  By extracting task-relevant information while maximally compressing the
input, the information bottleneck (IB) principle has provided a guideline for
learning effective and robust representations of the target inference. However,
extending the idea to the multi-task learning scenario with joint consideration
of generative tasks and traditional reconstruction tasks remains unexplored.
This paper addresses this gap by reconsidering the lossy compression problem
with diverse constraints on data reconstruction, perceptual quality, and
classification accuracy. Firstly, we study two ternary relationships, namely,
the rate-distortion-classification (RDC) and rate-perception-classification
(RPC). For both RDC and RPC functions, we derive the closed-form expressions of
the optimal rate for binary and Gaussian sources. These new results complement
the IB principle and provide insights into effectively extracting task-oriented
information to fulfill diverse objectives. Secondly, unlike prior research
demonstrating a tradeoff between classification and perception in signal
restoration problems, we prove that such a tradeoff does not exist in the RPC
function and reveal that the source noise plays a decisive role in the
classification-perception tradeoff. Finally, we implement a deep-learning-based
image compression framework, incorporating multiple tasks related to
distortion, perception, and classification. The experimental results coincide
with the theoretical analysis and verify the effectiveness of our generalized
IB in balancing various task objectives.
