A Lower Bound for the Optimization of Finite Sums

This paper presents a lower bound for optimizing a finite sum of $n$ functions, where each function is $L$-smooth and the sum is $\mu$-strongly convex. We show that no algorithm can reach an error $\epsilon$ in minimizing all functions from this class in fewer than $\Omega(n + \sqrt{n(\kappa-1)}\log(1/\epsilon))$ iterations, where $\kappa=L/\mu$ is a surrogate condition number. We then compare this lower bound to upper bounds for recently developed methods specializing to this setting. When the functions involved in this sum are not arbitrary, but based on i.i.d. random data, then we further contrast these complexity results with those for optimal first-order methods to directly optimize the sum. The conclusion we draw is that a lot of caution is necessary for an accurate comparison, and identify machine learning scenarios where the new methods help computationally.Comment: Added an erratum, we are currently working on extending the result to randomized algorithm

Unbiased Black-Box Complexities of Jump Functions

We analyze the unbiased black-box complexity of jump functions with small, medium, and large sizes of the fitness plateau surrounding the optimal solution. Among other results, we show that when the jump size is $(1/2 - \varepsilon)n$, that is, only a small constant fraction of the fitness values is visible, then the unbiased black-box complexities for arities $3$ and higher are of the same order as those for the simple \textsc{OneMax} function. Even for the extreme jump function, in which all but the two fitness values $n/2$ and $n$ are blanked out, polynomial-time mutation-based (i.e., unary unbiased) black-box optimization algorithms exist. This is quite surprising given that for the extreme jump function almost the whole search space (all but a $\Theta(n^{-1/2})$ fraction) is a plateau of constant fitness. To prove these results, we introduce new tools for the analysis of unbiased black-box complexities, for example, selecting the new parent individual not by comparing the fitnesses of the competing search points, but also by taking into account the (empirical) expected fitnesses of their offspring.Comment: This paper is based on results presented in the conference versions [GECCO 2011] and [GECCO 2014

Finite difference methods fengshui: alignment through a mathematics of arrays

Numerous scientific-computational domains make use of array data. The core computing of the numerical methods and the algorithms involved is related to multi-dimensional array manipulation. Memory layout and the access patterns of that data are crucial to the optimal performance of the array-based computations. As we move towards exascale computing, writing portable code for efficient data parallel computations is increasingly requiring an abstract productive working environment. To that end, we present the design of a framework for optimizing scientific array-based computations, building a case study for a Partial Differential Equations solver. By embedding the Mathematics of Arrays formalism in the Magnolia programming language, we assemble a software stack capable of abstracting the continuous high-level application layer from the discrete formulation of the collective array-based numerical methods and algorithms and the final detailed low-level code. The case study lays the groundwork for achieving optimized memory layout and efficient computations while preserving a stable abstraction layer independent of underlying algorithms and changes in the architecture.Peer ReviewedPostprint (author's final draft

Interprocedural Type Specialization of JavaScript Programs Without Type Analysis

Dynamically typed programming languages such as Python and JavaScript defer type checking to run time. VM implementations can improve performance by eliminating redundant dynamic type checks. However, type inference analyses are often costly and involve tradeoffs between compilation time and resulting precision. This has lead to the creation of increasingly complex multi-tiered VM architectures. Lazy basic block versioning is a simple JIT compilation technique which effectively removes redundant type checks from critical code paths. This novel approach lazily generates type-specialized versions of basic blocks on-the-fly while propagating context-dependent type information. This approach does not require the use of costly program analyses, is not restricted by the precision limitations of traditional type analyses. This paper extends lazy basic block versioning to propagate type information interprocedurally, across function call boundaries. Our implementation in a JavaScript JIT compiler shows that across 26 benchmarks, interprocedural basic block versioning eliminates more type tag tests on average than what is achievable with static type analysis without resorting to code transformations. On average, 94.3% of type tag tests are eliminated, yielding speedups of up to 56%. We also show that our implementation is able to outperform Truffle/JS on several benchmarks, both in terms of execution time and compilation time.Comment: 10 pages, 10 figures, submitted to CGO 201