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In this paper, we investigate single user throughput optimization in High Speed Downlink Packet Access (HSDPA). Specifically, we propose offline and online optimization algorithms which adjust the Channel Quality Indicator (CQI) used by the network for scheduling of data transmission. In the offline algorithm, a given target block error rate (BLER) is achieved by adjusting CQI based on ACK/NAK history. By sweeping through different target BLERs, we can find the throughput optimal BLER offline. This algorithm could be used not only to optimize throughput but also to enable fair resource allocation among multiple users in HSDPA. In the online algorithm, the CQI offset is adapted using an estimated short term throughput gradient without the need for a target BLER. An adaptive stepsize mechanism is proposed to track temporal variation of the environment. Convergence behavior of both algorithms is analyzed. The part of the analysis that deals with constant step size gradient algorithm may be applied to other stochastic optimization techniques. The convergence analysis is confirmed by our simulations. Simulation results also yield valuable insights on the value of optimal BLER target. Both offline and online algorithms are shown to yield up to 25% of throughput improvement over the conventional approach of targeting 10% BLER.