@inproceedings{10597, abstract = {We thank Emmanuel Abbe and Min Ye for providing us the implementation of RPA decoding. D. Fathollahi and M. Mondelli are partially supported by the 2019 Lopez-Loreta Prize. N. Farsad is supported by Discovery Grant from the Natural Sciences and Engineering Research Council of Canada (NSERC) and Canada Foundation for Innovation (CFI), John R. Evans Leader Fund. S. A. Hashemi is supported by a Postdoctoral Fellowship from NSERC.}, author = {Fathollahi, Dorsa and Farsad, Nariman and Hashemi, Seyyed Ali and Mondelli, Marco}, booktitle = {2021 IEEE International Symposium on Information Theory}, isbn = {978-1-5386-8210-4}, location = {Virtual, Melbourne, Australia}, pages = {1082--1087}, publisher = {IEEE}, title = {{Sparse multi-decoder recursive projection aggregation for Reed-Muller codes}}, doi = {10.1109/isit45174.2021.9517887}, year = {2021}, } @inproceedings{10053, abstract = {This paper characterizes the latency of the simplified successive-cancellation (SSC) decoding scheme for polar codes under hardware resource constraints. In particular, when the number of processing elements P that can perform SSC decoding operations in parallel is limited, as is the case in practice, the latency of SSC decoding is O(N1−1 μ+NPlog2log2NP), where N is the block length of the code and μ is the scaling exponent of polar codes for the channel. Three direct consequences of this bound are presented. First, in a fully-parallel implementation where P=N2 , the latency of SSC decoding is O(N1−1/μ) , which is sublinear in the block length. This recovers a result from an earlier work. Second, in a fully-serial implementation where P=1 , the latency of SSC decoding scales as O(Nlog2log2N) . The multiplicative constant is also calculated: we show that the latency of SSC decoding when P=1 is given by (2+o(1))Nlog2log2N . Third, in a semi-parallel implementation, the smallest P that gives the same latency as that of the fully-parallel implementation is P=N1/μ . The tightness of our bound on SSC decoding latency and the applicability of the foregoing results is validated through extensive simulations.}, author = {Hashemi, Seyyed Ali and Mondelli, Marco and Fazeli, Arman and Vardy, Alexander and Cioffi, John and Goldsmith, Andrea}, booktitle = {2021 IEEE International Symposium on Information Theory}, isbn = {978-1-5386-8210-4}, issn = {2157-8095}, location = {Melbourne, Australia}, pages = {2369--2374}, publisher = {IEEE}, title = {{Parallelism versus latency in simplified successive-cancellation decoding of polar codes}}, doi = {10.1109/ISIT45174.2021.9518153}, year = {2021}, } @article{9047, abstract = {This work analyzes the latency of the simplified successive cancellation (SSC) decoding scheme for polar codes proposed by Alamdar-Yazdi and Kschischang. It is shown that, unlike conventional successive cancellation decoding, where latency is linear in the block length, the latency of SSC decoding is sublinear. More specifically, the latency of SSC decoding is O(N1−1/μ) , where N is the block length and μ is the scaling exponent of the channel, which captures the speed of convergence of the rate to capacity. Numerical results demonstrate the tightness of the bound and show that most of the latency reduction arises from the parallel decoding of subcodes of rate 0 or 1.}, author = {Mondelli, Marco and Hashemi, Seyyed Ali and Cioffi, John M. and Goldsmith, Andrea}, issn = {1558-2248}, journal = {IEEE Transactions on Wireless Communications}, number = {1}, pages = {18--27}, publisher = {IEEE}, title = {{Sublinear latency for simplified successive cancellation decoding of polar codes}}, doi = {10.1109/TWC.2020.3022922}, volume = {20}, year = {2021}, } @inproceedings{9221, abstract = {Recent works have shown that gradient descent can find a global minimum for over-parameterized neural networks where the widths of all the hidden layers scale polynomially with N (N being the number of training samples). In this paper, we prove that, for deep networks, a single layer of width N following the input layer suffices to ensure a similar guarantee. In particular, all the remaining layers are allowed to have constant widths, and form a pyramidal topology. We show an application of our result to the widely used LeCun’s initialization and obtain an over-parameterization requirement for the single wide layer of order N2. }, author = {Nguyen, Quynh and Mondelli, Marco}, booktitle = {34th Conference on Neural Information Processing Systems}, location = {Vancouver, Canada}, pages = {11961–11972}, publisher = {Neural Information Processing Systems Foundation}, title = {{Global convergence of deep networks with one wide layer followed by pyramidal topology}}, volume = {33}, year = {2020}, } @article{6748, abstract = {Fitting a function by using linear combinations of a large number N of `simple' components is one of the most fruitful ideas in statistical learning. This idea lies at the core of a variety of methods, from two-layer neural networks to kernel regression, to boosting. In general, the resulting risk minimization problem is non-convex and is solved by gradient descent or its variants. Unfortunately, little is known about global convergence properties of these approaches. Here we consider the problem of learning a concave function f on a compact convex domain Ω⊆ℝd, using linear combinations of `bump-like' components (neurons). The parameters to be fitted are the centers of N bumps, and the resulting empirical risk minimization problem is highly non-convex. We prove that, in the limit in which the number of neurons diverges, the evolution of gradient descent converges to a Wasserstein gradient flow in the space of probability distributions over Ω. Further, when the bump width δ tends to 0, this gradient flow has a limit which is a viscous porous medium equation. Remarkably, the cost function optimized by this gradient flow exhibits a special property known as displacement convexity, which implies exponential convergence rates for N→∞, δ→0. Surprisingly, this asymptotic theory appears to capture well the behavior for moderate values of δ,N. Explaining this phenomenon, and understanding the dependence on δ,N in a quantitative manner remains an outstanding challenge.}, author = {Javanmard, Adel and Mondelli, Marco and Montanari, Andrea}, issn = {1941-7330}, journal = {Annals of Statistics}, number = {6}, pages = {3619--3642}, publisher = {Institute of Mathematical Statistics}, title = {{Analysis of a two-layer neural network via displacement convexity}}, doi = {10.1214/20-AOS1945}, volume = {48}, year = {2020}, } @inproceedings{8536, abstract = {This work analyzes the latency of the simplified successive cancellation (SSC) decoding scheme for polar codes proposed by Alamdar-Yazdi and Kschischang. It is shown that, unlike conventional successive cancellation decoding, where latency is linear in the block length, the latency of SSC decoding is sublinear. More specifically, the latency of SSC decoding is O(N 1−1/µ ), where N is the block length and µ is the scaling exponent of the channel, which captures the speed of convergence of the rate to capacity. Numerical results demonstrate the tightness of the bound and show that most of the latency reduction arises from the parallel decoding of subcodes of rate 0 and 1.}, author = {Mondelli, Marco and Hashemi, Seyyed Ali and Cioffi, John and Goldsmith, Andrea}, booktitle = {IEEE International Symposium on Information Theory}, isbn = {9781728164328}, issn = {2157-8095}, location = {Los Angeles, CA, United States}, publisher = {IEEE}, title = {{Simplified successive cancellation decoding of polar codes has sublinear latency}}, doi = {10.1109/ISIT44484.2020.9174141}, volume = {2020-June}, year = {2020}, } @inproceedings{9198, abstract = {The optimization of multilayer neural networks typically leads to a solution with zero training error, yet the landscape can exhibit spurious local minima and the minima can be disconnected. In this paper, we shed light on this phenomenon: we show that the combination of stochastic gradient descent (SGD) and over-parameterization makes the landscape of multilayer neural networks approximately connected and thus more favorable to optimization. More specifically, we prove that SGD solutions are connected via a piecewise linear path, and the increase in loss along this path vanishes as the number of neurons grows large. This result is a consequence of the fact that the parameters found by SGD are increasingly dropout stable as the network becomes wider. We show that, if we remove part of the neurons (and suitably rescale the remaining ones), the change in loss is independent of the total number of neurons, and it depends only on how many neurons are left. Our results exhibit a mild dependence on the input dimension: they are dimension-free for two-layer networks and depend linearly on the dimension for multilayer networks. We validate our theoretical findings with numerical experiments for different architectures and classification tasks.}, author = {Shevchenko, Aleksandr and Mondelli, Marco}, booktitle = {Proceedings of the 37th International Conference on Machine Learning}, pages = {8773--8784}, publisher = {ML Research Press}, title = {{Landscape connectivity and dropout stability of SGD solutions for over-parameterized neural networks}}, volume = {119}, year = {2020}, } @article{6750, abstract = {Polar codes have gained extensive attention during the past few years and recently they have been selected for the next generation of wireless communications standards (5G). Successive-cancellation-based (SC-based) decoders, such as SC list (SCL) and SC flip (SCF), provide a reasonable error performance for polar codes at the cost of low decoding speed. Fast SC-based decoders, such as Fast-SSC, Fast-SSCL, and Fast-SSCF, identify the special constituent codes in a polar code graph off-line, produce a list of operations, store the list in memory, and feed the list to the decoder to decode the constituent codes in order efficiently, thus increasing the decoding speed. However, the list of operations is dependent on the code rate and as the rate changes, a new list is produced, making fast SC-based decoders not rate-flexible. In this paper, we propose a completely rate-flexible fast SC-based decoder by creating the list of operations directly in hardware, with low implementation complexity. We further propose a hardware architecture implementing the proposed method and show that the area occupation of the rate-flexible fast SC-based decoder in this paper is only 38% of the total area of the memory-based base-line decoder when 5G code rates are supported. }, author = {Hashemi, Seyyed Ali and Condo, Carlo and Mondelli, Marco and Gross, Warren J}, issn = {1053-587X}, journal = {IEEE Transactions on Signal Processing}, number = {22}, publisher = {IEEE}, title = {{Rate-flexible fast polar decoders}}, doi = {10.1109/TSP.2019.2944738}, volume = {67}, year = {2019}, } @article{7007, abstract = {We consider the primitive relay channel, where the source sends a message to the relay and to the destination, and the relay helps the communication by transmitting an additional message to the destination via a separate channel. Two well-known coding techniques have been introduced for this setting: decode-and-forward and compress-and-forward. In decode-and-forward, the relay completely decodes the message and sends some information to the destination; in compress-and-forward, the relay does not decode, and it sends a compressed version of the received signal to the destination using Wyner–Ziv coding. In this paper, we present a novel coding paradigm that provides an improved achievable rate for the primitive relay channel. The idea is to combine compress-and-forward and decode-and-forward via a chaining construction. We transmit over pairs of blocks: in the first block, we use compress-and-forward; and, in the second block, we use decode-and-forward. More specifically, in the first block, the relay does not decode, it compresses the received signal via Wyner–Ziv, and it sends only part of the compression to the destination. In the second block, the relay completely decodes the message, it sends some information to the destination, and it also sends the remaining part of the compression coming from the first block. By doing so, we are able to strictly outperform both compress-and-forward and decode-and-forward. Note that the proposed coding scheme can be implemented with polar codes. As such, it has the typical attractive properties of polar coding schemes, namely, quasi-linear encoding and decoding complexity, and error probability that decays at super-polynomial speed. As a running example, we take into account the special case of the erasure relay channel, and we provide a comparison between the rates achievable by our proposed scheme and the existing upper and lower bounds.}, author = {Mondelli, Marco and Hassani, S. Hamed and Urbanke, Rüdiger}, issn = {1999-4893}, journal = {Algorithms}, number = {10}, publisher = {MDPI}, title = {{A new coding paradigm for the primitive relay channel}}, doi = {10.3390/a12100218}, volume = {12}, year = {2019}, }