Keerthi and Shevade (2007) proposed an efficient algorithm for constructing an approximate least angle regression least absolute shrinkage and selection operator solution path for logistic regression as a function of the regularization parameter. In this brief, their approach is extended to multinomial regression. We show that a brute-force approach leads to a multivariate approximation problem resulting in an infeasible path tracking algorithm. Instead, we introduce a noncanonical link function thereby: 1) repeatedly reusing the univariate approximation of Keerthi and Shevade, and 2) producing an optimization objective with a block-diagonal Hessian. We carry out an empirical study that shows the computational efficiency of the proposed technique. A MATLAB implementation is available from the author upon request.

To participate in sponsored search online advertising, an advertiser bids on a set of keywords relevant to his/her product or service. When one of these keywords matches a user search string, the ad is then considered for display among sponsored search results. Advertisers compete for positions in which their ads appear, as higher slots typically result in more user clicks. All existing position allocating mechanisms charge more per click for a higher slot. Therefore, an advertiser must decide whether to bid high and receive more, but more expensive, clicks. In this work, we propose a novel methodology for building forecasting landscapes relating an individual advertiser bid to the expected click-through rate and/or the expected daily click volume. Displaying such landscapes is currently offered as a service to advertisers by all major search engine providers. Such landscapes are expected to be instrumental in helping the advertisers devise their bidding strategies. We propose a triply monotone regression methodology. We start by applying the current state-of-the-art monotone regression solution. We then propose to condition on the ad position and to estimate the bid-position and position-click effects separately. While the latter translates into a standard monotone regression problem, we devise a novel solution to the former based on approximate maximum likelihood. We show that our proposal significantly outperforms the standard monotone regression solution, while the latter similarly improves upon routinely used ad-hoc methods. Last, we discuss other e-commerce applications of the proposed intermediate variable regression methodology.

Cache miss rates are an important subset of system model inputs. Cache miss rate models are used for broad design space exploration in which many cache configurations cannot be simulated directly due to limitations of trace collection setups or available resources. Often it is not practical to simulate large caches. Large processor counts and consequent potentially high degree of cache sharing are frequently not reproducible on small existing systems. In this article, we present an approach to building multivariate regression models for predicting cache miss rates beyond the range of collectible data. The extrapolation model attempts to accurately estimate the high-level trend of the existing data, which can be extended in a natural way. We extend previous work by its applicability to multiple miss rate components and its ability to model a wide range of cache parameters, including size, line size, associativity and sharing. The stability of extrapolation is recognized to be a crucial requirement. The proposed extrapolation model is shown to be stable to small data perturbations that may be introduced during data collection.We show the effectiveness of the technique by applying it to two commercial workloads. The wide design space contains configurations that are much larger than those for which miss rate data were available. The fitted data match the simulation data very well. The various curves show how a miss rate model is useful for not only estimating the performance of specific configurations, but also for providing insight into miss rate trends.

This article presents a technique for taking a sparse set of cache simulation data and fitting a multivariate model to fill in the missing points over a broad region of cache configurations. We extend previous work by its applicability to multiple miss rate components and its ability to model a wide range of cache parameters, including size, associativity and sharing. Miss rate models are useful for broad design exploration in which many cache configurations cannot be simulated directly due to limitations of trace collection setups or available resources. We show the effectiveness of the technique by applying it to two commercial workloads and presenting miss rate data for a broad design space with cache size, associativity, sharing and number of processors as variables. The fitted data match the simulation data very well. The various curves show how a miss rate model is useful for not only estimating the performance of specific configurations, but also for providing insight into miss rate trends. Furthermore, this modeling methodology is robust in the presence of corrupted simulation data and variations in simulation data from multiple sources.