ECE 498/598 Fall 2024, Homeworks 3 and 4
Remarks:
1. HW3&4: You can reduce the context length to ** if you are having trouble with the
training time.
2. HW3&4: During test evaluation, note that positional encodings for unseen/long
context are not trained. You are supposed to evaluate it as is. It is OK if it doesn’t
work well.
3. HW3&4: Comments are an important component of the HW grade. You are expected
to explain the experimental findings. If you don’t provide technically meaningful
comments, you might receive a lower score even if your code and experiments are
accurate.
4. The deadline for HW3 is November 11th at 11:59 PM, and the deadline for HW4 is
November 18th at 11:59 PM. For each assignment, please submit both your code and a
PDF report that includes your results (figures) for each question. You can generate the
PDF report from a Jupyter Notebook (.ipynb file) by adding comments in markdown
cells.
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The objective of this assignment is comparing transformer architecture and SSM-type
architectures (specifically Mamba [1]) on the associative recall problem. We provided an
example code recall.ipynb which provides an example implementation using 2 layer
transformer. You will adapt this code to incorporate different positional encodings, use
Mamba layers, or modify dataset generation.
Background: As you recall from the class, associative recall (AR) assesses two abilities
of the model: Ability to locate relevant information and retrieve the context around that
information. AR task can be understood via the following question: Given input prompt
X = [a 1 b 2 c 3 b], we wish the model to locate where the last token b occurs earlier
and output the associated value Y = 2. This is crucial for memory-related tasks or bigram
retrieval (e.g. ‘Baggins’ should follow ‘Bilbo’).
To proceed, let us formally define the associative recall task we will study in the HW.
Definition 1 (Associative Recall Problem) Let Q be the set of target queries with cardinal ity |Q| = k. Consider a discrete input sequence X of the form X = [. . . q v . . . q] where the
query q appears exactly twice in the sequence and the value v follows the first appearance
of q. We say the model f solves AR(k) if f(X) = v for all sequences X with q ∈ Q.
Induction head is a special case of the definition above where the query q is fixed (i.e. Q
is singleton). Induction head is visualized in Figure 1. On the other extreme, we can ask the
model to solve AR for all queries in the vocabulary.
Problem Setting
Vocabulary: Let [K] = {1, . . . , K} be the token vocabulary. Obtain the embedding of
the vocabulary by randomly generating a K × d matrix V with IID N(0, 1) entries, then
normalized its rows to unit length. Here d is the embedding dimension. The embedding of
the i-th token is V[i]. Use numpy.random.seed(0) to ensure reproducibility.
Experimental variables: Finally, for the AR task, Q will simply be the first M elements
of the vocabulary. During experiments, K, d, M are under our control. Besides this we will
also play with two other variables:
• Context length: We will train these models up to context length L. However, we
will evaluate with up to 3L. This is to test the generalization of the model to unseen
lengths.
• Delay: In the basic AR problem, the value v immediately follows q. Instead, we will
introduce a delay variable where v will appear τ tokens after q. τ = 1 is the standard.
Models: The motivation behind this HW is reproducing the results in the Mamba paper.
However, we will also go beyond their evaluations and identify weaknesses of both trans former and Mamba architectures. Specifically, we will consider the following models in our
evaluations:
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Figure 1: We will work on the associative recall (AR) problem. AR problem requires the
model to retrieve the value associated with all queries whereas the induction head requires
the same for a specific query. Thus, the latter is an easier problem. The figure above is
directly taken from the Mamba paper [1]. The yellow-shaded regions highlight the focus of
this homework.
• Transformer: We will use the transformer architecture with 2 attention layers (no
MLP). We will try the following positional encodings: (i) learned PE (provided code),
(ii) Rotary PE (RoPE), (iii) NoPE (no positional encoding)
• Mamba: We will use the Mamba architecture with 2 layers.
• Hybrid Model: We will use an initial Mamba layer followed by an attention layer.
No positional encoding is used.
Hybrid architectures are inspired by the Mamba paper as well as [2] which observes the
benefit of starting the model with a Mamba layer. You should use public GitHub repos to
find implementations (e.g. RoPE encoding or Mamba layer). As a suggestion, you can use
this GitHub Repo for the Mamba model.
Generating training dataset: During training, you train with minibatch SGD (e.g. with
batch size 64) until satisfactory convergence. You can generate the training sequences for
AR as follows given (K, d, M, L, τ):
1. Training sequence length is equal to L.
2. Sample a query q ∈ Q and a value v ∈ [K] uniformly at random, independently. Recall
that size of Q is |Q| = M.
3. Place q at the end of the sequence and place another q at an index i chosen uniformly
at random from 1 to L − τ.
4. Place value token at the index i + τ.
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5. Sample other tokens IID from [K]−q i.e. other tokens are drawn uniformly at random
but are not equal to q.
6. Set label token Y = v.
Test evaluation: Test dataset is same as above. However, we will evaluate on all sequence
lengths from τ + 1 to 3L. Note that τ + 2 is the shortest possible sequence.
Empirical Evidence from Mamba Paper: Table 2 of [1] demonstrates that Mamba can do
a good job on the induction head problem i.e. AR with single query. Additionally, Mamba
is the only model that exhibits length generalization, that is, even if you train it pu to context
length L, it can still solve AR for context length beyond L. On the other hand, since Mamba
is inherently a recurrent model, it may not solve the AR problem in its full generality. This
motivates the question: What are the tradeoffs between Mamba and transformer, and can
hybrid models help improve performance over both?
Your assignments are as follows. For each problem, make sure to return the associated
code. These codes can be separate cells (clearly commented) on a single Jupyter/Python file.
Grading structure:
• Problem 1 will count as your HW3 grade. This only involves Induction Head
experiments (i.e. M = 1).
• Problems 2 and 3 will count as your HW4 grade.
• You will make a single submission.
Problem 1 (50=25+15+10pts). Set K = 16, d = 8, L = ** or L = 64.
• Train all models on the induction heads problem (M = 1, τ = 1). After training,
evaluate the test performance and plot the accuracy of all models as a function of
the context length (similar to Table 2 of [1]). In total, you will be plotting 5 curves
(3 Transformers, 1 Mamba, 1 Hybrid). Comment on the findings and compare the
performance of the models including length generalization ability.
• Repeat the experiment above with delay τ = 5. Comment on the impact of delay.
• Which models converge faster during training? Provide a plot of the convergence rate
where the x-axis is the number of iterations and the y-axis is the AR accuracy over a
test batch. Make sure to specify the batch size you are using (ideally use ** or 64).
Problem 2 (30pts). Set K = 16, d = 8, L = ** or L = 64. We will train Mamba, Transformer
with RoPE, and Hybrid. Set τ = 1 (standard AR).
• Train Mamba models for M = 4, 8, 16. Note that M = 16 is the full AR (retrieve any
query). Comment on the results.
• Train Transformer models for M = 4, 8, 16. Comment on the results and compare
them against Mamba’s behavior.
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• Train the Hybrid model for M = 4, 8, 16. Comment and compare.
Problem 3 (20=15+5pts). Set K = 16, d = 64, L = ** or L = 64. We will only train
Mamba models.
• Set τ = 1 (standard AR). Train Mamba models for M = 4, 8, 16. Compare against the
corresponding results of Problem 2. How does embedding d impact results?
• Train a Mamba model for M = 16 for τ = 10. Comment if any difference.
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