We introduce and study the following natural total search problem, which we call the {\it heavy element avoidance} (Heavy Avoid) problem: for a distribution on $N$ bits specified by a Boolean circuit sampling it, and for some parameter $\delta(N) \ge 1/\poly(N)$ fixed in advance, output an $N$-bit string that has probability less than $\delta(N)$. We show that the complexity of Heavy Avoid is closely tied to frontier open questions in complexity theory about uniform randomized lower bounds and derandomization. Among other results, we show:

1. For a wide range of circuit classes C, including $ACC^0, TC^0, NC^1$ and general Boolean circuits, $EXP$ does not have uniform randomized C-circuits if and only if Heavy Avoid for uniform implicit C-samplers has efficient deterministic algorithms infinitely often. This gives the first {\it algorithmic characterization} of lower bounds for $EXP$ against uniform randomized low-depth circuits. We show similar algorithmic characterizations for lower bounds in $PSPACE$, $NP$ and $EXP^{NP}$.

2. Unconditionally, there are polynomial-time {\it pseudodeterministic} algorithms that work infinitely often for several variants of Heavy Avoid, such as for uniform samplers of small randomness complexity. In contrast, the existence of a similar algorithm that solves Heavy Avoid for arbitrary polynomial-time samplers would solve a long-standing problem about hierarchies for probabilistic time.

3. If there is a time and depth efficient deterministic algorithm for Heavy Avoid, then $BPP = P$. Without the depth-efficiency requirement in the assumption, we still obtain a non-trivial form of infinitely-often deterministic simulation of randomized algorithms. These results are shown using {\it non-black-box} reductions, and we argue that the use of non-black-box reductions is essential here.