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Title: Minimal Random Code Learning: Getting Bits Back from Compressed Model Parameters

Paper content:
While deep neural networks are a highly successful model class, their large memory footprint puts considerable strain on energy consumption, communication bandwidth, and storage requirements.
Consequently, model size reduction has become an utmost goal in deep learning.
A typical approach is to train a set of deterministic weights, while applying certain techniques such as pruning and quantization, in order that the empirical weight distribution becomes amenable to Shannon-style coding schemes.
However, as shown in this paper, relaxing weight determinism and using a full variational distribution over weights allows for more efficient coding schemes and consequently higher compression rates.
In particular, following the classical bits-back argument, we encode the network weights using a random sample, requiring only a number of bits corresponding to the Kullback-Leibler divergence between the sampled variational distribution and the encoding distribution.
By imposing a constraint on the Kullback-Leibler divergence, we are able to explicitly control the compression rate, while optimizing the expected loss on the training set.
The employed encoding scheme can be shown to be close to the optimal information-theoretical lower bound, with respect to the employed variational family.
Our method sets new state-of-the-art in neural network compression, as it strictly dominates previous approaches in a Pareto sense: On the benchmarks LeNet-5/MNIST and VGG-16/CIFAR-10, our approach yields the best test performance for a fixed memory budget, and vice versa, it achieves the highest compression rates for a fixed test performance.
With the celebrated success of deep learning models and their ever increasing presence, it has become a key challenge to increase their efficiency.
In particular, the rather substantial memory requirements in neural networks can often conflict with storage and communication constraints, especially in mobile applications.
Moreover, as discussed in BID4 , memory accesses are up to three orders of magnitude more costly than arithmetic operations in terms of energy consumption.
Thus, compressing deep learning models has become a priority goal with a beneficial economic and ecological impact.
Traditional approaches to model compression usually rely on three main techniques: pruning, quantization and coding.
For example, Deep Compression BID5 proposes a pipeline employing all three of these techniques in a systematic manner.
From an information-theoretic perspective, the central routine is coding, while pruning and quantization can be seen as helper heuristics to reduce the entropy of the empirical weight-distribution, leading to shorter encoding lengths BID15 .
Also, the recently proposed Bayesian Compression BID13 falls into this scheme, despite being motivated by the so-called bits-back argument BID8 which theoretically allows for higher compression rates.1
While the bits-back argument certainly motivated the use of variational inference in Bayesian Compression, the downstream encoding is still akin to Deep Compression (and other approaches).
In particular, the variational distribution is merely used to derive a deterministic set of weights, which is subsequently encoded with Shannonstyle coding.
This approach, however, does not fully exploit the coding efficiency postulated by the bits-back argument.In this paper, we step aside from the pruning-quantization pipeline and propose a novel coding method which approximately realizes bits-back efficiency.
In particular, we refrain from constructing a deterministic weight-set but rather encode a random weight-set from the full variational posterior.
This is fundamentally different from first drawing a weight-set and subsequently encoding it -this would be no more efficient than previous approaches.
Rather, the coding scheme developed here is allowed to pick a random weight-set which can be cheaply encoded.
By using results from BID6 , we show that such an coding scheme always exists and that the bits-back argument indeed represents a theoretical lower bound for its coding efficiency.
Moreover, we propose a practical scheme which produces an approximate sample from the variational distribution and which can indeed be encoded with this efficiency.
Since our algorithm learns a distribution over weightsets and derives a random message from it, while minimizing the resulting code length, we dub it Minimal Random Code Learning (MIRACLE).From
a practical perspective, MIRACLE has the advantage that it offers explicit control over the expected loss and the compression size. This
is distinct from previous techniques, which require tedious tuning of various hyper-parameters and/or thresholds in order to achieve a certain coding goal. In our
method, we can simply control the KL-divergence using a penalty factor, which directly reflects the achieved code length (plus a small overhead), while simultaneously optimizing the expected training loss. As a result
, we were able to trace the trade-off curve for compression size versus classification performance ( FIG4 ). We clearly
outperform previous state-of-the-art in a Pareto sense: For any desired compression rate, our encoding achieves better performance on the test set; vice versa, for a certain performance on the test set, our method achieves the highest compression. To summarize
, our main contributions are:• We introduce MIRACLE, an innovative compression algorithm that exploits the noise resistance of deep learning models by training a variational distribution and efficiently encodes a random set of weights.• Our method
is easy to implement and offers explicit control over the loss and the compression size.• We provide
theoretical justification that our algorithm gets close to the theoretical lower bound on the encoding length.• The potency
of MIRACLE is demonstrated on two common compression tasks, where it clearly outperforms previous state-of-the-art methods for compressing neural networks.In the following section, we discuss related work and introduce required background. In Section 3
we introduce our method. Section 4 presents
our experimental results and Section 5 concludes the paper.
In this paper we followed through the philosophy of the bits-back argument for the goal of coding model parameters.
The basic insight here is that restricting to a single deterministic weight-set and aiming to coding it in a classic Shannon-style is greedy and in fact sub-optimal.
Neural networks -and other deep learning models -are highly overparameterized, and consequently there are many "good" parameterizations.
Thus, rather than focusing on a single weight set, we showed that this fact can be exploited for coding, by selecting a "cheap" weight set out of the set of "good" ones.
Our algorithm is backed by solid recent information-theoretic insights, yet it is simple to implement.
We demonstrated that the presented coding algorithm clearly outperforms previous state-of-the-art.
An important question remaining for future work is how efficient MIRACLE can be made in terms of memory accesses and consequently for energy consumption and inference time.
There lies clear potential in this direction, as any single weight can be recovered by its block-index and relative index within each block.
By smartly keeping track of these addresses, and using pseudo-random generators as algorithmic lookup-tables, we could design an inference machine which is able to directly run our compressed models, which might lead to considerable savings in memory accesses.
This is shown by proving that q(w) − p i (w) ≤ q(w)(1 − p(w)) i for i ∈ N.In order to bound the encoding length, one has to first show that if the accepted sample has index i * , then E[log i * ] ≤ KL(q||p) + O(1) .Following
this, one can employ the prefix-free binary encoding of BID16 . Let l(n)
be the length of the encoding for n ∈ N using the encoding scheme proposed by BID16 . Their method
is proven to have |l(n)| = log n + 2 log log(n + 1) + O(1), from which the upper bound follows: DISPLAYFORM0

assistant: text: This paper proposes an effective method to compress neural networks based on recent results in information theory.