Deep Neural Networks (DNNs) are powerful models that have achieved excellent performance on difficult learning tasks. Although DNNs work well whenever large labeled training sets are available, they cannot be used to map sequences to sequences. In this paper, we present a general end-to-end approach to sequence learning that makes minimal assumptions on the sequence structure. Our method uses a multilayered Long Short-Term Memory (LSTM) to map the input sequence to a vector of a fixed dimensionality, and then another deep LSTM to decode the target sequence from the vector. Our main result is that on an English to French translation task from the WMT-14 dataset, the translations produced by the LSTM achieve a BLEU score of 34.8 on the entire test set, where the LSTM's BLEU score was penalized on out-of-vocabulary words. Additionally, the LSTM did not have difficulty on long sentences. For comparison, a phrase-based SMT system achieves a BLEU score of 33.3 on the same dataset. When we used the LSTM to rerank the 1000 hypotheses produced by the aforementioned SMT system, its BLEU score increases to 36.5, which is close to the previous state of the art. The LSTM also learned sensible phrase and sentence representations that are sensitive to word order and are relatively invariant to the active and the passive voice. Finally, we found that reversing the order of the words in all source sentences (but not target sentences) improved the LSTM's performance markedly, because doing so introduced many short term dependencies between the source and the target sentence which made the optimization problem easier.

/pdf/sequence-to-sequence-learning-with-neural-networks-4slgp125no.pdf

Sequence to Sequence Learning with Neural Networks

Deep Neural Networks (DNNs) are powerful models that have achieved excellent performance on difficult learning tasks. Although DNNs work well whenever large labeled training sets are available, they cannot be used to map sequences to sequences. In this paper, we present a general end-to-end approach to sequence learning that makes minimal assumptions on the sequence structure. Our method uses a multilayered Long Short-Term Memory (LSTM) to map the input sequence to a vector of a fixed dimensionality, and then another deep LSTM to decode the target sequence from the vector. Our main result is that on an English to French translation task from the WMT'14 dataset, the translations produced by the LSTM achieve a BLEU score of 34.8 on the entire test set, where the LSTM's BLEU score was penalized on out-of-vocabulary words. Additionally, the LSTM did not have difficulty on long sentences. For comparison, a phrase-based SMT system achieves a BLEU score of 33.3 on the same dataset. When we used the LSTM to rerank the 1000 hypotheses produced by the aforementioned SMT system, its BLEU score increases to 36.5, which is close to the previous best result on this task. The LSTM also learned sensible phrase and sentence representations that are sensitive to word order and are relatively invariant to the active and the passive voice. Finally, we found that reversing the order of the words in all source sentences (but not target sentences) improved the LSTM's performance markedly, because doing so introduced many short term dependencies between the source and the target sentence which made the optimization problem easier.

We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Random graphs

This beginning graduate textbook describes both recent achievements and classical results of computational complexity theory. Requiring essentially no background apart from mathematical maturity, the book can be used as a reference for self-study for anyone interested in complexity, including physicists, mathematicians, and other scientists, as well as a textbook for a variety of courses and seminars. More than 300 exercises are included with a selected hint set.

/pdf/computational-complexity-a-modern-approach-5gtk8urev9.pdf

Computational Complexity: A Modern Approach

A major consideration we had in writing this survey was to make it accessible to mathematicians as well as to computer scientists, since expander graphs, the protagonists of our story, come up in numerous and often surprising contexts in both fields But, perhaps, we should start with a few words about graphs in general They are, of course, one of the prime objects of study in Discrete Mathematics However, graphs are among the most ubiquitous models of both natural and human-made structures In the natural and social sciences they model relations among species, societies, companies, etc In computer science, they represent networks of communication, data organization, computational devices as well as the flow of computation, and more In mathematics, Cayley graphs are useful in Group Theory Graphs carry a natural metric and are therefore useful in Geometry, and though they are “just” one-dimensional complexes, they are useful in certain parts of Topology, eg Knot Theory In statistical physics, graphs can represent local connections between interacting parts of a system, as well as the dynamics of a physical process on such systems The study of these models calls, then, for the comprehension of the significant structural properties of the relevant graphs But are there nontrivial structural properties which are universally important? Expansion of a graph requires that it is simultaneously sparse and highly connected Expander graphs were first defined by Bassalygo and Pinsker, and their existence first proved by Pinsker in the early ’70s The property of being an expander seems significant in many of these mathematical, computational and physical contexts It is not surprising that expanders are useful in the design and analysis of communication networks What is less obvious is that expanders have surprising utility in other computational settings such as in the theory of error correcting codes and the theory of pseudorandomness In mathematics, we will encounter eg their role in the study of metric embeddings, and in particular in work around the Baum-Connes Conjecture Expansion is closely related to the convergence rates of Markov Chains, and so they play a key role in the study of Monte-Carlo algorithms in statistical mechanics and in a host of practical computational applications The list of such interesting and fruitful connections goes on and on with so many applications we will not even

/pdf/expander-graphs-and-their-applications-wbsb8gxfw2.pdf

Expander graphs and their applications

Special Issue In Memory of Misha Alekhnovich. Foreword

A well-known theorem by Tarsi states that a minimally unsatisfiable CNF formula with m clauses can have at most m-1 variables, and this bound is exact In the context of proving lower bounds on proof space in k-DNF resolution, [Ben-Sasson and Nordstrom 2009] extended the concept of minimal unsatisfiability to sets of k-DNF formulas and proved that a minimally unsatisfiable k-DNF set with m formulas can have at most (mk)k+1 variables This result is far from tight, however, since they could only present explicit constructions of minimally unsatisfiable sets with Ω(mk2) variables

In the current paper, we revisit this combinatorial problem and significantly improve the lower bound to (Ω(m))k, which almost matches the upper bound above Furthermore, using similar ideas we show that the analysis of the technique in [Ben-Sasson and Nordstrom 2009] for proving time-space separations and trade-offs for k-DNF resolution is almost tight This means that although it is possible, or even plausible, that stronger results than in [Ben-Sasson and Nordstrom 2009] should hold, a fundamentally different approach would be needed to obtain such results

On Minimal Unsatisfiability and Time-Space Trade-offs for k-DNF Resolution.

We continue a long line of research aimed at proving convergence of depth 2 neural networks, trained via gradient descent, to a global minimum. Like in many previous works, our model has the following features: regression with quadratic loss function, fully connected feedforward architecture, RelU activations, Gaussian data instances and network initialization, adversarial labels. It is more general in the sense that we allow both layers to be trained simultaneously and at {\em different} rates. Our results improve on state-of-the-art [Oymak Soltanolkotabi 20] (training the first layer only) and [Nguyen 21, Section 3.2] (training both layers with Le Cun's initialization). We also report several simple experiments with synthetic data. They strongly suggest that, at least in our model, the convergence phenomenon extends well beyond the ``NTK regime''.

Improved Convergence Guarantees for Shallow Neural Networks

Sergei Ivanovich Adian (on his eightieth birthday)

A computation or a proof is called feasible if it obeys prescribed bounds on the resources consumed during its execution. It turns out that when restricted to this world of feasibility, proofs and computations become extremely tightly interrelated, sometimes even indistinguishable. Moreover, many of these rich relations, underlying concepts, techniques etc. look very different from their “classical” counterparts, or simply do not have any. This talk is intended as a very informal and popular (highly biased as well) attempt to illustrate these fascinating connections by several related developments in the modern complexity theory.

Alexander A. Razborov

Papers

Special Issue In Memory of Misha Alekhnovich. Foreword

On Minimal Unsatisfiability and Time-Space Trade-offs for k-DNF Resolution.

Improved Convergence Guarantees for Shallow Neural Networks

Sergei Ivanovich Adian (on his eightieth birthday)

Feasible Proofs and Computations: Partnership and Fusion