Paper: Algorithms for Caching and MTS with reduced number of predictions (ICLR-2024)

Reviewer comment (Writing):
The writing and grammar needs improvement at certain points (see below for more specific comments).

Author response:
Thank you for your questions and comments. We have fixed the grammar and some
typos which we have encountered when revising our manuscript.

Reviewer comment (Experiments):
The motivation for well-separated queries is settings in which the waiting for the query to be produced may take a number of time steps to produce. Have you considered a setting where the prediction queries are explicitly delayed? For example, based on the state of the algorithm at time \( t \), we make a query to the predictor, but the result does not arrive until time \( t + a \). So predicted information is directly more costly in the sense that it may no longer be "fresh" or as useful when we receive it.

Author response:
We have not considered the case where the predictions are explicitly delayed,
  but we find it a very interesting question. This setting brings additional
  challenge to our analysis, where the prediction error should be computed
  with respect to the state of Belady at the time step when the predictor
  was queried rather than the time step when the prediction was produced
  (and used).

Reviewer comment (Theory):
For MTS, you are unable to give a bound on the number of predictions that scales with \( OPT \) since the adversary can scale the instance to make \( OPT \) arbitrarily small. Can the absolute number of predictions be bounded? For example, is it possible to use \( o(T) \) queries to the predictions for this setting, or is this impossible similar to the lower bound in Theorem 1.4?

Author response:
It is not possible to achieve non-trivial results for MTS with o(T) predictions.
  To illustrate this, imagine an input sequence composed of a large number $N$
  of difficult MTS instances of constant length one after another.
  Each of the difficult MTS instances requires a certain number of predictions
  and the total number of required predictions then needs to scale linearly with
  $N$. An additional note: although OPT might be much smaller than $T$ in some
  cases, it may also grow linearly in $T$.

Reviewer comment (Writing):
Please make general improvements to the grammar.

Author response:
Thank you for your questions and comments. We have fixed the grammar and some
typos which we have encountered when revising our manuscript.

Reviewer comment (Writing):
Page 6, first paragraph - "\( f(i) \leq 2^j - 1 \)" - should this be \( f(i) \leq 2^i - 1 \)? I don't see \( j \) defined nearby.

Author response:
On page 6, first paragraph, there should be $f(i)\leq 2^i-1$ as you suggest.

Reviewer comment (Theory):
For the synthetic noise a log-normal distribution is used. I believe you meant the parameters for this distribution are \( \mu = 0 \) and \( \sigma \in [0,50] \). The mean of a log-normal distribution cannot be 0.

Author response:
$\mu, \sigma$ are the parameters of the normal distribution. We have fixed our explanation in the manuscript.

Reviewer comment (Evaluation):
In the captions for Figures 1 and 2, please clarify if the standard deviation reported is for the competitive ratio averaged over many trials (and not the noise for the synthetic predictions), and how many trials were used for this average if so.

Author response:
The plots and tables in Section 5 contain averages of 10 independent
  experiments. Unfortunately, this line slipped out of the main version and was
  included only in Appendix. The standard deviation refers to the competitive
  ratio and the number of requested predictions respectively.
  We have added the omitted explanation.

Reviewer comment (Theory):
In Section 2.2, it is assumed that the predictor is trying to simulate Belady. Is this without loss of generality?

Author response:
* We assume that the predictor is trying to simulate Belady because the
  prediction error is measured with respect to Belady. Note that the current
  predictors do try to imitate Belady (Liu et al. 2020).
  Said that, it would be nice if
  we could extend our analysis to the case where we do not know
  which offline algorithm is simulated by the predictor. This would result
  in a stronger kind of guarantee as in Theorem 1.3.

Reviewer comment (Writing):
In Algorithm 2, is $S$ a set or a sequence? I don’t find S = [ xxx ] a standard notation. Also, S[I] does not look standard, either.

Author response:
* Questions about Algorithm 2: $S$ is a set. We define $F$ in line 4.
  $f$ in line 4 is a parameter of the algorithm. In case you meant
  to ask about $p$ in line 4 of Algorithm 1, this $p$ is chosen arbitrarily
  among pages in the algorithms cache not belonging to the predictor's cache
  $P$.

Reviewer comment (Writing):
In Algorithm 2, line 4, is this to define $F$?

Author response:
* Questions about Algorithm 2: $S$ is a set. We define $F$ in line 4.
  $f$ in line 4 is a parameter of the algorithm. In case you meant
  to ask about $p$ in line 4 of Algorithm 1, this $p$ is chosen arbitrarily
  among pages in the algorithms cache not belonging to the predictor's cache
  $P$.

Reviewer comment (Reproducibility):
In Algorithm 2, line 4, how is p chosen?

Author response:
* Questions about Algorithm 2: $S$ is a set. We define $F$ in line 4.
  $f$ in line 4 is a parameter of the algorithm. In case you meant
  to ask about $p$ in line 4 of Algorithm 1, this $p$ is chosen arbitrarily
  among pages in the algorithms cache not belonging to the predictor's cache
  $P$.

Reviewer comment (Presentation):
Page 6, there’s a paragraph about “implementation suggestions”. This paragraph is not directly related to the proof of the main theorem, so it is slightly out of context. Maybe move it to the end of the section, or somewhere in the experiments section?

Author response:
* We believe that the paragraph on implementation suggestions is related to the
  algorithm's description and we prefer to leave it in its current place.

Reviewer comment (Writing):
Page 7, 2nd paragraph. What is G_{i, I+1} exactly? In particular, what does “gap” mean? I’m not even sure if this is a number or a subset when I first read it.

Author response:
* gap $G_{i-1,i}$ denotes a time interval between two executions of Robust.
  Similarly, we meant $X$ to be a time period.

Reviewer comment (Writing):
Page 7, 2nd paragraph. What do you mean by “period” X? Maybe you simply say X is a sequence of requests.

Author response:
* gap $G_{i-1,i}$ denotes a time interval between two executions of Robust.
  Similarly, we meant $X$ to be a time period.

Reviewer comment (Theory):
I tried to read the proof of Lemma 3.3, but I didn’t find a clear place for “Robust_f uses at most f(log k) predictions”. Where is it? Also, does this bound depend on the randomness?

Author response:
* Algorithm Robust queries the predictor only on arrivals belonging to $F$
  (Algorithm 2, line 6) and the size of $F$ does not depend on randomness
  in the algorithm. When considering your question, we have noticed that we
  have missed a single query in the last window $W_{\log k +1}$ in our
  computation. So, $F$ is chosen in line 4 of Algorithm 2 to have size
  at most $f(\log k)+1$. Therefore, each execution of Algorithm 2
  requires at most $f(\log k)+1$ predictions.

Reviewer comment (Experiments):
One weakness is that the action prediction setting still looks a little bit weird. To me, particularly in online caching, predicting the next arrival time of the current element makes more sense and is more learnable.

Author response:
We do not want to argue that one type of predictions is better than another,
  but we would like to point out several nice properties of action predictions.
  First, they can be produced by any method which is able
  to generate next-arrival predictions. This is due to the simple conversion
  rule proposed by Antoniadis et al. (2020). Note that this does not
  work the other way. For example,
  a classification problem of determining whether a page is cache-friendly or
  cache-averse as in predictors by Jain and Lin (2016) and Shi et al. (2019)
  gives rise to action predictions but its solution does not contain enough
  information to reconstruct next-arrival time predictions. 
  Second, action predictions are usable beyond caching. Note that the effect of
  next-arrival time predictions is limited already for weighted caching,
  where the best competitive ratio achievable with perfect next-arrival time
  predictions is far from 1 and depends on the number of weight classes in the
  input instance, see Bansal et al. (2022).
  We believe that our paper adds a third item to this list, namely better
  consistency bounds achievable in a regime with a limited access to the predictor.

Reviewer comment (Theory):
It seems hard to learn the optimal actions from historical data in practice. But I understand that this is a setting proposed in the previous work, and the local information predictions (e.g. next arrival time) may not help in the restricted-learning setting.

Author response:
We do not want to argue that one type of predictions is better than another,
  but we would like to point out several nice properties of action predictions.
  First, they can be produced by any method which is able
  to generate next-arrival predictions. This is due to the simple conversion
  rule proposed by Antoniadis et al. (2020). Note that this does not
  work the other way. For example,
  a classification problem of determining whether a page is cache-friendly or
  cache-averse as in predictors by Jain and Lin (2016) and Shi et al. (2019)
  gives rise to action predictions but its solution does not contain enough
  information to reconstruct next-arrival time predictions. 
  Second, action predictions are usable beyond caching. Note that the effect of
  next-arrival time predictions is limited already for weighted caching,
  where the best competitive ratio achievable with perfect next-arrival time
  predictions is far from 1 and depends on the number of weight classes in the
  input instance, see Bansal et al. (2022).
  We believe that our paper adds a third item to this list, namely better
  consistency bounds achievable in a regime with a limited access to the predictor.

Reviewer comment (Theory):
The key algorithmic idea for algorithm 1 is built on observation 2.1 that Belady is somehow stable. If the predictor simulates other unstable algorithms, is there any non-trivial smoothness bound?

Author response:
Our smoothness bound for caching depends on the prediction error with respect
  to Belady. This is not the case in our result for general MTS, where the
  prediction error can be measured with respect to an arbitrary offline algorithm.
  This result covers caching as a special case of MTS, but gives only a linear
  smoothness.
  We believe that accommodating non-Belady predictors in our result for caching
  is a matter of a more refined analysis. Currently, we use properties specific
  to Belady (Observations 2.1 and 2.3) to detect incorrect predictions in the
  Follower part of our algorithm and estimate the prediction error in the anlysis
  of the Robust part. Overcoming these issues seems interesting but challenging.