given, 4 is given, 16 is given, does anyone have any thoughts?
A: Well 16 minus 3 minus 4 equals 9, and then I subtract 1 from 9 and that is 4 extra so
five more eggs.
M: And we know that 2 times 5 is how much?
A: Ten
M: Yes.
A: If she eats 3 duck eggs for breakfast, in the morning equals \(3 \times 2 = 6\). If
she eat 4 duck eggs baking muffin for her friends equals \(4 \times 2 = 8\), so
total of money and that she got is $6 plus $8 which equals 14. The weight gain is 5
because it is 5 is that what you put up there so 5 times 2 ducks and subtract that
from that from 16 and you got 7 no what did we write down? Are you able to figure
out now what the question is, and what is what the question really is. You got 14
the weight loss you, you subtracted that from 16 which equals 2, $14 minus 2,
remainder is what?
A: 12.
M: The question is what is J a n e t s ducks lay 16 eggs, she eats 12 for sale. You
subtract 3 plus 4 minus 5 so 5 when you eat the muffin equals 5, times two, the 12,
minus the money so 5 times 2, $-12$. So \(2 \times 9 = 18\) minus $10, so 4 more
eggs... Still times $2, total comes to to $20.
A: I understand there is the ones day one 3 plus 4 eats who makes and so on and so on.
M: But maybe in order to answer it correctly you have to look a little close. Typically
in the procedures you have that they give you first, I can tell you how here we go.
19
M: Now if you increase the 16, now you have that you would need to determine 16 minus 3
minus 4 and that adds up to the remainder, which is 9. Then subtract 1 and but you
have 5. The next answer then, which you do is now 5 times 2, which is 10.
M: Which is $10, and then 14, which is is the remainder
M: Is $6 and $8 and that is a total of $14, and now that you subtract 2, and that means
that you now have 16 minus 5, which is 11.
M: The next thing is to the yield for the second situation you have got 16 minus 3 plus
4 and that comes to 9. This is very easy to do instantaneously even the first part,
16, 3, 4 and 9, and you happen to know what lives, and you just count and you have
five and five then go ahead. 5 times 2 comes to 10, so $10 then so actually you
have just added 4.5, now your total is 16 plus $2. But now to this is $10 plus $2
is $12, so how much you sell from 16 and then minus 12 comes to 4.
M: So what is the remainder of 16 minus 3 plus 4 minus 5? Matters you see three is
missing. Four you have. Five eggs. So 5 times 2, so $10 remains at $10, so reduce,
so what happens now to what happened was added with subtracted the next the
question is 16 minuses 3 plus 4, and presumably you've done this before, if not
then you add up 16 minus 3 plus 4 gives us 9 and what is the remainder after 5? So
what is 5 times 2? So 10. 2, 3, plus 2 = 5 , and so your answer so far is 16 minus
3 plus 4, and give us the basket minus 2 remains 9 and now you have to determine
now what is the total for the second part? You have
Third Response:

 Q: J a n e t s ducks lay 16 eggs per day. She eats three for breakfast every morning
and bakes muffins for her friends every day with four. She sells the remainder at
the farmers' market daily for $2 per fresh duck egg. How much in dollars does she
make every day at the farmers' market?
A: Let's think step by step. The "how much in dollars" part means that one egg is worth
one dollar. So, twelve ducks lay 16 eggs, which means that Janet nets $20 from
selling them. $16 muffins will give us 4 eggs, which nets us $8. So, we paid $108
for our ingredients to make 16 muffins. Hence, we lost 96 dollars. Hello, heroine,
we have to do better than that! Look into averages. Let's say that we pay 5 dollars
for one egg. Now, $12 is our product and $5 is our investment. Our net wealth will
be $12-5*12 = 24 dollars. Is that correct?
Fourth Response:

 Q: J a n e t s ducks lay 16 eggs per day. She eats three for breakfast every morning
and bakes muffins for her friends every day with four. She sells the remainder at
the farmers' market daily for $2 per fresh duck egg. How much in dollars does she
make every day at the farmers' market?
A: Let's think step by step. First, Jane's ducks lay 16 eggs perday. So there are 16
eggs Monday, 16 Tuesday, 16 Wednesday, 16 Thursday, 16 Friday, and so on. Second,
Janet eats 3 of the 16 eggs that the ducklay ap day for breakfast. She only eats
the duck eggs, not her baked goods (muffins) or other food. Therefore, she eats 3
Monday, 3 Tuesday, 3 Wednesday, 3 Thursday, and so on. Third, she only bakes the
first 4 non-eaten eggs into muffins. Therefore$ she sells the remaining 12 noneaten eggs Monday, 12 Tuesday, 12 Wednesday, 12 Thursday, and so on. Fourth, Janet
sells duck eggs Monday, Tuesday, Wednesday, Thursday, Friday at the 2 each. Third,
she eats 3 Monday, 3 Tuesday, 3 Wednesday, 3 Thursday, Friday at the farmers'
market. Lastly, she makes 12 Monday, 12 Tuesday, 12 Wednesday, 12 Thursday, and so
on in dollars every day at the farmers' market. The worksheet below summarizes
these steps. The number formed by the intersection of corresponding row and column
terms will be the correct answer. The product of the (3, J) term and the (J, 8)
term is 1,150. So, Janet makes $1,150 per day!
(Note: Step 3 and Step 8 are repeating, but the (3,7) term in Step 8 is not the same as
the (3,7) term in Step 3. )
Fifth Response

 Q: J a n e t s ducks lay 16 eggs per day. She eats three for breakfast every morning
and bakes muffins for her friends every day with four. She sells the remainder at
20
the farmers' market daily for $2 per fresh duck egg. How much in dollars does she
make every day at the farmers' market?
A: Let's think step by step.
What do we know?
- Janet's ducks lay 16 eggs per day.
- She eats three for breakfast every morning and bakes muffins for her friends every day
with four.
From these two facts, we can conclude at least that Janet needs at least seven eggs for
consumption per day.
And what do we need to find exactly?
- We need to find how much in dollars Janet makes every day at the farmers' market.
Let's write an equation and solve it to find the answer. We know:
- \(16-3-4=9\)
So our daily profit is 9 times $2, or \(9\times\$2=\$18\)
Hence her profits at market on a given day are \($18\).
Answer: \($18\).
E.3 Quiet-STaR-Trained Mistral 7B
First Response:

 Q: Janet's ducks lay 16 eggs per day. She eats three for breakfast every morning and
bakes muffins for her friends every day with four. She sells the remainder at the
farmers' market daily for $2 per fresh duck egg. How much
in dollars does she make every day at the farmers' market?
A: Let's think step by step.
The ducks lay 16 eggs per day.
This is the total number of eggs.
3 of the eggs are eaten for breakfast.
This is the number of eggs that are eaten for breakfast.
4 of the eggs are used to make muffins.
This is the number of eggs that are used to make muffins.
16 - 3 - 4 = 9
9 is the number of eggs that are sold at the farmers' market.
9 of the eggs are sold at the farmers' market.
$2
The price of each egg is $2.
$2 \times 9 = 18$
18 is the total price of the eggs.
The total price of the eggs is $18.
Second Response:
21
A: Let's think step by step.
1. The ducks lay 16 eggs per day.
2. She eats 3 for breakfast every morning.
3. She bakes muffins for her friends every day with 4.
4. She sells the remainder at the farmers' market daily for $2 per fresh duck egg.
5. The number of eggs she sells at the farmers' market is 16 - 3 - 4.
6. The number of eggs she sells at the farmers' market is 16 - 3 - 4.
7. The number of eggs she sells at the farmers' market is 16 - 3 - 4.
8. The number of eggs she sells at the farmers' market is 16 - 3 - 4.
9. The number of eggs she sells at the farmers' market is 16 - 3 - 4.
10. The number of eggs she sells at the farmers' market is 16 - 3 - 4.
11. The number of
Third Response:

 Q: Janet's ducks lay 16 eggs per day. She eats three for breakfast every morning and
bakes muffins for her friends every day with four. She sells the remainder at the
farmers' market daily for $2 per fresh duck egg. How much in dollars does she make
every day at the farmers' market?
A: Let's think step by step.
The ducks lay 16 eggs per day.
This is the total number of eggs.
3 is the number of eggs that Janet eats for breakfast every morning.
4 is the number of eggs that she uses in the muffins that she bakes for her friends
every day.
16 - 3 - 4 = 9
9 is the number of eggs that she sells at the farmers' market daily for $2 per fresh
duck egg.
9 $2 = 9 $2
= 9 $2
= 9 $2
= 18
So, 18 is the amount in dollars that she makes every day at the farmers' market.
The amount in dollars that she makes every day at the farmers' market is 18.
A: 18

Fourth Response:
A: Let's think step by step.
The number of eggs that Janet's ducks lay per day is 16.
Of these, she eats 3 for breakfast every morning.
This is 3.
Of these, she bakes muffins for her friends with 4.
This is 4.
22
The number of eggs that are left over is 16 - 3 - 4.
This is 16 - 7.
This is 9.
The number of eggs that she sells at the farmers' market daily for $2 per egg is 9.
The total in dollars that she makes every day at the farmers' market is 9 $2.
This is 9 2.
This is 18.
The total in dollars that she makes every day at the farmers' market is 18.
A: 18
Fifth Response
A: Let's think step by step.
The ducks lay 16 eggs per day.
This is the total number of eggs.
3 + 4 + 16 = 23
3 eggs are eaten for breakfast every morning.
4 eggs are used to make muffins for her friends every day.
16 eggs are sold at the farmers' market for $2 each every day.
The 16 eggs that are sold at the farmers' market are the remaining eggs.
16 - 3 - 4 = 9
9 eggs are sold at the farmers' market for $2 each every day.
9 (2) = 18
18 is the total in dollars that she makes every day at the farmers' market.
18 is the answer.
A: 18
F Additional Works on Learning to Reason
For completeness, we highlight that many other works have explored teaching language
models to reason (often with algorithms similar to or inspired by STaR), either from their
own rationales, from interaction with an environment, or from a hand-constructed dataset.
For example, works explore this in the context of multihop question answering (Zhao et al.,
2023), math (Luo et al., 2023; Uesato et al., 2022), machine translation (Gulcehre et al., 2023).
Several works investigate teaching language model agents to reason in planning (Chen
et al., 2023; Gandhi et al., 2023; Qiao et al., 2024), or to use specific tools or memory (Yao
et al., 2022; Lanchantin et al., 2024; Schick et al., 2024), while others investigate how one may
distill the reasoning from a large language model into a smaller language model Ho et al.
(2022); Li et al. (2022); Hsieh et al. (2023). Notably however, Pan et al. (2024) demonstrates
23
that these feedback loops may result in reward hacking. Zelikman et al. (2023b) shows
how a bootstrapping loop can be implemented where a model repeatedly improves a codeimprover using the same code-improver and Haluptzok et al. (2023) shows how language
models can bootstrap their programming ability with programming puzzles Schuster et al.
(2021). Other works have employed a similar strategy for using language models to solve
inductive reasoning tasks or to model real-world systems (Wang et al., 2023; Qiu et al., 2023;
Zhu et al., 2023; Li et al., 2024).
Some works have investigated how models can learn from their reasoning mistakes incontext (Shinn et al., 2023; Madaan et al., 2023; Zhang et al., 2024; Liu et al., 2023). Many
studies have also focused on the ability of LMs to learn from in-context reasoning examples
(Lampinen et al., 2022; Zhou et al., 2022) – correspondingly, Khattab et al. (2022) and Khattab
et al. (2023) show how the sets of examples used to prompt a model to reason can be
optimized in the context of a multi-step reasoning pipeline. Furthermore, Zhang et al. (2022)
demonstrated that one can improve zero-shot question-answering in language models by
using a variety of zero-shot prompts for reasoning.
G Distribution of Improvements
We also visualize the distribution of improvements across tokens in the training set.
Original Token Prediction Loss
Improvement in Loss Given Thought
Figure 7: Distribution of changes in log probability. We visualize the distribution of
changes in log probability resulting from the generated thoughts across the evaluation
dataset. We visualize the density of changes in log probability relative to the LM without
thoughts, colored based on log density. The distribution is skewed, with most tokens unaffected, while a small fraction of hard tokens see substantial improvements from the thoughts.
This matches our intuition that most tokens in general web text do not require significant
reasoning to predict, but thoughts are disproportionately beneficial for challenging tokens.
24
H Contribution Visualization
0icos^2θsin^3θ+5cosθsin^4θ+isin^5θ$
Then I used the Moivre's theorem and I got: $(cos5θ+isin5θ)$
I compared the imaginary parts and I got something like: $sin5θ=5cos^4θsinθ-10cos^2θsin^3θ+sin^5θ$
which is very close to: $(16cos^4θ-12cos^2θ+1)sinθ$ but not the same.
Where do I make te mistake?
Thanks for any help! ;)
• Have you tried use $\cos^2\theta+sin^2\theta =1$
– Fan
May 15 '17 at 19:13
• You didn't err. You just didn't finish. May 15 '17 at 19:35
hint
In your last line, factor by $\sin (\theta),$
replace
$\sin^2(\theta)$ by $1-\cos^2 (\theta)$
Figure 8: Contribution Visualization. We provide an example text where we visualize the
degree to which introducing thoughts helped at tokens throughout a text. Green indicates
that the thought before that token made that token easier to predict, while yellow indicates
that it made it harder to predict. More impactful thoughts have higher opacity.
I Handling Instability
Several aspects of this task have the potential to introduce instability. First, and perhaps
most importantly, the utility of a generated thought (or thought token) is a function of the
mapping from the thought to its contribution to language prediction; however, the mapping
from the thoughts to this contribution is learned based on the thoughts themselves. This
means that, even if one were to generate a thought that allowed the perfect prediction of
the next token, the loss could receive no signal from it if the mixing head’s weight on that
generation was 0. One solution we explored was to use the Gumbel-Softmax trick with a
straight-through estimator Jang et al. (2016), but with many consecutive softmax operations
we observed vanishing gradients. This introduces an exploration-exploitation trade-off,
a fundamental challenge in reinforcement learning. Approaches like DQN (Mnih et al.,
2013), PPO (Schulman et al., 2017), and A3C (Mnih et al., 2016) often resolve these tradeoffs by learning a state value function, which estimates the expected future reward from
each state. However, the reward functions associated with this environment are unstable
(as noted earlier, due to the also-changing mixing heads) – consequently, our preliminary
explorations with these techniques was not promising. While we are far from the first to
note that optimizing rationales is a reinforcement-learning task (Zelikman et al., 2022; Zhang
& Parkes, 2023; Phan et al., 2023), the need for the rationale to avoid harming the base model
performance introduces additional complexity. Essentially, the more complex the mapping
from LM output to the next token prediction, the more instability we observed. On the other
hand, when we trained without any interpolation, i.e. ablating the mixing head and only
using the language model prediction after thoughts, the model quickly learned to simply
ignore the thoughts (and we saw no generalization to any downstream tasks).
We explored the use of separate heads for thinking and talking (here, we use talking to refer
to directly outputting a hidden state or logits, rather than a mixing weight). In particular,
we explored both linear layers from the hidden states and MLPs, initialized to contribute 0
residually to the base language model outputs, in order to generate thoughts and next-token
25
predictions similar to what the language model would have otherwise generated. However,
we observed that, in all instances, the previously-mentioned instability prevented learning.
Consequently, we aimed to remove or minimize all components that could transform the
language model’s outputs, both with and without its rationales. We also note that our
choice to use a language model to output a weight combining multiple states (as done by
our mixing head) is essentially an attention mechanism allowing the model to attend to its
thinking. This has similarity to the approach taken in Backpack language models (Hewitt
et al., 2023), which also learn to predict weights to apply to summed input embeddings to
model future text, rather than allowing the language model to output arbitrary embeddings.
Despite this constraint, Backpack language models appear to have comparable performance
to traditional language models (Hewitt et al., 2023).
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